Foundations for Machines
Foundations for Machines: Analysis and Design Shamsher Prakash University of Missq!)ri-Rolla
Vijay K. Puri Southern Illinois University, Carbondale
A Wiley-lnterscience Publication JOHN WILEY AND SONS
New York
Chichester
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To our friend the enlightened saint, humble philosopher, and friend of all mankind who speaks the language of the heart; whose religion is Jove; who always aspires to fill lives of one and all with spiritual bliss.
Copyright
© 1988 by John Wiley
& Sons, Inc.
All rights reserved. Published simultaneously in Canada. Reproduction or translation of any part of this work beyond that permitted by Section 107 or 108 of the 1976 United States Copyright Act without the permission of the copyright owner is unlawful. Requests for permission or further information should be addressed to ,., .:.t~~-·~,ermissions Department, John Wiley & Sons, Inc. {ibt-ary of'Congress Cataloging in Publication Data: .- ~. ,' · J?rakash~ Sham_s~~r. : Fo~ndations for machines. (Wiley series in geotechnical engineering) "A Wiley-Interscience publication." Includes bibliographies and index. 1. Machinery-Foundations. I. Pori, Vijay. II. Title. III. Series. TJ249.P618 1987 621.8 87-21678 ISBN 0-471-84686·4 Printed in the United State of America 10 9 8 7 6 5 4 3 2 1
Preface
The design of machine foundations involves a systematic application of the principles of soil eljgineering, soil dynarn,,i~s, and theory of vibrations-a fact that has been well recognised during tlie last three decades. Since the classical work by Lamb in 1904 and the paper on "Foundation Vibrations" by Richart in 1962, the subject of vibratory response of foundations has attracted the attention of several researchers. The state of art on the subject has since made significant strides. Methods are now available not only for computing the response of machine foundations resting on the surface of the elastic half space but also for embedded foundations and foundations on piles. Elastic half space analogs have further simplified the computation process and are a convenient tool for the designer. The linear spring approach of Barkan, which could previously be used only for surface footings, has also been extended to account for the embedment effects. Recent advances dealing with the determination of the dynamic soil prop- erties and rational interpretation of the test data are of direct application to the design of machine foundations. Information on several aspects of machine foundation design such as design of embedded foundations and pile supported machine foundations is either unavailable or only inadequately treated in the presently available texts. This text has been developed with the object of providing state-of-the-art ,,information on the analysis .,and design of machine foundations and is intended to cater to the interests of graduate students, senior undergraduates, and practicing engineers. Both authors have offered graduatelevel courses on the subject in the United States and India. They also organized many short courses for practicing engineers, including four by the senior author at University of Missouri, Rolla. The authors have also been engaged in the design and performance evaluation of machine foundations.
The feedback from the classroom and the professionals in the field has been of immense help in the planning and preparation of this text. vii
viii
PREFACE
The special features of this book are: (1) analysis of surface and embedded foundations by both the elastic half space method and the linear spring method; (2) analysis of pile supported machine foundations; (3) detailed discussion of the dynamic soil properties, methods for their determination, and evaluation of the test data; ( 4) detailed design procedure followed by examples; and (5) discussion of design of machine foundations on absorbers and vibration isolation. Knowledge of soil mechanics and elementary mathematics or mechanics is needed to follow the text. The reader is introduced to the problem of machine foundation and its special requirements in Chapter 1. In Chapter 2, the elementary theory of vibrations is discussed. Chapter 3 deals with the wave propagation in an elastic medium that provides an important basis for determination of dynamic soil properties as discussed in Chapter 4. Needless to say, soil properties play a critical role in the design of machine foundations. Chapter 4 thus forms a very important component of the text. Also included in this chapter is the procedure for rational selection of soil parameters for a given machine foundation problem. The determination of unbalanced forces and moments occasioned by the operation of a machine is reviewed in Chapter 5. The principal subject of the book, the analysis and design of machine foundations is introduced in Chapter 6, that deals with the design of rigid-block-type foundations for reciprocating machines. In this chapter the reader is made familiar with the concepts of elastic half space method and linear spring method for computing the vibratory response of surface footings. Foundations for impact-type machines such as hammers are discussed in Chapter 7. Foundations for high-speed rotary machines are discussed in Chapter 8 and for miscellaneous machines in Chapter 9. The principles of vibration isolation and absorption are considered in Chapter 10. The design of embedded block foundations for machines is described in Chapter 11 followed by pile supported machine foundations in Chapter 12. A few case histories are discussed in Chapter 13 and construction aspects in Chapter 14. Computer program for design of a block foundation based on principles discussed in Chapter 6 has been included in Appendix I, aud for design of a hammer foundation as in Chapter 7 has been included in Appendix II. A brief description of the commercially available programs PILAY for solution of piles and STRUDL for analysis of turbo-generator foundations is included in Appendix III. The subject matter has been developed in a logical progression from one chapter to the next. Every effort has been made to make the text selfcontained as far as possible. A comprehensive bibliography is included at the end of each chapter so that an interested reader may obtain additional information from published sources. Development in certain areas, particularly the determination of dynamic soil properties and analysis of embedded foundations and piles under
PREFACE
dynamic loads, is taking place at a very rapid rate. Analysis and design procedures may therefore undergo modifications. This fact has also been brought to the attention of the reader"' at appropriate places in the text. Thanks are due the American Society of Civil Engineers and National Research Council of Canada for permitting the use of materials from thefr publication. Acknowledgment to other copyrighted material is given at appropriate places in the text and figures. In preparing this text, several of our colleagues and graduate students have helped in a variety of ways. The authors wish to express their sincere thanks to them. Special mention must be made of Dr. Krishen Kumar, who read the entire manuscript and made useful suggestions, particularly on Chapter 12, and Dr. A Syed for his useful comments and suggestions and of Mr. Murat Hazinedarogulu for assistance in writing the computer programs. The manuscript was typed by Janet Pearson, Charlena Ousley, Allison Holdaway, and Mary Reynolds. The authors are most thankful to them for their care, painstaking efforts, and patience. John W. Koeing, technical editor at the University of Missouri, Rolla, provided editorial assistance and deserves our sincer~ !hanks. ->', <1"Acknowledgmenis are also due thiFpublishers for their cooperation during various stages of editorial and production work. A special mention must be made of the cooperation received from Everett Smethurst, David Eckroth and Linda Shapiro. It must also be mentioned that any suggestions or comments by the readers for making any improvements in the text will be highly appreciated. SHAMSHER PRAKASH VnAY
Rolla, Missouri Carbondale IL
K. PnRI
Contents
CHAPTER 1
INTRODUCTION 1.1 ;; Type of Machines 'l.~!l Foundations, 2 1.2 Design Criteria to Bi(Satisfied, 4 1.3 Relevant Codes, 9 1.4 Data Required for Design, 1 0 1.5 Significance of Soil Parameters, 1 0 References, 10
CHAPTER 2
THEORY OF VIBRATIONS 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14
Definitions, 12 Simple Harmonic Motion, 14 Free Vibrations of a Spring-Mass System, Hi Free Vibrations with Viscous Damping, 20 Forced Vibrations with Viscous Damping, 24 Frequency Dependent Excitation, 29 Systems under Transient Loads, 31 Rayleigh's Method, 34 Logarithmic Decrement, 38 Determiii!ltion of Viscous Damping, 39 Transmissibility, 41 Vibration Measuring Instruments, 42, Systems with Two Degrees of Freedom, 44 Multidegree Freedom Systems, 50 Practice Problems, 58 References, 61
12
CONTENTS
xii
CHAPTER 3 WAVE PROPAGATION IN AN ELASTIC MEDIUM ·
62
Wave Propagation in Elastic Rods, 63 3.1.1 Longitudinal Vibrations of Rods of Infinite Length, 63 3.1.2 Longitudinal Vibrations of Rods of Finite Length, 69 ,3,13 Torsional Vibrations of Rods of Infinite Length, 74 3.1.4 Torsional Vibrations of Rods of Finite Length, 76 3.2 Wave Propagation in an Elastic Infinite Medium, 76 3.3 Wave Propagation in a Semi-infinite Elastic Half Space, 84 ' 'i . ' . . ,, 3.4 ..•.Wave""Generated.. by a.Surface Footing, 91 3.5 Final Comments, 93 Practice Problems, 93 References, 93
4.9 Examples, 156 4.10 Overview, 177 Practice Problems, 182 References, 183
3.1
CHAPTER 4
DYNAMIC SOIL PROPERTIES
CHAPTER 5
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS 5.1 5.2 5.3 5.4
CHAPTER 6
6.7 6.8 6. 9 6.10
CHAPTER 7
189
Unbalanced Forces in Reciprocating Machines, 189 Unbalanced Forces in Rotary Machines, 201 Unbalanced Forces Due to Impact Loads, 205 Examples, 205 References, 211
FOUNDATIONS FOR RECIPROCATING MACHINES 6.1;;, 6.2 6.3 6.4 6.5 6.6
95
4.1 Triaxial Compression Test under Static Loads, 96 4.2 Elastic Constants of Soils, 100 4.3 Factors Affecting Dynamic Shear Modulus, 104 4.4 Equivalent Soil Springs, 118 4.5 Laboratory Methods, 122 4.5.1 Resonant Column Test, 123 4.5.2 Ultrasonic Pulse Tests, 127 4.5.3 Cyclic Simple Shear Test 128 4.5.4 Cyclic Torsional Simple Shear Test, 131 4.5.5 Cyclic Triaxial Compression Test, 133 4.6 Field Methods, 135 4.6.1 Cross-Borehole Wave Propagation Test, 135 4.6.2 Up-Hole or Down-Hole Wave Propagation Test, 136 4.6.3 Surface-Wave Propagation Test, 137 4.6.4 Vertical Footing Resonance Test, 140 4.6.5 Horizontal Footing Res6nance Test, 143 4.6.6 Free Vibration Test on Footings, 144 4.6.7 Cyclic Plate Load Test, 144 4.6.8 Standard Penetration Test, 145 4. 7 Evaluation of Test Data, 146 4.8 Damping in Soils, 147
xiii
CONTENTS
212
pesign Requirem~!'~s, 212 Modes of Vibration of a Rigid Foundation Block, 213 Methods of Analysis, 214 Elastic Half-Space Method, 214 Effect of Footing Shape on Vibratory Response, 234 Vibrations of a Rigid Circular Footing Supported by an Elastic Layer, 236 Linear Elastic Weightless Spring Method, 240 Design Procedure for a Block Foundation, 260 Examples, 268 Overview, 301 References, 303
FOUNDATIONS FOR IMPACT MACHINES
306
7.1 Methods of Analysis, 307 7.2 Design Criteria, 318 7.3 Design Procedure for Hammer Foundations, 319 7.4 Examples> 323 7.5 Overview, 328 References, 329 "ljq~'
CHAPTER 8
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES 8.1
Layout of a Typical Turbogenerator Unit, 331,
330
CONTENTS~
xiv
Loads on a Turbogenerator Foundation, 332 8.2.1 Loads Due to Normal Operation of Plant, 332 8.2.2 Loads Due to Emergency Conditions, 337 8.3 Design Criteria, 339 8.4 Design Concepts, 340 8.5 Methods of Analysis, 340 8.5.1 Simplified Methods, 341 8.5.2 Rigorous Methods, 357 8.6 Design Procedure, 363 8.6.1 Design Data, 364 8.6.2 Dynamic Analysis, 366 8.7 Examples, 371 Referentes, 374 FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES 9.1 9.2 9.3 9.4 9.5
376
Foundations for Low-Speed Rotary Machines, 376 Foundations for Machine Tools, 391 Foundations for Stamping, Forging, and Punching Presses, 392 Machines Supported on Floors, 394 Examples, 395 References, 398
CHAPTER 10 VIBRATION ABSORPTION AND ISOLATION
399
Principle of Vibration Absorption, 401 Common Vibration Absorbers, 404 10.2.1 Steel or Metal Springs, 404 10.2.2 Cork, 406 10.2.3 Rubber, 407 10.2.4 Timber, 408 10.2.5 Neoprene, 408 10.2.6 Pneumatic Absorber; 408 1 0.3 Design Procedure for Foundations on Absorbers, 41 0 10.4 Principles of Vibration Isolation with Wave Barriers, 413 10.4.1 Trench Barriers, 414 10.4.2 Pile Barriers, 420
XV
10.5 Design Procedure for Wave Barriers, 423 10.6 Methods of Reducing Vibration Amplitudes in Existing Machine Foundations, 406 10.7 Examples, 431 1 0.8 Final Comments, 436 References, 436
8.2
CHAPTER 9
.
CONTENTS
CHAPTER 11
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS. 11.1
11.:b
11.3 11.4 11.5 11.6
10.1 10.2
Elastic Half-Space Method, 439 11 .1 .1 Vertical Vibrations, 440 11 .1 .2 Sliding Vibrations, 443 11 .1.3 Rocking Vibrations, 448 11.1.4 Coupled Rocking and Sliding Vibrations, 451 11.1.5 Torsional Vibrations, 456 ·Linear Elastic Weightless Spring Method, 459 11.2.1 Vertical Vibrations, 459 11.2.2 Sliding Vibrations, 462 11.2.3 Rocking Vibrations, 464 11.2.4 Coupled Rocking and Sliding Vibrations, 468 11.2.5 Torsional Vibrations, 469 Design Procedure for an Embedded Block Foundation, 471 Examples, 4 77 Compliance-Impedance Function Approach, 482 Overview, 448 References, 490
CHAPTER 12 MACHINE FOUNDATIONS ON PILES 12.1
12.2 12.3 12.4 12.5 12.6
438
493
Analysis of Piles under Vertical Vibrations, 495 12.1.1 End-Bearing Piles, 495 12.1.2 .Friction Piles, 497 '~~ Analysis of Piles under Translation and Rocking, 517 Analysis of Piles under Torsion, 521 Design Procedure for a Pile-Supported Machine Foundation, 529 Examples, 532 Comparison of Measured and Predicted Pile Response, 541
xvi
CONTENTS
12.7 Final Comments, 547 Practice Problems, 550 References, 552 CHAPTER 13 CASE HISTORIES
554
13.1 Case History of a Compressor Foundation, 556 13.2 Case History of a Hammer Foundation, 569 13.3 Final Comments, 576 References, 576 CHAPTER 14 CONSTRUCTION OF MACHINE FOUNDATIONS
1 Introduction
578
14.1 14.2 14.3
Construction Aspects of Block Foundations, 579 Construction Aspects of Frame Foundations, 580 Erection and Interfacing of a Machine to the Foundation, 586 14.4 Gap around the Foundation, 589 14.5 Bonding of Fresh to Old Concrete, 589 14.6 Installation of Spring Absorbers, 589 References, 592 APPENDIXES
593 1 Computer Program for the De sign of a Block Foundation, 595 2 Computer Program for the Design of a Hammer Foundation, 61 0 3 Brief Description of Some Available Computer Programs, 620 4 Computation of Moment of Inertia, 624 5 Conversion Factors, 629
NOTATION
631
AUTHOR INDEX
647
SUBJECT INDEX
651
Machine foundations require the special attention of a foundation engineer. Unbalanced dynamic forces and momel),ts,are occasioned by the operation of a machine. The ;..achine foundation ilius transmits dynamic loads to the soil below in addition to the static loads due to the combined weight of the machine and the foundation. It is the consideration of the dynamic loads that distinguishes a machine foundation from an ordinary foundation and necessitates special design procedures. The foundation for the machine must therefore be designed to ensure stability under the combined effect of static and dynamic loads. In general, a foundation weighs several times as much as a machine, and the dynamic loads prod,•ced by the machine's moving parts are relatively small compared to the combined weight of the machine and the foundation (Prakash and Puri, 1969). Even though the magnitude of the dynamic load is small, it is applied repetitively over long periods of time. The behavior of the supporting soil is generally considered elastic. For the range of vibration levels associated with a well-designed machine foundation, this assumption seems reasonable. The vibration response of the machine-foundation-soil system defined by its natural frequency and the amplitude of vibration under the normal operating conditions of the machine are the two most important parameters to be determined in designing the foundation for any machine. In addition, the wave energy, which is transmitted through the underJ.¥:ing soil from the vibrating foundation, must not cause harmful effects on other machines, structures, or people io the immediate vicinity. This consideration and the operational requirements of the machine necessitate that the amplitudes of foundation vibration be limited to small values. Thus the local soil conditions and the foundationsoil interaction are important factors to be considered in the design of foundation for any machine. Satisfactory design of a machine foundation can be accomplished by systematic application of principles of soil mechanics, soil dynamics, and theory of vibrations.
2
INTRODUCTION
The initial cost of construction of a machine foundation is generally a small fraction of the total cost of the machine, accessories, and the installation, but the failure of the foundation as a result of poor design or construction can interrupt the machine's operation for long periods and cause heavy dollar losses. Great care should therefore be taken at all stages of the soil investigation and in the design and construction of these foundations to ensure their long-term satisfactory performance. There are many types of machines and each may require a certain type of foundation. The different types of machines, their special features, and the types of foundations commonly used to support them are briefly described now. The criteria used in design of these foundations, the relevant codes of practice, and the data required for their design are also discussed subsequently.
TYPES OF MACHINES AND FOUNDATIONS
3
------
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--.,
I
I
Foundation
I
--1_"///////
/
)'!_; L
Maximum vertical amplitude
Ia)
I
I
/
{.P)m Maximum rotation
(b)
Maximum translation
1.1
.::.- ~lf..
TYPES OF MACHINES AND FOUNDATIONS
II
There are many types of machines. All generate unbalanced exciting loads. In general, the various machines may be classified into three categories: 1. Reciprocating machines: This category of machines includes internal combustion engines, steam engines, piston-type pumps and compressors, and other similar machines having a crank mechanism. The basic form of a reciprocating machine consists of a piston that moves within a cylinder, a connecting rod, a piston rod and a crank (Fig. 5.1). The crank rotates with a constant angular velocity. The crank mechanism converts the translatory motion into rotary motion and vice versa. The operating speeds of reciprocating machines are usually smaller than 1200 rpm. The operation of the reciprocating machine or the crank mechanism results in unbalanced forces both in the direction of piston motion and perpendicular to it (Section 5.1). The magnitude of forces and moments will depend upon the number of cylinders in the machine, their size, piston displacement, and the direction of mounting. If one considers only the unbalanced force in the direction of piston motion in a machine with only one cylinder that is mounted centrally on a rigid foundation (Fig. l.la), the motion of the foundation will be only up and down. A two-cylinder reciprocating machine under similar conditions mounted centrally on a rigid foundation, will generate an oscillatory motion and no translation (Fig. 1.1b). Similarly, if a piston·is mounted horizontally, it will give rise to an unbalanced force and a moment on the foundation. The foundation will therefore undergo both translation and rotation simultaneously (Fig. l.lc). In the case of a two-cylinder machine mounted horizontally, the unbalanced forces in a plane parallel to the base of the foundation generate a couple (Fig. l.ld). This results in a motion that is similar to the motion of a torsional pendulum. It therefore becomes clear
II
1/ /
/
(c)
(d)
Figure 1.1. Types of motion of a rigid foundation due to unbalanced forces of reciprocating machines: (a) pure vertical translation; (b) pure rocking; (c) simultaneous horizontal sliding and rocking; and (d) pure torsional oscillations.
that the motion of a fpundation depends upon the resulting forces and moments imparted to it by the machine. Chapter 6 shows that the stresses at the base of a foundation supporting a reciprocating machine may be uniform compression as in Fig. l.la, nonuniform compression as in Fig. l.lb, or uniform or nonuniform shear depending on the nature of dynamic loads. Reciprocating machines are very frequently encountered in practice. A rigid block-type foundation is usually provided for these machines. The vibrations of such a foundation are essentially due to dynamic deformations in the soil. 2. Impact machines: Incluifed in this category are such machines as forging hammers, which produce impact loads. These machines consist of a falling ram, an anvil, and a frame (Fig. 7.1). Forging hammers are divided into two groups (Barkan, 1962): drop hammers for die stamping and forge hammers proper. Free forging operations are usually preformed by forge hammers. The anvil and the side frame, are generally mounted separately (Fig. 7.1b). The side frame, together with guides for the ram, contributes to the precision of
4
INTRODUCTION
the blows required in forging. The foundation block under the anvil serves as a support for the entire hammer. The speeds of operation of both these hammers are usually low and range from 60 to 150 blows per minute. Their dynamic loads attain a peak in a very short period of time and then practically die out. The unbalanced force occasioned by the impact lasts only a fraction of a second. In between two successive blows, the foundation and anvil vibrate freely. The analysis of the hammer foundation, therefore, proceeds along lines that are different from those for the analysis of a reciprocating machine foundation. A massive block foundation is usually provided for impact machines. Vibration absorber pads are placed between the anvil and the foundation to absorb some of the vibrations. 3. Rotary machines: High-speed machines, such as turbogenerators, turbines, and rotary compressors, may have speeds that exceed 3000 rpm and may even reach 10,000 rpm. Steam turbines have elevated pedestal foundations that may consist of an arrangement of a base-slab and vertical columns, which support at their tops a grid of beams on which skid-mounted machinery rests. Each element of such a foundation is relatively flexible (Fig. 8.1) as opposed to the rigid block-type foundation.
1.2
DESIGN CRITERIA TO BE SATISFIED·
A machine foundation should meet the following requirements in order to be satisfactory (Prakash, 1981). For static loads 1. The foundation should be safe against shear failure. 2. The foundation should not settle excessively.
These are standard requirements that are the same for all footings. For dynamic loads
1. There should be no resonance. That is, the natural frequency of the machine-foundation-soil system should not coincide with the operating frequency of the machine. In fact, a zone of resonance is generally defined, and the natural frequency of the soil foundation system must lie outside this zone (Fig. 1.2). The foundation may thus be designated as "high tuned" when its fundamental frequency is greater than the operating speed or as "low tuned" when its fundamental frequency is lower than the operating speed. This concept of a high or low tuned foundation is illustrated in Fig. 1.2.
DESIGN CRITERIA TO BE SATISFIED 2.5
2.0 ~
••c
c 0
I
5
I
I
~
I
I
-
Low tuned
tuned
~0 • Q
•• u.
·~
i5
1.5
cS ~·
~
c 0
1.0
-
)
-
"•u
~
"E "'
0.5 -
0.0 0.0
-
I 1.0
I
2.0
:~~~r~quency ratio ~ Figure 1.2.
3.0
4.0
oper"~trig speed of machine
5.0
6.0
fundamental··frequency of foundation
Tuning of a foundation.
2. The amplitudes of motion at the operating frequencies should not exceed the permissible values. These limiting amplitudes are generally specified by the machine's manufacturer. 3. The design should be such that the natural frequency of the foundation-soil system will not be a whole number multiple of the operating frequency of the machine to avoid resonance with higher harmonics (Section 5.1). 4. Vibrations occasioned by the machine operation should not be annoying to persons or harmful to other precision equipment or machines in the vicinity or to adjoining structures. In addition to the preceding criteria, geometrical layout of the foundation may be influenced by operational requirements of the machine. The failure condition of vibril;(ing foundations is reached when the motion exceeds a limiting value, which thay be expressed in terms of the velocity or acceleration of the movement of the foundation. For steady-state vibrations, these may be expressed in terms of allowable displacements at specified frequencies (Richart, 1962). Figure 1.3 illustrates the order of magnitudes that are involved in the criteria for determining the dynamic response. Five curves delimit the zones of vibrations to which persons are sensitive when
standing close to the vibrating machinery. These zones range from "not noticeable" to "severe." The boundary between "not noticeable" and
INTRODUCTION
6
+ From Reiher & Meister (1931)-(steady state vibrations) • From Rausch (1943Hsteady state vibrations) 6 From Crandell (1949)-(due to blasting) .OJ
1
r005
~
0.02
Frequency, cpm
Figure 1.3. Limiting amplitudes of vibrations for a particular frequency. (After Richart, 1962.)
"barely noticeable" in Fig. 1.3 is defined by a line that represents a peak velocity of about 0.01 in/sec (0.25 mm/sec), and the line separating the
100 rpm
zones of "easily noticeable" and "troublesome" represents a peak velocity
Figure 1.4. Criteria for vibrations of rotating machinery. Explanation of classes:
of 0.10 in/sec (2.5 mm/sec). The shaded area in Fig. 1.3 indicates the "limits for machines and machine foundations." This represents a peak velocity of 1.0 in/sec (25.5 mm/sec) below about 2000 cpm and corresponds to a peak acceleration of 0.5 g above about 2000 cpm. It should be noted that this shaded area indicates a limit for safety and is not a limit for the satisfactory operation of a machine.
AA Dangerous. Shut it down now to avoid danger. A Failure is near. Correct within two days to avoid breakdown. B Faulty. Correct it within 1 0 days to save maintenence dollars. C Minor faults. Correction wastes dollars. D No faults. Typical new equipment. "li:. This is a guide to aid judgment, not to replace it. Use common sense. Use with care. Take account of all local circumstances. Consider: safety, labor costs, downtime costs. (After Blake, 1964.) Reproduced with permission from Hydrocarbon Processing, January 1964.
The importance of a machine and its sensitivity to operational conditions
along with the cost of installation and losses due to interruption (down time) determine the limit of the motion amplitudes for which the foundation must be designed (Richart, 1976). Permissible amplitudes at operating speed can be established from Fig. 1.4 (Blake, 1964). The vibration amplitudes are generally specified at bearing level of the machine. The concept of "service factor" was introduced by Blake (1964). The
7
INTRODUCTION
8
service factor is an indication of the importance of a machine in an installation. Typical values of service factors are listed in Table 1.1. Using the concept of service factor, the criteria given in Fig. 1.4 can be used to define vibration limits for different classes of machines. Also with the· introduction of the service factor, Fig. 1.4 can be used to evaluate the performance of a wide variety of machines. The concept of service factor is explained by the following examples. A centrifuge has a 0.01 in (0.250 mm) double amplitude at 750 rpm. The value of the service factor from Table 1.1 is 2, and the effective vibration therefore is 2 x 0.01 = 0.02 in (0.50 mm). This point falls in Class A in Fig. 1.4. The vibrations, therefore, are excessive, and failure is imminent unless the corrective steps are taken immediately. Another example is that of a link-suspended centrifuge operating at 1250 rpm that has 0.0030 in (0.075 mm) amplitude with the basket empty. The service factor is 0.3, and the effective vibration is 0.00090 in (0.0225 mm). This point falls in class C (Fig. 1.4) and indicates only minor fault. General information for the operation of rotary machines is given in Table 1.2 (Baxter and Bernhard, 1967). These limits are based on peakvelocity criteria alone and are represented by straight lines on Fig. 1.4. The maximum velocity for the lower limit of the "smooth" range is 0.01 in/sec (0.25 mm/sec) in Table 1.2 and the lower limit of the range "barely noticeable to persons" is also 0.01 in/sec (0.25 mm/sec) in Fig. 1.3. The lower limits of "slightly rough" for machines is 0.16 in/sec ( 4.0 mm/sec) in Table 1.2 whereas the value for "troublesome to persons" is 0.10 in/sec (2.5 mm/sec) in Fig. 1.3. Also the danger limit of "very rough" is 0.63 in/ sec (15.75 mm/sec) in Table 1.2 whereas Rausch's limit for machines is 1.0 in/sec (25.0mm/sec) in Fig. 1.3 (Rausch, 1973). Tbus Table 1.2 and Fig. 1.3 are similar (Richart, 1976). Table 1.1. Service Factorsa
Single-stage centrifugal pump, electric motor, fan Typical chemical processing equipment, noncritical Turbine, turbogenerator, centrifugal compressor Centrifuge, stiff-shaftb; multistage centrifugal pump
Miscellaneous equipment, characteristics unknown Centrifuge, shaft-suspended, on shaft near basket
Centrifuge, link-suspended, slung a
1 1 1.6 2 2
0.5 0.3
Effective vibration= measured single amplitude vibration, in inches multiplied by the service
9
RELEVANT CODES
Table 1.2. General Machinery-Vibration-Severity Data Horizontal Peak Velocity
(in/sec) <0.005 0.005~0.010 0.010~0.020
0.020~0.040
Machine Operation Extremely smooth Very smooth Smooth
0.040~0.080
Very good Good
0.080~0.160
Fair
0.160~0.315
Slightly rough Rough Very rough
0.315~0.630
>0.630
Source: After Baxter and Bernhard (1967). Reproduced by permission of American Society of Mechanical Engineers, New York, NY.
1.3
RELEVANT CODES.. '""
The criteria for satisfactory design of a machine foundation are described in Section 1.2. Methods of analysis of foundations for different machines are described in Chapters 6 through 12. These enable the engineers to design safe and economical foundations. Because installation of heavy machinery has assumed increased importance throughout the world, their foundations have to be specially designed to take into consideration both the vibrational characteristics of the load and the properties of the supporting soil, which is subject to dynamic conditions. Although many considerations relating to the design and construction of such machine foundations are specified by the machines' manufacturers, other details must comply with the general design principles that govern machine foundations. With this objective in view, codes for the design and construction of machine foundations have been written in West Germany (DIN 4024, 4025), Russia (CH-18-58), Hungary (MSZ 15009-64), and India (Indian Stardards Institution, 1966, 1967, 1968, 1969, 1970). Unfortunately, no such codes have been written in the United States (1987). For design of turbogenerator foundations, leading manufacturers such as Westinghouse, General Electric, and Honeywjfii have their own design criteria. The designer must familiarize himselr with the relevant standards (code of practice) prevalent in the country in which he works. t
factor. Machine tools are excluded. Values are for bolted-down equipment; when not bolted, multiply the service factor by 0.4 and use the product as the service factor. Caution: Vibration is measured on the bearing housing, except as stated. " Horizontal displacement on basket housing. Source: After Blake (1964). Reproduced with permission from Hydrocarbon Processing, January 1984.
t American Concrete Institute is working on the codes for design of foundations subjected to dynamic machinery. But no codes have been finalized so far (1987). Naval Facilities Engineering Command (1983) describes only elementary criteria for design of machine foundations.
10
1.4
INTRODUCTION
DATA REQUIRED FOR DESIGN
To arrive at a satisfactory design for a machine foundation, all pertinent data must be procured. This data must include information on layout of the machine, operating speeds, unbalanced loads generated by the machine operation, point of application of the unbalanced loads, and permissible amplitudes of vibration. Details of the data required are discussed separately for each type of machine in Chapters 6 through 12. Besides the preceding information about the machine, detailed information on the static and dynamic properties of the supporting soil should form an essential part of the data that must be procured.
1.5
SIGNIFICANCE OF SOIL PARAMETERS
The reader must have realized by now that the design of a machine foundation essentially involves determination of the vibration characteristics (natural frequencies and vibration amplitudes) of the machine-foundationsoil system. Besides the machine and the foundation data, the soil properties are a rather significant input parameter governing the computed. response, i.e., the predicted behavior of this system. Depending upon the method of, analysis (the elastic half space or the linear spring theory, Chapter 6), the mode of inputting the soil parameters may vary. It will be shown in Chapter 4 that the number of parameters affecting the relevant soil properties are large and sometimes quite complex. Fortunately, the determination of soil properties for the design of machine foundations has reached a stage where fairly precise evaluations can be made for given loading conditions. The soil parameters can be determined in a realistic manner after a careful evaluation of the field or laboratory test data by following the procedure suggested in Chapter 4 (Section 4.7). The importance of soil parameters must always be kept in mind by an intelligent designer.
REFERENCES Barkan, D. D. (1962). "Dynamics of Bases and Foundations."
McGraw~Hill,
New York.
Baxter, R. L., and Bernhard, D. L. (1967). Vibration tolerances--for industry. Am. Soc. Mech. Eng. [Pap.] 67-PME-14.
Blake, M. P. (1964). New vibration standards for maintenance. Hydrocarbon Process. Pet. Refiner 43 (!), 111-114. Crandell, F. J. (1949). Ground vibrations due to blasting and its effects on structures. J. Boston Soc. Civ. Eng. 36 (2). Also reprinted in Contributions to Soil Mech. BSCE 1941-1953, pp. 206-229. CH-18-58 Soviet Code of Practice for Foundations Subjected to Dynamic Effects.
REFERENCES
11
DIN 4024 Stutzkonstruktionen fiir rotierende Machinen (Supporting structures for rotary machines). DIN 4025-1958 Fundamente fiir Ambo-Hiimmer (Schabotte-Hammer) Richtilinten fur die Konstruktionen-Bemessung und ausfuhrung (Criteria for the design and construction of foundations for anvil-hammer construction). Indian Standards Institution Construction of Machine Indian Standards Institution Construction of Machine
(1966). "Indian Standard Code of Practice for Design and Foundations," Part II, IS: 2974. lSI, New Delhi, India. (1967). "Indian Standard Code of Practice for Design and Foundations," Part III, IS: 2974. lSI, New Delhi, India.
Indian Standards Institution (1968). "Indian Standard Code of Practice for. Design and Construction of Machine Foundations," Part IV, IS: 2974. lSI, New Delhi, India. Indian Standards Institution (1969). "Indian Standard Code of Practice for Design and Constructi?n of Machine Foundations," Part I, IS: 2974 (rev.). lSI, New Delhi, India. Indian Standards Institution (1970). "Code of Practice for Design and Construction of Machine Foundations," Part V, IS: 2974. lSI, New Delhi, India. MSZ 15009-64 Hungarian Code for Design of Machine Foundations. Naval Facilities Engineering Command (1983). "Soil Dynamics, Deep Stabilization, and Special Geotechnical Construction," Design Manual 7.3, NAVFAC DM-7.3. Dept. of the Navy, Naval Facilities Engineering Command, Alexandria, Virginia. Prakash, S. (1981). "SoiliJ?ynamics." McGraw-Hi!l;;;)'~:cw York. Prakash, S., and Puri, V.K. (1969). Design of a "tYPical machine foundation by different methods. Bull.-Indian Soc. Earthquake Techno/._6, 109-136. Rausch, E. (1943). ''Maschinenfundamente und andere dynamische Bauaufgaben." VDI Verlag, Berlin. Reiher, H., and Meister, F. J. (1931). Die Empfindlinchkeit der Menschen gegen Erschiitterungen. Forsch. Geb. lngenieurwes. 2 (11), 381-386. Richart, F. E., Jr. (1962). Foundation vibrations. Trans. Am. Soc. Civ. Eng. 127, Part I, 863-898. Richart, F. E., Jr. (1976). Foundation vibrations. "Foundation Engineering Hand Book," Chapter 4. Van Nostrand-Reinhold, New York.
2
DEFINITIONS
I
Theory of Vibrations
13
Forced Vibrations: Vibrations that are developed by externally applied exciting forces are called forced vibrations. Forced vibrations occur at the frequency of the externally applied el
It was mentioned in Chapter 1 that machine foundations may be subjected
to either periodic loads or impact loads. A periodic load may be represented by a harmonic function, i.e., a sine or a cosine function. The problem of impact loads can be easily solved with an initial boundary value approach. For machine-foundation analysis it is only necessary to be familiar with the simple theoretical concepts of harmonic vibrations and with the methods needed to solve such problems. Although a block foundation may have six degrees of freedom, it is seldom necessary to solve for a system with more than two degrees of freedom. This simplifies the study of theory of vibrations. In analysis of flexible foundations, we have to use other solution techniques. This chapter is tailored to provide basic concepts on vibration problems of simple systems such as spring-mass-dashpot systems. These concepts provide the basis for attempting solutions to the machine-foundation problem. 2.1
=
m
1
X
Simple pendulum
n = 2
(a)
(b)
DEFINITIONS
Period of Motion: If motion repeats itself in equal intervals of time, it is called periodic motion. The time that elapses when the motion is repeated once is called its period. Aperiodic Motion: Motion that does not repeat itself at regular intervals of time is called aperiodic motion. Cycle: Motion completed during a period is referred to as a cycle. Frequency: The number of cycles of motion in a unit of time is c.alled the frequency of vibrations. Natural Frequency: If an oscillatory system vibrates under the action of
forces inherent in the system and no externally applied force acts, the frequency with which it vibrates is known as its natural frequency. 12
n
(d)
(c)
Figure 2.1. Systems illustrating degrees of freedom. (a) System with one degree of freedom (n = 1). (b) System with two degrees of freedom (n 2). (c) Systems with three degrees of
=
freedom (n = 3), and (d) Systems with infinite degrees of freedom (n --7 oo),
THEORY OF VIBRATIONS
14
position of this system is completely defined by the angle e only. Hence it is a system with one degree of freedom, that is, n is equal to 1. In Figs. 2.1 b and c, two and three independent coordinates are needed to fully describe the motion of the two systems respectively. Hence they constitute systems with two and three degrees of freedom. The number of coordinates necessary to completely describe the motion of an elastic simply supported beam is infinite. Hence the beam in Fig. 2.1d constitutes an infinite degree of freedom system. Resonance: If the frequency of excitation coincides with any one of the natural frequencies of the system, the condition of resonance is reached. The amplitudes of motion may be excessive at resonance. Hence, in the design of machine foundations, the determination of the natural frequencies of a system is important. Frequency Ratio: The ratio of the forcing or operating frequency to the natural frequency of the system is referred to as the frequency ratio. Principal Modes of Vibration: A system with n degrees of freedom vibrates in such a complex manner that the amplitude and frequencies do not appear to follow any definite pattern. Still, among such a disorderly array of motions, there is a special type of simple and orderly motion that has been termed the principal mode of vibration. In a principal mode, each point in the system vibrates with the same frequency, which is one of the system's natural frequencies. Thus, a system with n degrees of freedom possesses n principal modes with n natural frequencies. More general types of motion can always be represented by the superposition of principal modes. Normal Mode of Vibration: When the amplitude of motion of a point of the system vibrating in one of the principal modes is made equal to unity, the motion is called the normal mode of vibration. Damping: Damping is associated with energy dissipation and opposes the free vibrations of a system. If the force of damping is constant, it is termed Coulomb damping. If the force of damping is proportional to its velocity, it is termed viscous damping. If the damping in a system is free from its material property and is contributed to by the geometry of the system, it is called geometricalt or radiation damping.
2.2
SIMPLE HARMONIC MOTION
15
in which w is the circular frequency in radians per unit time. We can represent z by the vertical projection of a rotating vector of length Z that rotates with a constant angular speed of w, onto a vertical diameter (Fig. 2.2). Because the motion repeats itself after 2Tr radians, a cycle of motion is completed when wT=2Tr
(2.2a)
T= 2Tr
(2.2b)
or w
in which Tis the time period of motion. The frequency fis the inverse of the time period; hence 1 w (2.3) f= I' =z'lr In order to determine the velocity and acceleration of motion, we differentiate Eq. (2.1) with respect. to time, t: Velocity= i = wZ cos wt = wZ sin( wt +
~)
(2.4)
and Acceleration= i = -w 2 Z sin wt = w 2 Z sin(wt + Tr)
(2.5)
Acceleration= -w 2z
(2.6)
or
Equations (2.4) and (2.5) show that both velocity and acceleration are also harmonic and can be represented by the vectors wZ and w2 Z, which rotate at the same speed as Z, i.e., w rad/unit time. These, however, lead the displacement vector by Trl2 and Tr respectively.
r-1 cycle---1
SIMPLE HARMONIC MOTION
The simplest form of periodic motion is harmonic motion, which is represen· ted by sine or cosine functions. Let us consider the harmonic motion represented by the following equation:
z = Z sin wt I" For an explanation, see Section 3.4.
wt
(2.1) Figure 2.2.
Vectorial representation of harmonic motion.
THEORY OF VIBRATIONS
wZ
I I
Figure 2.3.
' \
\
I
I
/
\ , ....,'A....
Vectorial representation of displacement (z), velocity (Z), and acceleration (i).
17
FREE VIBRATIONS OF A SPRING-MASS SYSTEM
position of the system corresponding to this state is referred to as the equilibrium position. In Fig. 2.4c, the m~ss IS shown displaced by a d~stance z in the downward direction; the maximum downward deflection IS. Zmax (Fig. 2.4d). The double amplitude at any time is shown in Fig. 2.4e. Figure 2.4f shows the free body diagram. . If the mass is released from the extreme lower position (Fig. 2.4d), It starts to oscillate between the two extreme positions (Fig. 2.4e). If there IS no resistance to these oscillations, the mass will contmue to vibrate (theoretically) indefinitely. If we neglect the mass of the spring, the equation of motion can be written as (2.8a)
2, F= mi In Fig. 2.3, vertical projections of these vectors are plotted against the time axis t. The angles between the vectors are known as phase angles. Thus the velocity vector leads the displacement vector by 90°; the acceleration vector leads the displacement vector by 180° and the velocity vector by 90°.
in which E F is the sum of all forces in the vertical direction. If th~ sign convention shown in Fig. 2.4 is used and the inertial force acts opposite to acceleration, the equation of motion becomes
-(kli"" + kz) +wc=mg) 2.3
=
mi
(2.8b)
FREE VIBRATIONS OF A SPRING-MASS SYSTEM
Figure 2.4a shows a spring of stiffness kin an unstretched position. If a mass m of weight W is attached at its lower end, the mass-spring system occupies the position shown in Fig. 2.4b. The deflection li""' of the spring from the undeflected position is (2.7)
Because kli,"' is equal to W, we get (2.8c)
mi+kz=O
Equation (2.8c) is a second-order differential equation, and its general solution must contain two arbitrary constants, whtch can be evaluated from initial conditions. The solution of this equation can be obtained by substituting
in which k is the spring constant, defined as force per unit deflection. The Sign convention z,
z, z·
t+
(2.9) in which A and B are arbitrary constants, and wn is the natural circular frequency of the system. . . If we substitute the preceding solutiOn mto Eq. (2.8c), we get (2.10)
which gives 2-
lal
(b)
(c)
(d)
(e)
If)
Figure 2.4. Spring-mass system. (a) Unstretched spring; (b) equilibrium position; (c) mass in oscillating position; (d) mass in maximum downward position; (e) mass in maximum upward position; and (f) free-body diagram of mass corresponding to (c).
w~~-
k m
or
w = n
[k
\J;
(2.11)
When w, Tn is equal to 2"1T, one cycle of motion is completed. This yields the following expression for natural period:
18
THEORY OF VIBRATIONS
(2.12)
FREE VIBRATIONS OF A SPRING-MASS SYSTEM
19
Arbitrary constants A and B in Eq. (2.9) can be determined from the initial conditions. Let the initial conditions be defined by the following values:
The natural frequency of vibration is the number of cycles completed in unit time and is the reciprocal of the time period T,. Therefore, (2.13)
When t is equal to zero, z = Z0
and
i = V0
(2.16)
By substituting these values into Eq. (2.9), the solution can be obtained in the form
Equation (2.13) can also be written in the following form:
(2.17) (2.14a)
Other types of solutions of Eq. (2.8c) can be written in the following forms: (2.18)
Now and
W
mg
k
k
=
(2.14b)
=Ostat
z
= A
exp(iw,t) + B exp(- iw,t)
(2.19)
'';;:;:''~,-
Therefore,
EXAMPLE 2.3.1
(2.15)
A mass supported by a spring has a static deflection of 0.25 mm. Determine its natural frequency of oscillation. Solution
Equation (2.15) shows that natural frequency is a function of static deflection. When g is equal to 9810 mm/sec 2 and i5 stat is expressed in millimeters, the frequency in hertz can be shown in graphic form as in Fig. 2.5.
1 rg1 f,=y-;:---=2 2 7r
0 stat
7T
~9810 =31.541Hz 025 '
EXAMPLE 2.3.2
Determine the spring constant for the system of springs shown in Fig. 2.6. 40
30
"N
~ 20 ...§
10
~
......_
r--
0 0
2
4
6
8
10
Ostat, (mm)
Figure 2.5. Relationship between natural frequency and static deflection.
(a)
(b)
Figure 2.6. Equivalent spring constants: (a) springs in series; (b) springs in parallel.
20
THEORY OF VIBRATIONS
21
FREE VIBRATIONS WITH VISCOUS DAMPING
Solution (a) On application of a unit load to the system of springs in Fig. 2.6a, the total deflection is 1 1 kl + k, - + - = -7-c--'k, kl klk2
Hence, the equivalent spring constant is given by
k,,,
=
k
klk2 +k I
2
_l_ __
If k 1 = k 2 = k, the k,q, = k/2. (b) Let us consider that a unit load is applied at c to the system of springs shown in Fig. 2.6b. It is shared at a and bin the ratios of x 2 /(x 1 + x 2 ) and x 1 /(x 1 + x 2 ). The deflection of points a and b are x 2(x 1 + x 2 ) x 1/k 1 and x 1 1(x 1 + x 2 ) X k 2 , respectively. Therefore, the deflection of point c is
T' m m
Figure 2.7.
(a) Spring-mass-dashpot system; (b) free-body diagram.
!"~onstant
=
(x 1 + x 2 ) 2 (x71k 2 + x;lk 1 )
k)" 0
which gives us s
FREE VIBRATIONS WITH VISCOUS DAMPING
All real systems exhibit damping. When the force of damping Fd is proportional to velocity, it is termed viscous damping. Thus
2
c m
k m
(2.23)
+-s+-=0
Therefore, k
(2.20) in which c is the damping constant or force per unit velocity, FL-IT. Figure 2.7a shows a spring-mass-dashpot system. If the mass is displaced by a distance z below the position of static equilibrium, then the free-body diagram can be represented by Fig. 2.7b. By using the sign convention shown in this figure, the equation of motion can b_e written as
mi+ci+kz=O
By substituting this
" (s+ms+me= ' c
If x 1 = x, = x and k 1 = k, = k, then k,,, = 2k.
2.4
will~;;;::determined later.
in which s is that solution into Eq. (2.21), we obtain
Hence, the resulting equivalent spring constant at c is
k,,,
(b)
(e)
(2.21)
The solution to this equation may be written in the form
(2.22)
m
(2.24)
"' and the general solution can be written as follows: (2.25)
in which A and B are arbitrary constants depending upon the initial conditions of motion. If the radical in Eq. (2.24) is zero, the damping is said to be critical damping c,, and we obtain
= '5._ = w' (!..s.)' 2m m
n
Le.,
(2.26)
22
THEORY OF VIBRATIONS
The ratio of actual damping c to critical damping c, is defined as the damping factor g:
or
z =A exp(-gw,t) exp( +i~w,t) + B exp(-l;w,t) exp(-i~w,t)
(2.27)
=
exp- (gwnt)[A cos V(l- g')wnt + iA sin V(l- O'wnt)
V
+ B cos (1- g')w,t- iB sin
Now,
(2.28)
23
FREE VI ORATIONS WITH VISCOUS DAMPING
or
z
exp( -l;wnt)( C cos
=
V(1- /; )w,t] 2
V(1 -
1;) 2w,t + D sin
V(1 -
/;')w,t)
(2,33)
By substituting the preceding relationships into Eq. (2.24), we get in which Cis equal to A+ B, and D is equal to i(A- B). (2.29) 3
The nature of the ensuing motion depends upon the values of roots s 1 and s 2 , and hence on the magnitude of damping (in terms of critical damping) present in the system. Three different cases of interest are considered here. CASE I. /;
>1
/; =
~
~1---N·>" r-'~ 2
' -l:r
I
When I;> 1, both s 1 and s 2 are real and negative and z (Eq. 2.25) decreases as t increases but it never changes sign. Such a system is called overdamped or nonoscillatory. A typical solution for g = 2 is shown in Fig. 2.8a. If an initial displacement is given to such a system, the mass is pulled back by the springs and dampers absorb all the energy by the time the mass returns to the initial position. CASE II.
2
.. ,.,,_
"'--. r-:-. 't
0
I
2
3
5
4
6
7
8
w,t I
2
1
~ I
(a)
When I;= 1, Eq. (2.29) gives s 1 = s 2 = -w,. The solution becomes (2.30) The values of z for g = 1 are shown in Fig: 2.8a, from which it is seen that z decreases as t increases but never changes sign. Hence such a system does not oscillate. This system is known as "critically" damped and g = 1 is the minimum value of damping for no oscillations in the system. CASE III.
g< 1
When damping is less than the critical damping (I; < 1), the values of s1 and s, (Eq. 2.29) are obtained as
s 12 = (-1; ± i~)wn
(2.31) I
I
/
/
/
The general solution then becomes (b)
z =A exp[(-g + i~)w.t] + B exp[(-g- i~)w,t] (2.32)
Figure 2.8. (a) Free vibrations with g <1.0.
g=
2, and
g=
1.0. (b) Free damped oscillations for
24
THEORY OF VIBRATIONS
Thus, the natural circular frequency equals
in viscously damped vibrations
wnd
(2.34)
Sign convention
z·
z, Z,
Equation (2.33) can then be written as kz + cZ
(2.35) in which Z 0 and > are arbitrary constants depending upon the initial conditions. Figure 2. Sb shows typical damped oscillations when g is less than 1. 0.
2.5
Equilibrium position
_L_ _ _ T' m m
FORCED VIBRATIONS WITH VISCOUS DAMPING F0 sin wt
Figure 2.9a shows a spring-mass-dashpot system under the action of a force of excitation, F, such that
F= F0 sin wt
F0
Ia I
wt
(2.36) Z = wZo
in which w is the frequency of excitation. The free-body diagram is shown in Fig. 2.9b and the equation of motion is
mi + ci + kz = F0 sin w t
l
(b)
z· =
(2.37)
90"
<)Zo
90"
Zo
Directions
The solution to this equation is
(c)
z = Z 0 sin(wt- >)
(2.38)
Then
Fo
i= wZ0 cos(wt- >)
(2.39)
or
mw 2Zo a
+ 7T/2)
(2.40)
= w 2 Z 0 sin(wt- > + 7•)
(2.41)
i = wZ0 sin(wt- >
'l;j~-
and d
i
Figure 2.9c shows z, i, and
z vectors at any particular instant. The force
b Forces (d)
Figure 2.9. Forced
vib~ations
with viscous damping. (a) Spring-mass-dashpot. (b)
Free~body
diagram. (c) Vectorial representation of z, i and !i in space. (d) Vectorial solution of forces.
25
26
THEORY Of VIBRATIONS
in the spring is opposite to z, hence it can be represented by Oa in Fig. 2.9d. Similarly, the damping force, cwZ0 , acts in the opposite direction to that of the velocity and hence is represented by Ob. Similarly, Oc represents the 2 inertial force, mw Z0 , which acts opposite to acceleration. The resultant of these forces is Fb, which is represented by Od, and must be equal in magnitude and opposite in sign to F 0 • Thus, the displacement vector lags behind the force vector by >. From Fig. 2.9d, we get
FORCED VIBRATIONS WITH VISCOUS DAMPING
27
From Eq. (2.44b) we get Z0
1
ll,.
=
2 2 r )
[(1-
+ (2tr) 2 f
(2.44d)
12
Hence, 20
ast
= N = magnification factor
(2.45a)
1
or
N
zo=y (k-mw)22 +(cw) 2
(2.42)
=
Similarly,
and
>=tan >=tan
cw
-1
k-mw 2
(2.43)
Equation (2.42) can be expressed in non-dimensional terms as follows: F0 /k
Zo=-Y~(=1=-=m=w~~~k)~'=+=(~c=w=lk~)2
_1
2gr - -2
(2.44a)
180" ~0
Now, Fufk is equal to the static deflection ll,. of the system under the action of F0 . Also,
mw -=
(
-w )'
k
<>
3.0
w
"" goo
0.05 I 0.10
=r 2
~
~
0.15t_.£.'&.
1 -
w"
0.25 2. 0
in which r is the frequency ratio, and
'7 ~
~~ 1\
N
cc
L0
..............
I
-.........__
(2.44b) If there is no damping present, i.e., by
g = 0, undamped amplitude Z 0 is given
~ I~
0
LO
0.05 0.150.375-
fi/ ---r---- k', ~ f.-
LO
/
~ 0
03:75 0.50
(b)
1
2
4
3
5
Frequency ratio~ W0
/
Therefore,
(2.46)
1-r
Figure 2. t~· is a plot of N anct"
"versus frequency ratio r, for r varying from 0 to 5.
I
2
(2.45b)
-Vr=(=1-=r'""l""'=+""(2=g=rl""'
(a)
~
~
2.0
3.0
4.0
50
Frequency ratio .:::_ W0
Figure 2.10.
(2.44c)
(a) Magnification factor and (b) phase angle, versus frequency ratio in forced
vibrations. (After Thomson, 1972, p. 48. Reprinted by permission of Prentice-Hall, Englewood Cliffs, New Jersey.)
THEORY OF VIBRATIONS FREQUENCY-DEPENDENT EXCITATIONS
29
Effect of Frequency Ratio r for a Particular Case ( t = O)
When r is equal to 1, the phase angle > is 90' for all values of damping, except when I; is equal to 0. When r is less than 1, the phase angle is less than 90', and when r is greater than 1, the phase angle is greater than 90'. The maximum amplitude of motion when r is equal to 1 and I; is greater than 0 is expressed by Eq. (2.47):
From Eq. (2.45b), N=-1-2 1- r
When r=O
N=1
F,
When r = 1
(2.47)
cw
When
r->en
N=O
. When r is equal to 1, resonance occurs, and the amplitude tends to be mfimte. Introduction of damping reduces the resonant amplitudes to finite values. . The phase angle > is zero if r is less than 1; the displacement z is in phase With the exc1tmg force, F0 , and > is equal to 1800 if r is greater than 1. Effect of Damping
The corresponding vector diagram is shown in Fig. 2.11. The solution given by Eq. (2.44a) is a steady-state solution, which is important in most practical problems. However, there are transient vibrations initially that correspond to the solution given by Eq. (2.35). These vibrations, of course, die out in the first few cycles.
2.6
t
As the dampi~g increases, the peak of the magnification factor shifts slightly to the left. This IS due to the fact that maximum amplitudes occur in damped VIbratiOns when the forcing frequency w equals the system's damped natural frequency, wnd [Eq. (2.34)], which is slightly smaller than the undamped natural frequency, wn.
FREQUENCY-DEPENDENT EXCITATION
In many probl~Iils of machine fouii'iflitions, the exciting force depends upon the machine's operating frequency. Figure 2.12 shows such a system mounted on elastic supports with m 0 representing the unbalanced mass placed at eccentricity e from the center of the rotating shaft. The unbalanced force is F = (m 0 ew 2 ) sin wt. Therefore, the equation of motion may be written as follows: d 2z (M- m 0 ) - 2 dt
d2 (z dt 2
dz dt
.
+m0 -
+ e sm wt) = - kz- c -
0
'
Ia I
wt z
Fo M
9 "" goo
cwZo
Figure 2.11. Vector diagram at resonance in a damped system under forced vibrations. (a) Displacement, velocity, and acceleration. (b) Forces.
(b)
=
Fo
~k/2 0!
!'
~k/2
0!
//,/#lW##/mrw####J#m/,7/ Figure 2.12.
Force of excitation due to rotating unbalance.
(2.48a)
30
THEORY OF VIBRATIONS
SYSTEMS UNDER TRANSIENT lOADS
By rearranging the terms in the preceding equation, we get
Mi + ci + kz
m 0 ew 2 sin wt
=
h 3.0
(2.48b)
z=
Y(k- Mw
2 2 )
.
+ (cw ) 2
sm wt
(2.49a)
0.25
z0 =
m 0 ew
'"'
:;:
" 0
2 2
+(cw)
2
ro
•ro
0.15
90" t---
1.0
h
~
~
~
~
0
1.0
2.0
3.0
4.0
5.0
Frequency ratio~
(b)
""
~
k-:::0.50
.o
(2.49b)
and 0
cw tan¢= k-Mw 2
0
tv
K, ~
M
~
0.375
2
v(k-Mw)
0.50
•
~~
01 .
0.05 0.25
~
0.
1L
Therefore, the maximum amplitude Z 0 is given by
1 o.~5 t-
2. 0 2
180"
t-0.10
In this system, M also includes m 0 . Equations (2.48b) and (2.37) are 2 similar, except that m 0 ew appears in Eq. (2.48b) in place of F in Eq. 0 (2.37). The solution of this equation may therefore be witten as, m 0 ew
31
~
/Y v
~ 1--\,1
I
~~~1.0
1.0
3.0
2.0
4.0
5. 0
-~}:ic}~quency ratio~
(2.50)
Wn
-t,.
(a)
In nondimensional form, these equations can be arranged as follows:
me
. mZ, Figure 2.13. Response of a system with rotating unbalance. (a) versus 1requency ratiO wlwn. (b) Phase angle tfJ versus frequency ratio wlwn. (After Thoms~n, 1972, p. 50. Reprinted with permission of PrenticeRHall, Englewood Cliffs, New Jersey.)
or
2.7 SYSTEMS UNDER TRANSIENT LOADS Zo
,z
moe! M
Y(1- r')' +Ag'rz
(2.51)
and 2gr tan 1> = - -2 1-r
(2.52)
The value of MZ0 1m 0 e and 1> are plotted in Figs. 2.13a and 2.13b, respectively. These curves are similar in shape to those in Fig. 2.10 except that the peak amplitudes occur at (2.53)
Transient loads may be caused by hammers, earthquakes, blasts, and the sudden dropping of weights. In several such cases, the maximum motion may occur within a relatively short time after the application of the force. For this reason, damping may be of secondary importance m transient loads. CASE I. SUDDENLY APPLIED LOAD
Consider a spring-mass system (Fig. 2.14a) that is subjected to suddenly applied force represented by the forcing function F = F0 (Fig. 2.14b). The equation of motion of mass, m, is given by
..,...
mi + kz
=
F0
(2.55)
The solution for displacement, z, is
and the value of the ordinate when r is equal to 1 in Eq. (2.51) is (2.56) Z0
1
m 0 el M
21;
(2.54)
Initially, if at t = 0, z and i are equal to zero, then A is equal to- Fofk,
THEORY OF VIBRATIONS
33
SYSTEMS UNDER TRANSIENT LOADS F(t)
F(t)
?fLk Sboo
Fof-------
Fof---,
0
'
0
(b)
(a)
Ia I
(b)
dT (c)
Figure 2.15.
Dynamic amplification due to a square pulse. (a) One degree of freedom system.
(b) Square pulse,~.Jc) Magnification. f," .•
lcl Figure 2.14. Dynamic amplification due to suddenly applied load. (a) Single degree of freedom system. (b) Suddenly applied load. (c) Magnification factor versus time.
Wben t is equal to r, Z
and B is equal to 0. Thus, Eq. (2.56) becomes
z=
Fa
k
(1- cos w"t)
z
=
F lk
=
1- COS wnt
i
(2.57)
(2.58)
0
Magnification N versus time is plotted in Fig. 2.14c. The magnification has a maxtmum value of 2, wh1ch occurs when cos w"t is equal to minus 1. The first peak ts reached when wnt is equal to 7T or tis equal to Tl2, in which Tis the natural period of vibration of the system. CASE II. SQUARE PULSE OF FINITE DURATION
= -Fo
k
(1
)
(2.59)
)
(2.60)
-COS W" T
and
If the force F0 is applied gradually, the static deflection is F 0 I k. Thus the magnification N of the deflection z is '
N
'
Co.nsider. the system, shown in Fig. 2.15a, that is s~bjected to a pulse of umform mtenstty F(t) for a given duration r (Fig. 2.15b). . When tis less than r, the motion is governed by Eq. (2.55). The solution IS given by Eq. (2.57).
When t is greater than
T,
7
=
F0 w" (Slll . -k-
(t)n
T
the equation of motion is (2.61)
mi+kz=O The solution for displacement z is
(2.62) in which t' = t - r. The values of A and B in Eq. (2.62) are determined from the initial conditions when tis eqOO.l to r. By equating z, and i, from Eq. (2.62), with those in Eqs. (2.59) and (2.60), respectively, we get A= (F,/k)(l- cos w"r) and B = (F0 1k) sin w" r. Therefore, Eq. (2.62) becomes Z =
or
kFo
(1
)
-COS Wn T COS Wnt
'+k Fo Sill .
.
W 11 T Sill Wn
t'
THEORY OF VIBRATIONS RAYLEIGH'S METHOD
(2.63a)
2.8
in which cjJ =tan -t
1- cos w11 r sin w 11 'T
_
(2.63b)
F = ; Y2(1- cos wn r) sin( wnt' - cf>)
or F0 z=k
:t
T)
w • 2smz sin(wnt'-q,)
(
RAYLEIGH'S METHOD
According to Rayleigh's method, the fundamental natural frequency, i.e., frequency in the first mode of vibrations, of a continuous elastic system with infinite degrees of freedom can be determined accurately by assuming a reasonable deflected curve for the elastic system. If the true deflected shape of the vibrating system is not known, the use of tbe static deflection curve of the elastic system gives a fairly accurate fundamental frequency. In illustrating the application of this method, the energy principle is used. Expressions are developed for the kinetic energy KE, and potential energy PE. Because the total energy is constant, the sum of KE and PE is constant. Thus
Therefore,
Z
(2.63c)
Fa k
maximum KE = maximum PE .
(z sm. 2wn T)-- 2 kF, sm. T
7TT
Zmax _
.
Wn
'T
.
(2.64a)
Solution Let the displacement of the mass from the equilibrium position be Yo and y = y 0 cos w,t. If the extension of the spring is assumed to be linear, the displacement of the element of the spring at a distance z from the fixed support is y = (z I L )y 0 cos wnt, and the velocity of element dz is y = -(zl L )wnYo sin w,t. The maximum KE of the element with mass (wig) dz is then d(KE)m, = (w/2g) dz ((z!L)wnYof
'TTT
(2.64b)
The maximum value of N is 2 when r I Tn is equal to ~. (Fig. 2.15c.) . Constd~r the bmttmg case in Eq. (2.64a), if r!Tn is very small so that sm 7TT I Tn ts approximately equal to 7TT 1T n> then
z
= 2F0 7TT m"
k
Tn
(2.66b)
EXAMPLE 2.8.1
Hence, the dynamic magnification N is N- F/k-2sm-=2smo 2 Tn
(2.66a)
In Fig. 2.16, the weight of the spring of length L is w per unit length. Determine the natural frequency of the spring-mass system.
n
_
(KE + PE) = 0
or
The maximum value of z is
(zm" ) =
35
(2.64c)
Now,_k is equal to mw;, and Tis equal to 27T/w"" By substituting these quanl!hes mto Eq. (2.64c), we obtain
I I
f Now, F0 r is equal to the impulse I. Therefore,
Figure 2.16.
System with spring having weight.
36
THEORY OF VIBRATIONS
37
RAYLEIGH'S METHOD
By integrating this expression, we obtain the maximum kinetic energy of the spring
(KE)m,
=
W
g 2
(L )21L WnYo
2
Z
0
dz
or Figure 2.17.
(2.67)
Fundamental frequency determination of cantilever beam.
The maximum kinetic energy of the rigid mass m is
Solution: d The deflected curve of the cantilever beam may be assumed to cor~espon to that of a weightless beam with the concentrated load P actmg at 1ts end. Then and the total maximum kinetic energy is 1 (W+~wL)
2
g
PL 3
(2.71)
Yo= 3EI 2
2
(2.68)
WnYo
The maximum potential energy of the spring can be computed as follows: Maximum potential energy =
P;
-;,;i.f
in which EI 'is the flexural stiffnesii~of the beam. The stiffness k of beam at the free end is
P 3El k=-=-,Yo L
LYo ky dy 1
(2.72)
The expression for the deflected shape of the cantilever is 2
= 2 kyo
(2.69)
(2. 73) In a conservative system, the maximum kinetic energy equals the maximum potential energy. By equating the values of the two energies, we obtain
and Maximum potential energy=
(2.70a)
3 EI
1
2 2 kyo= 2 L'
2
Yo
(2.74)
If the weight of the beam is w per unit length and if a harmonic motion is
assumed,
Therefore, the natural frequency wn is given by
Maximum KE of (2.70b)
The effect of the spring's weight can thus be accounted for by lumping one-third of its mass with the concentrated mass of the system.
system=~ LL (wnYl
~~
=
~
(
W;Yo )'
2
dx
r Hf)'- (f)T
_~ (33wL)w' 140g nYo
- 2
2
dx
(2.75)
EXAMPLE 2.8.2
By using Rayleigh's method, determine the fundamental frequency of the cantilever beam shown in Fig. 2.17.
By equating the two energies from Eqs. (2.74) and (2.75), we obtain the fundamental frequency of vibration of the cantliever beam as
THEORY OF VIBRATIONS
38
Wn
=
PEl -
L3
g /gEl ) = 3.56 \I ~-wL wL 4 140
( 33
(2.76a)
The exact solution is
(2.81b) Therefore,
Wn
2.9
/iEi
=3.515 \I~ wL
1
2.10
LOGARITHMIC DECREMENT
zl
/5 =log -
' z,
(2. 77)
in which z 1 and z 2 are two successive peak amplitudes. If z 1 and z 2 are determined at times t1 and (1 1 + 27T) from Eq. (2.35) and substituted into Eq. (2.77), we obtain
27Ti;
(2.82)
DETERMINATION OF VISCOUS DAMPING
Viscous damping may be determined from either a free-vibration or a forced-vibration test on a system. In a free-vibration test, the system is displaced from its position of equilibrium, and a record of the amplitude of displacement is made. Then, from Eqs. (2.77) and (2.80b) /5 1 z <=-.=-log - 1 2'1r:->: 27T e z2
~
(2.83)
0"
or else, from Eqs. (2.80b) and (2.82)
(2.78)
/5 1 z i;=- =-log - 0 27T 2n7T e zn
(2. 79)
In a forced-vibration test, the system is excited with a constant force of excitation and varying frequencies, and a resonance curve is obtained (Fig. 2.18). When r is equal to 1, from Eq. (2.44d), we obtain
or /5 = log, exp , r:;--;z v1- i;
z
/l=-log-0 n e zn
(2.76b)
Logarithmic decrement is a measure of the decay of successive maximum amplitudes of free vibrations with viscous damping and is expressed (Fig. 2.8b) by
z, o, -
or (2.80a)
0.707 .
1 21;
(2.85)
1
-=-;==~~~ 2 2 21; r ) + 41;?
V(l-
(2.80b) in which i; is small. If the damping is very small, it may be more conyenient to measure the difference in peak amplitudes for n cycles. In such a case, if zn is the peak amplitude of the nth cycle, then
(2.84)
When the amplitude of motion is 0.707 (1121;), the frequency ratio r may be determined from Eq. (2.44d) as follows:
or
or
or r4
(2.81a) Also,
39
DETERMINATION OF VISCOUS DAMPING
or
-
2r 2 (1- 2/;') + (1- 81;
2
)
=0
40
THEORY OF VIBRATIONS
ri., = H2(1- 2t;') ± V4(1- 2eJ'- 4(1- Sg')]
v4 + 16t;
= ![2(1- 2t;') ±
4
2.11
41
TRANSMISSIBILITY
The system shown in Fig. 2.12 represents a practical case of a machine foundation that is subjected to rotating unbalance. The forces transmitted to the foundation through the spring and the dashpot can be easily computed. The maximum force in the spring is kZ0 and the maximum force in the dashpot is cwZ0 , the two forces are put out of phase by 90° (Fig. 2.9d). Hence, the force transmitted F, to the base is
16t;'- 4 + 32t;']
-
TRANSMISSIBILITY
= (1 - 2eJ ± 2t;'{l+T' or
(2.87a) if t; is small. Now,
or
ri _ ri = ti -,r; =(I'- t, )(!, + f,) -= 2 (1'- f,) .1 J,,
fn
fn
(2.87b)
/,,
If transmissibility T, is defined as the ratio of force transmitted F, to the
since (f, + [2 ) ![,,-=' 2. Therefore,
force of excitation m 0 ew 2, then by substituting for cw/ k = 2t;r and for Z 0 from Eq. (2.49b), we obtain (2.86)
\C'
F,
'/1 + (2t;r)
2
T, = -m-e_w_2 =, -Y-r(=l~-=r"'2c=)2'=+~(2=t;=r""')2
(2.87c)
0
This method for determining viscous damping is known as the bandwidth method.
4. 5
4. 0
1---E ~ 0
'~ o -
(
1--- ( ~
0.125-
~
0.1 6
0.125
Zmax = ~ 152 mm
Vertic11 vibratioL
3. 0
0.1 4
'E
0.1 2
.s •
~
.~ 0.
f-- 0.1 0
I
1/!
0.0 6 0.0 4 10
I ~
J!
0.08
14
18
/
!"
I
1-
=)oo
1. 0
"~',":...
I 26
v ,/' rx"'
~ o.5
~ 1.0
~
2.0
,I~:2~
( - 2.0-
1'---t.~
10
1'--<~os-
~.,t:o~-~
0 30
34
0
Determination of viscous damping in forced vibrations by bandwidth method.
1.0
2.0
3.0
Frequency ratio, wlwn
Frequency (cps)
Figure 2.18.
I/E
0
I
: 22
e
0.25
0
'-
1\
E
"'
1\ ~\-
~ =
'r
Figure 2.19.
Transmissibility (T,) versus frequency ratio (r).
42
THEORY OF VIBRATIONS
43
VIBRATION MEASURING INSTRUMENTS
The transmissibility T, versus the frequency ratio wlw" is plotted in Fig. 2.19. It will be seen that fort; equal to zero, the plot is the same as in Fig. 2.10a. Also, all curves pass through r = V2. When r is greater than V2, all the curves approach the x-axis asymptotically. The higher the frequency ratio, the better the isolation, and hence the smaller the force transmitted. But there may be excessive amplitudes at the time of starting and stopping a machine, because it will pass through the zone of resonance. Damping helps to reduce these amplitudes.
Equation (2.90) is similar to Eq. (2.48b). Hence the solution for Z 0 can be written as in Eq. (2.49b):
2.12
Equations (2.91) and (2.92) may be rewritten as
VIBRATION MEASURING INSTRUMENTS
Figure 2.20 shows the essential elements of a vibration measuring instrument. It consists of a seismic mass m which is supported by springs and a dashpot inside a case, which is fastened to a vibrating base. The motion of the base is to be monitored. Let the motion of the base be represented by x = X 0 sin wt
(2.88a)
The relative motion of the mass m of the instrument in relation to the vibrating base is monitored. Thus, we can let the absolute motion of the mass m of the instrument be y and, by neglecting transients,
y
=
Y sin wt
Zo
mw 2X 0
=
y(k-mw)
2 2
+(cw)
2
(2.91)
and > =
Z0
cw tan- 1 -=~2 k-mw
(w/wn)
Y(1- r
Xo
2 2 )
(2.92)
2
+ (2t;r) 2
(2.93a)
and _,_ "P
=tan
i,- ~nd
_1
2t;r - -2 1-r
(2.93b)
Plots of Z 0 / the frequency'fi\lio and the phase angle and frequency ratio are shown in Fig. 2.21.
(2.88b)
Then the equation of motion of m can be written as
my = If we let y - x
-
k( y - x) - c(.Y - i)
(2.89)
= z and y - i = i, we obtain mi
+ ci + kz =
mw 2X 0 sin wt
(2.90)
1.0
2.0
Frequency ratio ~
Wn
Base Figure 2.20.
base.
A vibration measuring instrument (seismic instrument) mounted on a vibrating
Ia I Figure 2.21. Response of a vibration measuring instrument to a vibrating base. (a) Amplitude. (b) Phase angle. (After Thomson, 1972, p. 60. Reprinted by permission of Prentice~Hall, Englewood Cliffs, New Jersey.)
45
SYSTEMS WITH TWO DEGREES OF FREEDOM THEORY OF VIBRATIONS
44
Sign conventions
--:-::--]
Displacement Pickup
z,
For large values of wlw", and for all values of damping, g, Z 0 1X0 is approximately equal to unity. Therefore, if the natural frequency of the instrument is low, such that the value of r is large, then the resulting relative motion Z 0 equals X 0 . Therefore, the instrument functions as a displacement pickup. One of the disadvantages of the displacement pickup is its large size. Because IZl is equal to IYl, the relative motion of the seismic mass will be as large as the amplitude of vibration to be measured.
k1
w X0
Figure 2.22.
I =
w;y/[1- (w/w")
1 2 2 ]
+ [2gw!wJ
2
w;vc
(2.94)
When g is equal to 0.69, the values of VC in the denominator for different values of w I wn are wlw,
0.0
0.1
0.2
0.3
0.4
1.000
0.9995
0.9989
1.000
1.0053
Thus Z 0 is proportional to the absolute acceleration, w 2X 0 , of the vibrating base. The instrument thus functions as an accelerometer. The frequency ratio wlwn in an accelerometer must be small. Therefore, the natural frequency of the instrument must be high.
t Zl
>
Z2
~ Fo sin wt 0!
Ia I
By rearranging Eq. (2.93a), we obtain 2
+
Z)
Acceleration Pickup
Z0
Z, Z
(b)
(a) Two degrees of freedom system. (b) Free-body diagram.
masses may be written in the following form:
or (2.95a) and
or (2.95b)
Phase Distortion
The instrument must be capable of reproducing a complex wave without changing its shape; that is, the phase of all harmonic components must be shifted equally along the time axis. This can be accomplished if the phase angle q, of the accelerometer output increases linearly with frequency. This condition is nearly satisfied when!; is equal to 0.70, and the phase distortion is practically eliminated.
2.13
SYSTEMS WITH TWO DEGREES OF FREEDOM
The natural frequencies of this system are obtained by considering its free vibrations. Making F0 = 0 in Eq. (2.95a), we obtain (2.95c) Let (2.96a) and (2.96b)
Figure 2.22 shows a two-mass-two-spring system, which has two degrees of freedom. Free-body diagrams of the masses are also shown. In a practical
system, the spring k 1 and the mass m 1 constitute the main system, and spring k, and mass m 2 a vibration absorber. The equations for motion of both the
By substituting the solutions from Eqs. (2.96) into Eqs. (2.95b and 2.95c), we obtain
46
THEORY OF VIBRATIONS
w
(2.97a)
_ nll-
I k, 'V m 1 +m 2
(2.99a)
f'§-
(2.99b)
and (J)n/2
=
2
(2.97b)
The values of the two natural frequencies wn 1 and wn 2 for this system are obtained by solving Eq. (2.98) as a quadratic in
w;.
From Eqs. (2. 97), we obtain
-m 1 w:+k 1 +k 2 k2
47
SYSTEMS WITH TWO DEGREES OF FREEDOM
k2
-m 2
w: + k
Amplitudes of Vibrations
2
For the force acting on mass m 1 , the vibration amplitudes are obtained by assuming the following solution for the principal modes:
Simplifying this we obtain
(2.100a) or
and
z 2 ~~;ji:J sin w t or
(2.100b)
By substituting the solution from Eqs. (2.100) into Eqs. (2.95a) and (2.95b), we obtain (2.10la)
or and (2.101b) Let
From Eq. (2.101b), (2.102)
Therefore, Substituting for Z 2 from Eq. (2.102) into Eq. (2.101a), we obtain
or or [m 1m2 w 4
or
=
(2.98) in which
or
k2 m 1 w 2
-
F0 (k 2
-
m2w
2
)
-
k 1 m2 u/ + k 1 k 2
-
k 2 m 2 w 2 + k;- k;]z 1
48
THEORY OF VIBRATIONS
SYSTEMS WITH TWO DEGREES OF FREEDOM
49
spring-mass system having its natural frequency given by Eq. (2.106). The negative sign in Eq. (2.107) indicates that Z 2 and F0 are in phase opposition. In fact, the amplitude of the main mass Z 1 , becomes zero at this frequency, because the force, k 2 Z 2 , exerted by spring 2 on mass m 1 is equal and opposite to the force of excitation F0 • The size of the absorber mass m 2 and its displacement, depend upon the magnitude of the disturbing force, F0 (=k 2 Z 2 ). For a given force F0 , the smaller stiffness of the absorber spring, the larger its amplitude Z 2 and vice versa. Figure 2.23a shows a plot of Z 1 /Z,, versus w/w" 12 (Eq. 2.103b) with
or
or
8
(2.103a)
Iii
or 6
-~~~-
in which
4
f_e·
2
Substituting for Z 1 from Eq. (2.103a) into Eq. (2.102), we obtain
Zz
=
m 1 [w
4
11
i\
II
(2.103b)
Fo w !,z 2 w (1 + ~t)(w~ 11 + w~ 12 ) + (1 + ~t)w~ 11 w! 12 ]
1::
lli_
I.
0~ 0
V
I !\
II
11
i\
J i \ r'~ 1I , I 018
1/ ItsI \
(2.105a)
= 2
From Eq. (2.103a), it is seen that
Fow ~12 2 m 1 A(w )
z, = 0,
..... 1.5
--
t---
2.0
2.5
w Wn/2
Ia I
or
z
ii
1.0
0.5
0.20
p. =
1.6
......
1.5
(2.105b)
1.4
if
/
I. 3
/
I. 2
(2.106) Then
/ Wnll
I. 0
~
1
Wn/2
\•i 9
"
8
:-.......
- r-.
7
(2.107)
/
v
/
1
or
..........
1-
6 0
.1
.2
.3
.4
.5
.6
.7
.8
Mass Ratio P. (b)
Equations (2.103, 2.105, 2.106, and 2.107) explain the principle of vibration absorbers that will be used in Chapter 10 (Section 10.1). The amplitudes of
Figure 2.23.
motion of mass m 1 can be appreciably reduced by attaching to it, a
versus
~J-(=m~
(a) Response versus frequency of a vibration absorber. (b) Natural frequencies I m,).
50
THEORY OF VIBRATIONS
JL = m 2 1m 1 of 0.20. Although the amplitude of the main mass m becomes 1 zero when w = W 1112 , there are two resonant frequencies at which the amplitude of mass m 1 becomes infinite. In Fig. 2.23b, (w/wn 12 ) has been plotted versus !L (Eq. 2.98) for a particular case of wn 11 = wn 12 • In Fig. 2.22a, if the forcing function F0 sin wt is acting on mass m 2 , instead of mass m 1 (as shown), it can be shown that the amplitudes of motion Z 1 and Z 2 are given by: 2
Z = I
Wnz2
m,~(w2)
F
(2.108)
o
and
Z _ (1 + JL)W~ll + JLW~ 12 -
m 2 ~(w2)
2-
w
2
F0
(2.109)
MULTIDEGREE FREEDOM SYSTEMS
51
Thus in a system with n masses, if the displacements of masses are denoted by xi (where j can take integral values from 1 ton) the total spring force on mass i due to displacements xi of all masses is r:;~, k,h, the summation of the spring forces. Applying Newton's law for the ith mass, n
m/ii +
2: kiixi = 0
(2.113)
j""l
There are n such equilibrium equations each corresponding to a mass. This procedure is called stiffness method. Let the system vibrate in one of its principal modes of vibration. Then its motion will be sinusoidal with a natural frequency of wn corresponding to that mode. Since the amplitudes of the masses may be different from each other, the motion of any mass i may be expressed as (2.114)
2.14
MULTIDEGREE FREEDOM SYSTEMS
in which i = 1, 2, ... , n. Substituting Eq. (2.114) into Eq. (2.113) we get
It has been shown in Section 2.1 that the number of independent coordinates of displacements in a vibrating system determines the degrees of freedom of the system. In this section we will discuss the techniques applicable to the solution of vibrations of multidegree freedom systems. Two approaches are commonly used for obtaining a solution: (1) stiffness matrix method; (2) flexibility influence coefficient method. The stiffness coefficient k 1i is defined as the force on the ith mass due to a unit displacement at the jth mass with all other masses held at their equilibrium position. With displacement x 1 , x 2 , and x 3 of points 1, 2, and 3, respectively, the principle of superposition can be applied to determine the forces in terms of stiffness coefficients as
ft f,
=
kuxt + kuxz + k13x3
=
k 21 x 1 + k 22 x 2 + k 23 x 3
!3 = k31X1
-;:·
n
'·:r
-~: ::
-miw~Ai + 2:. kiiAi = 0,
i=l,2, ... ,n
(2.115)
j=l
The frequency determinant corresponding to Eq. (2.115) is
=0 (2.116)
Expanding the determinant, we would get a frequency equation in the polynomial form as below: (2.110)
+ k32X2 + k33X3
In matrix notation, the equation is {f} = [k]{x}
(p --
(2.111)
in which
2 + (- 1)"w 2n 0 a0 +aw·+ .. ·+a n-1 w 2(n-l) 1 n n n =
(2.117)
Since the coefficient of w~ is not zero, Eq. (2.117) always has n roots. These roots would give the 1i~ natural frequencies of vibration, namely, w"" Wnz' . . . wnr ... wnn. Corresponding to each value of wn' there is an associated mode shape with amplitudes Air), Ar), ... , A~), which can be obtained by solving Eq. (2.115).
When a system vibrates in a principal mode, all the masses attain maxkll
[k]
=
k12
k13
k,, k,, k,,
k,, k,, k,,
f={J,}.
imum displacements simultaneously and also pass through their equilibrium
(2.112)
position simultaneously. When the number of degrees of freedom exceeds three, the problem of forming the frequency equation and solving it for determination of frequen-
52
THEORY OF VIBRATIONS MULTIDEGREE FREEDOM SYSTEMS
cies and mode shapes becomes tedious. Numerical techniques are invariably resorted to in such cases. The flexibility influence coefficient a,i is defined as the displacement at ith mass due to a unit force applied at jth mass. With forces f 1 , [ 2 , and f 3 acting at pmms 1, 2, and 3, the principle of superposition can be applied to determme the displacements m terms of the flexibility influence coefficients
x, +
53
2:" (mix)a,i = 0
(2.123b)
j=I
There are n such displacement equations each corresponding to a mass. Now, substituting Eq. (2.114) in Eq. (2.123b), we get n
(2.118a)
Ai-
2: m;w~A;aii = 0
(2.124a)
i""'I
(2.118h)
x,
=
a,,[,
+ a32f, + a,,[,
or
(2.118c) i=l,2, ... ,n
(2.124b)
In matrix notation the equation is
{x}
=
[a]{f}
(2.119)
The frequency determinant corresponding to Eq. (2.124b) is
in which 1) -( m-a :~,.. .,u z ·--'· wn
(2.120)
=0
is the flexibility matrix. If Eq. (2.119) is multiplied by the inverse of the flexibility matrix [ar' we obtain the equation '
[ar'{x} = {f} = [k]{x}
(2.121)
(2.125)
It is thus seen that the inverse of the flexibility matrix is the stiffness matrix [k], i.e.,
Expanding the determinant, we get a polynomial equation that is in the same form as Eq. (2.117). Then natural frequencies of the system can then he obtained.
[ar'
=
[kJ
(2.122a)
or
[a]= [kr'
Forced Vibrations
(2.122b)
<:;onside~ing flexibility coefficients defined above and applying D' Alemhert s pnnc1ple, ~he system can be considered to be in an instantaneous state of static eqmhhnum under the effect of reversed inertia forces acting on all masses. The displacement of mass i due to reversed effective inertia force - m .x. 1 1 actmg on all j masses is
Let an undamped n degree of freedom system be subjected to forced vibration and let F,(t) represent the force on mass i. Using Eq. (2.113), the equation of motion is given by n
miXi +
L,
ki;X; =
FJt)
(2.126)
j=l
where i = 1, 2, . .. , n. xi=-
or
2:" (m;X)aii
j=J
The amplitude of vibration of a mass may be taken as the algebraic sum (2.123a)
of the amplitudes of vibration in various principal modes, i.e., the individual
modal response would he some fraction of the total response with the sum of the fractions being equal to one. If the factors by which the modes of
54
THEORY OF VIBRATIONS
vibration are multiplied are represented by the coordinates mass 1,
t;, then, for any
X=A('lt +AC'lt!:.2 +···+AC'lc +···+A(nlt~JJ I I ~J 1 ~T t
I
or n
X="' I L..J AVlt I !:>r
MULTIDEGREE FREEDOM SYSTEMS
55
g, uncouples the n degree of freedom system into n of single degree of freedom systems. The t;'s are termed as normal coordinates and this approach is known as normal mode method, i.e., the total solution is thus expressed as a sum of contribution of individual modes. For determination of f,(t), multiply both sides of Eq. (2.130) by A~') (=A)'lT, the transpose of A)'l) and summing up for all the masses, we get It is seen that the coordinate
n
(2.127)
n
"' L.J FAC'l I I
r=l
n
n
n
reel
i=l
="'LJ AC'l "'LJ mAc'lt, ="'L.J /,rL.i"' m.AC'l AC'l I
i=J
i=l
I
I
I
T
T"'l
l
I
(2.133)
Where r represents the rth mode. Then from Eq. (2.126), we get n
L r=l
n
mA(•)i: 1 1 br
Using the orthogonality relationship,
n
+"' L.J
"'k A(')c = F(t) L.J 1 Sr i IJ
r=t j=l
n
(2.128)
"' AC'l = 0 LJ m.AC'l I I l
for
r o;6 s
i=1
From Eq. (2.115), the right-hand side of Eq. (2.133) reduces to n
f,
L
i=l
m,(A)~);,
when
r=s
~·
Hence . Hence n
n
L r=
2 mA(') < +"' t = F(t) 1 1 ~r L-1 w nr mAC') t 1 Sr 1
1
(2.134)
'"' 1
and Using Eqs. (2.132) and (2.134), the complete solution from Eq. (2.127)
n
L
r=I
m,A~')( {, + w;,t;,) = F,(t)
is (2.129)
n
F,(t) =
L
m,A~') f,(t)
"
1
X,--~ A, -;;; -
Since the left-hand side is a summation involving different modes of vibration, the right-hand side should also be expressed as a summation of equivalent force contribution in the corresponding modes. Let F, be expanded for convenience as
r-1
(r)
r
l' ""
L.. j=
0
l::"
1
Fj (T )AC'l j • (A('l)' sm w.,(t- r) dr
J=lm;
1
EXAMPLE 2.14.1
Figure 2.24 shows a three degrees of freedom system. Determine the stiffness matrix.
(2.130)
r=l
in whichf,(t), the modal force is derived subsequently as Eq. (2.134). Then from Eqs. (2.129) and (2.130), where
r = 1, 2, ... , n
(2.131)
This is a single degree of freedom equation and its solution can be written as
t;, = - 1wnr
l' 0
J,(r) sin w,(t- r) dr
where 0 < r < t
(2.132)
Figure 2.24.
Computation of stiffness matrix.
56
THEORY OF VIBRATIONS
Solution Let x 1 = 1.0 and x 2 = x 3 = 0. The forces required at 1, 2, and 3, considering forces to the left as positive, are
VERTICAL DISPLACEMENT
0
ft = kl + k, = k11 / 2
=
-k 2
[, = 0 =
=
57
MUlTIDEGREE FREEDOM SYSTEMS
+
-
0.445
0
+
1.247
k 21
k,l 0.802
Repeat with x 2 = 1, and x 1 = x 3 = 0. The forces are now
!1
=
-k,
=
k1z
[, = k, + k, = k,,
1.0
[,=-k,=k,,
First mode
Repeat with x 3 = 1, and x 1 = x 2 = 0. The forces are
Ia I
!I= 0 = k13 h = -k, = k,,
+ k,)
[
-k,
(k 2 + k 3 )
k2
-
-k 3
0
-k 0 +
(k,
k.J
l
EXAMPLE 2.14.2
For the system shown in Fig. 2.25a, solve for natural frequencies and mode shapes.
kl2
k21 = -k,
k,, = 2k,
k,, = -k
k,, = -k,
k,,
= 0,
k13 =0 d," k
The equations of motion of the system, from Eq. (2.113) are m 1i
1
+ 2kx 1 -
(2k-m 1 w;) -k 0
0
-k (2k- m 2 w!)
-k
-k
(k- m,w;)
=0
The frequency equation can be obtained by expanding the determinant. Letting m 1 = m 2 = m 3 = m (for simplicity),
(2k- mw!)[(2k- mw;)(k- mw!)- (- k)( -k)] - ( -k)[(- k)(k- mw;)- ( -k)(O)J + (0)[( -k)( -k)- (2k- mw;)(O)] = 0
- A' 3 + 5A' 2
k 11 =2k,
k31
(b)
Example 2.14.2 (b) Mode shapes.
which, on simplification, gives
Solution The stiffness coefficients K,i for the system are = ~k,
sylt~dls for 1
The stiffness matrix can now be written as (kl
(a) Spring-mass
The corresponding frequency determinant from Eq. (2.116) is
[, = k, + k, = k,,
K =
Figur-?~:25.
Third mode
Second mode
kx 2 = 0
m 2i 2 - kx 1 + 2kx 2 - kx 3 = 0
-
6A' + 1 = 0
in which A' = mw ;; k (A' will be equal to 1.0 for a single degree of freedom). The roots of frequencfequation can be determined by any of the standard techniques. In this case, the trial-and-error method and the plotting of the graph of the function will be used. The roots are A;= 0.198, A;= 1.555, A;= 3.247, and since A'= mw;lk, 2 Will=
k 0.198 m '
2
w 112 =
1.sss
k m
(I)
2 n3
= 3.247
k m
The amplitude coefficients for the three modes of vibration can be obtained from Eq. (2.125), from which we get
58
THEORY OF VIBRATIONS
1
1
- -w-; A 1 + m k
A1+ m
1
k
A2 + m
1
k
A3 =0
PRACTICE PROBLEMS
59
2.3 For the system represented by Eq. (2.48b), show that the peak amplitude occurs at a frequency ratio of
1 1 2 2 --,A,+m-kA 1 +m-A +m-A =0 w, k 2 k 3
1
r = -Y-,=1,;;_=2t;""'
1 1 2 3 - - , A 3 +m k A, +m k A 2 +m -A 3 =0 k
(f)n
and
-
z
Letting A'= (mlk)w;,, the preceding equations can be rewritten as (A' -1)A 1 + A'A 2 + A'A 3 = 0
A'A 1 +(2A'-l)A 2 +2A'A 3 =0 A'A 1 + 2A'A 2 + (3A' -l)A 3 = 0 It is more convenient to work with particular numerical values rather than with ratios. Therefore, let us arbitrarily set A 3 (1) = A 3 (2) = A (3) = 1. In 3 th1s manner, we will obtain two simultaneous equations for A and A from 1 2 above:
(A' -1)A 1 + A'A 2 =-A' A'A 1 + (2A' -1)A 2 = -2A' Substituting numerical values of A' determined above, we get
First mode Second mode Thrid mode
AI 0.445 -1.247 1.802
A, 0.802 -0.555 -2.247
A, 1.0 1.0 1.0
=-~~= Z~
If t; is greater than 0.707, r is imaginary. Discuss the significance of these values with the help of a diagram. 2.4 An unknown weight W is attached to the end of an unknown spring k and the natural frequency of the system is 1.5 Hz. If 1 kg weight is added to W, the natural frequency is lowered to 75 cpm. Determine the weight W and the spring constant k. 2.5 A body weighing 60 kg is suspended from a spring, which deflects 1.2 em under the load. It is subjected to a damping effect adjusted to a value O.QO times that requir~ct:for critical damping. Find the natural frequency of the undamped and damped vibrations, and, in the latter case determine the ratio of successive amplitudes. If the body is subjected to a periodic disturbing force with a maximum value of 25 kg and a frequency equal to one-half its natural undamped frequency, determine the amplitude of forced vibrations and the phase difference with respect to the disturbing force.
a
"
b
•
.I m
I'
These mode shapes have been plotted in Fig. 2.25b.
~k
//~7/////.////Jm. .
PRACTICE PROBLEMS 2.1
1
max
(a)
Determine the numerical value of viscous damping from the free vibrations record in Fig. 2.8b.
2.2 Show that, in frequency-dependent excitation, the damping factor t; is given by the following expression:
t;=Ht,-J;) !,,
T
!
~k
l
J;
in which [2 and are frequencies at which the amplitude is 1/"1/2 times the amplitude at r = 1.
(b)
Figure 2.26.
r~a=3~~
///?'"!ff.
y·'
6 (C)
Systems with one degree of freedom in Practice Problem 2.7.
REFERENCE
~k
61
2.6 An 8-cm diameter pole with an 8-m length is guided so that it floats vertically in water. The specific gravity of the pole's material is 0.79. Find the pole's natural frequency. 2.7 Set up the equations for motion of the systems shown in Figs. 2.26a, b, and c and determine the frequency equation and natural frequencies. Determine the expressions for critical damping in (a) and (b). 2.8 Write the equations of motion for the systems shown in Fig. 2.27 and determine their natural frequencies.
8 8~k :w)P//////#/#$#/#~,0 a.
Ia I
REFERENCE Thomson, W. T. (1972). "Theory of Vibration with Applications." Prentice-Hall, Englewood Cliffs, New Jersey.
(b)
21
(d)
Figure 2.27. 60
/
Systems with two degrees of freedom in Practice Problem 2.8.
63
WAVE PROPAGATION IN ELASTIC RODS
3 •
Wave Propagation an an Elastic Medium
convenient and useful method of determining the soil properties such as the d namic shear modulus for design of machme foundations. So.me of the y · t.ton o f the dynamic shear modulus are discussed m methods for d etermma Chapter 4 (Sections 4.5 and 4.6). . . 2. The waves generated due to a vibratmg footmg carry away a part o: the energy into the medium resulting in dampmg effect (geometncal damp ing) that helps in reducing the vibratiOn amplitudes. . . 3. The energy carried by the waves into the mediUm may mduce undesirable and harmful vibrations in adjoining structures and machmes, and . 4 Effective vibration isolation by wave barriers (Chapter 10, SectiOn can be achieved only if the depth of the trench barners IS adequate 10 compared with the wavelength of the propagatmg waves.
.4)
The principles of wave propagation are also used for subsoil exploration . h . b d the scope of the present text. . (geophysical exploration), wh JC IS eyon In this chapter, three problems will be stud1ed: When a continuous medium is disturbed from within or outside, the waves are generated. For example, when a pebble is dropped into a large, still pond, it generates waves, which travel in all directions. Now, if a small buoyant object such as a piece of wood is floating on the surface of water, it will oscillate about its original position as the waves travel under it away from the point where the pebble was dropped. Thus, it can be seen that the waves travel in one direction with a certain velocity, while the piece of wood and the particles of water beneath it oscillate to and fro with a velocity that is different from the velocity of the waves. Also, the waves are returned (reflected) from the edge of the pond, which is the reflecting boundary for the surface waves. This is a phenomenon that we frequently observe when we are children. In a similar manner, the sudden rupture of stressed rocks within the earth's crust will generate waves both at the surface and within the earth to produce what is known as an earthquake. Smaller but similar disturbances are artificially created by such means as blasting, aircraft landings, and the explosion of bombs. Machines in operation also generate waves at their bases and sides, and these waves travel in all directions in the soil. When a load is suddenly applied to a body, the entire body is not disturbed at the instant of loading. The parts closest to the source of disturbances are affected first, and the deformations produced by the disturbance subsequently spread throughout the b"ody in the form of stress waves. The phenomenon of wave propagation in an elastic medium is of great importance in the study of machine foundations due to the following reasons. 1. The velocities of wave propagation depend upon the elastic properties of the medium. The study of the wave propagation velocities thus provides a 62
1. .waJg propagation in elaS'tit rods.
2. Wave propagation in an elastic infinite medium. 3. Wave propagation in an elastic half space.
3.1
WAVE PROPAGATION IN ELASTIC RODS
Three inde endent kinds of wave motion are possible in. rods: longitudinal, . l p d flexural Only the longitudinal and torswnal waves are of torstona , an . . . importance to our study of machine-foundatwn-sml systems. 3.1.1
Longitudinal Vibrations of Rods of Infinite Length
Let us consider the free vibrations of a rod with a cro~s-section~l ~eat~, Youn 's modulus E, and unit weight y (Fig. 3.1) .. It IS ass.ume t at e g 'f over the area and each cross sectiOn remams undJstorted stress ts um orm · d the stress on during motion. The stress on a transverse plane at x IS ax' an . th + dx is crx + (acrx /ax) dx. The sum of forces m ex a transverse p lane a t x direction can then ~ written as follows:
2: Fx =
mass x acceleration
(3.1)
Now,
acr ) acrx "'F=-crA+ dxA ( cr+-xdxA=-a Li X X X ax X
(3.2)
64
WAVE PROPAGATION IN AN ELASTIC MEDIUM
a~ x rJx
65
WAVE PROPAGATION IN ELASTIC RODS
a'u at' -
+oa, - dx
ax
E
a'u
p ax'
(3.7b)
a'u- v' a'u2 at 2 - r ax
(3.7c)
or
in which
v' = '
Figure 3.1.
longitudinal vibrations of a rod.
If the displacement of the element in the direction of xis u, the equation for motwn_ of the element can then be written by applying Newton's second law of motiOn as follows:
au ax
_x
a'
dx A = dx A 1' ---':'. g at'
!:_
(3.8)
p
and V, is defined as the langitudinal-wave-prapagation velocity in the rod. Equation (3.7c) is called the one-diemsnional wave equation. It indicates that during longitudinal vibrations, displacement patterns are propagated in the axial direction at the velocity V,. If the wave propagation in a rod is considered at some intermediate point in the rod, it1can be easily seen th:at at the instant a wave is generated, there is compressive stress on the face in the positive direction of x and tensile
(3.3) dx
in which g is the acceleration due to gravity. Equation (3.3) can be written as
---...j
1------
!Jx::::] II
J
~~=~~jjj:=================rr---~-x (3.4) ~
The strain in the x direction in au/ax, and the ratio of stress u to strain is Young's modulus E; therefore, x
au ax
u =E-
"
I(
(a)
Substituting the value of aa)ax from Eq. (3.6) into-Eq. (3.4) and replacing the term y/g by p, the mass density, we get,
ax' -
Vrtn
----1 u I---
(3.5)
(3.6)
2
=
L
By differentiating Eq. (3.5) with respect to x, we get
Eau_
Xn
a
a,f----------,
2
au P at'
(3.7a) (b)
or Figure 3.2.
Velocity of wave propagation and particle velocity in a rod.
66
WAVE PROPAGATION IN AN ELASTIC MEDIUM
stress on the face in the negative direction of x. Thus, while the compressive wave travels in one direction, the tensile wave travels in the opposite direction. Initially, only small zones close to these cross sections feel the stress, but as time passes, larger zones undergo the stress caused by the displacement u. It is important to see clearly the distinction between the velocity of wave propagation V, and the velocity of a particle u, in the stressed zone. Let us consider the stressed zone at the end of the rod in Fig. 3.2a. When a uniformly distributed compressive-stress pulse of intensity ux and duration tn (Fig. 3.2b) is applied to the end of the rod, only a small zone of the rod will undergo compression initially. This compression is transmitted in time to successive zones of the rod. The transmission of the compressive stress from one zone to another occurs at the velocity of the wave propagated in the medium, that is, V,. During a time interval dt, the compressive stress travels along the rod for a distance of dx = V, dt. At any time after t,, a segment of the rod of length, xn = V,t,, constitutes the compressed zone. The amount of elastic shortening of this zone is given by Eq. (3.9a) and equals the displacement of the free end of the rod. Therefore, O"x
Ux
u=-x E n =-Vt E r n
which represents a function of x traveling at velocity V,. The derivatives of u with respect to x and t are as follows: 2
au -a = f'(x- V,t) , X .
a~ ax
au
aat2u = v'r f"(x- V,t)
=
f"(x - V,t) ,
2
at
= - V,f'(x-
V,t),
Substitution of the second derivatives in Eq. (3.7c) yields identical results on both sides, thus satisfying this equation. A more general form of the wave solution can be expressed by u=
f 1 (x - V,t) + f 2 (x + V,t)
(3.12)
In this equation, the first term, f 1 (x), represents the wave traveling in the . positive x <)irections, and the second term, f 2 (x), represents the wave traveling in ':'flre negative x directi~·
(3.9a) End Conditions
or
(3.9b) Now, the displacement u divided by time t" also represents the velocity of the end of the rod, or particle velocity. Therefore, (3.10) It is important to note, that (1) both wave-propagation velocity and particle
velocity are in the same direction when compressive stress is applied and (2) that wave-propagation velocity is in the opposite direction of particle velocity when tensile stress is applied. Another important consideration is that the particle velocity u depends on the intensity of the stress or strain induced [Eq. (3.10)], whereas the wave-propagation velocity V, is only a function of the material property. Solution of Wave Equation
The solution of Eq. (3.7c) for a one-dimensional wave may be expressed in the form
u = f(x- V,t)
67
WAVE PROPAGATION IN ELASTIC RODS
(3.11)
The conditions at the end of a bar may be studied by making use of the superposition of waves. This is possible, because the differential equation, Eq. (3.7c), is linear. Hence the sum of two solutwns 1s also a soluuon. Consider a wave whose form is described by a step functwn (Fig. 3.3a). In this figure, a compression wave is shown traveling in. the. positive x direction, and an identical tension wave in the negatlve x d1rect1on. In the crossover zone (Figs. 3.3b and c), where the two waves pass each other, the portion of the rod in which the two waves are superimposed has z~ro stress. However, the particle velocity is equal to tw.lce the partlcle veloc1ty m th1s zone. The particle velocity becomes double m the cr?ssover zone,. because the particle velocity is in the direction of wave tr~vel m a .compressiOn wave but is in the opposite direction of wave travel m a tenswn wave, and the waves are traveling in opposite directions. After the two waves have passed the crossover zone, the stress and velocity return to zero at the crossover point alo~g the centerline, and the compression and tensi'<').p waves return to their m1t1al shape and magmtude (Fig. 3.3d). It will thus be seen that on the centerline cross sectiOn, the stress is zero at all times. This stress condttion IS the same as that whtch exists at the free end of the rod. If half of the rod is removed, the centerline cross section can be considered a free end (Fig. 3.3e). Therefore, it can be seen that a compression wave is reflected from a free end as a tension wav~ of the same magnitude and shape. Similarly, it can be show~ that a tensiOn wave is reflected as a compression wave of the same magmtude and shape. Now , let us consider an elastic rod in which a compression wave IS
68
WAVE PROPAGATION IN AN ELASTIC MEDIUM
~
WAVE PROPAGATION IN ELASTIC RODS
--
GIL
GIL
v,
/r/{z : g
rrnrn;,_rTTITT1,
69
(a) _-'-'t.llll~III~III~III_T------rrmnn~T'"s~ion Compression I II II U!!J 1111
v,
~
(1
u
= =
0 0
/
Ia)
v,
I
o v,
{
,1--,
(bl----l..Lij'+--jqr---t'TTrz:_2_"- - -
L_L1uJ
(Jo
v, --1
r
It
I I
1111111
v,
uo
1111111
~
iDJIIIIIII (cl--nrllllrrrrlll..t---11 -H!~LW_{I.lllJ__ _ V
I I
(b)
v=O
U
=
2U
v, (d)
I IT I~Irnl~fln:.-1~1 rmltyJIIIL__ -ITTTITlTTTliTT~ens~ion~_j II Ill II Ill II I 1\{K :C~mpression
-Elastic waves in a bar with free end conditions.
=
0 0
(C)
v,
Figure 3.3.
t: ~
_(0 a
(d)
a
a
Figure 3.4. Elastic waves in a bar with fixed end conditions.
traveling in the positive x direction and an identical compression wave is traveling in the negative x direction (Fig. 3.4a). When these two waves pass each other, the cross section throngh the centerline has stress equal to twice the stress in each wave and zero particle velocity ·"(Fig. 3.4b). After the waves pass each other'. they return to their original shape and magnitude (Fig. 3.4c). The centerline cross sectiOn remains stationary during the entire process and, hence, behaves like the fixed end of a rod. Therefore, it is seen that a compression wave is reflected from a fixed end of a rod as a compression wave of the same magnitude and shape (Fig. 3 .4d) and that at the fixed end of a rod the stress is doubled (Fig. 3.4b).
In the preceding di~ussion, the waves of constant-stress amplitude are considered. Superposition and reflection of waves of any shape may be studied in a similar manner. 3.1.2
Longitudinal Vibrations of Rods of Finite Length
So far, it has been assumed that the rod is of either infinite or semi-infinite
length. In practice, the vibrations of rods of only finite length are of interest in the study of soil dynamics.
WAVE PROPAGATION IN AN ELASTIC MEDIUM
70
Therefore, if a bar of finite length l vibrates in one of its normal modes (Fig. 3.5a), the solution to the wave equation, (Eq. 3.7c), may be written as follows: U
= U(A
cos wnt + B sin wnt}
(3.13)
in which A and B are arbitrary constants, wn is the natural frequency of the rod, and U is the displacement amplitude along the length of the rod. U is a function of x and defines the mode shape of vibrations. By substituting Eq. (3.13) into Eq. (3.7c}, we get (3.14)
71
WAVE PROPAGATION IN ELASTIC RODS
The arbitrary constants, C and D, in Eq. (3.15} are determined by satisfying the boundary conditions at the ends of the bar. For a rod of finite length, the displacement amplitude, U, needs to be determined separately for the following three possible end conditions of the rod: 1. Both ends free (free-free).
2. One end fixed and one end free (fixed-free). 3. Both ends fixed (fixed-fixed). Free-Free Condition
For the rod of length, l (Fig. 3.5a), in the free-free case, the stress and strain on both faces must be zero. This means that dul d.x is equal to zero at both x = 0 and x = l. By differentiating Eq. (3.15} with respect to x, we get
The solution to Eq. (3.14} may be written as (3.15)
.. t;'
~
dU w ( wnx wnx) -=--" -Csin-.-+Dcos-V =0 dx vr .o;,,-';~ vr r
(3.16)
By substituting the preceding boundary conditions into this equation, we find that D is equal to zero, and
. w,J
Csm-= V,
(a)
0
(3.17)
For a nontrivial solution,
w)
-
U1
=
C cos
=
1)
V,
or w
2.x
U2 = C cos - - (n = 2) 1
3•x
U3 = C cos - - (n = 3) 1
(3.18a)
=n7T
n1rVr
n
=--
. in which n = 1,2,3.... Equation (3.18b} is the frequency equation from which the frequencies of the normal modes of vibrations of the rod for the free-hee. cas~ are determined. By substitlliiing Eq. (3.18b} into Eq. (3.15}, the d1stnbut10n of displacement along the rod can be found for _any harmonic. The first three harmonics are shown in Fig. 3.5b, and the displacement amphtude can be expressed by n1rx
Un = Ccos -[(b)
Figure 3.5. Vibrations of a rod of finite length with
free~free
end conditions.
(3.18b}
/
in which n = 1, 2, 3 ....
(3.19}
F
72
WAVE PROPAGATION IN AN ELASTIC MEDIUM
!
WAVE PROPAGATION IN ELASTIC RODS
!
The displacement amplitude may then be written as
73
~
Fixed-Free Condition
The end conditions for a rod in the fixed-free case (Fig. 3.6) are: at the fixed end (x = 0), the displacement is zero, i.e., U = 0; and at the free end du (x = 1), the strain is zero, i.e., dx = 0. By substituting these into Eq. (3.15), we get C=O
(3.20a)
and Dcos
! 1•.
I I
VX = D sin (2n-211)71"X
W
U, = D sin
(3.23)
'
The first three harmonics described by Eq. (3.23) are shown in Fig. 3.6b. Fixed-Fixed Condition
The end conditions for a rod in the fixed-fixed case (Fig. 3. 7) are U = 0 at x = 0 and at x =I. By substituting these conditions into Eq. (3.15), we get
w I
V =0
(3.20b)
C= 0 and
'
wnl Dsin- =0 V,
which gives wnl 7T = (2n -1)V, 2
-
(3.21)
(3.24a)
which gives (3.24b)
in which n = 1, 2, 3 .... wn is given by
in which n = 1, 2, 3 .... w =(2n-1) 71"V, "
(3.22)
21
lei (e)
D
--j_ D
T
l
U1 "" D sin ;~
-
~ (n = 1)
. 37rx U2 = D s1n 2"l 1n
D
T ----= 2)
v .,.
-..........
7
U1
=
D sin
U2
=
. 21rX I D sm - - n
U3
=
D sin
(n
1)
=
l
D
'l
/
1
~
2)
I
D
L_ ['.... (b)
-Y
3
;x
(n = 3)
(b)
Figure 3.6.
Vibrations of a rod of finite length with fixedMfree end conditions.
Figure 3.7.
Vibrations of a rod of finite length with fixed-fixed end conditions.
WAVE PROPAGATION IN AN ELASTIC MEDIUM
74
element are becomes
We can write Un = D
, n7TX Sill-/
75
WAVE PROPAGATION IN ELASTIC RODS
T and [T +(aT/ax) dx],
as shown in Fig. 3.8b. The net torque
(3.25) 2:;(torque)T=-T+(T+
in which n = 1, 2, 3 .... The first three harmonics described by Eq. (3.25) are shown in Fig. 3. 7b.
3.1.3 Torsional Vibrations of Rods of Infinite Length The equation for the motion of a rod in torsional vibration is similar to that for longitudinal vibrations of rods discussed above. A rod in Fig. 3.8 is acted upon by a torque T which produces angular rotation 11. The expression for the torque can be written as
T=GI ae
Pax
(3.26)
in which IP is the polar moment of inertia of the cross-sectional area of the rod, G, the shear modulus of the material of the rod; and ae/ax is the angular twist per unit length of rod. The torques on the two faces of the
~~ dx)
aT ax
(3.27a) (3.27b)
=-dx
By applying Newton's second law of motion to the vibration of the rod, we get
aT
y
a'e
- d x = - l d x -2 ax g P at
(3.28a)
or (3.28b) By substituting (3.28b), we get
aT/ax
obtained by differentiating Eq. (3.26) into Eq.
(3.29a) or
a'e _ 0 !I a'e ar' - y ax' (o)
(3.29b)
or (3.29c) in wbicb
g
G
z
y
p
'
G-=-=V
(3.30a)
or
v
'
(b)
Figure 3.8.
Torque acting on element dx of a rod.
=
(G
\/{)
and V is the shear-wave velocity of the material of the rod.
'
(3.30b)
76
WAVE PROPAGATION IN AN ElASTIC MEDIUM
77
WAVE PROPAGATION IN AN ElASTIC INFINITE MEDIUM
z
3.1.4 Torsional Vibrations of Rods of Finite Length The problem of torsional vibrations of rods of short length can be analyzed in the same manner as for the case of longitudinal vibrations (Section 3.1.2). The solution to Eq. (3.29c) may be written as O(x, t) = El(x)(A sin w,J + B cos w,t)
(3.31) I I
Txy . ,
in which El is the amplitude of angular vibrations, and A and B are arbitrary constants which can be determined from conditions at the ends of the bar. By substituting from Eq. (3.31) into Eq. (3.29c), the solution for the three types of end-conditions are obtained as given below.
-.-'(
I
t
I
)./'
arxy
/
(Txy
Free- Free Condition X
w
n1TV:
=-~
"
l
(3.32)
in which n=1,2,3 ....
(ux
+ Ty dy)
8ox
+ axdx)
Figure 3.9. Stresses on an element in an infinite homogeneous, isotropic, and elastic medium.
the x directfiln, the equilibrium~quation can be written by considering the sum of forces in this direction as follows:
Fixed-Free Condition
(3.33)
( (]'x
+
aax ax dX ) dy dz - (]'x dy dz
+
in which n=1,2,3 .... Fixed-Fixed Conditions
(3.34) in which n = 1, 2, 3 .... The concept of a natural frequency of a rod of finite length in a principal mode of vibration is used in determining the elastic properties of a soil in the laboratory. This is discussed in Chapter 4 (Section 4.5).
+ ( Txy +
(T
a(]' ( _x ax
aT aT) + ...__.2 + ----'-'ay
az
aTu dz) dx dy - Tn dx dy = 0
az
(3.35)
2
au dx dy dz = p(dx dy dz) - , at
(3.36)
or 2
p at' =
WAVE PROPAGATION IN AN ELASTIC INFINITE MEDIUM
The problem of wave propagation in an infinite medium will be considered now. It will be assumed that the infinite medium through which waves propagate is elastic, homogeneous, and isotropic. Let us consider a small element of dimensions dx, dy, and dz as illustrated in Fig. 3.9. The stresses
+
Similar equations can be written for the summation of forces in the y and z directions. By neglecting body forces and applying Newton's second law in the x direction, we get
J U
3.2
xz
aTxy ay dy ) dx dz - Txy d X d Z
Similar equations follows:
can!;~
Jcrx
ax+
iJTxy
ay
t3Txz
+
az
(3.37a)
written for the motion in the y and z directions as
(3.37b)
acting on the faces of this element have been shown in this figure. By
considering the variation in stresses on opposite faces of the element, the stresses on each face can be represented by sets of orthogonal vectors. In
(3.37c)
78
WAVE PROPAGATION IN AN ELASTIC MEDIUM
in which u, v, and w are displacement in the x, y, and z directions respectively. The right-hand sides of the three expressions of Eq. ·(3.37) may be expressed in terms of displacements with the help of the following relationships for an elastic medium:
au ae 2 p2 =(A+G)-+GVu 2
at
ax
a2 v
p -
at2
a2 w at 2
p -
G
=
2(1 + v)
(3.39a)
vE (1 + v)(1- 2v)
(3.39b)
cc-~~---::-c
in which v is the Poisson's ratio, A, G are the Lame's constants ( G is also termed the shear modulus), and
E.}. The strains and rotations may be defined in terms of displacements (Timoshenko and Goodier, 1951; Kolsky, 1963) as follows:
Ex= Jx '
av
ey = ay , E z
aw az '
=-
av au 'Yxy = ax + ay aw av 'Yy. = ay + az au
aw
'Yzx=az+ax
-
aw au x ay az - au aw 2w = - - Y az ax = av _au 201 ' ax ay
(3.43)
2
(3.44)
ax'
ay'
a')
(3.45)
az
Equations (3.4;~)_, (3.43), and (3.44). are the equations for motion of an infinite homoge'neous, isotropic, ari{L.~lastic medium. There are two solutions for the preceding equations. One solution • describes the propagation of an irrotational wave, whereas the other describes the propagation of a wave of pure rotation. The first solution is obtained by differentiating Eqs. (3.42), (3.43), and (3.44) with respect to x, y, and z, respectively, and adding all three expressions together. This gives 2-
p
(3.40a) (3.40b)
a~ at
= (A+ 2G)V 2<
(3.46a)
or (3.46b)
(3.40c) in which
Rotations 2w = - - -
ae az
=(A+ G)-+ GV w
a'+ a' + V'= ( 2
Strains
au
2
in which V2 (the Laplacian operator in Cartesian coordinates) is defined as
and A=
ae ay
=(A+ G)-+ GV v
and
(3.38c) E
(3.42)
Similarly, Eqs. (3.37b) and (3.37c) give, respectively,
(3.38a) (3.38b)
79
WAVE PROPAGATION IN AN ELASTIC INFINITE MEDIUM
v' = _A_+_2_G_ (3.41a) (3.41b) (3.41c)
in which Wx, WY, and Wz are rotations about the x, y, and z axes, respectively. Now, by substituting appropriate expressions from Eqs. (3.38), (3.39), and (3.40) into Eq. (3.37a), we obtain
'
(3.47a)
p
where V, is the velocit)li~of compression waves in the infinite medium. Equation (3.46b) is exactly of the same form as the wave equation (3.7c). Substitution of A and G from Eqs. (3.39a and 3.39b) into Eq. (3.47a) yields
£(1- v) 2 V' = -p(701-'=+C"--=;v)7( 1;-'---'-.,-2'v) in which E, is the constrained modulus.
E, p
(3.47b)
80
WAVE PROPAGATION IN AN ELASTIC MEDIUM
81
WAVE PROPAGATION IN AN ELASTIC INFINITE MEDIUM
v;
If _v is equal to zero, will be equal to Elp and V, is equal to the veloe1ty of the compressive wave propagation in the rod, V, [Eq. (3.8)]. For v greater than zero, V, is greater than V,. The second solution of the equations for motion can be obtained by differentiating Eq. (3.43) with respect to z and Eq. (3.44) with respect toy and then eliminating E' by subtracting these two equations. Proceeding in this manner, we get P
!.'._ (aw at'
_av) cv'(aw _av) ay az ay az =
and by using the expression for rotation
a2-wx p -,- = at
wx from Eq.
4
~ >
3
>I>'
-
0
(3.48)
w
2
0
"
>
(3.41a), we get
S waves
R waves
G'i/
2-
Wx
0
0.3
0.2
0.5
Poisson's Ratio, v
or
Figure 3.1 0. Rel~tion between Poisson's ~tj9- v and velocities of propagation of compression (P), shear (S), and Rayleigh (R) waves in a'
(3.49) Similar expressions can be obtained for w and rotation is propagated with velocity V, given' by,
V'= G '
p
w,,
which implies that
(3.30a)
By combining the Eqs. (3.47b) and (3.30a) and substituting for G in terms of E from Eq. (3.39a), we get
i
'
=
~z(r1~;J
(3.50)
A plot of V,IV, is shown in Fig. 3.10 (Richart, 1962). From the preceding analysis, it can be seen that in an infinite elastic medium, there are two kinds of waves: 1. Compression wave (also called primary wave, P wave, dilatational
wave, irrotational wave). 2. Shear wave (also called secondary wave, S wave, distortional wave, equi-voluminal wave). The two waves, which represent different types of body motions, travel at
different velocities. However, the particle motion associated with the compression wave in the rod and the dilatational wave in the infinite medium is the same, but the wave-propagation velocities are different. In the rod, = Elp, but in the infinite medium, = E,lp. Therefore, the compression wave travels faster in the infinite medium than in a rod. This is true because no lateral displacements are possible in the former but are possible in the latter. The second (distortional) type of wave propagates at the same velocity (V; = G/p) in both the rod and the infinite medium. It will be seen from Eq. (3.47b) that if v = 0.5, V,-> oo and E,-> oo. Because water is relatively incompressible compared to soil, measurement of the velocity of a compression wave in water-saturated soil is not a representative wave velocity for soil but for water. Because water has no shear strength and has a zero value of the shear modulus, the velocity of a shear wave in water-sal!l,rated soil represents the soil property only. This fact has to be kept in mind when one plans wave velocity measurements for determination of soil moduli. Figure 3.11 shows plots of shear wave velocity and void ratio at several confining pressures for sands (Hardin and Richart, 1963). Table 3.1lists the velocity of wave propagation for compression and shear waves for different materials at different strain levels and confining pressures (Prakash and Puri, 1981).
v;
v;
" N
Table 3.1. Velocity of Shear and Compression Waves through Different Materials
Soil Type
Range of Dynamic Shear Modulus G (kg/em') for Effective Confining Pressure of 1 kg/em'
Associated Strain Level
Shear Wave Velocity V, (m/sec)
Poisson's Ratio (Assumed)
Velocity V, (m/sec)
(1)
(2)
(3)
(4)
(5)
(6)
553 to 3146
3.9 X 10-' to 1.3 X 10-'
167.7 to 400.0
0.30
313.7
Stiff, brown-gray, silty clay (about 1m) underlain by medium to dense sandy silt and silty fine sand (about 9.0 m)
77.5 to 221.5
1.049 X 10-' to 1.1 X 10-'
63.3 to 107.0
0.35
Silty sand G, ~ 2.61 r ~ 1.80 e ~ 0.72 ¢ ~30.5 w, ~ 18.6%
186.9 to 587.0
1.63
101.0 to 179.0
0.35
210.0 to 373.0
Fine to medium sand with some silt G, ~2.47 ¢ =30' 1' ~ 1.75 w~. = 22.2% e ~ 0.72
131.8 to 306
87.0 to 133.0
0.33
173.0 to 264.0
82.0 to 218.0
0.30
164.0 to 433.0
79.0 to 335.0
0.30
148.0
131.5 to 328.0
0.35
Lateritic soil silty sand G, ~2.67 r ~ 1.93 e ~ 0.59
¢
~32'
w,
~
Compression Wave
to
748.3
15.2%
X 10-' to 1.5 to w-'
1.8
10-' to 4.2 x w-• X
131.8 to
222.70
·---··--·-····
Medium sand G, =2.58 r ~ 1.79 e ~ 0.71
¢
~30
w,
~
3.0 X 10-' to 3.0 X 10- 5
136.3 to 2442.0
1.03
...,,,
19.2
Boulder deposits witft'cmatrix of medium to coarse silty sand: Properties of the matrix material G, ~ 2.70 1' ~ 2.15 e ~ 0.605 ¢ ~ 32.5 We =28%
Poorly graded fine silty sand up to 5.0m G, ~2.63 r = 1.83 e
122.9 to 867.3
X 10-' to 1 X 10- 6
to
627.0
....
,j."
'~.
322.2 to 200
1.5 x 10_' to 1 X 10- 6
~0.69
¢ =30' w, ~ 17.6% a Gs =specific gravity of particles; y =bulk density of soil; e =void ratio; ¢ =angle of internal friction; we =water content.
Source: After Prakash and Puri (1981).
"w
274.0 to
683.0
84
WAVE PROPAGATION IN AN ELASTIC MEDIUM
~
.2 0
u
0
E
o~~~~~L-~--~ 0.3
Void ratio
(b)
space, another type of wave, the Rayleigh wave shows up. The motion of a Rayleigh wave is confined to a zone near the boundary of the half space or the ground surface. The solution for this wave was obtained by Rayleigh (1885) and later described in detail by Lamb (1904). The study of waves propagating in a zone close to the surface is of practical interest in the study of the machine foundations. The effect of the free surface of the soil medium on the propagation of waves in soil will now be described. The half space may be defined in the xy plane only with z axis assumed positive in the downward direction (Fig. 3.12). For a plane wave traveling in tbe x direction, particle displacement is independent of y. If the displacements in the directions of x and z are represented by u and w, respectively, and v is equal to 0, then
--Round grains -Ottawa sand
au= aw =0
---Angular grains- Crushed quartz
ay
" .......... ,....,...._
.............................. _
'...f:.s~
................................ ..............
-.........
_ ----
-.. ........
'.fl..0 Pst
_
-.. .......
If tbe action of body forces is neglected, the equations of wave propagation are the,.same as Eqs. (3.42) and (3.44). ":?' ?.~'.:'t - .l: 2 a. a u 2 (A+ G) -a + GV u = p - 2 (3.42) x at
ae
.3Q
-....._Pst
-..-.Iooa
(3.51)
ay
+'
-'--fooo
........... -..
85
WAVE PROPAGATION IN A SEMI-INFINITE ELASTIC HALF SPACE
2
a2 w
(A+ G) az + GV w=p at'
(3.44)
-...._Pst
Sao
--
-.._pst
400,~-n~-n~-n~~~~~~~~L_~L__j~--~~ 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
Plane wove front
Void ratio
Ia I Figure 3.11 . Variation of shear wave velocity and shear modulus with void ratio and confining pressure for dry round and angular·grained sands. (After Hardin and Richart, 1963.)
3.3 WAVE PROPAGATION IN A SEMI-INFINITE ELASTIC HALF SPACE In a. practical situat_ion, the solutions for wave propagation in an infinite
dastlc body are of httle value. Because machine foundations are supported m the soil at a shallow depth, the boundary conditions approximating this SituatiOn are those of a semi-infinite half space. It will be assumed that the ?'led~um
is homogeneous, isotropic, and elastic. For the case of an elastic
mfimte medium (Section 3.2), it was shown that there are two types of body waves: the compressiOn wave and the shear )'lave. In case of an elastic half
Figure 3.12. Wave propagation in elastic half space, coordinate convention.
86
WAVE PROPAGATION IN A SEMI-INFINITE ELASTIC HALF SPACE
WAVE PROPAGATION IN AN ElASTIC MEDIUM
a
alj;* az
u=--+--
(3.57a)
f3'=x'-k'
(3.57b)
f (3.52b)
in which * and if;* are analytic functions. Since the z axis is perpendicular to the soil surface and is positive downwards, then for all points in the soil, z is greater than zero. If the steady-state propagation of the waves has a frequency of w, the functions * and if;* can be expressed as
-
and
(3.52a)
and
a
h'
a 2 =x 2
A general solution of these equations may be expressed as
The arbitrary constants A, B, and x are determined from the boundary conditions. If the boundary surface, z = 0, is subjected to the action of external normal forces distributed over the entire plane, these forces induce normal stress given by (3.58) It is assumed that the tangential stresses on the border plane are zero, that is
(3.53a) and
(3.59) The stresses JZ,.and "'
T"
may be expr'"ssed as functions of > and if; as follows: -':<;--·-<
(3.53b) T,
(3.54a)
=
G
The equations that should be satisfied by the functions and if; then are
(V' 2 + h 2 )
(]', =
G
(3.54b)
2 a' _ k'iJ; _ 2 a'v;
(3.60a)
-k'- 2 a'- 2 a'if; ax' axaz
(3.60b)
axaz
ax
and
and
(V' 2 + k 2 )1j; = 0
87
By substituting the expressions for > and if! from Eqs. (3.56) into the right-hand sides of Eqs. (3.60) at z = 0, we get
in which
-2ixaA + (2x'- k')B
=
o
(3.61a)
(3.55) and where w is equal to 2Tr!T, in which Tis the period of the propagated waves. Then his equal to 2Tri(VJ) and k is equal to 2Tri(V;T). Now, VJ and V,T r~present the wavelengths of lo~gitudinal and transverse waves, respectively. Therefore, h and k are reciprocal values of the wavelengths, and k is always larger than h. Particular solutions of Eqs. (3.54a and b) may ~:)!' assumed as (3.56a)
2
Uo = G
(3.61b)
Therefore, we obtain (3.62a) and
and
B=(~t~)(~)
(3.56b) where
2
(2x - k )A + 2ix(3B
where
(3.62b)
88
WAVE PROPAGATION IN AN ELASTIC MEDIUM
89
WAVE PROPAGATION IN A SEMI-INFINITE ELASTIC HALF SPACE
F(x)
=
(2x
2
k')'- 4x'af3
-
(3.63)
By using Eqs. (3.52), (3.56), and (3.62), we get u = ix
(2
2
X -
k') -a' 2 {3 -#< ;(X) - a e e;xx ~
F(x)
=
o
(3.67)
x may (3.64a)
be determined from Eq. (3.63). Instead of Eq. (3.67), which contains irrational expressions, we may consider the following equation, which does not contain radical signs (Barkan, 1962):
and
F(x)f(x) w =a
-(2x'- k')e-"' + 2x'e-~' · u 0 e'xx _ F(x) G
J+- x[(2x
2
k') e-"'- 2a{3e-~']e;x., F(x) dx
-
-00
(3.65a)
__!__
J+" a[ -(2x
2.,.0 -·
2
-
k') e-"' + 2x 2 e-~']e;xx F(x) dx (3.65b)
f(x)
=
k')B
=
0
(3.66a)
and
o
(3.66b)
Equations (3.66a and b) will give nontrivial solution for A and B only when the determmant of this system equals zero. This leads to
k')x'
4
( h2) K_6J= 0
K_4 - 16 1 - k k'
k
6
(3.69)
Since k is greater than h, one of the roots of Eq. (3.68) lies between 1 and +oo. It be shown that th<\:"other two roots, if real, lie between zero and h 21k'. · The first root corresponds to positive values of a and {3; and therefore, it does not satisfy the condition f(x) = 0. The last two roots make a and {3 positive and imaginary and, therefore, they do not satisfy the equation F(x) = 0. This equation has only one root, x' =A 2;t which is greater than one. Therefore, A2 is greater than k'. For a Poisson's ratio of 0.5, the real root of Eq. (3.68) is Alk = 1.04678. When vis equal to 0.25, all roots ofEq. (3.68) are real and are equal to
gn.
A2
1 4
(3.70a)
=
~ (3- YJ)
(3.70b)
=
4 (3+YJ)
and 1
(3. 70c)
Of these roots, onl~;the last one satisfies the conditions of this problem. Its value is A
(2x'- k')A + 2ixf3B =
-
(2x'- k') + 4x'af3
Velocity of Wave Propagation
-
2
in which
k'
-2ixaA + (2x 2
2
h )(x
(3.68)
Equations (3.65a) and (3.65b) correspond to the waves induced by an excihng force acting along the line x = 0, z = 0.
Free surface waves occur where they are induced by some initial excitement on the border surface. Assuming, u 0 = 0, for this case, the values of constants A and Bin Eqs. (3.61a) and (3.61b) may be determined from the followmg expressiOns:
-
k'
k
and w = _
(2x'- k 2 ) 4 -16(x 2
= k 8 [1- 8 K_22+ (24- 16 -h2)
(3.64b)
In_ order to tra~sform the exciting force into one that acts along the line x- o, z= 0 (Fig. 3.12), assume that u 0 is equal to - P(dx/2.,.). By subshtutmg th1s expressiOn mto the right-hand side of Eq. (3.64) and mtegratmg With respect to x from +oo to -oo, we obtain the following expressiOns for the displacements u and w:
u = __!!'__ 2.,.0
=
1
k = 2 V3+ Y3 = 1.087664 Similarly, if we designate ·1· A is a root of Eq. (3.68) satisfying the required conditions.
(3.71)
90
WAVE PROPAGATION IN AN ELASTIC MEDIUM
WAVES GENERATED BY A SURFACE FOOTING
(3.72) Ia)
s -wave
P-wave
u
A=_":_ VR
91
(+away)
~
R -wave
I
I
r
./'>-
in which Vu is the velocity of propagation of the Rayleigh wave under consideration, it may be seen that
vR =Ak v; for v
= 0.5,
for v
=
0.25,
(3.73) Minor tremor
VR
=
0.9553V:
VR
=
0.9194V,
Thus, it is seen that the velocity of surface waves propagation is somewhat smaller than that of shear or transverse waves. Therefore Vn is less than v;. A plot of VRIV: for different values of vis sketched in Fig. 3.10 (Richart, 1962). It is thus seen that there are three principal waves in an elastic half space. These ':av~s tra~el at different velocities. Knowing their velocities, it is easy to prediCt m which order the waves will arrive at a given point as a result of a disturbance at another point.
~
t
Major tremor--\
w
(b)
{+down)
I
I
v
1
~
\_
t
Particle motion
lei
Particle Motion at the Surface
Lamb (1904) studied in detail the surface motion that occurs long distances away from a pomt source at the surface of an elastic half space. When a pomt source acts at the surface, the disturbance spreads out in the form of ~ymmetncal annular waves. The initial form of these waves depends on the mput Impulse. When the input is of short duration, the characteristic waves shown m Fig. 3.13a develop (Richart et al., 1970). These waves have three salient features which correspond to the arrivals of the P-wave, S-wave, and Raylmgh (R) wave. The horizontal and vertical components of particle motwn are shown m Figs. 3.13a and b, respectively. At the ~urface, a particle first undergoes an oscillatory lateral dismen! on arnval of the P-wave. This is followed by another oscillation at the arnval of the S-wave after a relatively quiet period. This is then followed by an oscdlatwn of much larger magnitude when the R wave arrives. With mcreasmg distance from the source, the time interval between wave arrivals becomes greater and the amplitude of the oscillations becomes smaller. In additiOn, P-wave and S-wave amplitudes decay rapidly compared to that of an R-wave. Therefore: the R-wave is the most significant disturbance along the surface of an elas!Jc half space. At large distances from the source, this wave may be the only clearly distinguishable wave. If ~he h~rizo?tal and vertical components of the particle motion starting at pomt 1 m Figs. (3.13a and b) are combined, the locus of the surfaceparticle motion for the R wave can be drawn as shown in Fig. 3.13c. The path of the particle motion is a retrograde ellipse.
Direction of wave propagation --;.-
w
Figure 3.13. Wave system from surface point source in ideal medium. (After Richart, Hall and Woods, 1970, p. 90, Reprinted by Permission of Prentice Hall, Englewood Cliffs, N.J., After Lamb, 1904.
3.4
WAVES GENERATED BY A SURFACE FOOTING
In practice, a machine foundation generates waves in the soil. To illustrate this condition, an ideal case of a circular footing undergoing vertical oscillations at the surface of an elastic half space will be considered. The energy of the oscillating footing is carried away by a combination of P, S, and R waves. The esse\)tial features of this wave field at a relatively large distance from the source'are shown in Fig. 3.14 (Woods, 1968). The distance from the source of the waves to each wave in Fig. 3.14 has been drawn in proportion to the velocity of each wave for a medium with v equal to 0.25. T_l!e body waves(!:. a!!d S J~f.()p_ag~l_"..Qll~."'~r.IVC propagates radmlly outward
3.Iofii~-"~ ~EY-.lindjkf!l wav£_~ r~~!:...~U
of
tN.~~~es____~-~~ou~-~~r -~t:J:_. i-~~~~~!~-g~y
larger volum()( ofthemed\um as they tra el ouiwara. Therefore, ilie energy density ofep: wave.decr~ases with distiihce from the footing. This decrease
i~-~!!_~gy_q._fu!~]-~~~ or decn\ase m dtsplac~ment amplitude, ts ca1l_~~-.S~t-
'
!
·.
ii .
\
.
t"·! .Q,_, "
92
WAVE PROPAGATION IN AN ELASTIC MEDIUM
r-2
r-2
3.5
Circular footing
Geometrical damping law r-0.5
-c--ov =
0.25
Relative amplitude
Percent of
Wave Type (a)
PRACTICE PROBLEMS
.--
P·-
FINAL COMMENTS
Solutions for the velocity of the P wave and the S wave (body waves) and the propagation of the R wave (the surface wave) have been described. Importance of the wave propagation phenomenon to the study of machine foundations bas been discussed. In studying wave propagation caused by a vibrating footing, it has been assumed tbat the footing is circular and placed at the surface of a semi-infinite elastic half space. In practice, a footing is more often rectangular than circular. Also, a footing is always embedded a certain depth below the ground level. Nevertheless, the study of the problem as a simplified case, as in the preceding discussion, does not reduce its practical value. In fact, the departure in the actual results as compared with the analytical predictions based upon simplified assumptions helps in advancing the state of the art in two ways:
total energy
~ ~ r-----1----'~=:::.__j S
93
Rayleigh
67
Shear Compression
26 7 (b)
~igure .3.14. ?istribution of displacement waves from a circular footing on a hom g rsotropic, elastrc half space. (After Woods, 1968.) o eneous,
rica/ damp!ng. The amplitude of the body waves decreases in proportion to the rat1o of 11 r m wh1ch r 1s the distance from the. input source However along the surface of the half space, the amplitude decreases a~ 1fr2 Th~ amphtude of the Rayleigh wave decreases as 1 /vr (Woods, 1968). The partlcl~ motion a~sociated with the compression (P) wave is a push-pull motion m the d1rect10n of the wave front. The particle motion associated w1th the shear (S) wave is a transverse displacement that is norm~! to the d1rectwn of the wave front. However, the J>article motion f ] ....... l..f·----------.--assoc1ated w1th the R wave at the surfac re-trograde ellis(; p{....jlj-;,-Te ····-······ e o_an_eastl<;_ha .. ~p.a:<_Oe__ts~ .li____b ............... J'......J;-_K __ . .J. h shaded zones along the wave fronts for t e ?dy waves md1cate the relative amplitude of particle displacement as a function of the dip angle (the angle measured downward from the surface at the ~enter of the source). The R-wave can be described by two components, vertical and honzontal, each of which decays with depth but according t 0 separate laws. For a vertically oscillating circular energy source, e.g., a vibrating fo?tmg, on the surface of a homogeneous, isotropic, and elastic half space M11ler and Pursey (1954, 1955) determined that the distribution of total mput energy among the three elastic waves was 67 percent for the R-wave 26 per~ent for the S-wa_ve, and 7 percent for the P-wave. The fact tha; two-thuds of the total mput energy is transmitted away from a surface energy source by the R-wave, which decays much more slowly with distance than the body waves, indicates that the R wave is of primary concern for foundatiOns on or near the surface of the ground.
l. The analytical tools are refined by making more realistic assumptions
and more sophisticated analysis, and 2. Test data that may be uS<;d:in developing correction factors for the analytical solutions are generated. It is important to note, however, that most soils are neither homogeneous and isotropic nor elastic. Therefore, gaps between prediction and actual behavior can be explained partially on this count. Further, many soils occur in layers. Analytical solutions for elastic waves in layered systems have been obtained (Zoeppritz, 1919; Ewing et al., 1970). The reflection and refraction of waves are used to determine the depth of overburden in exploring soils for civil engineering projects, but the corresponding solutions have limited application in machine foundations.
PRACTICE PROBLEMS
3.1
What do you understand by (a) compression wave (b) shear wave a,nd (c) Rayleigh wa'lie? 3.2 Describe the particle motion associated with compression, shear, and Rayleigh waves propagating in a semi-infinite, homogeneous, isotropic 3.3
and elastic half space. 3 In a deposit of dry sand with G = 2. 70 and dry density of 112lb/ft , estimate the shear wave velocity at 10, 20, and 30ft below ground
level. 3.4 If the Poisson's ratio of sand is 0.3, estimate the compression wave velocity in Problem 3.3.
94
WAVE PROPAGATION IN AN ELASTIC MEDIUM
3.5 If the sand gets fully submerged, will tbe shear wave velocity be altered? Justify your answer. 3.6 Compute the new shear wave velocities in Problem 3. 3. 3.7 Discuss the effect of saturation on tbe compression wave velociiy in soils. 3.8 (a) A circular footing is vibrating in the vertical direction on the surf~ce of elastiC half space. Describe schematically the dispersion of v1bratwns and several wave fronts generated (b) Explain what you understand by geometrical tktmping.
. .
I·.
'
4 Dynamic Soil Properties
REFERENCES Ba~kan, D. D. (1962). "Dynamics of Bases and Foundations." McGraw-Hill, New York. Ewmg, W. M., _Jardetzky, W. S., and Press, F., (1957). "Elastic Waves in Layered Media" McGraw-Hill, New York. ·
Hardin • B. 0. • and Richart.' F · E ., J r. (1963) . Elastic · wave velocJhes ·· m · granular soils. J. Soil Mech. Found. Eng. Dtv., Am. Soc. Civ. Eng. 89 (No. SM-1), 33-65. Kolsky, H. (1963). "Stress Waves in Solids." Dover, New York. Lamb, H. (1904). On the propagation of tremors over the surface of an elastic solid Ph ·t 1 Trans. Soc. London, Ser. A 203, 1-42. · as. Miller, _G. F., and Pursey, H. (1954). The field and radiation impedance of mechanical radtators on the free surface of a semi-infinite isotropic· solid Proc R Soc Londo S A 223, 521-554. · · · · n, er.
Mille~,
~··.and ~ursey, H. (1955). On the partition of energy between elastic waves in a semt-m ntte sohd. Proc. R. Soc. London, Ser. A. 223, 55-69.
?:
Prak~h, s... and R
Pori, V. K.. (1981). Dynamic properties of soils from in situ test. J. Geotech ng. Drv., Am. Soc. Ctv. Eng. 107 (No. GT-7), 943-963. '
ay!Leighd, L. M(l885). On waves propagated along the plane surface of an elastic solid Proc · · on on ath. Soc. 17, 4-11. Richart, F. E .• Jr. (1962). Foundat· 'b ,. T 863-898. ton Vl ra tons. rans. Am. Soc. Civ. Eng. 127, Part 1, Richart, F. _E., Jr., Hall, J. R., and Woods, R. D. (1970). "Vibrations of Soils and . Foundattons." Prentice-Hall, Englewood Cliffs, New Jersey. Ttmoshenko, S., and
Goodie~,
G. N. (1951). "Theory of Elasticity." McGraw-Hill, New York. J. Soil Mech Found v· A S Gv. Eng. 94 (No. SM-4), 951-979. . . lV., m. oc.
Wood~, R. D. (1968). Screemng of surface waves in soils.
Zoeppritz, K. (1919). Nachr. Konigl. Ges. Wiss. GOttingen Math._ Phys., Berlin, pp. 66--
94
.
Several problems in engineering practice require a knowledge of dynamic soil properties:, "In general, proble,p~involving the dynamic loading of soils are divided into small and large strain amplitude responses. In a machine foundation, the amplitudes of dynamic motion and, consequently, the strains in the soil are usually low, whereas a structure that is subjected to an earthquake or blast loading may undergo large deformations and thus induce large strains in the soil. A large number of field and laboratory methods have been developed for determination of tbe dynamic soil properties. The principal properties that are determined by many of these methods are: 1. Shear strength, which is evaluated in terms of strain rates and stressstrain characteristics; 2. Liquefaction parameters, such as cyclic shearing stress ratio, cyclic deformation, and pore-pressure response; 3. Dynamic moduli, sucb as Young's modulus, shear modulus, bulk modulus, and constrained modulus with corresponding spring constants; 4. Damping; and 5. Poisson's ratio. In machine foundations, an understanding of dynamic soil moduli (with corresponding elastic spring constants) and damping is frequently required. Poisson's ratio is also needed, even though it is frequently not determined. In this chapter, triaxial tests under static conditions are summarized first, followed by a detailed discussion of the laboratory and field methods used to determine dynamic soil properties. Typical values of dynamic soil moduli and damping are also presented. Shear strength determination and liquefac95
96
DYNAMIC SOIL PROPERTIES
lion parameters, are beyond the scope of this text and are discussed in detail elsewhere (Prakash, 1981).
4.1
TRIAXIAL COMPRESSION TEST UNDER STATIC LOADS
A typical triaxial test set up is shown schematically in Fig. 4.1. A cylindrical sample of soil Is placed between porous stones or metal discs. The length of the sample IS g~nerally kept as twice its diameter. The soil sample is enclosed m a thm, Impermeable membrane, which is secured to the base and top caps with rubber rings (0-rings). The 0-rings and the rubber m~mbrane Isolate the sample from the fluid in the chamber. If appropriate flmds are used to pressurize the chamber, it may not be necessary to use the membrane. The porous stones provide access to the sample for either pore-water dramage or pore-pressure measurements. Fluid pressure is applied within the chamber containing the sample. The chamber pressure IS controlled by a "cell pressure control" and is measured with a pressure gauge or a pressure transducer. The chamber pressure acts umf~rmly on the surface of the sample, including the top and bottom loadmg caps. Alternatively, the chamber pressure may be activated laterally, and the vertical pressure applied independently. . A vertical load can be applied through a loading ram, which is equipped With a Jackmg arrangement and measured with a proving ring or load cell.
Loading
mm Pressure
gauge
lL.~s,nwlc enclosed i'n
a
rubber membrane
Water
Pore-pressure gauge
TRIAXIAL COMPRESSION TEST UNDER STATIC LOADS
97
The axial deformation is usually measured by a dial gauge or an LVDT attached to the bottom of the proving ring and abutted against the top of the chamber. Triaxial shear tests permit a better control of stresses and volume changes during shear and drainage conditions as compared to other shear tests. There are basically three different types of tests that may be performed in a triaxial apparatus depending upon the drainage conditions. Drainage Conditions during Shear
Most soils are saturated at some time during the design life of a structure. Drainage conditions before and during shear influence the shear characteristics of saturated soils. In shear tests, soils are first subjected to normal or confining stress, which is usually maintained at a constant level. An increasing deviator (shear) stress is then applied. Shear tests have been devised to measure the shear characteristics of soils under three different drainage conditions as follows: 1. Unconsolidated-undrained test or "quick test": In these tests, no drainage is ~ermitted under conlli)fng pressure or during shear. Thus, the normal load is not transferred to the soil grains as intergranular pressure but exists as hydrostatic excess pore pressure. It cannot, therefore, mobilize any frictional resistance. Preventing drainage during shear prevents volume changes that might otherwise take place. 2. Consolidated-undrained test or "consolidated quick test": In these tests, soils are allowed to drain (consolidate) under applied confining pressure, and no drainage is permitted during shear. Although volume changes can occur during normal loading, these are not possible during shear. This, however, leads to development of pore pressures during shear. 3. Drained test or "slow test": In these tests, full drainage (consolidation) is allowed under confining pressure. Free drainage is also permitted during shear, so that excess hydrostatic pressure does not exist in the pores of the soil, and all stresses are integranular throughout. With the triaxial apparatus, volume changes in the drained tests are measured by the amount of water that flows into or out of a calibrated burette. In undrained tests, Jhe volume change tendencies of the soil result in generation of pore wiler pressures that can be measured by an electronic pore pressure transducer, manometer, or other suitable type of pressure gauge.
To cell pressure control Mercury
Figure 4.1. Triaxial apparatus.
Different Types of Tests in Triaxial Apparatus
The aforementioned tests can be performed in a triaxial test machine. Test interpretations are as follows:
-
200
DYNAMIC SOil PROPERTIES
98
160 Effective stress circle
'\;/
I
80
I
4
0~ 0
2
0
40
Total
~'Dense
/
120
stress circle
0
..L
~
1r
Consolidated-Undrained Test on Saturated Cohesive Soil. In a consolidated-undrained test, all-around pressure,
20
=
_
0.85
25
0 35
30
1
0
5
/
--
Loose
oens: Unconsolidated-Undrained Test on Saturated Soils. When an all around incremental stress Ll.
ILoose
10
-40 -80
0.75
0
15
'-...
Figure 4.2. Mohr plot for undrained test.
=
r
I
10
5
e0
/
~
i
15
lQ.
20
f 30
25
35
Axial strain (%)
StreS~._strain curves for undrained triaxial compression of a saturated sand. (After . _ ,, . " F1gure 4 . 3 . Leonards, 1962; Bishop and Henkel, 1957)."
~ I
I
"'
(a)
- - - Effective stress
- T o t a l stress
'
a
u31
1132
(b)
Figure 4.4. (a) Void ratio vs. confining pressure; (b) Mohr's plots for consolidated undrained tests.
99
100
DYNAMIC SOil PROPERTIES
25 20
15
,,.
..1'1'
/
iil 0'3
l7.Pore-water pressure, u..,
Chamber pressure 10 f-- varied between 30.5 1-_ and ~2.0 I~ per sq in. 5 35
v
0
sl'
I....- ......,
0
ol\
'
5
( 4.1)
4
/
;::.;::;:~dtr3
and
•
J
(4.2)
II F_igure 4.~. Plot of undrained cylindrical compresSIOn test m which pore-pressure values are observed. (After Taylor, 1948.)
2 0
son's ratio are not constants for a soil but, rather, are quantities that approximately describe the behavior of a soil for a particular set of stresses, loading conditions, and geometry. The different values of the modulus and Poisson's ratio apply for any other set of loading conditions. The terms tangent modulus and secant modulus are used frequently. Tangent modulus is the slope of the tangent to a stress-strain curve at a particular point on the curve (Fig. 4.6). The value of a tangent modulus will vary with the point selected. The tangent modulus at the initial point of the curve is known as the initial tangent modulus. Secant modulus is the slope of a straight line connecting two separate points of a stress-strain curve. Based on a linear stress-strain relationship, the following elastic constants can be defined. If a uniaxial stress a, is applied to an elastic cylinder (Fig. 4.7a), there will be a vertical compression and a lateral expansion such that
.,
5
101
ElASTIC CONSTANTS OF SOILS
5
10
15
Axial strain, AL/L 0 in percent
nitude of the pore pressure at failure, f!.ud, is approximately equal to the The inclination of the envelope of th e Mo hr c1rc · 1es of total deviator stress. . stresses give the apparent angle of internal friction, cf>,, and that of the Mohr circles the "effective" angle of interna1' · t.wn, 'Po "' F or llof effectiVe I'd stresses, d Jnc no rma Y canso~ ate clays, both envelopes pass through the origi~. In figure 4.5, the tnaxial test data for a consolidated undrained test for Boston Blue clay (Taylor, 1948) has been shown.
Drained Tests on Saturated Cohesive Soils. In a fully drained triaxial test ~he mduced pore pressures are allowed to dissipate before the stresses ar~ mcreased. Therefore, the pore pressures are zero, and the stresses in this test at all stages are effective stresses.
in which ex, eY, and ez ar~ the strains in the x, y, and z directions, respectively (considered positive when compressive), E is Young's modulus of elasticity, and vis the Poisson's ratio. If shear stresses r,x are applied to an elastic cube (Fig. 4.7b), there will be a shear distortion Yu such that
(4.3) in which G is the shear modulus.
Tangent modulus
Secant modulus
4.2
ELASTIC CONSTANTS OF SOILS
Concepts from the Theory of Elasticity
The ?ehavior of a soil is nonlinear from the time a stress is applied. For f.~~c!Ical p~rposes, the actual nonlinear stress-strain curves of a soil are
zneanzed , t.e., replaced by straight lines. Therefore, modulus and Pais-
Strain ~
Figure 4.6.
Definitions of secant and tangent moduli.
102 DYNAMIC SOIL PROPERTIES
103
ELASTIC CONSTANTS OF SOILS
( 4.5d) Uniaxial
Young's
loading
modulus
E•!!l
(4.5e)
••
(a)
and Simple
Shear modulus G Tu -"f..
shear (b)
(4.5f) The volumetric strain is given by
Isotropic compression
3<,
For a special -·case in which volumetric strain equals
av
V
Constrained
compression
modulus
..
ux:'£'5.:£TY
= 3e, =
= uz = cr0
3CTo
E
= Tyz = Tzx
= 0, the
(1- 2v)
"
The bulk modulus, Eb (Fig. 4.7c) is defined as
E
2(1 + v)
(4.4)
O"o Eb = 3e, =
ao
aviV
=
E 3(1- 2v)
1
E [O'x- v(O'y + O'J]
(4.5a)
(4.7) which gives
1
= E [O'y- v(O', + O'xl]
(4.5b)
1
E [0',- v(O'x + O'y)]
(4.5c)
( 4.6)
Still another special modulus is the constrained modulus E,, which is the ratio of axial stress to axial strain for confined compression or zero lateral strain (Fig. 4.7d). This modulus can be computed from Eq. ( 4.5c) by letting ex = ey = 0 in Eqs. 4.5a and b. Thus,
ca:or an elastic and isotropic material with all stress components acting, we employ the pnnc1ple of superposition to obtain the various strain components:
e, =
Txy
Various types of moduli.
G=
eY
and
~-,,..,-,-.,-fil Ec .!!...•
theEquations (4.l) through ( 4.3) defined the three basic constants of the odrydofbelaslJcJty: E, G, and v. Actually, only two of these constants are nee e , ecause they are related as follows:
ex=
(4.5g)
Z
s.!!
Confined
Figure 4.7.
X
modulus
(c)
(d)
av
-=e +eJ +e v
Bulk
E,
=
£(1- v) (1 + v)(1-2v)
(4.8)
Uniaxial loading and confined compression involve both shear strain and
volume change as shown below:
105
FACTORS AFFECTING DYNAMIC SHEAR MODULUS 104
DYNAMIC SOIL PROPERTIES
Condition
plasticity in terms of effective stresses. On this basis, the maximum value of 6 the shear modulus Gm" (at low shear strain of 10- ) is expressed by Eq. (4.9) (Hardin and Black, 1969):
Shear
Volumetric
Uniaxial loading
G
~(1-2v) u< E u<
Confined compression
(1
=
2
'Ymax
=
2G
+ v)(1- 2v) E(1- v)
The volumetric strain becomes zero if v
(1- 2v)
Tmax
u< u<
~
1-
1J
(1- 2v) 1- ll
max
~ 1230 OCRk ( 2 .973 - e)' 1+ e
.rg' psi
( 4.9)
in which OCR is the overconsolidation ratio, 0'0 the effective all-ar~u~d stress in psi, e the void ratio, and k a factor that depends up~n the plasUcity index of clays. Hardin (1978) recommended tha~ thiS _equatiOn be used for an anisotropic state of stress by taking u 0 = ( ui + u, + u,) /3, the mean effective confining stress. The parameter k, given in Table 4.1 is related to the plasticity index PI.
1/2. Table 4.1. Values of k k
Plasticity Index PI
4.3
FACTORS AFFECTING DYNAMIC SHEAR MODULUS
Based on the study of dynamic elastic constants, the factors on which these depend are (Hardin and Black, 1968): 1. Type of soil and its properties (for example, water content and yd) and state of disturbance; 2. Initial (sustained) static stress level or confining stress; 3. Strain level; 4. Time effects; 5. Degree of saturation; 6. Frequency and number of cycles of dynamic load; 7. Magnitude of dynamic stress; and 8. Dynamic prestrain.
In machine foundations, the magnitude of dynamic stress is usually small. The tests may, therefore, be performed at small strain levels. Also, the number of dynamic stress cycles is very large. To avoid buildup of any residual strains in the soil due to the operation of the machine, it is necessary to ensure that the vibrations of the soil-foundation system are in the elastic range. This is achieved by restricting the vibration amplitudes to small values. The effect of different factors on the''value of dynamic shear modulus is now discussed. Type of Soil and Confining Pressure. The large amount of data on the values of soil constants that had been collected was analyzed by Hardin
(1978). He developed a mathematical formulation of soil elasticity and soil
0 0.181 0.30 0.41 0.48 0.50
0 ' ,. 20 ";""·
40 60 80 >100 Source: Hardin 1978.
Equation ( 4.9) may be expressed in a more convenient form as follows: = A(OCR)k ( Gmax
F(e)
Pa
)I-n(U: )"
( 4.10)
0
By introducing p, (the atmospheric pressure), the par~meter A is dimensionless whereas U: and G in Eq. (4.9) are m lb/m., and the constant 1230 ha~ the dimen;ions (lbli';;'.') 0 ' 5 It is also desirable to change the form of the void ratio function in Eq. ( 4.9) by letting
F(e)
=
0.3 + 0.7e
2
(4.11)
in Eq. (4.10). The function F(e) is more convenient to use than the void ratio function in Eq. (4.9), but it gives about the same effect as em the range 0.4 < e < 1.2. For very large values of e, Eqs. (4.9) and ( 4.10) g~e monotonically decreasing values of Gm" whereas Eq. ( 4.9) g1ves Gm" -0 for e=2.973, and Gm"' increasing for e>2.973. Equation (4.10) will approximate Eq. (4.9) for 0.4 < e < 1.2 if one makes n = 0.5 and A = 625 ·
106
DYNAMIC SOIL PROPERTIES 0 5 ) " ]
Figure 4.8t shows a plot of [ Gm,I(OCR)k(p,ii0 versus the void ratio from laboratory and field measurements (Hardin, 1978). The elastic parameters reqmred for computation of the soil constant are k, n, and v. For most purposes, it is sufficient to use v = 0 12 =05 d 1 fk" Tbl · ,n .,an vaues o 1~ a e 4.1. For a preliminary analysis, Fig. 4.8 can be used as a gmde (Hardm, 1978). For. clean sands, Richart (1977) found that G depends on 0'0 and e. His analyl!cal express10ns for the shear modulus of clean sands are G max
= 700 (2.17- e)2 - o.s 1+e (uo)
( 4.12)
FACTORS AFFECTING DYNAMIC SHEAR MODULUS
for round-grained sands (e < 0.80), and G
= max
max
-4 -
(2.97- e) 1+ e
2 ( _
)o.s
lTo
( 4.13)
= 900 (2.17- e) l+e
2 (-
)o.3s
Uo
(4.14)
For shearing strains of 10- 4, their results agree with those of Eq. (4.12). The precoiding expressions inyatiably point out the fact that Gm, is proportional to ( 0'0 )". The value of'n has been recommended as 0.5 by most investigators. Therefore, if the shear modulus is determined at a mean effective confining pressure of ( 0'0 ) 1 , its value at any other mean effective confining pressure (0'0 ) 2 can be determined from Eq. (4.15):
l'I 3 ~0
326
for angular sands. In Eqs. (4.12) and (4.13), G and 0'0 are expressed in terms of kg/ cm 2 Both equations were originally established to correspond to shearing strains of 10- 4 or less. Equation (4.12) yields values slightly lower than those obtained by pulse tests (Whitman and Lawrence, 1963). Iwasaki and Tatsuoka (1977) determined experimentally from tests on clean sands (0.61 < e < 0.86 and 0.2 < 0'0 < 5 kg/ cm 2 ) at shearing strain amplitudes of 10- 6 that G
r;;gl li2l f712 I2J
107
2000
0
!i
•a:
(4.15)
(.)
0
....1
Effective overburden pressure only.
CJ
0.5
1.0 Void ratio
Figure 4.8.
Elastic stiffness from laboratory and field measurements· 1
lab
"It
d
"I
andclays·2 1 b 1 d b · ,sr ysan s,srts, . ' :-a ' c ean san s; 3- 1a , dense, well-graded graveiMsand with some fines; 4-lab ~~labvely u~rform clean gravels; F~-field, silty. sands, silts, and days at Ferndale, Cholame,
and
Centro srtes by Shannon and Wdson-Agbabran Associates (1976)· F field d "It d claysatsit AB dCAd '2,sans,srs,an es , , an · n erson et al. (1978}. (After Hardin, 1978.) i" In this figure, S is a dimensionless elastic stiffness parameter in Hardin's (1978) d stress-sdtrai n relation ~or inherent isotropy. For clean sands, s varies from 1200 to lr~:;~~r 1 st 1ts an cays, S vanes from 700 to 2000. '
150
a, may be used in place of 0'0 in Eq.
(4.15)
Strain Level. The other important factor affecting soil modulus is strain level. Ishihara (1971) presented Fig. 4.9, which indicates strain levels associated with different phenomenon in the field and in corresponding field and laboratory tests. Typical variations of G versus shear strain amplitude for different types of in situ tests are shown in Fig. 4.10. The soil modulus values may vary by a factor of 10, depending upon the strain level. It is customary to plot a graph between normalized modulus (defined as G-value at a particular,,~train, divided by Gm, at a strain of 10- 6 ) and shear strain. One such plot is shown in Fig. 4.11. The shear strains induced in soil may not be precisely known (Prakash and Puri, 1981). In the case of wave propagation tests, the shear strain amplitudes are low and are assumed to be of the order of 10- 6 The shear strains induced in soil essentially depend upon the amplitude of vibration or settlement, which in turn depends upon superimposed loads, the foundation
contact area, and soil characteristics. The measured values of amplitude or settlement take care of the factors affecting them. The shear strain am-
109
FACTORS AFFECTING DYNAMIC SHEAR MODULUS
1.2,-------,,---------,-----,------,------, Magnitude of strain
ta,·'
ta,-'
w-4
10-3
10- 2
w-'
I
Phenomena
Wave propagation, vibration
Cracks, differential settlement
Slide, compaction, !iquifacation
Mechanical characteristics
Elastic
E!astic·plastic
Failure
Constants
Shear modulus, Poisson's ratio, damping ratio
Angle of internal friction cohesion
Seismic wave method
•e a" 0
in situ vibration test
·;;; f
.s
~ •E
--,
Repeated loading test Wave propagation test
~~
Dynamic shear strain 1
o E
Resonant column test
~0 f,
.0~
j~
Figure 4.11.
Repeated loading test
Figure 4.9.
No~;lized
shear modulus
(t;)·gmaJ
vs. shear strain. (After Prakash and Puri,
1980; Prakash, 1981.)
Strain levels associated with different in situ and laboratory tests. {After Ishihara,
1971.)
plitude y, may be considered equal to the ratio of the amplitude or settlement to width (Prakash, 1975; Prakash and Puri, 1977). For the case of a vertically vibrating footing, the ratio of amplitude to width yields the normal strain e, and the shear strain y, may be computed as follows: (4.16)
"oo·r---r-~--r-rr--~--,-Tl-r--,---~~==~==~---,,---.---,-,-, 0 fo(t:edv'lbrotloot•" 6 F,..e vibration to•t 0 Cyclic plate lood tost
,ooo~'---+--4-~-4-~-+--~---f-tf---~~~·~·~M;"~·oo~"~'"~"~'"~,_f-1~--~--~~tj
l;-t---r-1
1600
j
For an axisymmetric ·case, ax
S:::.i::--- r-- ...
•
= aY,
and ez is given by
1
r-
8'
=
2G(1 + v) (
(4.17)
0
O'z -
'Ye =
O'x
2G
·•:~ y, (
( 4.18)
( 4.19a)
or 'Yo E,
Figure 4.1 0. 108
Dynamic shear modulus vs. strain. {After Prakash and Puri, 1980; Prakash, 1981.)
=
( 1- "x)(l + v) ----".2'-------
(4.19b)
DYNAMIC SOIL PROPERTIES
110
If the then the If one principal
loading condition corresponds to the at rest condition (K0 case), ratio of o")a, is given by Eq. (4.7) and y0 /e, = 1. considers an extreme case of incipient failure with a, as the major stress, then ( 4.20)
and the value of y 0 /e, will depend upon the value of the angle of internal friction of soil,>, and the Poisson's ratio, v. Typical values of y0 /e, are listed in Table 4.2. Table 4.2. Typical Values of y0 /E, Angle of Internal Friction, cf> (I)
v = 0.25
v = 0.33
v = 0.45
v = 0.5
(2)
(3)
(4)
(5)
25 30 35 45
0.9317 !.00 !.054 !.097
1.014 1.142 1.183 1.2!6
1.35 1.38 1.398 1.41
!.5 1.5 1.5 1.5
Source: Prakash and Puri, 1981.
111
FACTORS AFFECTING DYNAMIC SHEAR MODULUS
It was found that for most soils the time-dependent behavior at low strain levels can be characterized by an initial phase when a modulus changes rapidly with time. This is followed by a second phase when the modulus increases almost linearly with the logarithm of the time (Fig. 4.12). For the most part, the initial phase results from the void ratio changes and iucrease in effective confinement during primary consolidation. The second phase, in which the modulus increases almost linearly with the logarithm of time, is believed to result largely from any decrease in void ratio and changes in the soil structure due to a strengthening of the physico-chemical bonds in the case of cohesive soils and an increase in particle contact for cohesionless soils. This phase proceeds at a constant confining stress and is referred to as the long-term time effect. The long-term effect represents the increase in the modulus with time that occurs after primary consolidation is completed. Two methods are used to describe the long-term time effects. The long-term effect is expressed in an absolute sense as a coefficient of shear modulus increase with time, 10 . That is,
( 4.21) in which t 1' t2 are the times after primary consolidation, and !:J.G is the . change in low-amplitude shear modulus from t 1 to t 2 (Fig. 4.12). Numerically, I a equal the value of G for one logarithmic cycle of time. The long-term effect is also expressed in relative terms by the normalzzed shear modulus increase with time, Na· That is,
For values of > and v in the range of interest, it is reasonable to assume that Yo= e,.
Time Effects. The effect of duration of confinement at a constant pressure on the magnitude of shear moduli was first reported by Richart (1961). It was observed later that when specimens were confined at a constant confining pressure, shear moduli measured at shearing strain amplitudes below 0.001 ,percent (commonly referred to as low-amplitude moduli at a strain of 10- or less) increased with the time of the specimen's confinement. These studies showed that the shear moduli of artificially prepared soil specimens indeed increase with the length of time a specimen is confined. More recently, sustained-pressure studies on undisturbed specimens of sands and clays have shown that this time-dependent behavior is also characteristic of natural soils (Anderson and- Stokoe, 1977). The time dependence of moduli has significant implications. In the first place, it means that duration of confinement at a constant confining pressure must be considered when performing laboratory tests. Also, if the laboratory values of tbe moduli of a given soil are to be compared, they should be compared after equal confinement times for similar drainage conditions, and these times should be equal to or greater than the time of primary consolidation.
( 4.22)
~
~ ~
1
u
-
/';G
G- LOG 10 (t 2tt 1l ao "' CONSTANT y < J0-3%
.,; ~ ~
c~ c ~
oc
~
w
~
~
w
c
~
:::; ~
2
~
"'
PRIMARY CONSOLIDATION--+- LONG-TERM TIME t 1000 EFFECT
c
~
0
10'
102
103
t1
10 4
DURATION OF CONFINEMENT (LOG SCALE) Figure 4.12. Phases of rnodulus4irne response. (After Anderson and Stokoe, 1977, copyright ASTM. Reprinted with permission.)
"'
Table 4.3. Typical values of /G and NG
Specimen Type
(kN/m')'
Low-Amplitude Shear Modulus 2 G 1000 (kN lm Y
Vacuum extruded Compacted by raining and tamping
200 to 300 70 to 280
140000 to 190000 50 000 to 180 000
Kaolinite Bentonite
Vacuum extruded
70 to 550
4000 to 170 000
Agsco sand Ottawa sand
Compacted by raining and tamping
70 to 280
50000 to 110000
Confining Pressure
Soil Type
EPK kaolinite Ottawa sand Quartz sand Quartz silt
Typical
r
G
Typical
N'G
(kNim')'
(%)
24 000 to 35 000 1400 to 5500
17 to 18 1 to 11
1000 to 8500
5 to 25
Marcuson and Wahls (1972)
2000 to 10 000
1 to 17
Afifi and Richart (1973)
Reference Hardin and Black (1968) Afifi and Woods (1971)
Dry clay
Air-dried EPK Kaolinite
Saturated EPK Kaolinite
Vacuum extruded
~~~",-,.~~~
i''""~"'c''S,--- --'-
Undisturbed a
70 to 220
SO 000 to 2600 000
2000 to 22 900
1 to 14
Silty sand Sandy silt Clayey silt Shale Boston blue clay 9 Clays 1 Silt Clay fills Decomposed marine limestone San Francisco Bay mud Dense silty sand Stiff OC' clay
Undisturbed a Undisturbedd
70 to 700 35 to 415
32 500 to 54 000 13 000 to 235 000
=7000 26 OOO!'to 23 500
15 to 18 2 to 40
Undisturbedd Undisturbedd
35 to 70 325 to 830
50 000 to 200 000 365 000 to 1300 000
4200 to 15 000 28 000 to 102 000
7 to 14 3 to 4
Stokoe and Abdel-razzak (1975) Yang and Hatheway (1976)
17 to 550 220 to 620 1280 to 1300
7600 to 150 000 45 000 to 180 000 300 000 to, 320 000
725 to 32 000 5000 to 17 000 14 000 to 26 000
8 to 22 4 to 10 4 to 8
Lodde (1977) Fugro, Inc, (1977) Fugro, Inc, (1977)
#
ltJndisturbedd Undisturbedd Undisturbedd
Source: Anderson and Stokoe, 1977, copyright ASTM. Reprinted with permission. "I a defined by Eq, (01), "NG defined by Eq, (422),
1 kN/m 2 = 0.145 psi. Nominally undisturbed. e Overconsolidated.
c
d
-"'
-~~,""""'""'''
;;1, 'i
Stokoe and Richart (1973a, b)
Trudeau et al. (1974) Anderson and Woods (1975, 1976)
115
FACTORS AFFECTING DYNAMIC SHEAR MODULUS
114
DYNAMIC SOIL PROPERTIES
in which G 1000 is the shear modulus measured after 1000 minutes of constant confining pressure (after completion of the primary consolidation). The duration of primary consolidation and the magnitude of the longterm time effect vary with such factors as soil type, initial void ratio, undrained shearing strength, confining pressure, and stress history. Figure 4.13 shows typical time-dependent modulus responses for different soils. Typical values of Ic and Nc are given in Table 4.3. The shape of tbe low-amplitude modulus-time graph at a constant confining pressure depends primarily on whether the soil is predominantly fine-grained (silts and clays) or coarse-grained (sands). Figure 4.14a illustrates typical changes in the shear modulus of a clay with time at a constant confining pressure. The two distinct phases of modulustime response are clear from this figure. First, during primary consolidation, values of the shear modulus are initially constant. They then increase rapidly and finally begin to level off. Second, during the long-term time effect, values of the modulus increase linearly with the logarithm of time. Figure 4.14b shows the vertical height change of the clay specimen during the constant-pressure confinement. By comparing the height change results with the modulus-time graph, it can be seen that the end of the initial phase in the modulus-time graph coincides with the end of primary consolidation. The point of transition in the modulus-time graph for this loading sequence is defined as the end of primary consolidation. Therefore, modulus values determined at a time before the end of the primary consolidation is complete are at an effective stress less than that assumed, because excess pore-water pressures still exist in the specimen. During the long-term time effect phase of modulus response, the shear
6G DUE TO CHANGE IN VOID RATIO
iG
,, - - - ·151
Gtooo
--+-
lONG-TERM TIME EFFECT
0,5
~
4
'' 8
L~.::.:..:..:_.:.,-.__
I
101
'
___._,__
(b) HEIGHT CHANGE
102
-:'::,---~
10 3
10
2.0 4
tO-WEEKS
DURATION OF CONFINEMENT, t (MIN) NG
SOl LS
at constant confining F•rgure 4 .14 · Typical modulus and height changes with time for day · d ·h · · )
<•>
NC CLAYS
OC CLAYS CLEAN SANOS
pressure. (After Anderson and Stokoe, 1977, copyright ASTM. Reprmte wrt permrssron.
5-20 3-10 .c
1-3
'
:' SANDS :
OL-----~----~----~----~-----' to 1 to2 toJ to4 to5
DURATION OF CONFINEMENT (MIN)
Figure 4.13. Effect of confinement time on shear modulus. (After Anderson and Stokoe, 1977, copyright ASTM. Reprinted with permission.)
modulus increases almost linearly with the logarithm of time. In Figure 4.14a, this increase was monitored for 10,000 minutes or about o~e week (Anderson and Stokoe,_ }.977). The coefficieJ~t of shear modul~s mcrease ith time I is aboui~900 psi (6200 kN I m ) , and the normalized shear W ' G' modulus increase with time Nc is about 15 percent. The value of I generally increases as the confining pressure increases. Values of N de~rease with increasing undrained shearing strength and increase with increasing void ratio for fine-grained soils (Anderson and Woods, 1976). Stress history also affects values of I 0 and Nc· Figure 4.15 shows this effect for a series of modulus-time tests conducted on one specimen. Values of the modulus were determined over approximately a
116
DYNAMIC SOIL PROPERTIES
12~-------,--------r--------r--------,
..
FACTORS AFFECTING DYNAMIC SHEAR MODULUS
117
28
AIR-DRY OTTAWA SAND 130- 50)
X
<
"'
w
.,;
y
0-
"'~ ~ ~ < 0 w z %<
w
~
190
4
~
~
10-3,
~
ON
26
c,. ,.,
JD-
~ 61000
180
~
~
"'<,l.
~
co
i::'J. G DUE TO CHANGE IN YOlO RATIO
25
Is
0
6G
N lo G .. G1000 ~"
.. LOG 10 (t 21t 1)
~
~X ~~ ~ ~
2
<
170
24 1
101
102
DURATION OF CONFINEMENT.
o,'--------~~--------~--------~.-------_j 1 1 10 10 2 10-4 103 DURATION OF CONFINEIIENT, t (MIN)
"''
0
~
"' 1725 KN/12
DETROIT CLAY
%z w~
~
::; ~
~~ We ~<
0 0% ~
"'
w
.,;
e0 = 0.5
27
~ ~ ~
0
~
X
<
0 0 .. 207 KN/111 2
'"'
t (min)
'"'
Figure 4.16. Typical modulus change with time for sand. (After Lodde, 1977; Anderson and Stokoe, 1977, cbpyright ASTM. Reprinted;;,~~th permission.)
Fi~ure 4.15. Effect _of stress history on shear modulus~time relationship. (After Anderson and Stokoe, 1977, copynght ASTM. Reprinted with permission.)
one-week period of confinement at each pressure in the following sequence· 17, 34, 60, 34, and 17psi (117, 235, 414, 235, and 117kN/m 2 ). It can b~ observed that m the overconsolidated state values of I and N d d · · ' a 0 were re uce relative to value of Ia and N 0 in the normally consolidated state. A typiCal modulus-time graph of a coarse-grained soil is shown in Fig. 4.16. It can be se~n that the shape of the modulus-time response for the ~ohes10nless soil differs significantly from that of the fine-grained soil shown m Fig. 4.14. For the sand, the primary consolidation phase is not evident Rather, the long-term time effect had begun by the time the first mea: sur~ment was made. The long-term time effect is, however, similar to that which occurs for clays. In Fig. 4.16, the linear increase in modulus with the logarithm of time was momtored for ~bout 10,000 min. Values of I a and Na for this sand are 250 psi (1725 kN /m ) and 1.0 percent, respectively. These values are much smaller than those shown for the clays in Figur<> 4.14a. Moduli m~asured -~t sheari ng strain amplitudes between 0.001 and 0.1 3 percent (stram of 10 to also increase with time. The results of a number of tests show that long-term modulus increases occur at low to mtermediate strain levels (0.001 to 0.1 percent) for stiffer clays (Lodde, 1977). Preliminary results from long-term, high-amplitude modulus tests on sand seem to indicate that long-term modulus increases occur m clean, dry sands at strain amplitudes up to 0.1 percent as well.
w- )
Because of the general similarity between the increase in moduli with time at low and high-shearing strain amplitudes, it seems reasonable to conclude that many of the factors that affect the low-amplitude modulus time response also affect the high-amplitude modulus-time response (at the start of high-amplitude cycling) (Anderson and Stokoe, 1977). Anderson and Stokoe (1977) also proposed a method, which can be used to predict the in situ shear moduli from laboratory tests after allowing for time effects.
Degree of Saturation. Biot (1956) showed that the presence of fluid exerts an important influence on the dilatational (longitudinal) wave velocity but produces only a minor effect on the shear wave velocity. The fluid affects the shear wave velocity only by adding to the mass of the particles in motion. A study of the influence of degree of saturation on the shear wave velocity for a sample of Ottawa sand shows that much of the difference between the values for the dry and saturated conditions can be accounted for by the effect of the weight of water. Therefore, it is sufficient for an evaluation of V or G fi)J cohesionless soils to consider the in situ unit weight ·' . and the effective pressure. Frequency and Number of Cycles of Dynamic Load. The effect of number of cycles and frequency of dynamic loading was investigated by Hardin and Black (1969). For number of cycles between 1 and 100, the dynamic shear modulus of dry sands was observed to increase slightly with number of cycles whereas for cohesive soils a decrease in modulus with number of cycles was observed. Low strain shear modulus was found to be practically unaffected by the frequency of loading.
119
EQUIVALENT SOIL SPRINGS DYNAMIC SOIL PROPERTIES
118
Magnitude of Dynamic Load. The magnitude of dynamic load controls the shear strain levels induced in the soil and hence the dynamic shear modulus should be expected to decrease with an increase in the dynamic load. Dynamic Prestrain. The effect of strain history on the dynamic shear modulus of sands was investigated by Drnevich, Hall, and Richart (1967) using torsional-vibration-type resonant column equipment. The soil samples were first subjected to large amplitude vibrations (dynamic prestrain) for a predetermined number of cycles and then the low-amplitude vibration modulus was determined. It was observed that the value of the dynamic shear modulus generally increased with the number of prestrain cycles, as shown in Fig. 4.17. No data is available of the effect of dynamic prestrain on the dynamic shear modulus of clays and silts.
pressure under working conditions in the field. The. sample is then subjected to an axial stress equal to the anticipated statlc stress under workmg conditions. Positive and negative values of a small mcrement . of load corresponding to the loading levels in the .field should then be apphed (Fig. 4 18) It will be seen that initially the stram mcreases after each apphcat10n . t· le But after 6 to 10 cycles the additional axial stram generally of s ress eye . ' . d' 1 · becomes negligible, and a closed loop on the loadmg-:-unloa mg eye e !S obtained. The value of the modulus may then be determmed from the slope of the line "aa" by using Eq. ( 4.23): change in unit stress Modulus= corresponding change in unit deformation
The soil behaves as an elastic material in the sense that there is a reduction in deformation when the stress is removed, but the matenal absorbs energy and hence provides material damping. . The load-deformation behavior of soils may also be represented m ter~s of equivalent ;~oil springs which_ar_~ essentially, related to th~ Young s modulus E, tlie shear modulus c;, ·and Pmsson s rat10. v. Thts concept considerably simplifies the solution of many problems a?d ts commonly used ' in analyzing the machine foundation problem and ts d~scussed belo':. The spring constant k is defined as the load per umt deflectiOn, 1.e., p
(4.24a)
k = LI.Z
ao o-0 "" 1188 psf 'Yox = 1.6 X lQ-4
=
(4.23)
612 psf
'YOx = 1.6 X
w- 4
Cyc!es of high-amplitude torsional vibration
Figure 4.17.
Effect of number of cycles of high-amplitude vibration on the shear modulus at
I I
low amplitude (C-190 Ottawa Sand, e0 =0.46, Hollow Cylindrical Specimens). (Drnevich, Hall, and Richart, 1967. © 1968 The University of New Mexico Press.)
"'
For analysis of the dynamic behavior of foundations-soil system, the soil is usually represented by equivalent springs. This concept will now be explained.
4.4
Static working stress
EQUIVALENT SOIL SPRINGS Axial strain
By way of illustration, let a triaxial test be performed on a sample of soil such that the confining pressure is as near as possible to the confining
Figure 4.18.
Ax·,·al stress vs. axial strain in a triaxial test using repeated static loading.
120
DYNAMIC SOIL PROPERTIES EQUIVALENT SOIL SPRINGS
in which P is the load on the spring (Fig. 4.19b) and I!.Z is the change in length of the spring. Next, consider a plate of area A and thickness t that is subjected to a pressure of umform intensity, p (Fig. 4.l9c). The change in the thickness of the plate, l!.t, is given by ( 4.24b) Because the total load P is pA, the above equation may be rewritten as
or P EA -=-=k
l!.t
may undergo vertical oscillations, horizontal translation only, rocking only, and yawing (torsion about the vertical axis). The corresponding conditions induced in the soil are (1) uniform compression, (2) uniform shear, (3) nonuniform compression, and (4) nonuniform shear. Therefore, the soil constant, characterizing the soil reaction below the foundation block and the corresponding elastic deformation are different in each case. The value of the equivalent soil spring for any mode of vibration may be computed from the dynamic shear modulus or Young's modulus, Poisson's ratio, and the geometry of the foundation contact area as explained in Chapter 6 (Sections 6.4 and 6.7). Barkan (1962) defined a set of soil coefficients to represent the soil reaction for any particular mode of vibration. These soil coefficients are defined below: 1. Coefficient of Elastic Uniform Compression (C,J:
(4.25)
t
in which P is the total load and k the spring constant. It ts explamed m Chapter 6 (Section 6.2) that a rigid machine foundation
121
c" =
uniform compression ( p) elastic settlement (s")
( 4.26)
';o,- '7
From the definition, the spring constant k, is then
k'
=
load elastic deformation
Therefore,
(4.27) in which A is the area of the test plate or the foundation base. 2. Coefficient of Elastic Uniform Shear ( CJ:
(a)
uniform shear
(b)
C,
=
Tav
elastic shear displacements,
( 4.28a)
p
As for the spring constant k, the corresponding spring constant kx is A
(4.28b)
~~~~~~~~
+ r'========-----V'; T (c)
Figure 4.19.
Co~cept of spring constant in soil. (a) Unloaded spring· (b) static deflecti
{c) plate as a sprrng.
'
A .
on z,
3 & 4. Coefficient of Elastic Nonuniform Compression ( C1 ) and Coefficient of Elastic Nonuniform Shear ( C",). Because the elastic deformation is not uniform over the base of the block as in cases 1 and 2, no simple definitions for C and C", can be given. However, definitions in terms of mathematical quantities and overall displacements (rotations) of the block and its geometry are given in Chapter 6 (Section 6.7). The value of C, can be determined from Eq. (4.29) (Barkan, 1962)
122 DYNAMIC SOIL PROPERTIES
c
=
u
1.13£ _1_ I - vz
VA
( 4.29)
LABORATORY METHODS
123
ultrasonic pulse test, (3) cyclic simple shear test, ( 4) cyclic torsional simple shear test, and (5) cyclic triaxial compression test.
in which 4.5.1
E =Young's modulus and
A = Area of the footing v =Poisson's ratio
The value of C" varies inversely with the square root of the f h alrea o contact of the foundation with the soil. Thus if C and C c ffi · t f 1 · . ' "' u2 are t eva ues of oe helen s o e asttc umform compression corresponding to areas A and A 2 , t en 1 ( 4.30) However, for areas greater than 10m 2, A, is taken as !Om' only. This is duhe to the fact that for large areas, soil rigidity and area effects cancel each ot er. The values of the different soil coefficients are approximately related to each other as g1ven m Eq. (4.31) (Barkan, 1962)
c,- = ! cu
( 4.3!a)
C 4, = 2Cu
( 4.3!b)
C1,=0.75C"
( 4.3lc)
and
Th~ vaflue of the ratio ratio o 2,
c.;c" depends upon the shape of area and for (alb) Ci> =!.73Cu
( 4.3ld)
may be used in calculation. Also these coefficients will vary with area as
does.
c u
It may be pointed out that the equivalent soil springs essentially depend ohn ttheffvaluEes of E and G. Therefore, they will be influenced by all factors t a a ect and G. ···
4.5
LABORATORY METHODS
The following laborato~y methods are used to determine the dynamic elasl!c constants and dampmg values of soils: (!)resonant column test, (2)
Resonant Column Test
The resonant column test for determining the modulus and damping characteristics of soils is based on the theory of wave propagation in prismatic rods (Richart et a!., 1970). Either compression waves or shear waves can be propagated through the soil specimen from which either the Young's modulus or shear modulus can be determined. Such a device consists essentially of a coil-magnet drive system, an accelerometer or velocity transducer to monitor the motion of the drive system, a linear variable differential transducer (LVDT) or other type of displacement transducer to detect a change in the vertical height of the soil specimen, and a confining chamber. In the test setup, the coil-magnet drive system is attached to a top cap, which is seated on a membrane-encased, cylindrical soil specimen. The soil specimen can,be either hollow or sol~d, depending on the capabilities of the particular test" device. The bast'''"pedestal, upon which the specimen is placed, is connected to a drainage line. Filter paper strips may be used along the length of the specimen to accelerate specimen consolidation. The top cap and bottom pedestal are usually serrated or roughened in some manner to assure good mechanical coupling between the soil and equipment. The system is generally set up so that only a hydrostatic confining pressure can be applied, although anisotropic loading conditions can be simulated in some devices. An electrical system is employed to operate and monitor. the resonantcolumn equipment. To obtain accurate shear wave velocity measurements, it is necessary to use electronic equipment (Fig. 4.20). In such a system, a signal generator supplies a sinusoidal voltage to the coils in the coil-magnet drive system. The magnetic field induced by the current in the coils interacts with the magnetic field from the permanent magnet, thereby resulting in a torsional oscillation of the drive cap and specimen. By varying the frequency of the input signal, the amplitude of vibration can be varied. An accelerometer (or velocity transducer) located on the top cap generates a voltage proportional to the amplitude of vibration of the soil-top-cap system. This signal is c~nditioned and then displayed on an oscilloscope. The amplitude and freqlency of the signal at resonance are monitored. An LVDT is used to monitor the changes in specimen height. The purpose of the test is to vibrate the soil-top-cap system at the first-mode resonance at which the material in a cross section at every elevation vibrates in phase with the top of the specimen. The shear wave velocity and shear modulus are then determined on the basis of system constants and the sizes, shape, and weight of the soil specimen (Drnevich, 1977; Drnevich et al., 1977).
DYNAMIC SOIL PROPERTIES
124
125
LABORATORY METHODS
r::::=~;t7'0(1,t)
SINE-lAVE
r--------1
VOLTitiE.TER
SINE-WAVE
GENERATOR
GENERATOR
X
X
O(l,tl
AHD AMPLIFIER
FREQUENCY COUNTER
f----.--j
J
(a) J/Jo =
{b) J!Jo = 0.5
00
"-
.J
Weightless spring
Dnvmg force '--""'' '--~-j
CHARGE AMPLIFIER
~~O~,
OSCILLOSCOPE
Rigid mass
Dashpot
~I'
ORAl NAGE
Figure 4.20. Typical electronics for resonantMcolumn device.
The resonant column technique was first applied to testing soils by the Japanese engineers Ishimoto and !ida (1937) and Iida (1938, 1940). About 20 years later, Shannon et al. (1959) and Wilson and Dietrich (1960) described new applications of the resonant column principle (Woods, 1978). It is possible to develop several versions of the resonant column test by using different end conditions to constrain the specimen. Some common end conditions are shown schematically in Fig. 4.21. Each configuration requires a slightly different type of driving equipment and methods of data interpretation. The fixed-free apparatus is the simplest configuration in terms of equipment and interpretation. Apparatus of this type was described by Hall and Richart (1963). In the fixed-free apparatus shown in Fig. 4.21a, the distribution of angular rotation 0 along the specimen is ~ sine wave, but by adding a mass with mass polar moment, 10 at the top of the specimen as in Fig. 4.21b, the variation of 0 along the sample becomes nearly linear. Later models of the fixed-free device (Drnevich, 1967) take advantage of endmass effects to obtain uniform strain distribution throughout the length of the specimen. The apparatus in Fig. 4.21d has a fixed base and a top cap that is partially restrained by a spring, which in turn reacts against an inertial mass. The configuration of the apparatus shown in Fig. 4.21c can be described as the spring-base model. The apparatus of the Shannon-Wilson device is of this type, if the spring is considered stiff compared to the specimen's stiffness. For a condition in which the spring is weak compared to the specimen, the
Specimen, nonrigid distributed mass
Rigid mass
weightless spring
,J;
0/'Fixed
Orivingfo* (c)
(d)
tic of resonant column end conditions (After Woods 1978.) Figure 4.21. Schema
. n of Fl·g 4 21c could be called free-free. In such a case, a node confi gurat10 · · · d' 'b t' o ld . 'dh . ht of the specimen and the rotatwn lstn u IOU w u will occur at m1 e1g ' . . · · 1 be 1 • ve By adding end masses, the rotatwn d1stnbut10n can a so . . · b 1 d by a b e a - sme wa . ' I ]' For K = 1 0 tests the mertml mass IS a ance made near y mear. o · ' . · 11 d be counter weight, but if !!>he changes the counterweight, an axm oa can nt applied to the specimen. . · Th hearing strain on a circular cross section Ill a torswna1 resona colun:'n ~est varies from zero at the center to a maximum at the outer edge. To stud the influence of shearing strain amplitude on she~r modulus and . y D . h (1967 1972) developed the hollow cyhnder apparatus dhampm_g, F" rne4v2IC2 the co,nfiguration of which is similar to the schematic of · l t' · not s own m Ig. · • . 4 21b The average shearing strain on any honzonta cross s~c wn IS. . F Ig. · · · · d the sheanng stram IS greatly different from the maximum or mtmmum, an
127
LABORATORY METHODS DYNAMIC SOIL PROPERTIES
126
Permanant
Soil spec'! men
Top cap
No. 28 gauge
winding wire
(b)
W
AA -:+-~· . • f t cOfutfin adapted for attenuation measurements. ( er Figure 4.23. s·chematrcs o resonan ,, -Woods 1978.)
Bottom cap
Figure 4.22. Hollow specimen resonant column and torsional shear apparatus. (After Drnevich, 1972.)
. (W d 1978) but the technique still needs to be oo s, ' shear wave attenuation perfected. .b d calibration procedure and aids for that propagates either a rod Drnevich et a!. (1977) descn e a t reducing data derived from an appara u~ compression wave or a shear wave or bot . 4 .5.2
uniform along the height of the specimen. Drnevich also increased the torque capacity of his device to produce "large" shearing strain amplitudes. Anderson (1974) used a modified "Drnevich" apparatus to test clays at shearing strain amplitudes up to 1 percent. Woods (1978) tested dense sands on the same device at shearing strain amplitudes up to 0.5 percent at 40 psi (276 kN 1m 2 ) confining pressure. Lord et a]. (1976) suggested a technique to measure. attenuation as a function of frequency in the resonant column apparatus. In the apparatus, two straight lengths of coil winding wire are embedded a vertical distance d apart in a soil specimen (Fig. 4.23a), which is excited longitudinally by short pulses of different carrier frequencies. As the waves travel past the wires, the wires move with the velocity of their neighboring soil particles. Because the wires are in a magnetic field, they generate a voltage proportional to their Velocities. The voltage is measured at two points a known distance
apart making it possible for the operator to evaluate the coefficient of attenuation. A similar concept (Fig. 4.23b) was developed to determine
Ultrasonic Pulse Test
. . enerate and receive ultrasomc waves m Piezoelectric crystals are used ~~Juce either compression or shear waves. soils, and some of them can p th t includes a pulse generator, an Stephenson (1977) described a s:J~~s (t:ansmitter and receiver). The pulse oscilloscope, ~nd two ultraso~iCt direct current pulse to the transmittmg enerator dehvers a vanable vo tage, . ulse to the time base of the g d . 1 •th a 7-V tngger P probe simultaneous y WI h dueled tests on unconfine . oscilloscope. With t~\s . apparatus, e con specimens of cohesive 'soils. . h . e is that it is difficult to identify and One of the drawbacks of this tee mqu More importantly the strain · a1 f es of the waves. ' . h interpret exact arnv lm . d with this pulse technique, are only m t e f the technique is that tests can be amplitudes, which can be achieve . -n. ·mary advantage o . · very low region. "'e pn d. t while they are still retamed m a ft seafloor se Imen s . l f performed on very so . h . e is not currently used routme y or core liner (Woods, 1978). This tee mqu the measurement of soil properlies.
llH
4.5.3
DYNAMIC SOil PROPERTIES
Cyclic Simple Shear Test
A soil element at xx, as indicated in Fig. 4.24, may be considered to be subjected to a series of cyclic shear stresses, which may reverse many times during the life of a machine foundation. In the case of a horizontal ground surface, there are no shear stresses on the horizontal plane before the foundation of a machine is installed. However, shear stresses are introduced when soil is excavated for a foundation and when the machine is installed. Normally, static stresses are also not constant at the base of an embedded foundation. Then, when the machine is operated, cyclic shear stresses are introduced. Thus, the actual problem involved with machine foundations is that there are normal as well as initial shear stresses that act on the horizontal plane. Oscillatory shear stresses are also introduced. With a surface footing, there are no initial shear stresses on the soil at the base of the foundation, but these stresses are present on all other horizontal planes below the ground surface. These conditions are different for earthquake loading, because the normal stresses on the plane remain constant while cyclic shear stresses are induced during the period of shaking. A simple shear device simulates all these loadings and consists of a sample box, an arrangement for applying a cyclic load to the soil, ·and an electronic recording system. The Roscoe (1953) device has a box for a square-shaped sample with side lengths of 6 em and a thickness of about 2 em. This box is provided with two fixed side walls and two hinged end walls so that the sample may be subjected to deformations of the type shown in Fig. 4.25. The schematic diagram in Fig. 4.25 illustrates how the end walls rotate simultaneously at the ends of the shearing chamber to deform the soil uniformly (Peacock and Seed, 1968). Kjellman (1951), Hvorslev and Kaufman (1952), Bjerrum and Landra (1966), and Prakash et al. (1973) have described this type of apparatus. Typical shear-stress-shear-strain relationships obtained during cyclic simple shear tests are shown in Fig. 4.26a. A soil exhibits nonlinear stress-strain characteristics from the very beginning of the loading cycle. For purposes of high stress-high strain loading such as that due to an earthquake analysis, this behavior can be represented by a bilinear model, as shown in Fig. 4.26b (Thiers and Seed, 1968). This bilinear model is defined by three parameters: a
a
t
t
\ \((0\'0000\\ _, Figure 4.24.
earthquake.
X
L
I v..-:;
II I
111111 A
I W//,
ru
r
l
D
'lll
Soil sample
Shearing chamber Plan view
I
I
Soil deformation Elevation
Figure 4 . 25 . Schematic diagram illustrating rotation of hinged end plates and soil deformation in oscillatory simple shear. (After Peacock and Seed, 1968.)
Shear stress
•
_L X
Shear strain
/;?5777747/ H
Idealized stress condition for an element of soil below ground surface during an
(a)
(b)
Figure 4.2&. (a) Stress-strain curve of a soil. (b) Bilinear model (After Thiers and Seed, 19&8.) 129
DYNAMIC SOIL PROPERTIES
130
(1) modulus G, until a limiting strain 'Y, is reached, (2) modulus G 2 beyond strain 'Yy, and (3) strain 'Yy· If the direction of strain is reversed, the behavior can again he determined hy using modulus G 1 until a strain change of 2y, is developed, and the modulus G 2 again controls the behavior. This pattern is continued throughout the cycle. Typical stress-strain plots taken from simple shear tests of San Francisco Bay mud were made from the records of deformation and loads versus time data for different cycles of loading. Figure 4.27 shows such plots for cycle 1, cycle 50, and cycle 200, with about 4 percent shearing strain. The decrease in peak load as the number of cycles increases is reflected by the progressive flattening of the stress-strain curves. Similar tests were performed at different peak strains, and plots of the dynamic moduli G 1 and G 2 versus peak strains are shown in Fig. 4.28. However, in machine foundations, the magnitude of dynamic (oscillatory) load is small compared to the static load (Fig. 4.18). Hence, the stress-strain loops stabilizes after 6 to 10 cycles, and no further irrecoverable deformations occur. Consequently, the modulus corresponding to that condition is adopted in machine foundation design. However, corrections for confining pressure and other factors need to be applied, as described in Sections 4.3 and 4.7. A major drawback of most of the cyclic simple shear apparatus is that they do not permit measurement or control of lateral confining pressures during cyclic loading; therefore, it is impossible to investigate in detail the effects that K0 -consolidation has on the behavior of a given soil.
40r-----.--,------,----,
"
~ 20~~~~~----+-------r------1
'0
E u
E
i
r----~~~::;;r~~~~~~~
(a) Dynamic modulus G 1 ~,·
1u
"'
20
~
'
".2" 0 "0
10
0
E u
'[' ro
>-
0
Shear
10
Peak strain,%
c
stress
131
LABORATORY METHODS
00 Peak strain,% (b) Dynamic modulus G 2
(kgfcm 2)
Figure 4.28. Effect of cyclic loading on dynamic moduli. (a) Dynamic modulus G,. (b) Dynamic modulus G2 • (After Thiers and Seed, 1968.)
4.5.4 Cyclic Torsional Simple Shear Test
(a) Cycle 1
(b) Cycle 50
(c) Cycle 200
Figure 4.27. Stress-strain curves and bilinear models for San Francisco Bay mud. (a) Cycle no. 1. (b) Cycle no. 50. (c) Cycle no. 200. (After Thiers and Seed, 1968.)
In an attempt to provide the capability of measuring confining pressur~ and controlling K 0 conditions, several investigators have developed torswnal simple shear devices. ]$ihara and Li (1972) modified a tn.axml apparatus to provide torsional straining capabilities. Although this device permits lateral stress control it has a distinct disadvantage in that the sheanng strams of a specimen bei~g tested can range from zero at the center of the specimen t? a maximum at the outside radius (Woods, 1978). Other mvestigators mcludmg Hardin (1971), Drnevich (1972), Yoshimi and Oh-Oka (1973), Ishibashi and Sherif (1974), Ishihara and Yasuda (1975), Cho et al. (1976), and Iwasaki et a!. (1977) developed hollow cylinder torsional shear apparatus. The devices developed by Yoshimi and Oh-Oka, Ishibashi and Sherif, and Cho et al. are
132
lABORATORY METHODS
DYNAMIC SOil PROPERTIES
4.5.5
based on the same concept but use much shorter specimens, and the height at the outside and inside diameters can be varied. . The apparatus (Fig. 4.22) designed by Drnevich (1972) has an advantage m that both resonant column and cyclic torsional shear tests can be performed in the same device. Ishihara and Yasuda (1975) also used a long, hollow, cyhndncal sample configuration as did Iwasaki et al. (1977). It is difficult to prepare specimens for the long, hollow, cylindrical devices, and, obviously, "undisturbed," cohesionless soils cannot be tested in this device. However, Woods (1978) was able to trim hollow, cylindrical specimens of many different types of soils. The cyclic torsional simple shear devices of Yoshimi and Oh-Oka Ishib~shi and Sherif, and Cho et al. were designed to generate uniforO:: sheanng strams throughout the wall of a cylindrical specimen whose bottom or top surface was tapered (Fig. 4.29) so that the taper would be exactly proportional to the inside and outside radii of the hollow cylinder. "Undisturbed" sp~cimens can be formed more easily for the size and configuration of this device than for the devices which require taller hollow cylindrical specimens. Although the initial shearing stress conditions in a short hollow cylindrical specimen were described as uniform by Ishibashi and Sherif (1974), Ladd and Silver (1975) noted that, because the shearing stresses on each boundary are not the same, all initial stresses cannot be uniform throughout a sample.
133
Cyclic Triaxial Compression Test
Cyclic triaxial tests have been extensively used to study the stress-deformation behavior of saturated sands and silts (Puri, 1984; Seed et al., 1986). Also, Young's modulus E and the damping ratio ' have often been measured in cyclic triaxial tests (Fig. 4.30) when strain-controlled tests have been conducted. These tests are performed in essentially the same manner as the stress-controlled tests for liquefaction studies. A servo-system is used to apply cycles of controlled deformation. Young's modulus is determined from the ratio of the applied axial stress to axial strain, and the shear modulus is computed from Eq. ( 4.4). As in all laboratory attempts to duplicate dynamic field conditions, cyclic triaxial tests have limitations, among which are the following: 1. Shearing strain measurements below 10- 2 percent are generally difficult. 2. The extension and compression phases of each cycle produce different results.(Annaki and Lee, 1977); therefore, the hysteresis loops are not symmt!'fric in strain-controlleji tests, and samples tend to neck in stress-controlled tests. '· 3. Void ratio redistribution occurs within the specimen during cyclic testing (Castro and Poulos, 1977).
Vertical stress
---X
!t
A
Compression
l
~ Boundaries indicated by heavy lines
I
~//~>-~
0H 0H~u,
""
•·
aH
I
IJH
"" '"
~
Figure 4.29. Specimen, cross section and initial stress distribution in a hollow cylindrical sample with tapered ends. (After ladd and Silver, 1975.)
I D _ _!___ Area of Hysteresis Loop - 21r Area of Triangle OAB & OA'B'
Figure 4.30. Equivalent hysteretic stress-strain properties from cyclic triaxial test. (After Silver and Park, 1975.)
134
DYNAMIC SOIL PROPERTIES 135
FIELD METHODS
4. Stress concentrations occur at the cap and base of the specimen being tested. 5. The principal stress changes direction by 90° during the test. Void ratio distribution is common to all cyclic shear tests, whereas the other limitations are related mostly to the cyclic triaxial test. It is generally agreed that (Seed et al.. 1978; Annaki and Lee, 1977) for an earthquake-type excitation of soils, cyclic simple shear or cyclic torsional shear techniques are more appropriate than cyclic triaxial compression. Wolfe et al. (1977) reported on their adaptation of the cyclic testing of a cubical device developed by Ko and Scott (1968). Their results compared favorably with cyclic triaxial compression tests, but problems remained concerning specimen preparation and appropriate testing methods to simulate _field conditions correctly with this apparatus (Woods, 1978). Silver (1981) prepared a table (Table 4.4) indicating the relative quality of each test technique for measuring dynamic soil properties.
4.6
FIELD METHODS
The following methods for determining dynamic properties of soil are in use in different parts of the world. 1. 2. 3. 4. 5. 6. 7. 8.
Cross-borehole wave propagation test Up-hole or down-hole wave propagation test Surface wave propagation test Vertical footing resonance test Horizontal footing resonance test Free vibration test on footings Cyclic plate load test Standard penetration test
Brief descriptions of these tests along with the typical setups and methods of interpretation of data are presented in the following pages. {'''[ .
Table 4.4. Relative Quality of Laboratory Techniques for Measuring Dynamic Soil Properties
4.6.1 Relative Quality of Test Results
Technique (1)
Shear
Young's
Material
Modulus
Modulus
Damping
Effect of No. of Cycles
(2)
(3)
(4)
(5)
Good
Good
Good
Good
Resonant column with adaptation Ultrasonic
pulse Cyclic
Fair Fair
triaxial
Poor
Good
Good
Good
Good
Good
Good
Good
Good
Cyclic torsional shear
Fair
Good
Cyclic simple shear
Attenuation (6)
Source: Silver, 1981.
There are several available field methods with which the dynamic soil properties and damping of soils can be determined. Salient features of these methods will now be described.
'"!;-~.:7
Cross-Borehole Wave Propagation Test
In the cross-borehole method (Stokoe and Woods, 1972), the velocity of wave propagation is measured from one borehole to another. A minimum of two boreholes are required, one for generating an impulse and the other for the sensors. In Fig. 4.31, the impulse rod is struck on top, causing an impulse to travel down the rod to the soil at the bottom of the hole. The shearing between the rod and the soil creates shear waves that travel horizontally through the soil to the vertical motion sensor in the second hole; the time required for a shear wave to traverse this known distance is monitored. There are four important considerations in conducing a cross-borehole shear wave propagation test: (1) the boreholes, (2) the seismic source, (3) the seismic receiver, and (4) the recording and timing equipment. Although a minimum of two boreholes are required, for extensive investigations and for increased accuracy, three or more boreholes should be preferred whenever possible. If boreholes are installed in a straight line, wave velocities can be calculated from the intervals of time required for passage between any >J:,wo boreholes. Thus, the necessity for precisely recording the triggering time is eliminated (Stokoe and Hoar, 1978). In addition, the boreholes must be vertical for the travel distance to be
measured properly. In general, any borehole 10m or deeper should be surveyed with an inclinometer or another logging device for determining its verticality (Woods, 1978). Impulsive sources are more often used, although both impulsive andsteady-state seismic sources are in use. The major criteria for a seismic
137
FIELD METHODS 136
DYNAMIC SOIL PROPERTIES
/
S = Source R = Receiver
(a) Up hole
(b) Down hole
/
Figure 4.32. (a) Up-hole and (b) down-hole techniques for measurement of velocity of wave propagation.
Transducer
Figure 4.31. Sketch showing propagation.
(((((~itcross~bore
Impulse rod Recorder
~~~en
hole technique for measurement of velocity of wave
~1:)"
Trigger geophone
~
\
Wooden
source are: (1) It must be capable of generating predominantly one kind of wave. (2) It must be capable of repeating desired characteristics at a predetermined energy level. Velocity transducers (geophones) that have natural frequencies of 4 to 15 Hz are adequate for detecting (receiving) the shear waves as they arrive from the source. The receivers must he oriented in the shearing mode and should be securely coupled to the sides of the boring. The recording equipment should be able to resolve arrival times of up to 0.2 msec or 5 percent of the travel time. Storage oscilloscopes are also often used ..
plate
Rubber expander
plate
3-component geophone
Figure 4.33.
4.6.2
Back
Equipment and instrumentation for down~hole survey. (After Woods, 1978.)
Up-Hole or Down-Hole Wave Propagation Tests
Up-hole and down-hole tests are performed by using only one borehole. In the up-hole method, the receiver is placed at the surface, and shear waves are generated at different depths within the borehole Fig. 4.32a. In the down-hole method, the excitation is applied at the· surface·, and one or more receivers are placed at different depths within the hole (Fig. 4.32b). Both the up-hole and the down-hole methods give average values of wave velocities for the soil between the excitation and the receiver if one receiver is used, or between the receivers, if more than one is used in the borehole
(Richart, 1977). Figure 4.33 is a schematic diagram of a down-hole survey with all principal elements included (Imai and Yoshimura. 1975).
4.6.3
Surface-Wave Propagation Test
Rayleigh waves and L'llve waves can be used to determine the shear moduli of soils near the surface. The Rayleigh wave (R-wave) (Section 3.3) travels in a zone close to the surface. An electromagnetic or other harmonic vibrator can be used to generate a steady-state R-wave, and the ground surface can be deformed as shown in Fig. 4.34. A mechanical oscillator is usually set to work at approximately 10Hz. One ray is drawn away from the centerline of the oscillator. One of the geophones connected to the horizon-
138
DYNAMIC SOIL PROPERTIES
FIELD METHODS
139
Qo sin wt
(4.34) and G=V:p
f--_,._x
Figure 4.34.
( 4.35)
in which p is the mass density of the soil and v, the Poisson's ratio of the soil. Table 4.5 gives repesentative values for Poisson's ratios that can be used in lieu of the measured values.
Deformed -shape of a half~space surface. (After Woods, 1978.)
tal plates of the oscilloscope is fixed 30 em away from the oscillator along the ray so that the sensmg ax1s of the geophone is vertical. A similar geophone, connected to the vertiCal plates of the oscilloscope, is moved along this ray away from the oscillator. The second geophone is moved until the Lissajous figure on the oscilloscope screen becomes a circle. The two signals are at the same frequency and 90° out of phase. However, if the phase angle is d1ffere~t than 90~, the Lissajous figure is an ellipse, and for a zero phase angle, 1t 1s a strmght hne (Doebelin, 1966). The distance between the two geophones is measured. This distance is then a measure of the wavelength of the generated Rayle1gh wave. The test is repeated at other frequencies. The test can also be conducted by using a phase meter in place of an oscilloscope. In case of uniform soil up. to infinite depths and the Lissajous figure of a cncle, the wave length, A", of the propagating waves is given by (4.32) in which S is the measured distance between geophones. The veloCity of the Rayleigh waves v, is then given by ( 4.33a) or (4.33b) in which f is the frequency of vibration at which the wavelength has been measured. In case the phase angles corresponding to different distances between
g~ophones are recorded, a curve is plotted of th€ phase angles versus the dt~tances. From the ~urve, the distance S between the geophones is deter-
mmed for a phase difference of 90°. The remaining computations remain unaltered as above. Shear wave velocity may then be obtained from Eq (4.~. . The elastic modulus E and the modulus G of the soil medium are calculated as follows:
Table 4.5. Representative Values of Poisson's Ratio
Type of Soil Clay Sand
v
0.5 0.3 -0.35 0.15-0.25
Rock
-
~ .•
--------:c:-;;:"_ _ _ _ _ _ _ _ _ __
c:-.
':£ ',,-
The effective depth of penetration of the R-wave has been empirically related to one-half of Rayleigh's wavelength (Fry, 1963; Ballard, 1964). A procedure for monitoring the shear wave velocities with depth based on an impact of a falling weight and Spectral Analysis of the Surface Wave (SASW) has been the subject of investigation by Nazarian et al., 1983. In the field, two vertical velocity transducers are used as receivers. The receivers are placed securely on the ground surface symmetrically about an imaginary centerline. A transient impulse is transmitted to the soil by means of an appropriate hammer. The range of frequencies over which the receivers should function depends on the site being tested. To sample 50 to 100ft deep materials, the receiver should have a low natural frequency, in the range of 1 to 2Hz. In contrast, for sampling shallow layers, the receivers should be able to respond to high frequencies of 1000Hz or more. Several tests with different receiver spacings are performed. The distance between the receivers after every test is generally doubled. The geophones are always placed symmetrically about the selected, imaginary, centerline. Nazarian et al. (1983,~have shown that use of this setup reduces scatter in data collection due to the fact that the distances covered in the previous tests are always included in the next tests. In addition, at each receiver spacing, two series of experiments are performed. First, the test is carried out from one direction (forward profile) and then without relocating the receivers the same test is performed with the source on the opposite side of the receivers (reverse· profile). By running forward and reverse profiles and by averaging the data of these two tests, the effect of any internal phase shift between receivers is minimized.
140
DYNAMIC SOil PROPERTIES
The raw data obtained from the impact test is reduced with the help of a dynamic signal analyzer (DSA), and the inversion curve is obtained. From the different DSAs that are being commercially manufactured, the DSA supplied by Hewlett Packard (HP 3562A) has been used successfully, A typical shear wave profile for a site in which the velocity profiles have been determined both by !he crosshole method and SASW method show good agreement (Nazarian and Stokoe, 1984). The tally between the values measured by the two methods is strikingly close. The SASW method is economical, and less time consuming than the crossbore hole method and has the advantage of complete automation (Nazarian and Stokoe, 1984; Woods and Stokoe, 1985; Prakash, 1986). However, proper inversion techniques must be used.
4.6.4 Vertical Footing Resonance Test The block (footing) resonance test can be used for determining modulus and damping values. According to IS 5249 a test block 1.5 x 0.75 x 0.70 m high is cast either at the surface or in a pit 4.5 X 2.75 mat a suitable depth (Fig. 4.35a) and is excited in vertical vibrations. Two acceleration or displacement transducers are mounted on top of the block (Fig. 4.35b) such that they
Motor oscillator assembly
Concrete IM 1501 Depth to be
Tied
Ia) 4 50 m 1 mfmin
[§]
i'=;:J[O] 'tr
l ~
,. 2.75 m
1m min
(b)
Figure 4.35.
Setup for a block~resonance test. 141
142
DYNAMIC SOIL PROPERTIES
FIELD METHODS
sense vertical motion of the block. A mechanical oscillator, that works on the principle of eccentric masses mounted on two shafts rotating in opposite directions (Prakash, 1981) is mounted on the block so that it generates purely vertical sinusoidal vibrations. The line of action of the vibrating force passes through the center of gravity of the block. After a suitable dynamic force value is chosen, the oscillator is operated at a constant frequency, and the acceleration of the oscillatory motion of the block is monitored. The oscillator frequency is increased in steps of small values from one cycle up to the maximum frequency of the oscillator, and the signals of monitoring transducers are recorded. The same procedure is repeated for different dynamic force levels used in a test. At any force level and frequency, the dynamic force should not exceed 20 percent of the total mass of the block and motor-oscillator assembly. A form for recording results of this test is shown in Table 4.6.
because different forces cause different strain levels below the block. This is accounted for when the appropriate design parameters are being chosen. The coefficient of elastic uniform compression, C, of the soil is then determined from Eq. (4.37), (Refer to Chapter 6, Section 6.7), ( 4.37) in which fnz is the natural frequency in vertical vibrationS, m the mass of the block, oscillator, and motor, and A the contact area of the block (footing) with the soil. The value of C" from Eq. ( 4.37) corresponds to the area of the test block and for other areas; its value may be obtained by using Eq. (4.30). Knowing the value of C", E may be obtained from Eq. (4.29). Shear modulus can then be determined by using Eq. ( 4.4).
Determinaton of Coefficient of Elastic Uniform Compression of Soil C".
143
The amplitude of vibration A, at a given frequency f, is given by
A=~ 2 2 '
4tr
I
in which a, represents the vertical acceleration of the block in mm/sec 2, and f is the frequency in Hz. Amplitude vs. frequency curves are plotted for each force level to obtain the natural frequency of the soil and the foundation system tested (Fig. 4.36). The natural frequency, fu., at different force levels is different
0.20
'E
-""'
f--
0.12
"'
0.08
I
For a horizontal footing (block) resonance test, the mechanical oscillator is mounted on the block so that horizontal sinusoidal vibrations are generated in the direction of the longitudinal axis of the block. Three acceleration or displacement transducers are mounted on the side of the block with one near the top, a second near the bottom, and the third in the middle along the vertical centerline of the transverse face of the block to sense horizontal vibrations (Fig. 4.35a). The oscillator is excited in several steps, starting from rest. The signal of each acceleration pickup is amplified and monitored. The remaining procedure is the same as for vertical resonance test. Similar tests may be performed by exciting the block in the direction of transverse axis.
Determination of Coefficient of Elastic Uniform Shear of Soil, C,.
I
\
VJ
0.04
~
~~ r--o:
70'
r
"" 20
25
140'_
30
35
40
45
Frequency, (Hz)
Figure 4.36. Amplitude vs. frequency plot from vertical resonance test at raw mill site in Bhutan. (Prakash et al., 1976.)
( 4.38)
105°
__ ,y ,/ ~ 15
In a
horizontal-vibration test, the amplitude of horizontal vibrations, A"' is determined by the equation
~
I !f ·~
E
0 10
f
Horitontal Footing Resoilarice Test
1 r;
0.16
.s u .0..e"
4.6.5
( 4.36)
/
in which ax is the horizontal acceleration in the direction under consideration, and f the frequency of the horizontal vibrations in hertz. Amplitude vs. frequency curves are plotted for each force level to obtain the natural frequency, fnx, of the soil and block tested as for the case of vertical vibrations. A plot of the amplitude with the height of the block determines its mode of vibrations. The coefficient of elastic uniform shear (C,) of the soil is then determined by using Eq. ( 4.39). (For details, see Section 6.7.)
144
C ~
'
145
FIELD METHODS
DYNAMIC SOIL PROPERTIES 2
87T Yf~x (Ao + Io ± (Ao + Io) 2 - 4yA 0 I0
Y
(4.39) Load intensity
in which Y is equal to Mml Mmo• /,,x is the horizontal resonant frequency of block soil system, A 0 is equal to AIM, I0 is equal to 3.46 (II Mm 0 ), Mm is the mass moment of mertia of the block, oscillator, and motor about the horizontal axis passing through the center of gravity of the block and perpendicular to the direction of vibration, Mmo the mass moment of inertia of the block, oscillator, and motor about the horizontal axis passing through the center of the contact area of the block and soil and perpendicular to the direction of vibration, and I the moment of inertia of the foundation contact area about the horizontal axis passing through the center of gravity of the area and perpendicular to the direction of vibration. The coefficient of elastic uniform shear, C, should be corrected for area effects as for the case of C".
\
\ I I
Ses
-r
,___ _ _ ___J
Elastic rebound -
4.6.6
Free Vibration Test on Footings
(a)
Figure 4.37. (a) Load intensity vs. settlement' ill a cyclic-plate-load test and (b) Load intensity vs. elastic rebound from cyclic-plate-load test. (After Prakash, 1981.)
Free vibration tests may be performed by pulling the block and releasing it in a longitudinal direction or by hitting it with a hammer for vertical excitation. From the observed natural frequency, the C,, C,., E, and G values can then be determined. 4.6. 7 Cyclic Plate Load Test The equipment for a cyclic-plate load test is similar to that used in a static-plate load test. It is assembled according to details given in the American Society for Testing Materials (1977) or Barkan (1962) or in textbooks on foundation engineering. After the equipment has been set up and arranged, the initial readings of the dml gauges are noted, and the first increment of static load is applied to the plate. This load is kept constant for some time until no further settlement occurs or until the rate of settlement becomes negligible. The final readings of the dial gauges are then recorded. The entire load is removed and the plate is allowed to rebound. When no further rebound occurs, the readings of the dial gauges are again noted. The load is then gradually increased until its magnitude is equal in value to the next higher proposed stage of loading; the load is maintained constant and the final dial gauge readings are noted. The entire load is then r~ducect' to zero and final dial gauge readings are recorded when the rate of rebound' becomes negligible. The cycles of loading, unloading, and reloading are continued until the estimated ultimate load has been reached; the final values of dial gauge
readings are noted each time. The magnitude of the load increment is such that the ultimate load is reached in five to six increments.
Se
(b)
r
I i I
i
The elastic rebound of the plate corresponding to each intensity of loading can be obtained from the data obtained during cyclic-plate load tests, as shown in Fig. 4.37a. The load intensity versus the elastic rebound is plotted as shown in Fig. 4.37b. The value of C" can be calculated from Eq. ( 4.26). This equals slope of the plot in Fig. 4.37b. 4.6.8
Standard Penetration Test
In the standard penetration test (SPT}, a standard split spoon sampler is driven with a 140-lb hammer that falls freely through a distance of 30 in. The number of blows for 12 in of penetration of the split spoon sampler is designated as theN value. This is Nmcasured· In a design problem using N values, a correction for e~ective overburden pressure is applied (Peck et al., 1974). Although the test iS designated as a "standard" test, there are several personal errors as well as errors that are equipment based. Therefore, the use of SPT to measure any soil property has been questioned by many engineers (Woods, 1978). The "uses and abuses" of SPT have been described by Fletcher (1965}, Mohr (1966), and Ireland et al. (1970}. De Mello (1971} presented an extensive review of SPT from which it is evident that although SPT is used extensively in soil investigations, there has been no documented, carefully controlled research conducted on it. Schmertmann
146
DYNAMIC SOil PROPERTIES
{1975) described uses of SPT in estimating soil properties and pointed to some hope for more accurate uses of SPT in the future with improvements in the performance and understanding of the test. Recent careful studies by Kovacs (1975), Kovacs et al. (1977a, b), Palacios (1977), and Schmertmann (1975, 1977) have thrown new light on the potential of SPT for obtaining consistent and useful soil properties. Seed (1976, 1979) and Seed et al. {1983) presented correlations between SPT and observed liquefaction. It has been reported that Chinese engineers also use this approach in their attempts to assess the liquefaction potential of sand deposits. Skempton (1986) has presented a more recent review on the effects of overburden pressure, relative density particle size, aging and overconsolidation on the measured SPT values. Schmertmann {1977) suggested that with SPT, either only those dynamic properties for which SPT provides a direct model of the phenomenon can be measured, if at all, or the factors which control the behavior of SPT similarly control the correlative dynamic behavior. As an example of the former, pile driving is an obvious direct model; of the latter, it has been shown that the factors which affect cyclic liquefaction also affect SPT in a parallel manner (Seed, 1976). Additional advantages of using SPT to evaluate dynamic effects, according to Schmertmann (1977), are that SPT is a dynamic test for modeling a dynamic phenomenon, and it is essentially undrained for each blow in that it generates principally a shearing energy. Seed et al. (1984) have correlated the energy input of different types of SPT-hammers used in the United States, Japan, and China and have proposed correction factors for the energy input of different hammers. Imai (1977) reported a correlation between (uncorrected) N and shear wave velocity, V,(m/sec) in 943 recordings at four urban locations in Japan and was able to establish the following relationship: V
'
=
91N°. 337
( 4.40)
In arriving at the above relationship, be converted the N values over 50 or under 1 from the penetrating length at the time of 50 or 1 blows into the number of blows necessary for penetration as deep as 30 em. Prakash and Puri (1981, 1984) successfully applied the above relationship in predicting dynamic soil properties at different depths.
4.7
EVALUATION OF TEST DATA
Inthe twoprevious sections, laboratory and field methods for determining soil moduli have been described. This leaves us the task of ascertaining the strains and the confining pressures at which the moduli values have been
determined. It is customary to make a plot of G versus shear strain. It is necessary to
147
DAMPING IN SOILS
determine G at a mean effective confining pressure corresponding to the mean effective confining pressure below the prototype foundation and at a shear strain which may be induced in the soil when the foundation is subjected to dynamic load. Prakash (1981) and Prakash and 2Puri (1981) used a mean confining pressure a01 of 1 kg/em' or (1000 kN/m ) to reduce the data from different tests to a common confining pressure for comparison purpose only using Eq. ( 4.15) (4.15)
( 4.41) and _ _ vcrz ux = u = - -v y 1-
( 4.42)
The sam/;.~marks apply to arf"ivaluation of C", C
4.8
DAMPING IN SOILS
Types of Dgmping in Soils. The motion of the footing-soil system can be damped by two specific energy losses: (1) the ab_r;~_!fl~I()IIOf"nerg)'':"It~m the soil mass and (2) the dissipation of energy associ:~t.,dwitli tlie geometry of the fo"iii1dationc-soi!..J;ystem:-Thefol"mer 1S known as material or internal 4
·
~ta~~~~~~:~;n;~i~~~~~;';;~~~;:~i"t;~~-i~t;;;;~i~1~i~!~ L fo ~, Materi7d Dam in8] Damping in a freely vibrating system reduc~s the pea amplitudes after each successive cycle. If the fore~ of dampmg IS considered proportional to the velocity of the mollon, It IS called vtscous damping (Section 2.4). . . . When a cylindrical s_a,mple of soil is set into a state of free vibratiOn, as m a resonant column test:1lhe vibration decreases in amplitude and eventually disappears. This reduction in amplitude of vibration is caused by internal damping within the soil mass. The decay of vibration is similar to that described for the free vibration of a viscously damped system. It should, however be understood that the internal damping in soils is not considered to be
the
result of a viscous behavior; nevertheless, the theory for a
single-degree-of-freedom system with viscous damping is useful for describing the effect of the damping that occurs in soils.
149
DYNAMIC SOIL PROPERTIES
148
DAMPING IN SOILS
=
The decay of free vibration of a single-degree-of-freedom system with viscous damping is described by the logarithmic decrement, which is defined as the natural logarithm of two successive arriplffiides..of motion, or
z a= log, _l z,
;:
f::
I
(2.77)
II
or
a=
2-rrt;
v ' I
(2.80a)
~
I II
I
I•
;:
f::
The logarithmic decrement may be obtained experimentally from a resonant-column test by setting a soil sample into steady-state forced vibration, then shutting off the driving power and recording the decay of amplitude with time. Figure 4.38a shows a typical vibration-decay curve obtained from a resonant-column test for Ottawa sand (Hardin, 1965; Hall, 1962). The logarithmic decrement was obtained from the decay curve by plotting each amplitude against a cycle number on semilog graph paper (Fig. 4.38b). If the damping in the material produces an effect similar to that occasioned by the viscously damped free vibrations, the plot is a straight line on the semilog graph. Hall (1962) found that the damping determined from the decay of steady-state vibrations in resonant-column samples of rounded granular material is similar to viscous damping. The values of logarithmic decrement varied from 0.02 to approximately 0.20 in his tests. Hardin (1965) described further studies on resonant-column tests for evaluating the ;!
1::
f\h
I'
v
Ia I
to J-,--.-~-,-,,.,--r-ITIII 9 8 7
No. of cycles (b)
·
(4.43)
T ical free vibrationwdecay curves obtained from resonantwcolumn tests on 4 38 6~::=a ~and (a/!mplitude-time decay curves and (b) amplitude vs. cycle number plot. (After Hardin, 1965.)
in which t!:...i.L£S?.~!!lsi~!!L'!Ls)1~~~--~~s
a=9-rr(y,) \
Cr .... ,t- .,n
in which Yo is the
(4.44)
>!'··•
shearing~~{~;it~ ~~piit'~d~ ---and iT0
the mean confining
pressure (expressed in lblft'). He also suggested that Eq. ( 4.44) be used only within the limits of the shearing-strain amplitudes of 10- 6 to 10- 4 for
2
•
f
confining pressures of 500 lbift' < (T0 < 3000 lblft and for frequencieS o less than 600 cycles/sec. . . Drnevich (1967) studied the effect of high-amplitude sh~anng strams on the damping in sands. He found that: (1) no change occ\l~s m dampmg With cycles of prestrain for prestrain amplitudes less than 10 ; (2) the loga~ith. d t vari·es with (0' )- 113 within the range of 400 to 2000 lb/ft for mtc ecremen o _5 -4. d (3) an m Y all shearing-strain amplitudes between 10 and 6.0 X 10 , an
151
DYNAMIC SOil PROPERTIES
150
cycles of high amplitude prestrain increase damping in some cases to twice the original value. Significant increases in damping may be related in part to the test procedure used to control the shearing-strain amplitude. As the shear modulus increases because of the prestraining, the procedure of maintaining a constant amplitude develops a larger strain energy at each cycle. It would be expected that the hysteresis loop would then include a larger area to represent the increased damping. Richart et al. (1970) stated that the values of logarithmic decrement for sands may be as large as 0.20 and that these can be estimated from Eq. ( 4.44 ). Some additional data on internal damping for several types of soils are shown in Table 4.7. ~--------~-----······--·-"· · · · ....
Figure 4.39. Stress-strain curves for a system with hysteresis damping.
Equivalent
g
Dry sand and gravel Dry and saturated sand Dry sand Dry and saturated sands and gravels
Clay Silty sand Dry sand
Reference
0.03-0.07
Weissmann and White (1961)
0.01-0.03 0.03
Hall and Richart (1963) Whitman (1963)
0.05-0.06 0.02-0.05 0.03-0.10 0.01-0.03
Barkan (1962) Barkan (1962) Stevens (1966) Hardin (1965)
Sour~e:
Richart, Hall, and Woods, "Vibration of Soils and Foundations," Repnnted by permission of Prentice-Hall, Englewood Cliffs, New Jersey.
_
ilEu
1-l.cs-£ "
The condftion for a decaying '\ibFation is illustrated in Fig. 4.39b. Point 1 corresponds to the maximum stress in a cycle, which starts at pomt 1 and ends at point 2. It is seen from this figure that the value of 11, depends o~ whether the steady-state A" or the decaying-vibration cond1t1?n A,d IS considered when the damping values are large. In case of decaymg vibrations, the relationship between the lqgarith!Il_ic_.~_e_c_!:~~_!lt and the !l'_ecJ~C damping capacity is (Richar.!_.."!__'l~~,_]_~!O~
----- ----------"
©
( 4.46)
1970, p. 398.
Material damping in soils is also sometimes studied in terms of specific damping capacity, which is defined as the ratio of the energy absorbed in one cycle of vibration to the potential energy at maximum displacement in that cycle. The damping capacity may be expressed as a percentage or as a decimal. In a stress-strain diagram, the specific damping capacity may be represented as the ratio of (1) the area enclosed by the hysteresis loop and (2) the total area under the hysteresis loop. For the steady-state condition in Fig. 4.39a, t the specific damping capacity ll" is given by A
'
'
(a) steady-state vibrations.
Table 4.7. Some Typical Values of Internal Damping in Soils
Type Soil
DAMPING IN SOilS
( 4.45)
The term E. in Eq. (4.45) represents the strain energy described by the area under the hysteresis loop. t The horizontal scale in Fig. 4.39a is greatly exaggerated for simplicity of illustration.
in which k" represents the proportionality factor between strain energy and the square of the displacement amplitude for the nth cycle of decaymg vibration. It needs be emphasized that there is no general relat1o~sh1p il,d, and the ralio of between Acs and 11 cd• but for small values of a, A"= • 1 the proportionality constants, K~+ 1 1 K", is approx1~ately .. Sometimes the decrease in amplitude of vibratiOn w1th distance fro~ _a source that is caused by energy losses in the soil is also evaluated. This IS designated as attenuation (the energy loss as a function of distance) and IS measured in terms of -W,e coefficient of attenuation a (1 /ft). The coeffic1ent of attenuation is related to the logarithmic decrement by
z,. Va
{j= - - =Aa
(4.47)
w
in which V is the wave velocity, w the circular frequency, and A the
wavelength of the propagating wave (Richart et al., 1970). . . Attenuation needs to be distinguished from JI:~":'".t!..'.~~~~~ll_'!'f!~? (SectiOn
153
DYNAMIC SOIL PROPERTIES
152
DAMPING IN SOILS
3.4), which occurs in elastic systems because of the dispersion of wave energy from a source. Internal damping in materials may also be evaluated by measuring the angle at which the strain lags the stress in a sample undergoing sinusoidal excitation. If the soil is assumed to be a linear viscoelastic solid, the complex shear modulus G may be considered to be composed of a real and an imaginary component, each of which is a function of frequency, such that G*(w) = G 1 (w) + iG2 (w)
(4.48a)
In Eq. (4.48a), G 1 (w) is the elastic component and G 2 (w) the viscous component. The loss angle 15L is defined by
G,
tan 15 = L Gl
( 4.48b)
and it is related to the logarithmic decrement 15 and to the ratio p.w I G [see Eq. (4.43)]. Thus,
2
3 ':;~if;, Bx, B
or By,
~
( 4.49)
ratio for .,~scillation of rigid circular footing on the elastic Figure 4.40. EquJValen ampl g d © 1970 226 Reprinted by permission of Prentice~ half-space. (Richart, Hall, and Woo s, C ' P· · Hall, Englewood Cliffs, New Jersey.)
From the above discussion, it can be seen that there are several methods for measuring and describing the damping liisOITS. ..F~rthermore:··because darniJTiigin~oil~~~ii"~~i!ses·;;;;mt~I11~Iiiii5lli~~if~·o(vf6rationJEq. 4. 44], i!_~ay be convementto use different .method.s for differ~nt r"f!ges ()f amplitude ('Rlcilarlet al., 1970). Beneath machine foundations, the order llrinagnii.llde of vibrations encountered in soils is such that the logarithmic decrement should be less than 0.2.
damping constants obtained in the analog solutions (See Section 6.4) and
·
15
= 1T
tan 8L
~~9metl:i(;(J]i5a~P.i;~. It was shown in Chapter 3 (Section 3.4) that in an elastic medium, energy is dispersed as it recedes from the source of disturbance. This dispersion of energy produces response curves that have a finite amplitude of motion at resonance. This indicates that damping is present in the system. However, the assumption of an ideal elastic halfspace precludes loss of energy because of inelastic behavior of the material, which constitutes the half-space. ;This geometrical distribution of elasticw~~~_ll..~_::!\l:Jt.."s·_l.>=~-de~igf!"t"d as gf()'!'etr!f.(ll_!f:q!!'J!iiJg. · - ... ·... ·- From each solution for vibration of a footing on a half-space, -",Yi!]tle of the equivalent damping ratio !; is derived. This iS' then used in a lumpedanalys!S:-Aconvenient method for evaluating !; is to equate the peak amplitudy of motion from the half-space solution to the peak amplitude obtain(ld from the mass-spring-dashpot system and then to solve for g. Figure 4.40 wasprepared by using this method (Richart et al., 1970). Approximately the same results can be obtained by calculating !; from the
parameter..
t d
·0
the expression for critical dampmg, c = 2v'kiii
(2.26)
'
With this approach, the damping ratio is c !; = -
(2.27)
c,
Expressions for the d~mping ratio (Section 6.4) are For vertical osctllauon, 0.425 !;, =
'[lJ;
( 4.50a)
For horizontal oscillation, (4.50b)
155
DYNAMIC SOIL PROPERTIES
154
DAMPING IN SOILS
For rocking oscillations, X
( 4.50c} For torsional oscillations, 0.50 1 + 2B" in which
(4.50d}
B =(1-v} m
=
B x
(7- 8v} .!!!._ 32(1- v) pr~
= 3(1- v}
B "' and
pr~
4
z
8
( 4.50f} w
Mmo pr~
B = Mmz
( 4.50g}
2. Determine the asymptote
It must be noted that if the response of an actual machine foundation or a vibrating footing is monitored, the measured ~mp.li1J,I_d.e.s.r_e!!.ectJhe effu£!.!?!. both the material ~nd geol!l.e_t!i£lJLc.!am£i!l.g. Barkan (1962) showed that an irrcre1!se "inflie·-value of Poisson's ratio of soil leads to an increase in dissipation of energy from a vibrating foundation into the soil. Further, the damping (combined material and geometrical} properties of soil are deter· mined not only by its characteristics (inertia and elastic properties) but by the size and mass of the foundation. A large amount of the data in the literature shows that the damping constant increases with the contact area of the footing for vertical vibrations. Lorenz (1953) recommended a graphical method for determining damp· ing (combined material plus geometrical} from a forced amplitude frequency graph by exciting a footing with vertical vibrations. For frequency-dependent excitation, the maximum amplitude of motion zmax is 2
z m"'
= -c===m~0:=;ec=w~===, 2 2 2 V(k- Mw
}
+ (cw)
(2.49b}
and the procedure for determining damping is as follows: I. Draw a tangent from the origin of the coordinate system to the
amplitude frequency curve (Fig. 4.41} so that it touches the curve at w =
,
• ce c rve for the damped forced vibrations. (After Lorenz, 1953, Figure 4.41. Ty~•ca1 resonan u copyright ASTM:' Reprinted with permissionp:
(4.50h}
pr~
l/1
( 4.50e)
If
"V "M
(4.51}
(4.52) · M 1·n Eq . (4 .52} , because Z "' is measured in Fig. 4.41 and 3. D etermme (m 0 e) is a known quantity. 4. Determine k from Eq. ( 4.51). 5. Then damping constant c from ( 4.53) in which Z, is the amplitude at w
=
w,.
Factors Affecting Damping. There are several factors that affect damp~ The more important of these are: strain level, confimng press~re, vot ra~io number of cycles of oscillatory stress, and frequency of motton. In' Table 4.8, the ini'~ortance of each variable is indicated by the symbols ~ R d L (Hardin and Drnevich, 1972a). . . . , Theandam~ing factor !; increases generally with an mcreasmg stra~~ amplitude, decreasing confining pressure, '?creadsm~ vmd :a(t~~72~ldp~o. b of cycles of loadmg. Hardm an rnevtc creasmg num e~d ex ressions for determining damping in a partic~lar P~~~fe~h~~~s~derabl~ judgment is still needed to select ~value of da?'pmg P . d . blem Fortunately the chmce ts not that dtfficult to be taken m a estgn pro · •
in
and varied in machine foundation design.
156
DYNAMIC SOIL PROPERTIES
157
EXAMPLES
Table 4.8. Parameters Affecting Damping for Complete Stress Reversal Damping a Parameter
Clean Sands
(I)
(2)
Strain amplitude IT2. .··.Effective mean principal stress !
. 3. 4. 5. 6. 7. 8. 9.
:Void ratio
Number of cycles of loading
Cohesive Soils (3)
/v
v v v v
(v
\ v '
\_vj
L R L L R R
u
gradation, mineralogy 12. Soil structure 13. Volume change due to shear strain
R R
R R
(for strains less than 0.5%)
u
R
Degree of saturation Overconsolidation ratio Effective strength envelope Octahedral shear stress
Frequency of loading (above 0.1 Hz) 10. Other time effects (thixotrophy) II. Grain characteristics, size, shape,
L L L L L Figure 4.42.
Soil profile (Example 4.9.1 ).
Dynamic shear modulus at point' A is given by (3.30)
G=V ,,2 p
Source: Hardin and Drnevich, 1972a. a V means Very Important, L means Less Important, and R means Relatively Unimportant except as it may affect another parameter; U means relative importance is not clearly known at this time.
= G
120' x 1750 = 256.8 kg/em' 9.81 X 100 X 100
The effective overburden pressure at point A is given by 4.9
EXAMPLES
cr,A:
5 X 1750 = 0.8750 kg/ em' 100 X 100
EXAMPLE 4.9.1
The soil profile at a site is shown in Fig. 4.42. A cross borehole test was conducted at this site to determine the value of shear wave velocity in a small area surrounding point A (Fig. 4.42) and its average value was observed to be 120 m/ sec. Calculate the value of dynamic shear modulus G for point A. Also determine the values of G for points B, C, and Din the profile. If subsequently the water table rises to the ground surface, what will be the values of G at A, B, C, and D. Assume 'YJ = I. 75 g/ em 3 and y,., = 2.05 g/cm 3 Solution Dynamic Shear Modulus-water table 10m below ground level (as shown in Fig. 4.42}. Observed value of shear wave velocity V, =120m/sec
Dry unit weight of soil 'Yd = 1.75 g/cm 3 = 1750 kg/m
3
The values of effective overburden pressure at B, C, and D may be calculated in a similar manner. These values of cr, for different pomts are shown in column 3 of Table 4.9. The value of dynamic shear modulus at B may be ca~culated from the known value at A by using Eq. ( 4.15} and cr, in place of 0'0aud replacmg 1 and 2 by B and A, respectively: ( 4.15} or 1.75 ) (G) B = 256 ' 8 ( 0.875
0 5 '
= 363.2 kg/em
2
The value of G at points C and D may be calculated in a similar manner. These values are shown in column 5 of Table 4.9.
159
EXAMPLES
DYNAMIC SHEAR MODULUS-WATER TABLE AT GROUND SURFACE
"
-'0
c "
.0
" 0" [-< ~
~
N
2
~
(j
u
~
" t: ~ ~
s "-"' " u :;:: "
0\('f")V)OO
ocirl~~
0\00'70\ ,......, N M('f")
~
.,~
all these calculations. The values of G corrected for the effect of rise in water table at different points may be obtained by modifying the previous values for the change in effective overburden pressure by using Eq. ( 4.15). These corrected values are shown in Table 4.9, column 6.
.c ""
"'~
-~ ~ ....c " o-6
0
" 2 ::: g ..oo't:
-" '"""' 2 s "g
::E " ~a:) ~
0
~
OONM-.;:t 0('1')~~
lf)\O,.....,lrl N('f")~"i"
~
- {.)2
~
.,;
...
,., ,.,
"
tnOI.rlc),_;~r-i
E
"'>< ~
'0
:1 c
"'~ .0 ""
..a
~
s
~.,.
~~ lb"'
0
~ ~ ~
ao =
~
"'
.c"
"' '5"'
....,;
" :;:; ,_"'
158
3
=
KoO\
The values of K for different soil layers are calculated as follows: 0 For sand layers 1 and 2 K 0 = (1- sin>)
~
"" ~
0\ + ii2 + ii3
iiz = ii3
"""
0;
(4.9)
The values of e, OCR, k, PI and 0'0 are needed for using this equation. The values of e, PI, and OCR are given in the problem. These values are also listed in Table 4.10, columns 2, 3, and 4, respectively. The value of 0'0 is calculated as
u
~
"'>-c Q
2
05 29 3 - e) (0'0 ) psi +e
Gm"' = 1230(0CR)k ( · ;
"~
" " i5 t:::" " ~0:: ~
~
Solution .•;_.,_,, The values of low strain dynamiC shear modulus for the soil profile can be calculated by using Eq. (4.9).
O)N
- ·B " c
EXAMPLE 4.9.2
The'soil profile for a site is shown in Fig. 4.43a. Calculate the values of dynamic shear modulus at different depths in the profile. Also plot the variation of shear modulus with depth. Assume shear strain amplitude 'Yo= 10- 6 (i.e., low-amplitude vibration case).
NV)f:-..0
Q.
~
Due to rise in water table, the effective overburden pressure decreases. The new values of effective overburden pressure for points A, B, C and D 3 are shown in Table 4.9, column 4. ')',, = 2.05 g/ cm has been used in making
oooo tri 0 tr)' 0 -
-N
For normally consoli~ted clay (layer #3) K 0 is given by the equation below (Brooker and Ireland,' 1965) K = 0.4 + 0.007(PI) for 0 s; PIs; 40 0 For the normally consolidated clay and PI= 30, K 0 is determined to be 0.61. (See also Fig. 4.49.) For over consolidated clay (layer #4), the value of K 0 is obtained from charts given by Brooker and Ireland (1965). For PI= 55 and OCR= 2, the value of K 0 is determined to be 0.95 (Fig. 4.49).
0
,..
_,.
~
. ...,...,...,...,:::;,_, 0" " ::r .
Ci(ll~(tl
(D<=:In
;.. .
" ~
2
~"
" 0.
" e:.
><-,.,..,
(!>
"
0.
<
...,..,_ ~
~. ~
0
(!> 0 -O> 0o-· "' ~ "
or """0 s· _q,
" ~
0
""" 0>1
"0. Nq[ ....S' i:t' 0 s "0. "" " "' -l " o'lr 82l ";....._,. ".... ".... "" ~· 0" 0.. " :E"" "".... :r-l "s· ~
(!>
0"
(!>
(!>
~
(!>
~
"" _,." g." " 0
0"
:E
" s·
..... 0
"
0
" s0<= 8g. '-0
"
~
"'
e
E.
Or:n'"1-
. og:=; """ " '
. ...
(!>
" 0. " +:-.-oo-
s
"
v. 0
""" -l
"='
~
0"'
tl:l
-
~
0
~
i
-c
Depth (ftl
..
N
"'
k-"'~"'
. I.
~
a ~
. if .
~§
/
e...
...
-< ~
3• ;;·
0 0 0
,.. ~
"'
0 0 0
~
!;
...3 0
"'
0 0 0
!;;.
c < ~
0 0
~
~
0 0
c
... ~
~
~
0
~ 3
•
15' ~ ~
ro ~
3
0
~
0
c ~
·"'
u
~
N 0 0 0
"¥-
ro~r~~~ , 5"00~: «9,_.<::;:; ~0 "'0 ~0~-o
..
;;; ~
•3 -c
~
"00
': . •• ' '
0
'
f' '· o o
'
•
<
,
0
"3.,.
'
·:. ~ . • ~ :'. . .~ :-:::: ~·:. ·:. < ,·.: :. .:_: . :.
0 0 0
..."
'".,_
ocoU.'. ·, ~
;.
~
;=;
V>
0 ;::
. "'.
"':::!m
~
Lateral
:::
;::
m
"
OCR
K,
Effective Vertical Pressure r:T, (psf)
Void Ratio e
z >
0
"
0"
Table 4.10. Values of Dynamic Shear Modulus at Different Depths (Example 4.9.2)
Depth (ft)
0
-<
fl- PI
':7'
Stress (7"3
=
=
(T2
Ko iii (psf)
(1)
(2)
(3)
(4)
(5)
(6)
(7)
0 2.5 5.0 5.0 10.0 15.0 15.0 17.5 20.0 20.0 22.5 25.0
1.10 1.10 1.10 0.85 0.85 0.85 1.125 1.125 1.125 0.80 0.80 0.80
0 0 0 0 0 0 30 30 30 55 55 55
1 1 1 1 1 1 1 1 1 2 2 2
0.50 0.50 0.50 0.455 0.455 0.455 0.61 0.61 0.61 0.95 0.95 0.95
0 250 500 500 763 1026 1026 1157.5 1289 1289 1458 1627
0 125 250 227.5 347.1 466.8 625.9 706.1 786.3 1224.5 1385.1 1545.7
;f~ '~}-
Mean Effective Confining Pressure r:T0 (psf)
k
Dynamic Shear Modulus G (psi)
Remark
(8)
(9)
(10)
(11)
0 166.7 333.3 318.3 485.8 653.2 759.2 856.5 953.8 1246.0 1409.4 1572.8
0 0 0 0 0 0 0.24 0.24 0.24 0.38 0.38 0.38
0 2210.4 3126.0 4455.6 5504.4 6381.6 4538.4 4821.6 5088.0 12348.0 13136.4 13876.8
Top of layer #1 In layer #1 In layer #2 (top) In layer #2 In layer #3 (top) In layer #3 In layer #4
162
DYNAMIC SOil PROPERTIES
The value of G may now be calculated. The calculated values of G at different depths are shown in Table 4.10, column 10, and are plotted in Fig. 4.43b. EXAMPLE 4.9.3
The following tests were performed at a proposed factory site to determine the in situ dynamic properties of the soil. 1. Four steady-state vibration tests were conducted on a 1.5 x 0.75 x 0. 7 m high concrete block. The tests were conducted by subjecting the block to a vertical unbalanced force and a horizontal unbalanced force. In both cases, the tests were repeated at different settings of eccentric masses. The data on observed resonant frequencies and corresponding amplitudes of vibration are shown in Table 4.11, columns 1 through 4. The tests were conducted in an open pit, and the base of the block was at 4.0 m below ground level. The density of concrete was 2.4 t/m 3• 2. Two cyclic plate load tests were performed on 60 X 60 em plate in a pit 3 X 3 X 4.0 m deep. The typical cyclic plate load test data for Test-5 are plotted in Fig. 4.44 and the values of the load intensity and corresponding settlements are listed in Table 4.12, columns 1 through 3.
•I •I •I .i~r
-
•• I I
•I •I
.. I
I
~
~
s-s;"~
00
X
X X
X X
X X
"'v-:oo
"' "' """' ("!')vi
"'"' "'"' -::r..-.:
X-'_X
~~Vl
r-.i("<')ff)
~
~
"' 0
00
00
~~
load intensity (kg/cm 2) 0
0
2
3
4
5
--"':::::- :'--.... ~f'::...
6
~
4
8 E E c
12
w
E w
Bw
"'
16
20
24
Figure 4.44.
., ~ ~ "'',I'.,,, ~-..., ~ ' ' ' "', ~[\ ' ' -...,
s
.,. 0
"-""~>
' ',
'-....,
''
''
II
j
~~
0 ~
~
X
E 0 ;-:;
1\ ' ',
"
-....\
Settlement vs. load intensity {cyclic plate load test) (Example 4.9.3).
0 0
0
0
" <'>t--O ' 0 "' ~
0
0
"' 0 <'>t--
0
0
"' 0
"' r--
0
0
"' 0
"' r--
X
E
"'
~
II
~
'E
Jj" 163
165
EXAMPLES N values
-::t-Mt-ov;
.,.)000<")("") ViMOOO\t;'"--
('1'")7('1,....,,......,
:6:6!;:~M
rl"
~rl~NN
"'I ..,I "'I "'I "'I
--,.--1-00000
x X X X X
l.rJ\O,....;om
Figure 4.45.
'7"1.00\000
' Typical bore log of the site.(Prakash and Puri, 1984) (Example 4.9.3).
rlrlNv)v-;
l- t-- -
\0
'('f)
~ci~ciN
\ONO,.....,O\ 7V1"1"("1'")N
~ 00 t-l.rlN\00
v-ioONr-..:0
NNN--
v;
t--OV'>
0001."--0tn
o,....;,....;triM
0000 v; Mtrl'7NO
N~t--=>.0~
~
"'
o..:,....; "" OO'T ..,...,.
3. A standard penetration test was performed, and a typical borehole log for the site and the SPT values vs. depth are shown in Fig. 4.45. Water table was not encountered in the borehole, which was 12m deep. The average in 3 situ density of the soil was 2.05 g/ cm Determine from the above data 1. a plot of dynamic shear modulus vs. shear strain at a depth of 4.0 m.
2. a normalized plot of G/Gm, vs. shear strain. 3. the variation of G vs. shear strain at the depth of 2.4 m. Solution (1) Block vibration test Size of tbe block= 1.5 X 0.75 X 0.70 m 2 Area of the block A= 1.5 x 0.75 = 1.125 m Weight of the bloc~.= (1.5 x 0.75 x 0.7) x 2.4 = 1.89 t Weight of vibrator ='0.11 t Weight of block and vibrator= 2.00 t
Vertical Vibration Test: C = 4'1T
v;
v; N000\,......;\0
"
NNMviv-i
c 164
2
2 2
f nz m
A
t/
m
3
4'1T (2.00) 2 3 2 3 "= l.lZ5 x 9 _81 f"' tim = 7.154!"' tim
( 4.37)
166
DYNAMIC SOIL PROPERTIES
or
167
EXAMPLES
0', 1 = Effective overburden pressure at the depth under consideration 0', 2 = Increase in vertical pressure occasioned by the static weight of the
C" = 0. 007154[~, kg/ em 3
block
for
0'" 1 = 37.5 x 2.05
fu, = 29.75 Hz
= 0.076875 kg/ cm 2
C" = (0.007154)(29.75) 2 = 6.33 kg/cm 3
2
4q [ 2mnv'm 2 +n 2 +1 m 2 +n +2
_
These calculated values of C" are shown in Table 4.11, column 5. The values of G are calculated from Cu as follows·•
avz = 41T
mz .
c
=
"
1.13£ 2 (1- v )
G=
+ sm
1
\Ill
(4.29)
2mnv' m + n + 1 J 2 z 2 z m +n +1+mn 2
2
(Taylor, 1948).
(4.4)
+ v)
- L/2- 75 -2 m37.5-
-z-
Substituting for E in terms of G from Eq. ( 4.4) into Eq. ( 4.29), we get
B/2
C = 1.13(2G(1 + v)] _1_ " (1- v 2 ) \Ill
n=
Z
37.5
= 37.5 = 1
•S<': 2
2
G = (1- v)(VA)C" 2 X 1.13 (1- 0.33) G= 2. (Y150 x 75)C" 26
2
2
2
= 134 g/cm = 0.134 kg/cm
2
2
0'" = 0.076 + 0.134 = 0.210 kg/cm 2 Mean effective confining pressure, 0'01 , letting K 0 = 0.5
C" = 6.33 kg/ cm 3 G = 31.44 X 6.33 = 199.01 kg/cm
_
2
The calculated values for G are shown in column 7, Table 4.11.
Correction for confining pressure
=0.21
(2X0.5+1) 2 =0.14kg/cm 3 2
The value of G for the mean effective confining pressure of 1 kg/cm may be obtained as follows:
. The mean effective confining pressure 0'01 at a depth of one-half the w1dth = 75/2 = 37.5 em below the center of the blo'ck is given by for
_ _ _ (2K0 + 1) in which
(T'v
2
, _1 2 X 2 X 1V2 + 1 + 1] +sm 2 2 + 1 2 + 1 +2 2 1 2
For
Uoi -
2
0' = 4(2.4 X 70) [ 2 X 2 X 1V2 + 1 + 1 2 + 1 + 2 2 " 4'1T 2 2 + 1 2 + 1 + 2 2.1 2 2 2 + 1 2 + 1
or
= 31.44C" kg/cm
_1
+ nz + 1 + m2n2 mz + nz + 1
Figure 4.46 may be used for computing 0'" 2
E
2(1
= 76.875 g/ cm 2
3
G 1 = 199.01 kg/cm
2
1.0 )0.5 G 2 = 199.01 ( 0. 14 = 531.9 kg/cm
2
169
EXAMPLES
The values of G corrected to a mean effective confining pressure of 1 kg/cm 2 are shown in Table 4.11, column 9. Shear strain: Shear strain 'Yo is given by
U JO Ui
;
":
~
~ "
"
~
~
f"!
f"!
~
N
"!
~
~
'Yo=
~' ~~~~ ~
for amplitude= 0.185 mm 'Yo =
'-( ~
II"',.
~'
I= 0.75
~V~ VI
~; ~'<~r-~'~if~~h rh '); II I/ I I II I I 'll_.f/~/., rT I. 11111111 I 1 1 I "'~'l,.,ff$) ~
:·<(J//,
.
r; '/. I. I
II
L
I 1 I I I
X
10
.. ~
'-'<..'//
~
J//'1///.
/
"'-'<'/ / / ·'"
~
;..;;.
..0
"' N
N
// r////-
T""¥ /7 ,..'0 ::« /
~
":
'//.1/ ///_/j : /~ %~W& ~
.
_v
r::::
~
'"'"
..
- %'k" 0: ~~;
~
0 0
0
0
1,{}
0
1,{}
~
M
M
N
U JO Ui
~·
'.. "":-:."'~.'<
Mm t
=
X 0.85
20
= O 37 5 . 7 m
1.89 ( 1.5' + 0.70') 1.89 (0 3775-0 35)' 0.11 (0 85- 3775)' 9.81 12 + 9.81 . . + 9.81 · O.
= 0.04669 t m/sec-
~ ~
.
...,
2
Mass moment of inertia M mo:
N
v,.~·
0
.
Mass moment of inertia M m about an axis passing through the combined center of gravity and perpendicular to tbe plane of vibration is given by
~ ~~1~~ v 'k~@~~
~
Mmo
t
=
2
1.89 (1.5 + 0.7') 1.89 (0 35)' 0.11 (0 85)' 9.81 12 + 9.81 · + 9.81 . ,-~~.
= 0.07569 t m/secMm
2
0.04669
'Y = Mmo = 0.07569 = 0.6168
0
oi
The values of C" may be obtained as follows: t For calculations of Mm, M, 0 , and /, see Appendix 4.
168
-4
(\r:~= 0.2109 m'
= 1.89 X 0.35 + 0.11
WI/I 'I I 7 I II I I
~W,tC1rt,lfk1Vj r/ I I ///; f7"""<4·//;~~'/ '/ I ~Ill
"'
0
2.46
Height of combined center of gravity of block and vibrator L.
f
~0
=
(2) Horizontal vibration test Moment of inertia of the base contact area about axis of rotation
N
"
0.185 750
The calculated values of 'Yo are shown in column 6 of Table 4.11.
oi
5 ~
Amplitude Width
170
DYNAMIC SOIL PROPERTIES
2 C = 87T Yf:x " (A 0 + I 0 ) ± Y(A 0 + I 0 ) 2
s, = 2.38- 1.505 = 0.875 mm
-===-- = 5.5181
p 2.25 2 c.=;.:= (0. 875110 ) = 25.71 kg/em
2.0 ) ( 9.81
=3 46 (-I-)= 3.46 (0.2109) _ 0 6408 Mmo 0.07569 - · •
Substituting the values of A 0 , I0 , and yin Eq. (4.39), we get C =
8( 7T
) X
(0.6168)/~x
(5.5181 + 9.6407) ± Vc5.5181 + 9.6408)'
"
4 X 0.6168
c
2
X
5.5181
48.7005[~
t-
X
( 4.26)
The values of C,. for other data in the cyclic plate load test may be calculated in the same way and these are shown in Table 4.12, column 4. The values of G for the test and the corresponding values at a mean effective confining pressure of 1 kg/ em' may be computed in the same way as for the case of block vibration test. Similarly, the strain level, Yo, may be obtained as a ratio of the elastic deflection to the width of the plate. All these values are shown in Table 4.12, columns 5 through 9.
X
0
For, p = 2.25 kg/cm 2
1.125
A= A o m
I
( 4.39)
4yA 0l 0
-
171
EXAMPLES
( 4) Standard penetration test The Standard Penetration Test (SPT) value or N-value at 4.0 m is 13 • (Fig. 4.45). The,shear wave velocity, V, is given by
9.6408
3
" - 15.1589 ± 9.9267 tim
( 4.40)
= 9.30 f~ tim 3 = 0.00930[~
= 91 (13) 0 "337 = 215.9 m/sec
kg/ cm 3
or
2
C,. = 2(0.0093 f~) = 0.0186 f~ kg/ cm
G
3
For
2
(V,) y/g =
(215.9) X 2.05(1000) k I 2 9.81 X 100 X 100 g em
( 4.35)
= 974.0 kg/ em' fu
= 16'
c.= 4.76 kg/cm
3
_ 4 X 2.050 2 u,. = 100 X 100 = 0.82 kg/ em
The calculated values of C,. are shown in Table 4.11, column 5. The value of c. may be calculated and corrected for confining pressure in the same manner as for the case of vertical vibrations. Shear strains may also be calculated in the same manner (column 6) Table 4.11.
(3)
=
Cyclic plate load test
From the load intensity versus settlement plot for the cyclic plate load test (Frg. 4.44), we can compute the elastic settlement as follows:
_
u 01
= 0.82
(2K0 +1) 3
= 0.546 kg/ em
2
2
Gat 1 kg/cm 2 = 1317 kg/cm The associated strain level in the wave propagation test is low. It is in the • -6 range 10 . a. G vs. 'Yo·
in which
The values of G vs. shear strain, y0 , are plotted in Fig. 4.47 (Curve A). b. G/Gm, vs. y6 , i.e., normalized G/Gm, vs. Yo plot:
s, =Total settlement for a given applied pressure increment s P = Residual moved.
settlement
after the
load
increment
t Since fn is the first natural frequency, negative sign has been used.
has
been
re-
The values of G for different strain levels in plot A (Fig. 4.47) were divided by Gm,, i.e., 1317 kg/cm 2• These values for different strain levels are shown in Table 4.13, column 3, and are plotted in Fig. 4.47 (Plot B).
172
DYNAMIC SOIL PROPERTIES
EXAMPLES
173
151()0
The mean effective confining pressure 1.0
N~
E 1250
Legend 6 Block vibration test
u
" 0
1000
"'
o Cyclic plate load test
~ 3.28 tim'
o From N values
~
0
3
0.6
750
~
0
E
"
<'•i
0.4
u
i"ro
0
~
~ 2.05 X 2.4( 0.5 X32 + 1)
0.8
250
0
Shear strain 'Yo
Figure 4.47.
G vs. 'Yo and G/Gma~ vs. Yo (Prakash and Puri, 1984) (Example 4.9.3);.
a
1 '
(3)
( 4)
1 X 10 6 5 X 10' 6 1 X 10' 5 5 x 10··' 1 x 10·• 5 x 10·' 1 x 10·' 5 X 10·' 1 X 10·'
13!7 1225 1137 837 740 425 355 237 200
1.0 0.930 0.863 0.635 0.562 0.323 0.269 0.180 0.152
1121.3 1042.8 967.7 712.0 630.2 362.1 301.6 201.8 170.4
The value of G corresponding to the mean effective confining pressure of 1 kg/cm 2 ~ 642.2 (110.328) 0 ' 5 ~ 1121.3 kglcm 2• This value of G is at a shear strain level of 10· 6 and is Gm" for the depth of 2.4 m. The value of G at 2.4-m depth may be obtained by multiplying Gm" at 2.4 m with ordinates of G/Gm"x plot (Plot B, Fig. 4.47). The values of G so obtained are shown in Table 4.13, column 4, and are also plotted in Fig. 4.47, plot C. This plot shows the variation of G vs. 'Ye at 2.4-m depth for a mean effective confining pressure of l.Okg/cm 2• 4.9.~-
.
-;;..
~
1. In order to determine the in' situ dynamic properties of a soil at a proposed site for a machine foundation 10m X 8 m, the following tests were performed: (a) Vertical vibration tests on a block 1.5 m x 0.75 m x 0.70 m with different settings of the oscillator's eccentric masses. The data obtained are shown in Table 4.14, column 1 through 4. · (b) Cyclic plate load on a plate 30.5 em X 30.5 em. The elastic settlement corresponding to a load intensity of 1.0 kg/ cm 2 is 2.4 mm, which gives a c" value of 4.20 kg/cm 3 (c) Wave propagation test which gives an average value of the dynamic shear moduli, G ~ 335.5 kg/cm 2 The distance between the geophones is 5.0m. 2. The permissible amplitude of vibration is 1.0 mm. Static stress below 2 the foundation is 0.7kg/cm • 3. The water table is 2.5 m below the ground level. The unit weight of soil at the site is 1.7 g/cm 3, and the submerged unit weight is 1.05 g/cm 3 Find the value of C" for the design of the machine's foundation. The 3 assumed density of the•.,concrete in the test block is 2.4 tim •
Gmax
(2)
"'
~ 0.328 kg/cm 2
EXAMPLE
Table 4.13. G and Yo Values (Example 4.9.3) Shear G kg/em' Strain at 4.0-m G kg/em' at G depth" 1', 2.4-m depth'' (1)
~ C3
Gmax -1317 kg/cm 2 at 4.0-m depth. Gmax = 1121.3 kg/cm 2 at 2.4-m depth.
c. G vs. 'Ye at 2.4-m depth
'
N value at 2.4-m depth ~ 7
Shear wave velocity V, ~ 91(7) 0 ' 337 ~ 175.3 m/sec 2
G ~ (175.3) X 2.05 _ , - 6422 tim 9 . 81 ~ 642.2 kg/ cm 2
Solution
( 4.40)
(1) Block vibration test. By following the procedure illustrated in Example 4. 9 .3, one can analyze the block vibration test data. The values of C" for the area of the test block can be computed from the observed natural frequencies of the vertical vibrations. These values are shown in Table 4.14, column 5. The values of
175
EXAMPLES
the shear strain levels are shown in Table 4.14, column 7. The mean effective confining pressure ii0 below the test block can be computed in the2 same way, (as in Example 4.9.3) and its value is found to be 0.14 kg/cm The values of C" for ii0 = 1 kg/cm 2 are shown in Table 4.14, column 8.
Nt'--0\\0tn
--,.....,
oci0~MN ,.....,....,
"'
1
'
I
""
I
,....._,....,
""
I
(2) Cyclic plate load test. 3 The value of C" from the cyclic load test data is 1/0.24 = 4.2 kg/ cm The value of C" corresponding to the area of the test block can now be obtained.
.;-
I
,.-.,
0000 0 ,.....,.,....,
t:,
xxxxx
0
X
0'
~
N
c"'=~ C~.~1 Az
-----
3 Cuz = ~ 0.305 X 0.305 _ (1. X 0 _75 ) or , Cu 2 = 1.20 kg/em 5 42 The mean effective confining pressure at a depth of 30.5/2 = 15.25 em is found to be 0.732 kg/cm 2• The corresponding shear strain level is equal to 2 2.4/(30.5 X 10) = 7.86 X 10- 3• The value of C" for ii0 = 1 kg/cm is equal to 3 112 (1/0.732) (1.2) = 1.46 kg/cm
tnNOO-::tt---
00
00000 «::t«::t-::1"-::t«::t
00000
(4.30)
7
(3) Wave prbpagation test ·s· 2 From the wave propagation test, G = 335.5 kg/cm • The value of C" corresponding to the area of the test block is obtained from
ooc:!ifn~b
00o.rio.ri-.,i
C"
=
1.13£ (1- vz)
1
vA
( 4.29)
= 1.13 X 2G(1 + v) _1_ k /em' [1 - (0.33) 2 ] vA g
1!2 "'
Q,....;('f)('f)tr)
c:ic:iOOO
1.13
X
2 X 335.5 X (1 + 0.33) 1 2 (1- 0.33 ) Y150 X 75
= 10.67 kg/ cm 3
E
X
E u U
. o.
0<'>
"'
For the wave propagation test, the value of the dynamic shear modulus is representative of a depth of L/2, in which L is the distance between the geophones. The average effective confining pressure at a2 depth of 1/2(5/2) = 1.25 m is 1.7 x 1.25 X 100(2 X 0.5 + 1)/3 = 141.7 g/cm =i0.1417 kg/cm 2 2 The value of C" at a,~confining pressure of 1 kg/ cm is given by C" = 3 (1/0.1417) 0 · 5 x 10.67 = 28.41 kg/ cm • The shear strain level associated with the wave propagation is taken as 10-'.
(4) C" vs. 'Yo 2 The values of C" for a mean effective confining pressure of 1.0 kg/cm and for the area of the test block from the different tests and the corresponding shear strain levels are plotted in Fig. 4.48. 174
176
"'
DYNAMIC SOIL PROPERTIES 30
E
"' M
6
"" '"
~I
I
I
I
~
0
20
0
......
0
15
.E
~
0
. ro
10
·o
"'
w 0 u
o Resonance test
A Cyclic plate lo~d test
0
cw
I
I
I
I
A = 1.5 X 0.75 m 2
u
u
I
11o = 1 kg/cm 2
E
~
I
5
~
c---"- W~~~,P'opagat,on
0 10 6
I
177
and the value of C" corresponding to 0'0 = 0.726 kg/cm 2 is 17.5(0.72611)0 · 5 = 14.9 kg/cm 3•
c
E
I
25
·;;;
.E
I
OVERVIEW
I
I
-
10 5
I
I
(7) Correction for area The area of the machine foundation= 10 x 8 =80m 2• This is more than 10m2 The value of C., corresponding to 10m2 will therefore be used in the design (Barkan, 1962) and is given by
.~
I
w-3
lQ-4
C" = 14.9 (
I~w-2
1.5X0.75) 10
112
= 4.99 kg/em
3
The reader may note that no direct correction has been applied to the value of C" for effects of water table as is usually done when calculating the bearing capacity. However, this has been accounted for in calculating the mean effective confining pressure.
Strain level 'YO
Figure 4.48.
C., vs. 'Yo (Prakash and Puri, 1977) (Ex. 4.9.4).
4.1 o
(5) Value of C" needed for the design of the machine foundation The shear strain level associated with the machine operation, 'Yo, is (Prakash and Puri, 1977, 1981). 'Yo=
Amplitude Width
1.0 -4 'Yo- 8 X 1000 = 1. 25 X 10 From Fig. 4.48 the value of C" for 'Yo= 1.25 x 10- 4 is 17.5 kg/em' and corresponds to 0'0 = 1.0 kg/ em 2• This value of C" has to be corrected for the mean effective confining pressure below the foundation and the area of the block. ( 6) Correction for the confining pressure The ~ean effective confining pressure at a depth of 8/2 = 4.0 m below the foundatiOn block can be calculated as follows:
eTa= [(250 X 1.7 + 150 X 1.05) + 4 X 700 x ..
!J[ 2Ko3+ 1 ]
in which I is the influence factor obtained from Fig. 4.46, m = LJ z = 10/4=2.5, and n=814=2. The value of /=0.181. Therefore, rTo = ((250 X 1.7 + 150 X 1.05) + (4 X 700 X 0.181)][ 2 X 0; 5 + 1 2
= 726 g/cm = 0.726 kg/cm 2
J
OVERvfEw
Soil moduli under dynamic loads depend on soil characteristics, such as void ratio, relative density, stress history, preconsolidation pressure, confining pressure, and strain level. In machine foundations, the initial static stress level and pulsating stress level are generally low, and the number of stress pulses are very large. Hence, the combination of all three factors needs to be such that the soil will not experience plastic deformations or else the machine foundation will undergo progressive settlements and tilt and fail. Simple equations have been developed for use with available data to make preliminary estimates of soil mgsJ)l]i ..at.loJ1U.tmin.qmplitudes for sands and c~_,J§9..:..HJJJ In this equation, the value of K 0 , th~~o~fflclent of eaml pressure at rest, which is a function of the plasticity index and overconsolidation of clays, may be determined from Fig. 4.49. Therefore, depending upon the strain value associated in a particular machine faun· dation, a reasonable estimate of the soil modulus can be made. If the values · are determined at one confining pressure, the corresponding values at any other confining pressure can be determined with the help of Eq. (4.15). There is no universaLand unique relationship between soil moduli and strain. However, a nonillllized plot of moduli versus strain is an adequate guide for deciphering values from one site to another in similar type of soil conditions. Several laboratory and field methods for determining soil moduli and damping have been described. Time effects oo the moduli in clays have been described briefly, but more information on this particular aspect is needed.
Several comparisons have been made among the Gm,. values obtained from different tests. Cunny and Fry (1973) determined the values of Gm,.
DYNAMIC SOIL PROPERTIES
178
179
OVERVIEW
(4.54}
3. 0
v 2. 5
•
•
.............
/
';- ~32
1/ F
0
5~
v.
c
~
B 0.5
A
~
113
1---1----
--
) K,= 2o(N1''' N = Corrected value of N measured in SPT test delivering 60% of the I theoretical free-fall energy, ii0 = mean effective confining pressures psf.
r---
8
•
D
c
-~ 1.0
where
!-....
....
4
2 I
0
0
10
20
40 50 30 Plasticity index, P.l.
60
70
80
Figure 4.49. K0 as function of overconsolidation ratio and plasticity index. (After Brooker and Ireland, 1965; Lambe and Whitman, 1968.) Reprinted by permission of John Wiley, New York.
A correlation of standard penetration values with low-amplitude shear modulus in sands is particularly welcome, since the standard penetration test is invariably performed in all soil investigations. SPT values have been successfully correlated with liquefaction potential of soils. . The use of such correlations necessitated a considerable effort m Standardizing the otherwise non standard standard penetration test. (Seed et al. 1985; Skempton, 1986}. . . The detailed discussion on material and geometncal dampmg brought out the fact thah as in the resonant..•oolumn test, only material damping is obtained from laboratory tests, whereas in footing resonance tests, combined material and geometrical damping are obtained. . . Hardin and Drnevich (1972b} demonstrated that the matenal_<'."...l11_P!~g factor I; may be expressed as --~-----------~---.....-------·-<·--""
for 14 sites from laboratory and field tests. They used the steady-state, surface vibration method to evaluate G m"' in the field, but applied the resonant column test in the laboratory. The laboratory-determined shear and compression moduli were found to range within ±50% of the in-situ moduli. They observed that the cross-hole method would have given better values of shear wave velocity V, at the depths from which undisturbed samples were taken and that inclusion of the secondary time effect would have brought the laboratory cohesive-soil values nearer to the field values. The secondary time effect is negligible for sands. Stokoe and Richart (1973a, b) and Iwasaki and Tatsuoka (1977) found agreement between the resonant column and the cross-hole field test values. Prakash and Puri (1981), who obtained in-situ data on dynamic soil constants by making resonance tests on blocks, the shear modulus test, the wave propagation test, and the cyclic-plate-load test at several sites, reduced the modulus G values to a mean effective confining pressure ii0 of 100 kN /m 2 for purposes of comparison and suggested a method for rational evaluation of test data. They suggested that the value of dynamic shear modulus for analysis and design should be selected after taking into consideration the effect of important parameters influencing its value. Standard penetration values have been related with low strain shear
modulus (Gm,,) in psf for sands by Seed et al. (1986) by Eq. (4.54)
I;= l;m"(1-
cf-)
(4.55)
m"'
in which l;m,, is equal to 2k 1 17T and k 1 is ratio of the hatched area show? in Fig. 4.50 and the area of triangle abc, A (obc): There. are ma?y questiOns concerning the validity of this equatiOn, and 1ts use m a design problem. This indicates that more work needs to be done in this direction. Machine foundations generally are partially embed~ed when they are installed. This reduces the._ampiltuifeol motio-natThe resonant peaks,
increases the value of the resonant frequency, and i~creases t~e _e~~:~!~~e
damping. However, the effects on amplitude and frequency~~n··me tests depeiiaupon the mode of vibration and m~gmtude of the. motiOn as well. For motions within the range of des1gn cntena for machmery, 1t appears that thi~ reduction Jn '.l.~!?!~tude_~esulting from partial embed~e~t.Is~~-t-he order of 10 to 25 percent. The ti~~·-effe(iis- for' damping in clays were studed by Marcusson and Wahls ( 1977) with a Hardin oscillator in the laboratory part1~ularly to determine the time-dependent characteristics of the dampmg rat1o of !SOtropically consolidated specimens of kaolinite and calcmm bentomte. They used a steady-state method and the decay of free .vib~ations to obta~n the1r results. After completion of the primary consolidatiOn, they stud1ed the
180 DYNAMIC SOIL PROPERTIES
'
Gmax
a
a
'
-';+----__j d
Figure 4.50. Geometric relationship betw h and Drnevich, 1972b.) een s ear modulus and damping ratio. (After Hardin
d . . yna~mc response as a function of time for both drained d d . conditiOns. an un ramed
fo/~:~,J~i~~d a~hdat ;~ep~:~~:n1orat~o ~ecr~ased approximately 12 percent
dimensionless time ratio during se~on~n omte per logarithmic cycle of a mended that to evaluate th ~ry compressiOn. Thus, they recomshould be continued t fie efftects of lime m c!aye~ soils, at least one test . . o ve o ten limes the lime after th . e. ~nmary consohdatwn. Particular attention should be .d Instrument error when performi'ng I t pm to the possihJhty of . ong- erm tests. . The questiOn of dynamic prope t' k . . r Ies, I.e., strength parameters for earthqua e type loadmg has been discussed by Prakash (1981) No senous efforts have been d' t d · b Irec e tow~rd the determination of the Poisson's ratio of soils AI bet . . so etter correlatiOns need to be established nee~~:nbda~pmg dand the different factors that affect it, and further studies . Id e Irecte towards evaluatiOns of the . to vibrations of footings supported by 1 d_g~_<:>,:U"!!:!gt__al!lJllllg_I_e~ supported by soils that vary in stiffnes:y:~t~ ;.:pe t~a o:s welfil ~s of footings 1 · d (F' con mng pressure As alread field result ydeffxp ame . Igure 4.9), different tests in the laboratory and m t erent strams In tri · 1 t -can be developed Eff t h . b axia ests, generally intermediate strains or s ave een made to extend the strain ra . .
~=:~n~~::~:t~r:a:cE:::t: !:~! s~~ll ~~er~ediate
~~:~i~~
to values and in profile with depth from the spec! u,es. I e. etermmatwn of shear wave ra ana ysis of surface waves (SASW) appears to be a promising field method for the future. , Ladd and Dutko (1985) presented detailed testing procedure which was
OVERVIEW
181
used to determine the cyclic properties of soils at very small strains in a triaxial apparatus. Results of tests conducted using Monetery 0 sand were compared to test results obtained using resonant column apparatus. In 14 cases involving various soil types, the authors found that the ratio of the maximum shear modulus determined by either shear wave velocity measurements or resonant column apparatus to that determined using this testing methodology ranged between 0.55 to 1.26, with an average value of 0. 92. The authors have concluded, however, that the moduli and damping data obtained from resonant column apparatus are more reliable than those from cyclic triaxial apparatus, except for very dense soils. The unique benefit of using cyclic triaxial apparatus is that a complete stress-strain relationship as a function of strain can be obtained along with post-cyclic monotonic behavior. Dyvik and Madshus ( 1985) described in detail the installation and use of piezoceramic bender elements in a variety of standard geotechnical laboratory testing equipment, triaxial, direct simple shear and oedometer devices to measure G of the specimens during consolidation and/ or shear. For purposes of cg>p1parison, piezoceramiP bender elements were also installed in a resonant column device (Itardin oscillator). Simultaneous measurements of G m" were taken on five different clays at different confining stress levels and durations both for loading and unloading. The results of both techniques over a range of Gm., from 10 to 150 MPa (145.0 to 2175.0 psi) were in excellent agreement. By incorporating bender elements into these laboratory testing devices, more information is obtained on the variation of shear modulus at large strains.
In Equation (4.9), the shear modulus has been expressed as a function of iJ-0 , the mean effective confining pressures. However, Stokoe et a!. (1985) studied the effect of state of stress on the shear wave velocity and shear modulus of a dry sand in a large-scale triaxial device in which cubic soil samples measuring 7 ft on a side were loaded in isotropic, biaxial, and triaxial states of stress. Stiffness of the sand skeleton was evaluated by propagating shear waves along each principal stress axis of the sample at pressure of 10 to 40 psi. All testing was performed at !ow-amplitude strains (less than 0.001 percent) and at wave frequencies less than 3000Hz. Shear wave velocity was found to depend about equally on only two principal stresses; (1) in the direction of wave propagation and (2) in the direction of particle motion, and it,ras determined to be relatively independent of the third principal stress. ·· · Yu and Richart (1984) tested three sands under six different stress paths to study the effect of stress ratio on shear modulus of sands. They recommend modifying Hardin's equation [Eq. ( 4.9)] by replacing u 0 = (j-1 + a-, + a-, )1 /3 by
182
DYNAMIC SOIL PROPERTIES
in which a-. and ii6 are principal stresses in the direction of shear wave propagation and along the particle motion respectively. They also proposed two new equations for Gm", which depended on a function of principal stress ratios (ii1 /ii3 or ii,lii,). Thus, it would appear that our understanding of the factors affecting shear modulus is gradually improving, and some of the concepts presented in this chapter may need a revision as more information becomes available. Machine foundations may be supported on piles in some cases. The corresponding spring constants will depend upon the: (1) soil properties; (2) pile properties; (3) pile group geometry; and (4) mode of vibrations of piles. These properties are examined in detail in Chapter 12. PRACTICE PROBLEMS
1.
In a uniform deposit of sand with G = 2.7 and e = 0.7, estimate the variation of Gm" with depth from 1m to 6 m. Assume Poisson's ratio of 0.33 and water table at great depth.
2.
If the water table in problem 1 rises to the ground level, will the values
3.
of Gm., estimated above be altered? Estimate the new values. For a depth of 5 m and from the computed values of Gm" in problem 1, estimate and plot the variation of G with shear strain for strain range of 10- 6 to w - l
4.
What do you understand by (a) material damping (b) Coulomb damping (c) viscous damping, and (d) critical damping?
A clay has liquid limit of 60% and plastic limit of 30%. Estimate the variation of G m" with overconsolidation ratio of this clay at a standard confining pressure of 1 kg/ cm 2 if OCR varies from 1 to 32. 6. The shear wave velocity determined by torsional vibration of uniformly graded dry sand specimen in a resonant column device was 750 ftl sec. The longitudinal wave velocity of this soil is 1200 ft/sec. Determine: (a) Young's modulus and shear modulus if the void ratio and the specific gravity of soil solids of the specimen were 0.7 and 2.7 respectively. (b) Poison's ratio. 5.
Assume that the confining pressures (ii0 ) in both determinations was
1 t/ft 2•
7. List and discuss briefly laboratory and field methods for determination of shear modulii of soils at different strains. 8. Discuss the factors affecting soil modulii and damping. 9. In Figure 4.13, determine Ic and N 0 for 3-soils.
REFERENCES
10.
183
( ) If you had a choice to order either laboratory tests or field tests for
a determination of soil modulii for design of a machme foundatmn, . what would be your preference? Justify completely. (b) Write a small note ~n the future role of standard penetratiOn test in estimating the soil moduh1.
REFERENCES . h ar, t F · E ., Jr · (1973) · Stress history effects on shear modulus of soils. Afifi, S. E. A., an d R lC Soils Found. (Jpn.) 13(1), 77-95. "Is shear modulus of sot . Afifi S E. A., an d W 0 ds, R · D · (1971) . Long-term pressure effects 5on1460 j Mech. Found. Div. Am. Soc. Civ. Eng. 97(SM-10), 144 . . . . . . (1977) "St ndard test method for beanng capactty of , As~-D1l 4-(?2) (reapproved 1977). ASTM, American Societ~ for Testmg Matenals_ soil for static load on spread footmgs,
Soil
°
9
Philadephia, PA. . · An derson, D . G . (1974) . Dynamic modulus of cohesive soils. Ph.D. Dtssertatton,
u ·
't of
mverst y
Michigan, Ann Arbor. . --~ . d d ·1 on D G<~:· .and Stokoe, K. H ., 'Jr·-·11977) ,,, · Shear modulus: A ttme- epen en1 sot .. ASTM Spec. Tech. Pub/. STP 654, Dyn. Geotech. Test., 66-8. d r 9 d W d R D (1975). Comparison of field and laboratory shear mo u I. Anderson, D. G., an oo_ s, . .R l. h NC Am Soc Civ. Eng., Vol. 1, pp. 69-92. · · f Proc -In Situ Meas. Sod Prop., a etg ' . Woods R. D. (1976). Time dependent increase in shear modulus o Anderson, D. G., and .' c· E 102 (GT-5) 525-537. '. . . cla . I. Geotech. Eng. Dtv., Am. Soc. tv. ng. A d y D G Espana C and Me Iore, V· R · (1978). Estimatmg m sttu shear modulus p d at n erson, . ., , ., . S . lty Conf Earthquake Eng. Soil Dyn., asa ena, .. y, I 1 181-197. competent sites. Ge?tech. Eng. Dtv. pecta CA Am. Soc. Civ. Eng., June, Proc. Am. Soc. Czv. Eng. o. ' . . . , d Le K L (1977) Equivalent uniform cycle concept of soil dynamtcs. I. Annak1, M., an e, · · · ) 549 564 Geotech. Eng. Div., Am. Soc. Civ. Eng. 10 ~ (GT-6 ' ~ . t b in-situ vibration II d R .F. , Jr. (1964). Determination of sml Baar, . N shear 4691moduh at dep h y techniques. Waterways Exp. St. Mtsc. Pap. o. . . N y k . " McGraw-Htll, ew or .. D D. (1962). "Dynamics of Bases an d Foun dat tons. Barkan, . a fim'd-saturated porous sohd. I. . of elastic . waves m Biot, M. A.· {1956). Theory of propagatiOn A d n
~~~p;rty
Acoust Soc Am. 28, 168-191. . . h T . . 1 . . and . H enk e,I D · J · (1957) • "The Measurement of Soil Properties m T e naxxa Btshop, A. W., Test" Edwin Arnold, London, U.K. . ·k 1 . ,L., an d L an d ra, A . (1966). Direct simple shear tests on a Norwegtan qmc cay. BJerrum, Geotechnique 26(1), 1-20. h' t Brooker, E. W., and Irelan d-··H '*'•·~- · 0 · (1965). Earth pressures at rest related to stress IS ory. Can. Geotech. I. 2, No. L . · bTt J I S J (1977) Factors affecting liquefactiOn and cychc mo l l y. . (GT-6) 501-516. .· E Castro, G., and Pou_os, · · Geotech Eng. Dtv., Am. Soc. Ov. ng. 103 , . . . H h . W K (1976) Saturated sand and cyclic dynamtc tests. Cho, Y., Rizzo, P. C., and ump nes,E . . Ph"l d~lphia PA Meet. Prepr. 2752, 285-312. 1a Am. Soc. Civ. Eng., Ann. Conv. xpo., ' ' . . d dF Z B (1973). Vibratory in-situ and laboratory sml moduh compare .1. Conny, R. W., an rdy, D·. . Am Soc Civ Eng. 99{SM-12), 1055-1076. Soil Mech. Foun · tv., · · · p Pan De Mello, V. B. F. (1971). The standard penetration tes~: A state-of-the-art report. roc. Am. Conf. Soil Mech. Found. Eng., 4th, Puerto Rtco, VoL 1, 1-86.
1H5
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DYNAMIC SOil PROPERTIES
Doebclin, E. 0. (1966). "Measurement Systems: Application and Design." McGraw-Hill, Ne~ York. Drnevich, V. P. (1967). Effect of strain history on the dynamic properties of sand. Ph.D. Dissertation, University of Michigan, Ann Arbor. Drnevich, V. P. (1972). Undrained cyclic shear of saturated sand. J. Soil. Mech. Found. Div., Am., Soc. Civ. Eng. 98(SM-8), 807-825. Drnevich, V. P. (1977). Resonant column testing-Problems and solutions. ASTM Spec. Tech. Publ. STP 654, Dyn. Geotech. Test., 384-398. Drnevich, V. P., Hall, J. R., and Richart, F. E. (1967). Effects of amplitude of vibration on the shear modulus of sand. Proc. Int. Symp., Wave Propag. Dyn. Prop. Earth Mater., Albuquerque, NM, pp. 189-199. Drnevich, V. P., Hardin, B. 0., and Shippy, D. J. (1977). Modulus and damping of soils by the resonant~column method. ASTM Spec. Tech. Pub{. STP 654, Dyn. Geotech. Test., 91-125. Dyvik, R., and Madshus, C. (1985). Lab measurement of Gma~ using bender elements. Adv. Art Test. Soils Under Cyclic Dyn. Cond., Am. Soc. Civ. Eng. Conv. Detroit, 186-196. Fletcher, G. (1965). Standard penetration test: Its uses and abuses. J. Soil. Mech. Found. Div., Am. Soc. Civ. Eng. 91(SM-4), 67-75. Fry, Z. B. (1963). Development and evaluation of soil bearing capacity, foundations of structures. Waterways Exp. St. Tech. Rep. No. 3~622, Repr. No. 1. Fugro, Inc. (1977). "Sustained~Pressure studies," Fugro Tech. Dev. Program (unpublished). Hall, J. R., Jr. (1962). Effect of amplitude on damping and wave propagation in granular materials. Ph.D. Dissertation, University of Florida, Gainesville. Hall, J. R., Jr., and Richart, F. E., Jr. (1963). Dissipation of elastic wave energy in granular soils. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 89(SM-6), 27-56. Hardin, B. 0. (1965). The nature of damping in sands. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 91(SM-l), Part I, 63-97. Hardin, B. 0. (1970). Suggested methods of test for shear modulus and damping of soils by the resonant column. ASTM Spec. Tech. Pub/. STP 479, 516-529. Hardin, B. 0. (1971). Program of simple she
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(1969). Closure to vibration modulus of normally consolidated Div., Am. Soc. Civ. Eng. 9S(SM-6), 1531-1537. V. P. (1972a). Shear modulus and damping in soils, meaeffects. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng.
Hardin, B. 0., and Drnevich, V. P. (1972b). Shear modulus··and damping in soils, design equations and curves. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 98(SM-7), 667-692. Hvorslev, M. J., and Kaufman, R.I. (1952). Torsion shear apparatus and testing procedure. USAE Waterways Exp. Stn., Bull. 38, 1-76. Iida, K. (1938). The velocity of elastic waves in sand. Bull. Earthquake Res. Inst., Tokyo Imp. Univ. 16, 131-144. Iida, K. (1940). On the elastic properties of soil particularly in relation to its water content. Bull. Earthquake Res. Inst., Tokyo Imp. Univ. 18, 675-690.
Imai, T. (1977). Velocities of P~ and S-waves in subsurface layers of ground in Japan. Proc.lnt. Conf. Soil Mech. Found. 9th, Tokyo, Vol. 2, 257~260. . . M (1975) "The Relation of Mechanical Properttes of Sods toP- and · d y h' Ima 1 T an os tmura, · · U R 1 t Qya $:Wave Velocities for Soil Ground in Japan,'' Rep. RD-477, TN-07 . rawa es. ns ., Corp., Tokyo, Japan. . . Indian Standards Institution (1978). "Indian Standard Method of Test _for ~etermmatton of Dynamic, Properties of Soil," IS 5249-1978, 1st rev. lSI, New Delhi, Indm. Ireland, H. 0., Moretto, o., and Vargas, M. (1970). The dynamic penetraton test-A standard . that is not standardized. Geotechnique 20(2), 185. Ishibashi, 1., and Sherif, M. A. (1974). Soil liquefaction by torsional simple shear dcvtce. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. lOO(GT-8), 871-888. . Ishihara, K. (1971). Factors affecting dynamic Properties of soils. Proc. Asian Reg. Conf. Sotl Mech. Found. Eng. 4th, Bangkok, Vol. 2. . 'h K d L' s (1972) Liquefaction of saturated sand in triaxial torston shear test. t, . . Isht ara, . , an Soils Found (Jpn.) 12(2), 19-39. . . · d y d s (1975) Sand liquefaction in hollow cylinder torston under asu a, . · Ishthara, K., an irregular excitation. Soils Found. (Jpn.) 15(1), 45-59. . · M d rd K (1937) Determination of elastic constants of sotls by means of U · 15 67 Ishtmoto, ., an t a, · · vibration methods. Bull. Earthquake Res. lnst., Tokyo 1mp. mv. : · . . k' T d .T._atsuoka F (1977). Dynam .. ic soil properties with emphas1s on companson of Iwasa t, . , an ~ • · .··" '.:.."" h k E 6th New , laboratory t'eSts with field measuremetfis:, Proc. World Conf. Eart qua e ng., Delhi, Vol. I, 153-158. · F . d T k · y (1977). "Shear Moduli of Sands Under Cyclic M' · t Iwasaki T Tatsuoka, . , an a agt, · 1 · ' ·' 1 Sh L ear oa d•'ng ," Tech . Memo . No · 1264. Public Works Res. Inst., mts ry o T orstona Construction, Chiba-Shi, Japan. Kjellman, w. (1951). Testing of shear strength in Sweden. Geotechnique 2, 225-232. Ko, H.-Y., and Scott, R. F. (1968). Deformation of sand at failure. J. Soil Mech. Found. Div., Am. Soc. Civ: Eng. 94(SM-4), 883-898. Kovacs, w. D. (1975). Discussion of On dynamic shear moduli and Poisson's ratios of soil deposits Soils Found. (Jpn.) 15(1). . Evans 1 c and Griffith A. H. (1977a). Towards a more standardized SPT. , · ., ' 6 K ovacs, W . D ., Proc. Int. Conf. Soil Mech. Found. Eng., 9th, Tokyo, Vol. 2, 269-27 · 'ffith A. H. and Evans, J. C. (1977b). An alternate to the cathead and rope , , Kovacs, W. D ., Gn for the SPT. Am. Soc. Test. Mater. Geotech. Test. 1 1(2), 72-81. Ladd, R. s., and Dutko, P. (1985). Small strain measurements using triaxial app~ratus. Adv. Art Test. Soils Under Cyclic Dyn. Cond., Am. Soc. Civ. Eng. Conv., Detrott. 120-147. Ladd, R. s., and Silver, M. L. (1975). Discussion of Soil liquefaction by torsional simple shear device. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. lOl(GT-8), 827-829. Lambe, T. w., and Whitman, R. V. (1968). "Soil Mechanics," Wiley, New York. A (1962). "Foundation Engineering." McGraw-Hill, New York,. . d L eonards, G . · Lodde, P. J. (1977). Shear ·Bl,oduli and material damping ratios of San Francisco Bay mu · Master's Thesis, University of Texas, Austin. Lord, A. F., Jr., Curran, J. w., and Koerner, R. M. (19~6) ..New_ transducer system for determining dynamic mechanical properties and attenuatiOn m sotl. J. Acoust. Soc. Am.
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DYNAMIC SOil PROPERTIES
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!he
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Peacock,."'!· H., and .seed, H. B. (1968). Sand liquefaction under cyclic loading simple shear conditions. J. Sod Mech. Found. Div., Am. Soc. Civ. Eng. 94(SM-3), 689~708. Peck, R. ~··Hansen, W. E., and Thornburn, T. H. (1974). "Foundation Engineering" 2nd ed. Wtley, New York. ' Prakash, S. (1975). Analysis and design of vibrating footings. Symp. Recent Dev. Anal. Soil . Behav., Sydney, 295-326. Prakash, S. (1981). "Soil Dynamics." McGraw-Hill, New York. Prakash, S. (1986). Future trend~ in geotechnical earthquake engineering research. Paper presented to the VIII Symposmm on Earthquake Engineering, Roorkee, December. Prakash, S., and Puri, V. K. (1977). Critical evaluation ofiS-5249-1969. J. Indian Geotech Soc 6(1), 43-56. . . . Prakash, S.; and Puri, V. K. (1980). "Dynamic Properties of Soils from In Situ Tests" (unpubhshed report). University of Missouri-Rolla, Rolla. Prakash, S.,.and Puri, V. K. (1981). Dynamic properties of soils from in situ tests. J. Geotech. . Eng. Dw., Am. Soc. Civ. Eng. 107 (GT-7), 943-963. Prakash, S.,. and Puri, V. K. (1984). Design of compressor foundations: Predictions and observattons. Int. Conf. Case Hist. Geotech. Eng., St. Louis, MO, Vol. 4, 1705~1710. Prakash.' S., Nandkumaran, P., and Joshi, V. H. (1973). Design and performance of an osclllatory shear box. J. Indian Geotech. Soc. 3(2), 101 ~ 112. Prakash, S., .Ran~an, <:'"·· .saran, S., Srivastava, L. S., and Singh, B. (1976). Report on geotechmcal. mv~stlgatiOn for Penden Cement Authority cement factory at Gornto, . Bhutan, Umverslty of Roorkee, Geotech. Eng. Stud. No. 110, Roorkee, May. Pun, V. ~· (1~8.4). Li~uefa.ction behavior and dynamic properties of Ioessial (silty) soils. Ph.D. . Thests, Ctvtl Engmeenng Department, University of Missouri-Rolla, Rolla. Rtchart, F. E., Jr. (1961). Closure to Foundation vibrations. J. Soil Mech Found n· A Soc. Civ. Eng. 87(SM-4), Part I, 169-178. · · IV., m. Richart, F. E., Jr. (1977). Dynamic stress-strain relations for soils, state-of-the-art report Proc. Int. Conf. Soil Mech. Found. Eng., 9th, Tokyo, Vol. 2, 605-612. . Richart, F ..E., .~r., H~ll, 1. R., and Woods, R. D. (1970). "Vibrations of Soils and Foundattons. Prentice-Hall, Englewood-Cliffs, New Jersey. Roscoe, K. H. (1~53). An apparatus for the application of simple shear to soil samples. Proc. Int. Conf. Sozl Mech. Found. Eng. 3rd, Zurich, VoL 1, 186~191.
Schmertmann,. J. H. (1975). Measurement of in situ shear strength. Proc. Spec. Conf. In Situ Meas. Soz/ Prop., Am. Soc. Civ. Eng., Rayleigh, NC, Vol. 2, 56-138. Schmertmann, J. H. (1977). Use the SPT to measure dynamic soil propertiesLYes but ASTM Spec. Tech. Pub/. STP 654, Dyn Geotech. Test., 341-355. . ' . Seed, H. B. (1976). Evaluation of soil liquefaction effects on level ground during earthquakes.
REFERENCES
187
State-of-the-Art Paper. Proc. Geotech. Eng. Div. Specialty Sess. Liquefaction Problems in Geotech. Eng. Am. Soc. Civ. Eng., Philadelphia PA Sept., 1~104. Seed, H. B. (1979). Soil liquefaction and cyclic mobility evaluation for level ground during earthquakes. J. Geotech. Eng. Div. Am. Soc. Civ. Eng. 105(GT~2), 201~255. Seed, H. B., and Peacock, W. H. (1971). Test procedure for measuring soil liquefaction characteristics. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 97(SM-8), 1099~1119. Seed, H. B., Idriss, I. M., and Arango, I. (1983). Evaluation of liquefaction potential using field performance data. 1. Geotech. Eng. Div. Am. Soc. Civ. Eng. 109(GT-3), 458-482. Seed, H. B., Pyke, R. M., and Martin, G. R. (1978). Effect of multidirectional shaking on pore pressure development in sands. J. Geotech. Eng. Div. Am. Soc. Civ. Eng., 104, (GT-1), 27-44. Seed, H. B., Tokimatsu, K., Harder, L. F., and Chung, R. M. (1984). "The Influence of SPT Procedures in Soil Liquefaction Resistance Evaluations," Rep. No. UCB/EERC 84-15. Earthquake Eng. Res. Cent., University of California, Berkeley. Seed, H. B., Tokimatsu, K., Harder, L. F. , and Chung, R. M. (1985). Influence of SPT procedures on soil liquefaction resistance evaluations. J. Geotech. Eng. Div. Am. Soc. Civ. Eng. IU(GT-12), 1425-1445. Seed, H. B., Wong, R. T., Idriss, I. M., and Tokimatsu, K. (1986). Moduli and damping factors for &Y.namic analysis of coheslp_11~ess soils. J. Geotech. Eng. Div. Am. Soc. Civ. Eng. 112(GT-11), 1016 to 1032. .. Shannon and Wilson~Agbabian Associates (1976). Geotechnical and strong motion earthquake data from U.S. accelerograph stations, Ferndale, Cholame, and El Centro, California. U.S. Nuclear Regulatory Commission, NUREG-0029, Vol. 1. Shannon, W. L., Yamane, G., and Dietrich, R J. (1959). Dynamic triaxial tests on sand. Proc. Pan Am. Conf. Soil Mech. Found. Eng., 1st, Mexico City, Vol. 1, 473-486. Silver, M. L. (1981). Load deformation and strength behavior of soils under dynamic loading. State-of-the-Art Paper. Proc. Int. Conf. Recent Adv. Geotech. Earthquake Eng. Soil Dyn., St. Louis, MO, VoL 3, 873-896. Skempton, A. W. (1986). Standard penetration test procedures and the effects of overburden pressure, relative density, particle size, aging and overconsolidation. Geotechnique, XXXVI(3), 425-448. Stephenson, R. W. (1977). Ultrasonic testing for determining dynamic soil moduli. ASTM Spec. Tech. Pub!. STP 654, Dyn. Geotech. Test., 179~195. Stevens, H. W. (1966). Measurement of the complex moduli and damping of soils under dynamic loads. U.S. Army Cold Reg. Res. Eng. Lab., Hanover, N. H., Tech. Pub!. No. I73. Stokoe, K. H., II, and Abdel-razzak, K. G. (1975). Shear moduli of two compacted fills. Proc. In Situ Meas. Soil Prop., Am. Soc. Civ. Eng., Raleigh, NC, Vol. 1, 422-447. Stokoe, K. H., II, and Hoar, R. J. (1978). Variables affecting in situ seismic measurements. Proc. Am. Soc. Civ. En~, Spec. Conf. Earthquake Eng. Soil Dyn., Pasadena, CA, Vol. 2, 919-939. . "• Stokoe, K. H., II, and Richart, F. E., Jr. (1973a). Shear moduli of soils: In situ and from laboratory measurements. Proc. World Conf. Earthquake Eng, 5th, Rome, Vol. 1, 356-359. Stokoe, K. H., II, and Richart, F. E., Jr. (1973b). In situ and laboratory shear wave velocities. Proc. Int. Conf. Soil Mech. Found. Eng., 8th, 1973, Moscow, Vol. 1, 403-409. Stokoe, K. H., II, and Woods, R. D. (1972). In situ shear wave velocity by cross-hole method. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 98(SM-5), 443-460. Stokoe, K. H., II, Lee, S. H. H., and Knox, D.P. (1985). Shear moduli measurements under
188
DYNAMIC SOIL PROPERTIES
true triaxial stresses. Adv. Art Test. Soils Under Cyclic Dyn Co d A S · n · m. oc. Civ. Eng. Conv., Detroit, 166--185.
Ta~lor, D. W, (1948). "Fundamentals of Soil Mechanics." Wiley, New York.
5
· Thters, G. R. and Seed H B (1968) C r p, d •. • · : · yc tc stress-stram characteristics of clay. 1. Soil Me h c · oun · Dtv., Am. Soc. Ctv. Eng. 94(SM~6), 555-569.
Trud::~· !~:i~e~~ytm~nBR. v., sand cc~ristian, J. T. (1974). Shear wave velocity and modulus
. · . aston oc. tv. Eng. 61(1), 12-25. Wetssman, G. F., and White S R (1961) D · · Spec Tech Pub[ STP 305 . S . S .,· ampmg capactty of some granular soils. ASTM . · · · , ymp. 01 Dyn., 45-59. Whtt~an, R. V. (1963). "Stress"Strain Time Behavior of Soil in One-D· · !·Co ston," Rep. R. 63-2S. De C' . tmensiOna mpres~ . P· tv. Eng., M.I.T., Cambndge, Massachusetts. WhitCm.an,ER. V., and Lawrence, F. V. (1963). Discussion. J. Soil Mech. Found Div A S IV. ng. 89(SM-5), 112-115. · ., m. oc.
Wils~~~n~thD~~o~~~ti~i~;i~~y.Rp,!~. ~~O)S Ef~~t ~ consolidation Soils, Boulder, CO, 419~
435 .
· oc.
tv.
pressure on elastic and ng. Res. Conf. Shear Strength Cohesive
Wolfe, W. E. Annaki M and Le K L ( 1977) · · ' ., a ' e, · · · Sotl liquefaction in cyclic cubic test pparatus. Proc. World Conf. Earthquake Eng. 6th, New Delhi Vol. 3 2151-2156 Woods R D (1978) M ' · ' So~ E "i. easurement of dynamic soil properties-State-of-the-Art. Proc Am · rv. ng. ,pee. Conf. Earthquake Eng. Soil Dyn., Pasadena CA Vol 1 9i-180. Woods, R. D. and Stokoe K H II (1985) Sh · · ' ' . ' · . ' · ·' · allow setsmtc exploration in soil dynamics R· h 1 C' zc ar ommemorattve Lectures R D Wood Ed"t p · · ~~~· c~~c~:dgmgDs otfa_tseossionbsponsored by the Geotechnical Engine~rin.g Division 120-156. ' · · · ·• e rm, cto er 23,
C· .
SAm
YangGe~~~c~n~~gat~:'a~ A.SW. (1~76).
Dynamic response of tropical marine limestone. J. m. oc. Ctv. Eng. 102(GT-2) 123-138 Yoshimi, Y. and Oh-Oka H (1973) A · · ' . ' nng torston apparatus for simple shear tests. Proc. Int. Conf. S~il Mech Fo , d. E · un · ng., oscow, Vol. 1 Pt 2 501-506 Yu P d R" · ' . ' . , G~o~~h ~char~· F. AE. (1984). ~tress ratio effects on shear modulus of dry sands. J. · ng. zv., m. Soc. Ctv. Eng. 110(3), 331-345. ·
·
IV.
M.
Unbalanced Forces for Design of Machine Foundations An essential requirement for adequate design of a machine foundation is that the motiqn. amplitudes underoperating conditions do not exceed the specified values. The vibration am!'ifit~des depend upon the natural frequency of the vibrating system, the operating frequency, and the magnitude of the applied dynamic forces and moments (Section 2.5). The information on magnitude and characteristics of the dynamic loads imposed by the machine on the foundation is thus vital for a satisfactory design of the machine foundation system. This information is generally supplied by the manufacturer of the machine and should be procured from him. This presents difficulty sometimes since the interests of the client and the manufacturer of the machine are not in unison, and the manufacturer of the machine may not like to admit that large unbalanced forces may occur from operation of the machine supplied by him. The situation is of course different for certain types of equipment such as compaction machinery, where unbalanced forces are purposely developed. The process of computation of unbalanced forces due to operation of reciprocating and rotary machines is discussed in this chapter.
5.1
UNBALANCED FORCES IN RECIPROCATING MACHINES
The simplest examp1~ of a reciprocating machine is the basic crank mechanism (Fig. 5.1) consisting of a piston, piston rod, connecting rod, and the crank rod. The crank mechanism is used to convert reciprocating motion into rotary motion or vice versa, and, is used in internal combustion engines, steam engines, reciprocating pumps, and compressors. The reciprocating machines may be of single-cylinder type or multicylinder type in which case several cylinders are mounted according to a definite pattern on a common crankshaft. Forces generated due to a single-cylinder reciprocating machine are considered first. 189
191
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
190
UNBALANCED FORCES IN RECIPROCATING MACHINES
(5.2)
OB = OC + CB = r cos 8 + l cos>
where > =Angle between the connecting rod and the X axis, and 8 =Angle between crank rod and X- aXIS. t Substituting for OB from Eq. (5.2) into Eq. (5.1), we ge x = r(1-cos 8) + 1(1-cos >) p
Cylinder
ll II
B
or
--X
II JJ0
p
From triangles AOC and ACB (Fig. 5.1)
%
I•
Xp
AC = r sin 8 = l sin>
'I Letting 8 = "!I
-'
_t:--
r---c--~~----------l+c----------~
·:r
-;to'_,_~
sin
=
r . sm wt
Single-Cylinder Machines
The motion characteristics of the different components of the crank mechanism (Fig. 5.1) are as follows:
1 "'(1.!__ sin2 wt-- r4 sin4 cos '" 8 I 2 1' 2
axis
h. Different points on the crank rod undergo a rotational motion about the axis of rotation through 0. c. The connecting rod AB undergoes a complex motion, the end A has a circular motion, end B moves linearly and points between A and B move along elliptical paths.
OB may be calculated from Eq. (5.2)
(5.1)
(5.5)
)
4
wt
+ ...
)
may be
(5.6)
Neglecting higher powers of r/1 in Eq. (5.6), we get, cos > = ( 1- ;;, sin' wt)
(5.7)
"' f E (57) into Eq. (5.3), we get Substituting the value of cos '"' rom q. .
Unbalanced forces associated with motion of each of these components of the crank mechanism will now be considered. Forces Due to Piston Motion. Let the crank rotate With a constant angular velocity w as shown (Fig. 5.1) causing displacement xP of the piston with reference to its extreme outward position. Let the length of the connecting rod be l and crank radius r. The displacement of the piston is given by
2 112
The value of r/1 is usually small (around 0.25) and Eq. (5.5) written in a series form [Eq. (5.6)]:
a. The piston and the piston rod undergo an oscillatory motion along X
= (r +I)- OB
(5.4)
l
_ ( _ . 2 "')'/2 = (1- (~ sin wt) :.cos>- 1 sm '" I
Figure 5.1. Motion of a crank mechanism.
xP
(5.3)
x =r(1-coswt)+l(1-cos>)
~P = r(1- cos
r2 . 2 wt) + 21 sm w1
(5.8)
Equation (5.8) may be rewritten as 2
xP = r( 1 _cos wt) +
~~ (1- cos 2wt)
or xP = r( 1 +
;I) _r(cos wt + ;I cos 2wt)
(5.9)
193 UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
192
UNBALANCED FORCES IN RECIPROCATING MACHINES
(5.14) Equation (5.8) shows that the piston motion is periodic but not necessarily harmonic. Equation (5.9) expresses the piston motion in terms of a primary component that varies with the frequency of rotation and a secondary component that varies at twice the frequency of rotation. The magnitude of the secondary component depends upon the r II ratio and if the length of the connecting rod is large compared to the crank radius r, the influence of the secondary component becomes negligible and the piston motion may be considered harmonic. The velocity xP and acceleration xp of the piston may be obtained by differentiating Eq. (5.9):
.
t. n of radius 0 A and may be resolved into compo-
F acts along t h e dnee IO n~nts along the directions of X and
z.
.
Forces Due :a
R d
The unbalanced forces due to
Mo:i~:e 0:a~::c~~~:~~d ~;y be determined with reasonable
;l sin 2wt)
(5.10)
complex mo wn o. f the connecting rod by a concentrated accuracy by replacmg the mass M" 0 M and M should be M at A and a mass Mh at B. Masses "' b . . :e~~rmi~~d such that the resultant unbalanced force due ~o th;~mou~n~s the same as due to motion of the connecting rod. Magmtu es. o "'an b may be determined from Eq. (5.15) and (5.16) respectively.
.. = rw '( cos wt + 7 r cos 2wt) xP
(5.11)
(5.15)
iP = rw (sin wt +
The inertia force FP due to translatory motion of the piston and piston rod is thus given by Eq. (5.12):
and
-;~-:-.·7
(5.16)
Ma2-f.:·MIJ=Mcr (5.12) where I,, I, in which MP =mass of piston, piston rod and cross head . Force Due to Motion of Crank Rod. The unbalanced force due to rotary motion of the crank will now be determined. The unbalance in the crankshaft may be replaced by an equivalent mass at the crankpin A so as to produce the same inertia force as due to motion of the crank. Since all points on the crank rod move in a circular path, equivalent mass can be determined from Eq. (5.13):
(5.13)
=
distances of A and B' respectively' from the mass centre M of
the connecting rod. · d ay be at A will be undergoing a rotary motiOn an m M h b 1 f e F due to T e mass a2 in combined with the mass of the crank M,,. The un a ance ore ..; M rotation of masses at A may be obtained by replacmg M,, by (Ma, ,,) Eq. (5.14): 2
FA =(Ma1 +Ma 2 )rw =MArw
wh~~~i~rly: :~!a~:~!a~:gm~;~e a:o~~idered along with th;,rto~rna(~_;;~ the inertia force Fn may be obtamed by addmg Mb to Fn = (Mb
in which M , 1 = Equivalent mass of crank rod lumped at A M, =mass of crank rod acting at its centre of gravity K r = crank radius
and r 1 = distances of center of gravity of crank rod from 0
The inertia (centrifugal) force F, due to rotation of mass M , 1 is given by Eq. (5.14):
(5.17)
2
P
m
q.
+ MP)rw'( cos wt + ~cos 2wt)
= "'fo.1 8 rw '(cos
wt +
I cos
2wt)
(5.18)
where M = total reciprocating in ass at B. . . . The f~ce FA at any time can be resolved into Its honzontal and vertical components given by 2
FAX = MArw cos wt
and
(5.19a)
194
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
Fz = M,\rw 2 sin wt
UNBALANCED FORCES IN RECIPROCATING MACHINES
z
(5.19b)
F,
~h';Je ~x atd F, are, A·
195
respectively, the horizontal and vertical components esn tmg force Fx m the direction of X is then given by
Fx =FAX+ FB - M
-
= MArw 2 cos wt + Msrw 2 ( cos wt + y cos 2wt) 2
2
Arw cos wt
2
r w + MBrw 2 cos wt + MB-1cos 2wt
(5.20)
It is observed from Eqs (5 19b) d ( 5 20 ) crank mechan. s . .. . an . that the operation of a single I m gives nse to unbalanced forces in the direction of piston motion F motion ~ a~ also. m a directiOn perpendicular to the direction of piston · (d . , · orce m the directiOn of piston motion has both bee;~~::~:/fr~~ ~ and secondary (depending on 2w) component1'~;';:~ of the iston m . q. (5.20). The force m a direction perpendicular to that . Pd f otwn has only a pnmary component [see Eq. (5 19b)] The magmtu e o pnmary and secondary components of F can be . . . obtamed by separatmg the terms containing w and 2w in Eq. (5.20):
pl)
Counterweights
F,
Figure 5.2.
F'x - M A rw 2 cos wt + M Brw 2 cos wt
(5.21)
--,. };,.,..
Principle of counterbalancing (both horizontal and vertical components due to
rotating mass and counterweights balance).
and
(5.22) · where F'x -- th e pnmary component of F and F"- th
of Fx.
x
x -
d
e secon ary component
The unbalanced force due to rotation f h t e mass M A can be balanced by installing a mass M' known • . A as counterwe 1ght at f . dtrectwn of radius 0 A (Fig 5 2 ) Th . an ang1e o 180° Wtth the e magmtude of M' d · d. · · · from 0 are selected so that it A an tts tstance 1 opposite to that produced bya "J.tays ~~duces a centrifugal force equal and forces is known as counterbalanci::. d s process of reducmg unbalanced reducing vibrations in certain c g a~ ts used as a remedial measure for resulting unbalanced force in thea~es. . hen co~nterbala~cing is done, the (5.23) [by making MA,; 0 in Eq. (~~~~)]~ of piston motiOn ts given by Eq.
°.
2
2
Fx-M rw cos 2wt srw 2 cos wt + Ms-I
(5.23) Multicylinder Engines
The primary component of Fx is then given by Eq. (5 .24): F; = Msrw 2 cos wt
The secondary component F; is unchanged and is given by Eq. (5.22). The unbalanced force in the direction of Z becomes zero after counterbalancing. As the primary unbalanced force in the direction of piston motion has the same frequency as that of the crank rotation Eq. (5.23), it is possible to install a mass on the crank to completely eliminate or minimize this force. However, the rotation of this counterbalancing mass will increase the force in a direciton perpendicular to the piston motion. Therefore, it is impossible to completely eliminate the unbalanced forces due to operation of a single cylinder reciprocating engine which is thus inherently unbalanced, and its operation gives rise to unbalanced forces both along and perpendicular to the direction of piston motion. The force in the direction of piston motion has a primary as well as secondary component, whereas the force perpendicular to the direction of piston motion has only the primary component. In the case of multicylind11r engines, however, it is possible to mount the cylinders so as to completely eliminate the unbalanced forces. The unbalanced forces due to operation of multicylinder engines are now considered.
(5.24)
A schematic layout of a horizontal multicylinder engine is shown in Fig . 5.3a. The cylinders are located in a plane parallel to xy plane and have a linear arrangement. The cylinders are parallel to each other. The crankshaft
197
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
196
UNBALANCED FORCES IN RECIPROCATING MACHINES 2
Y T
I I
1
Direction of
rShaft
Cylinder # i
tJJe---,---j
:
1'1
Y,
X
Y!
~q---L-i-n
F, = w'
y
x-j
cos 2(wt +.a,) (5.26)
2:"
(5.27a)
r,M Ai sin ( wt + a,)
i=l
Cylinder #1
I
~
in ':hichbalanced force in the z direction due to operation of ith cylinder, F,, =~~balanced force in the x direction due to operatiOn of tth cybnder. i~sultant exciting forces transmitted to the foundatiOn due to operatlon of all the n cylinders are given by
or1 ----~~~-------1~----------~
y
r,w' cos (wt +a,)+ MBi r,
z
--3]
I
ill
+ Ms,)
Fxi = (MA,
crank rotation
2
Fx = w'
i
r,[ (MAi
+ MsJ cos(wt +a,)+
1
Ms, cos 2(wt +a,)]
(5.27b)
i~l (a)
(b)
Figure 5.3. Schematic layout of (a) multicylinder horizontal engine in xy plane (piston motion in x direction, Engine shaft is parallel to y axis.) (b) Orientation of axes.
. rs ·27) may be written as · 1, E qs,,_~: . are 1"denuca If all cylinders F = rw'MA z
is parallel to Y axis and the crank rotation is in XZ plane with the origin of coordinates at 0, the combined mass center of gravity of the engine and the foundation. The arrangement of multicylinder engine under consideration will generate unbalanced forces in the directions of X and Z. The magnitude of unbalanced force due to any cylinder will depend upon the crank angle a which defines the relative position of pistons in different cylinders at any time. For example, a = 180" for a two-cylinder engine implies that when the piston in one of the cylinders is in the extreme right piston, the piston in the other cylinder is in the extreme left position. That is, they operate out of phase. The unbalanced forces due to operation of a multicylinder engine can be computed by following the principle that was used for the case of a single-cylinder engine and then determining the resultant force in a given direction as the algebraic sum of the components of forces due to all the cylinders in that direction. Depending upon the mounting arrangement of different cylinders in a multicylinder engine, its operation may result in the generation of either the unbalanced forces or unbalanced moments or both. Let the total number of cylinders be n. The unbalanced forces due to operation of the ith cylinder may be obtained from Eqs. (5.19b) and (5.20) by replacing MA with MA,, Mn with Mn 1, r by r, and changing wt to (wt +a;). This substitution gives (5.25)
2:" sin (wt +a,)
(5.28a)
i=l
and F =rw'f(MA +Ms) X
l
i; cos(wt+ a,)+ l Ms ~1 cos2(wt+ a,)} i~l
'
(5.28b)
The exciting moments due to these unbalanced forces are given by
"
(5.29a)
M="'FY ~ X
Zl
l
i=l
"
My ="'Fx L..J Zl
(5.29b)
I
i=l
"
M z ="'FY L..J XI I
i=l
where
M = Exciting moment about the axis of X X
M = Exciting moment about the y axis y
and
M = Exciting moment abont the Z-axis
'
(5.29c)
198
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
!he exciting moments will have primary and secondary components dependmg upon the ~ature of unbalanced forces generated by the machine. The conditiOns for balancing the fo~ces and moments generated by operatiOn ~fa honzontally mounted multiCylmder engine can be established by exammmg Eqs. (5.28) and (5.29). The primary component of exciting forces Will be balanced if [from Eqs. (5.28a and b)]
199
UNBALANCED FORCES IN RECIPROCATING MACHINES
Unbalanced moments and forces in any particular case may be calculated from the mounting details of cylinders and crank angles a. The computations become simple when all cylinders are identical and crank and connecting rods are of uniform cross section. When counterweights are used to balance the exciting force due to rotating mass M A, Eqs. (5.28) become (5.33a)
F, =0
n
2.: sin (wt +a,)= 0
t=l
(5.30a) Fx= rw
and
t=l
~
[
M 8 cos(wt+ a,)+
(5.30b)
8
cos2(wt+
a,)]
(5.33b)
n
F~ = rw 2 M 8
The secondary component will be balanced if
2.: [cos (wt +a,)]
(5.33c)
i=l
2 2
n
.2.: cos 2(wt +a,)= 0 i=l
IM
The primary and secondary components of Fx are then given by
n
2.: cos (wt +a,)= 0
2
n
"' F: = -rw- M 8 ,~[cos 2(wt +a,)] 1
(5.30c)
(5.33d)
-~;. ,' :>"
The primary component of the exciting moments will be balanced if Eqs · (5.31) are satisfied:
Substituting the value ofF, from Eqs. (5.33a) in Eqs. (5.29a,b), we get (5.34a)
n
.2.:
i=l
X, sin ( wt + a ) = 0
(5.31a)
Y, cos (wt +a,)= 0
(5.31b)
I
M y =0
n
.2.: 1=1 n
2.:
Y, sin (wt +a,)= 0
1=1
(5.31c)
Substituting the values ofF; and F: from Eqs. (5.33c,d) in Eqs. (5.29c), we get M;=LF;iyi
Y,cos2(wt+ a,)=O
(5.32)
The unbalanced exciting loads due to operation of reciprocating machine may be balanced dependmg on geometerical location of the equipment arid the crank angle. · So far, we have c6nsidered the unbalanced exciting loads due to operatiOn of honwntally mounted smgle or multicylinder engines. For vertically mounted engmes (piston motwn along Z axis), the exciting loads can be computed by mterchangmg X with Z in Eqs. (5.19)-(5.24) for singlecylmder engmes and m Eqs. (5.28) and (5.29) for multicylinder engines. For piston motion along Y axis, replace X by y in the above equations.
(5.35a)
i=l
n
M"="' Z L.,; F".Y XI I
n
1=1
(5.34c)
Also
n
In the case under consideration, only the force Fx has a secondary component and the secondary component of moment will be balanced if
.2.:
(5.34b)
(5.35b)
i=l
in which, M ~ = the primary component of M z and M~ = the secondary component of M,. . .It is observed from l!qs. (5.33) and (5.34) that in case of multicylinder engines in which the rotating masses have been counterbalanced we get only the exciting forces and moments given below: 1. Exciting force along the direction of piston motion, this force has both primary and secondary components 2. Exciting moment about an axis perpendicular to the plane of the engines; this moment has both primary and secondary components
200
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS UNBALANCED FORCES IN ROTARY MACHINES
It is also observed from Eqs. (5.33c) and (5.33d) that the maximum secondary force is equal to r/1 X (maximum primary force in that direction). The absolute values of the exciting forces and moments depend upon the cr~k :ngle a. For example, for the two-cylinder engine with crank angle a- 90 (Fig. 5.4) and rotatmg mass counterbalanced, we have, from equatiOnS (5.33), F, = 0
F~ = rw Mn[cos wt +cos (wt + 90°)] = rw 2Mn[cos wt- sin wt] 2
= V2rw 2M 8 cos (wt + 45°) = 1.41 x (primary force due to single cylinder) 2
201
Unbalanced primary and secondary forces and couples due to single and multicylinder engines for different crank arrangements are summarized in Table 5.1 (Newcomb, 1951). In case all the cylinders of a multicylinder engine are not identical, the exciting loads should be calculated using Eqs. (5.27) and (5.29). Besides the forces due to engine operation, there will be additional forces due to machines being operated by the engine or vice versa. The resultant exciting loads will influence the dynamic response of the foundation. The horizontal forces due to engine operation will also give exciting moments about the combined center of gravity of the machine foundation system.
(5.36a)
2
F"x = -r w- Mn[cos 2wt +cos 2(wt + 90)] 1 rzwz = - - M 8 [cos2wt- cos2wt] = 0
5.2
UNBALANCED FORCES IN ROTARY MACHINES
Exciting force due to rotation of an unbalanced shaft (Fig. 5.5a) is given (5.36b)
1
~~milarly, it can be shown that component of primary moment
by
M; is given
M; = 1.41 x (primary component of force due to single cylinder) x D
(5.36c)
(5.37) in which m =mass of rotor and e =effective eccentricity, that is, the distance of the mass center from the axis of rotation. The horizontal and vertical components of the unbalanced force are given by
where D =distance between center line of the pistons, and 2
2
Fx
7T)J
"- r w-M8 [ 2cos2wt-2cos2 D D ( wt+z M,--
1
2
= mew cos wt
(5.38a)
2
(5.38b)
and
2
r w2 D = - -2 M 8 [cos2wt+cos2wt] 1 rzwz = - - M nD cos 2wt 1
•
F z =mew sin wt
=(secondary force due to a single cylinder) x D'
(5.36d)
-----+D/2 ---0
High-speed rotary machines are well balanced and the eccentricity e is generally very small. However, due to their high speed of rotation, the magnitude of exciting loads may be significant. Further, the effective eccentricity may increase due to wear and tear, and balancing may be necessary to keep the unbalance within tolerable limits. When two rotary machines of the same capacity and having the same unbalance are coupled together, the unbalanced mass in each of them may be in-phase as shown in Fig. 5.5b, out-of-phase'•by 180° as shown in Fig. 5.5c, or at any phase, as shown in Fig. 5.5d. The unbalanced force for the case shown in Fig. 5.5b is given by
X
F= 2mew 2
D/2
(5.39a)
The vertical and horizontal components of the unbalanced force are given
by Figure 5.4.
Two cranks at 90°,
Fx = 2mew 2 cos wt
(5.39b)
:::"'
Table 5.1.
Unbalanced Forces and Couples for Different Crank Arrangements Crank Arrangements
Forces
Couples
Primary
Single crank
_A_
Two cranks at 180° In-line cylinders Opposed cylinders
Two cranks at 90°
k
~ 1~
y
Two cylinders on one crank Cylinders at 90°
Two cylinders on one crank Opposed cylinders
~
Three cranks at 120°
). Four cylinders Cranks at 180° Cranks at 90° Six cylinders
~
r J,llt ~
+~
~~
F' without
"' "' 0
Primary
Secondary
None
None
F"
counter wts. (0.5) F' with counter wts. 0
2F"
0
0
F'D without counter wts. F' T D with counter wts. Nil
(1.41) F' without counter wts.
0
(1.41) F' D without counter wts.
(0.707) F' with counter wts. F' without counter wts. 0 with
None
Nil
F'D
(0.707) F' D without counter wts . .
(1.41) F"
Nil
Nil
counter wts.
2F' without counter wts. F' with counter wts.
0
0
'"' .. 0
0
0
0
0
. 0
Source: Newcomb, 1951. Reprinted with the permission of the ASME. D = cylinder-center distance (in) F' = Primary force F" = Secondary force
Secondary
,.~,
None
(3.46) F' D without counter wts. (1.73) F' D with counter wts. 0 (1.41) F' D without counter wts. (0.707) F' D with counter wts.
Nil
(3.46) F' D
0
4.0 F"D
''-~
0
0
0
204
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
EXAMPLES
205
The components of the moment M in horizontal and vertical directions are given by Mx
= mew
2
l cos wt
2
Mz =mew 1sin wt
(5.39e) (5.39f)
X
When masses have an orientation as shown in Fig. 5.5d, the machine operation will give rise to both an unbalanced force and a moment. For design, the worst combination of loads should be assumed to be acting on the foundation. The unbalanced force will be given by Eq. (5.39a) and unbalanced moment by Eq. (5.39d). For more than two rotors on a common shaft, combined unbalanced forces and moments may be obtained in a similar manner.
(a)
(b)
5.3
Machines such as forging hammers, punch presses, and stamping machines produce inwact or pulse-type loads. There is no method to date (1987) to define the fbrces imposed on tile\"foundation due to an impact and the variation of these forces with time. This information may however be obtained experimentally. Experimental data on impact tests on 30-cmdiameter instrumented model footing (Drnevich and Hall, 1966; Lysmer and Richart, 1966) indicates that the shape and duration of the loading pulse varies with the energy of impact and energy absorbing characteristics of supporting medium such as elastic pads. The necessity to acquire such data on prototype foundations of this type cannot be overstressed. We have so far considered the nature of unbalanced exciting loads due to operation of different types of machine. The process of calculation of exciting loads will now be illustrated with some typical examples.
m m
.t'
UNBALANCED FORCES DUE TO IMPACT LOADS
lei
m
..._ '
•
. t' (d)
5.4
Figure_ 5.5. Unbalanced forces due to rotary machines. (a) single rotor; (b) two rotors with •: ual m phase u~balance; (c) two rotors with equal unbalance with a phase difference of 180°· 1 1 two rotors with equal unbalance at any phase. '
and (5.39c) For the case of rotors shown in Fig. 5.5c, the
res~ltant unbalanced forces
due to the two masses at any time cancel out, but there is a resulting moment M gtven by M = mew 2 (l)
(5.39d)
in which l = distance between the mass center of gravities of rotors as shown m Ftg. 5.5c.
EXAMPLES
EXAMPLE 5.4.1
A horizontal single cylinder reciprocating engine is mounted on a foundation block as shown in Fig. 5.6. The center line of the piston is 30 em above the top of the block and the center of the crank lies vertically above the mass center of the combined machine foundation system. The data on the engine is as follows~'' Weight of the piston and piston rod= 10.5 kg Weight of crank assumed concentrated at the crank pin= 4.6 kg Weight of connecting rod= 5.3 kg Length of connecting rod l = 45 em Crank radius r = 15 em Operating speed of engine = 600 rpm The connect;ng rod is of uniform cross section.
206
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
(a)
When counterweights are not installed
1. F~
207
EXAMPLES
, F"x
Force in the vertical direction (perpendicular to direction of piston motion) F, (5.19) Fz = MArw 2 sin wt
(FJm" = MArw 2 = 0.739(0.15)(62.83) 2 = 436.6 kg= 0.4366 t 2.
m
Force in horizontal direction (along the direction of piston motion) Fx The primary component of force F~ in the horizontal direction is given by (5.21) (F~)m" = (MA + M 8 )rw = (0.739 + 1.340)(0.15)(62.83) 2
Foundation block
Combined
e.G.
= 1231.0 kg= 1.231 t F~
The secondary component givenc?Y
in the horizontal direction is
-~·;·-::M r 2 w~
F" =- ·
Figure 5.6. Schematics of Problem 5.4.1.
X
2
(F~)m, =
Compute the unbalanced exciting forces and moments for the design of the foundation assuming:
3.
(a) No counterweights are installed (b) Counterweights are installed. Also find the magnitude of the counterweights Solution Total rotating mass at M A due to crank and connecting rod
cos 2wt 2
2
[
(0.45)
8
10.5 + H5.3) = ,.------;~';-'---'9.81
w
600(21T) = 62.83 rad/sec 60
(b)
(5.22)
k = 264 .5 g 0 .2645 t
Unbalanced excited moments The force F, passes through the mass center of gravity 0 and has no moment about it. The unbalanced force FY along the direciton of Y is zero and therefore the exciting moment Mx is zero. The only exciting moment in this problem is MY due to the primary and secondary components of Fx and tries to cause rotation about Y axis through the center of gravity. The primary component of moment M~ is given by (Fig. 5.6) =
F~h
= (1.231)(0.3 + 1.0) = 1.60 t m
M; = F:h = (0.2645)(0.3
= 1.340 kg sec 2/m
=
l
The secondary component of the moment
=0.739kgsec 2 /m . d T o t a I mass at B due to ptston an crank rod M
2
B
M 8 r w = (1.340)(0.15) (62.83) =
M~
4.6+ t{5.3) 9.81
Operating speed of the engine
2
M; is given by
+ 1.0) = 0.344 t m
When counterweigH'ts are installed 4. The mass of the counterweight depends upon its distance from the center of rotation. If the counterweight is installed at the same distance from the center of rotation as the mass M A, then M = 2 1 MA =0.739kgsec /m5.
Exciting force Fz
F, =0
208
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
.·. .!
209 EXAMPLES
6. Unbalanced force Fx in the direction of X The primary component
F~
z
is given by 2
F; = M 8 rw cos wt
(5.24)
(F~)m, = M 8 rw = (1.34)(0.15)(62.83) = 793.5 kg= 0.7935 t 2
2
(')
The secondary component F~ is unchanged and its value is equal to 0.2645 t as calculated in part (a). 7.
Exciting moment The primary component of exciting moment M ~ is given by M; = F~h = (0.7935)(1.3) = 1.0315 t m. The secondary component of the exciting moment remains unchanged and is equal to 0.344 t m as calculated in part (a).
M;
1.0 m
j_
EXAMPLE 5.4.2
In Example 5.4.1 if the engine was mounted vertically with XZ as the plane of crank rotation and the counterweights are installed, determine the exciting forces and moments for design of the foundation. Assume that the line of motion of the piston lies along the Z axis through the center of gravity of the engine and the foundation.
(b)
Figure 5.7.
Solution Data is as given in Example 5 .4.1. The direction of the piston motion is vertical and along Z axis and counterbalancing has been done:
Unbalanced force along the direction of piston motion: F,. The primary component of unbalanced force F, is obtained from Eq. (5.24) by replacing X by Z, F; = M 8 rw 2 cos wt
(a)
2
(F;)m, = M 8 rw = (1.34)(0.15)(62.83) 2 = 793.5 kg= 0.7935 t F~
The secondary component of force 2 (M 8 r w 2/l) cos 2wt [Eq. (5.22)] "
2
r w
2
(FJm" = MB - 1- =
2
is similary given by
F~
=
2
(1.340)(0.15) (62.83) (0.45) = 264.5 kg= 0.2645 t
(b)
Unbalanced force in a direction perpendicular to piston motion
(c)
Because of counterbalancing, Fx = 0 Unbalanced exciting moments. As F; and F~ pass through 0 and Fx = 0, no unbalanced moments are generated and Mx = My = M, = 0.
EXAMPLE 5.4.3
A two-cylinder engine is mounted on a foundation so that the direction of piston motion is parallel to Z axis and plane of crank rotation is XZ as shown in Fig. 5.7. The engine is mounted symmetrically with respect to the
t f
(a) Arrangemen o era
nks a ,
= 180°·
(b) Schematic sketch showing line of action
of exciting loads.
foundation. The two cylinders are identical in all respects. The data for the · d ers ts · as gtven · below · The connecting rods are of umform cross cyhn section. The crank angle a = 180". Weight of piston and piston rod= 5.2 kg Weight of crank assumed at crank pm = 2.1 kg Weight of connecting rod= 4.2 kg Length of connecting rod l = 45 em Crank radius r = 15 em Operating speed of engine = 1500 rpm Compute the unbalanced exciting loads for design of the foundation for the engine. Assume n~~~counterbalancing has been done. Solution
4.2) 2.1+ ( T ' · M=0428kgsec/m Total rotatmg mass A 9.81 · 4.2) 5.2+ ( 2 2 · · =0 ·744kgsec /m Total rectprocatmg mass M n -9.81
210
UNBALANCED FORCES FOR DESIGN OF MACHINE FOUNDATIONS
Operating speed of engine=
w
=
2
'lT(~~OO)
= 157.07 sec- 1
REFERENCES
Unbalanced forces Fx 1. Unbalanced force Fx 1 due to cylinder #1
(c)
2.
Crank angle a = 1800 (a)
Fx 1 = M Arw sm
Unbalanced forces and moments
Fx 1 = (0.428)(0.15)(157.07) sin wt = 1583.8 (sin wt) kg
The direction of piston motion is along Z axis. The unbalanced forces replacmg X With Z m Eqs. (5.21) and (5.22). Similarly the unbalanced forces Fx perpendicular to direction of piston motion are obtained by mterchangmg Z and X in Eq. (5.19b).
= 1.5838(sin wt) t 2.
1.
Primary component
F;, = (MA + Ms)rw
2
F;,
cos wt
(From 5.21) (cos wt) kg
= 4.3771 (cos wt) t 2.
F;,
Secondary component F 11
zl
2
=
r w MB - 1
due to cylinder #1
(d) Unbalanced moments The only exciting moments in this case results from the fact that F;, and F' are equal in magnitude but act in opposite directions and thus form :couple that causes rotation about y axis. The moment of this primary couple M; is given:l>r' (Fig. 5.7) M; = F;,(D) = 4.377(0.5) cos wt = 2.188 cos wt t m
2
COS
Resultant forces Fx Fx = Fx 1 + Fx 2 = 1.5838 (sin wt- sin wt) = 0
due to cylinder #1
= (0.428 + 0.744)(0.15)(157.07) 2 cos wt = 4337.1
Unbalanced force Fx 2 due to cylinder #2
Fx 2 = (0.428)(0.15)(157.07)' sin (wt + 180") = 1.5838(-sin wt) t 3.
F;
Unbalanced forces
(From 5.19b)
wt 2
F, along the direction of piston motion in this case are obtained by
(b)
211
The foundation should be designed for the following exciting forces and
2wt
(From 5.22)
2
2 0 ) (0.15) = ( .744 (0. 4S) (157.D7) cos 2wt = 9.17 (cos 2wt) kg
moments: F, = F; = 1.8354 t (secondary)
MY= M; = 2.188 t m (primary)
= 0.9177 (cos 2wt) t 3.
Primary component F::Z due to cylinder #2
REFERENCES
F;, = (0.428 + 0.744)(0.15)(157.07) 2 cos(wt + 180") Drnevich, V. P., and Hall, J. R., Jr. (1966). Transient loading tests on a circular footing. J. Soil
= 4.3771( -cos wt) t 4.
Secondary component F'::,_ due to cylinder #2 2
F'::,_ = (0.744)
~~·.~~))
(157,07) 2 cos 2(wt + 180")
=0.9177 (cos2wt)t 5.
Resultant primary component
F;
F; = F;, + F;, = 4.3771 6.
(cos wt- cos wt) = 0
Resultant secondary component F~ = F~ 1 + F~
=
0.9177(cos 2wt) + 0.9177 cos 2wt
= 1.8354 cos 2wt t (F;)"'"' = 1.8354 t
Mech. Found. Div., Am. Soc. Civ. Eng. 92 (SM-6), 153-157. Lysmer, J., and Richart, F. E., Jr. (1966). Dynamic response of footings to vertical loading. I. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 92 (SM-1), 65-91. Newcomb, W.K. (1951). Principles of foundation design for engines and compressors. Trans. ASME 73, 307-312, 313-318.
MODES OF VIBRATION OF A RIGID FOUNDATION BLOCK
6
Reciprocating machines are probably the oldest machines used by mankind. The classical example is a crank mechanism (Fig. 5.1) which is used to convert translatory motion into mtary motion and vice versa (Section 5.1). Typical examples of reciprocating machines are steam engines, internal combustion engines (e.g., petrol, diesel, and gas engines), pumps and · compressors. These machines may consist of a single cylinder and a piston which may be single acting or double acting, may consist of multicylinders with pistons operating in a regular pattern and mounted on a common crank. The pumps and compressors belonging to the category of reciprocating machines may be of the single-stage or multiple-stage type depending on whether the total compression is developed in one or more than one operation. The multiple-stage arrangement may consist of either several piston-cylinders operated by a common engine or several different engines. The direction of piston movement may be horizontal or vertical. Most reciprocating machines have operating speeds that are smaller than 12001500 rpm. Reciprocating machines operating at higher speeds are sometimes encountered. The foundations for reciprocating machines usually consist of rigid concrete blocks that have openings for mounting the machines. The machines may be mounted directly on a concrete block or on suitably designed elastic pads. Block foundations resting on springs are also sometimes used.
DESIGN REQUIREMENTS
The general criteria for ensuring long-term satisfactory performance of a machine foundation with respect to static and dynamic stability are set forth
in Chapter 1 (Section 1.2}. The foundation requirements for reciprocating machines with respect to dynamic stability are as follows: 212
There should be at least 30 percent difference between the operating speed of the machine and the natural frequency of the soil foundation system. 2. The amplitudes of vibration must be less than the specified permissible values. 3. In the soil, the resultant stresses occasioned by the combined action of static and dynamic loads should not exceed the permissible values.
1.
Foundations for Reciprocating Machines
6.1
213
The natural frequency of the foundation soil system is strongly influenced by the base contact area of the foundation, its geometry, mass, depth of embedment, and stiffness and damping properties of the soil. Besides these factors, the amplitudes of vibration are influenced by the unbalanced forces and moments associated with the machine's operation. It is mentioned in Chapter 4 that soils undergo increasing strains when subjected to combined static and dynamic loads, but if the magnitude of dynamic load is very small compared to the static load, as is the case with most reciprocating machines, the increase in strain after the first few applications of the dynamic load is negligible. Therefore a soil may b.e·:'!ssumed to behave as an elastic material under such loading conditions (Fig. 4.18). The fact that permissible static soil pressures below machine foundations are generally smaller than the corresponding soil pressures beneath ordinary footings, helps to ensure the above behavior. Therefore, the residual or plastic settlement of a soli below an adequately designed machine foundation is generally negligible.
6.2
MODES OF VIBRATION OF A RIGID FOUNDATION BLOCK
It may be assumed that rigid foundation experiences only rigid body displacements. Therefore, under the influence of superimposed forces and moments a rigid concrete block can vibrate in six dtfferent modes (Ftg. 6.1): 1.
2. 3. 4.
5. 6.
Translation along the Z axis Translation along the X axis Translation along the Y axis Rotation around the X axis (pitching) Rotation around Jhe Y axis (rocking) Rotation around"the Z axis (yawing)
Any movement of the block can be resolved into these six independent displacements. Hence, the block has six degrees of freedom (or modes of vibration) and six corresponding natural frequenctes. Of the stx modes, translation along the Z axis and rotation around the Z axis can occur independently of any other motion. However, translation along the X or Y axis and corresponding rotation about the Y or X axis, respectively, always
214
FOUNDATIONS FOR RECIPROCATING MACHINES Vertical
ELASTIC HALF-SPACE METHOD
215
V'
t
Torsion (yawing)
0'
0'
•o
~+X
Longitud ina I
z Figure 6.1.
METHODS OF ANALYSIS
Several methods are used in analyzing the vibration characteristics of block foundations, they are as follows: Elastic half-space method, 2. Linear elastic weightless spring method 3. Linear elastic weighted spring method, and 4. Empirical methods 1.
Because only the first two are used in practice, these are described in detail here. ··
6.4
~
~
00
Modes of vibration of a rigid block foundation.
occur together and are called coupled modes. Therefore, in analyzing rigid block foundations, one is concerned with four types of motion of which two (translation along the Z axis and rotation around the z axis) are independent, and two (translation along the X axis and rotation around the Y axis and vice versa) are coupled and occur simultaneously. The nature of soil reactions that come into play for the different modes of vibration are different, as already discussed in Section 4.4. 6.3
w
Figure 6.2. Lamb's problems for steady-s!ate o~cillating force or ~ulse loa_ding acting at a point (three-dimensional) or along a line (two-dimensional). (a) For verhcalload•ng at the surfa.ce. (b) For horizontal loading at the surface. (c) For vertical loading within the body. (d) For horazon~al loading within the body. (After Lamb, 1904. After Richart, Hall, and Woods, "Vibrations of S?ds and foundations/'© (1970, p. 193. Reprinted by permission of PrenticeRHall, Englewood Chffs, New Jersey.)
tropic, sellf\cinfinite body, whi~~_,is referred to as an elastic half-spac~. Mathematical solutions for computJng the response of a footmg vibratmg m different modes have been obtained by several investigators. Lamb's (1904) study of the response of an elastic half-space excited by a periodic vertical force acting along a vertical axis is the first investigatiOn m this area. This problem, which is also known as "dynamic Boussinesq loading", was analyzed as a two-dimensional wave propagation case. The study was then extended to cover conditions of vibratiOn occasiOned by a honzontal oscillating force acting on the surface and by a vertical or horizontal line load acting at any point within an elastic medium. These cases are Ill~strated m Fig. 6.2. Lamb also demonstrated how a series of vertical penodic forces havmg different frequencies could be combined and replaced by a single force acting on the surface. Lamb's solution covered hoth steady-state .and transient cases for calculation of surface displacement. Lamb also obtamed solutions for the response of a vibrating foundation by considering it as a problem of three-dimensional wave propagation. · Vertical Vibrations of a Footing
The problem of a ~brating rigid, circular footing resting on the_ surface of an elastic half-space was exammed by Reissner (1936, 1937). His solution for the vertical displacement at the center of the footing was based upon the solution obtained by Lamb in 1904. In this problem, the vertical displacement can be expressed by
ELASTIC HALF-SPACE METHOD
(6.1)
The elastic half-space method idealizes the machine foundation as a vibrating mechanical oscillator with a circular base resting on the surface of the ground. The ground is assumed to be an elastic, homogeneous, iso-
in which
216
FOUNDATIONS FOR RECIPROCATING MACHINES ELASTIC HALF-SPACE METHOD
P, =the magnitude of the oscillating force, w =forcing frequency (rad/sec)
3. Parabolic
G =the dynamic shear modulus of the medium r, = radius of the footing f1 and [, = Reissner's displacement functions.
(6.6)
Reissner (1936) observed that the displacement functions t, and /; were v of the medium and the fre~uenc/ of the exc1tmg force. The dimensionless terms were defined by him as follows:
depe~dent on the Poissons' ratio a = wr
,
/p = wro- 2wfro
oVa
v -1.7""" ·'
and
m
'
(6.2)
W
b=-=P r'o Ysro'
217
(6.3)
in which
ao =dimensionless frequency ratio
V, = shear wave velocity m =mass of the footing (including the mass of the machine) b =mass ratio y,. = unit weight of soil p = unit mass density of soil
Figure 6.3 shows the amplitude frequency response of a typical footing for the three types of pressure distribution for b = 5 and v = ~. Parabolic and uniform pressure distributions produced a higher displacement than a rigid base. This shows the importance of pressure distribution below the base of the footing (Richart and Whitman, 1967). Improvement on computed response can be made if the amplitude response is based upon a weighted average vertical displacement (Housner and Castellani, 1969). The effect of change in the Poisson's ratio of the elastic half-space on tbe steady-state vibration response of the footing for the case of a rigid base pressure distribution is shown in Fig. 6.4 (Richart and Whitman, 1967). The valups of the displacementJunctions, [, andf2 , Eq. (6.1) for values of v = 0.25, 0~33 and 0.5 for tbe ihr"ee types of pressure distribution and for a0 = 0 to 1.5 were obtained by Sung (1953b). The amplitude frequency response for the case of constant force excitation and frequency-dependent excitation and for different values of mass ratio b, and for v = 0.25 are shown in Fig. 6.5 (Richart, 1962). Figure 6.5 brings out the effect of mass ratio on the peak response amplitude. A high mass ratio (greater height of footing and smaller contact radius) implies a large amplitude of vibration for
It can be seen from Eq. (6.3) that the mass ratio essentially describes the relatiOn between the mass of the vibrating footing and a certain mass of the elastic half-space. Reissners' solution was extended by Quinlan (1953) and Sung (1953a), both of whom considered the effect of three types of vertical contact pressures bel?w the base occasioned by an oscillating vertical force. The pressure distnbutiOns considered were
1. Rigid base (T
z
=
p
eiwt 0
27TroVr~-r2
az=O
(6.4)
for r>r 0
2. Uniform (6.5) Figure 6.3. Effect of pressure distribution on the theoretical response of vertically vibrating rigid footing (After Richart and Whitman, 1967.)
ELASTIC HALF-SPACE METHOD
Rigid
b
~I~
~
5
1.0
o~------t;------~------~-0.5
1.0
1.5
Figure 6.4. Effect of Poisson's rati th . I (After Richart and Whitman, 1967.)o on eorebca response of a vertically vibrating footing.
Vibration amplitude = Az
b~4o ~4
a given set of conditions. Another significant point revealed when one compares Figs. 6.5a and b with Figs. 2.10a and 2.13a is that the general shapes of the response curves are similar. The curves for the low values of mass ratio b correspond with the curves for the high damping ratios. This implies that the vertical vibrations of a rigid body on an elastic half-space are damped. This damping occurs as a result of the energy being dissipated into the elastic half-space by the elastic waves radiating away from the vibrating body. This loss of energy is referred to as geometrical damping (see Fig. 3.14). The values of the displacement functions, {1 and { 2 , were computed by Sung on the assumption that the pressure distribution remained unchanged with frequency. Bycroft (1956) evaluated the weighted average of dis, placements beneath the footing and established displacement functions for rigid base pressure distribution (Fig. 6.6). For the static case, a0 is equal to zero, and f 2 also equals zero, The static displacement, Z,, due to a vertical load P0 is given by
l
(6.7a)
~T
1//7/.////7/ Po
219
Equation (6.7a) helps to define the value of the equivalent spring constant k, for vertical vibrations as follows:
= constant
.;:I~
0
4Gr 0
(6.7b)
k, = (1- v)
".Ef u
"'•
~
..§
"E ro ~
0.3 . . - - - - . . - - - - - , - - - - - - - , 0.3
c•
.Q ~
c
•E
0.2
~ I
0
< •c
"g_
0.]
0.2
,
0
__..;>.._-----.j]/4
E 0
"
0 1.5
.___.::::..-c-~:-1112 0
114 112
1.5
(,)
. ~ ;•g~!e 6.5. Ampli~ude versus frequency relations for vertical oscillation of a rigid oo .;~g ~n an elast~c halfMspace (v = ! ). (a) For constant amplitude of exciting force exc1 mg orce amphtude dependent on exciting frequency. (After Richart, 1962 .) 218
circular (b) For ·
Frequency ratio, ao
Figure 6.6. Displacement functions for a rigid circular footing vibrating vertically on the surface of an elastic half~space. (After Rycroft, 1956.)
220
FOUNDATIONS FOR RECIPROCATING MACHINES
The basic solutions of Reissner were modified by Hsieh
Hsieh's Analog.
(19~2) for the purpose of obtammg an equation for vertical vibrations simllar to the equation for damped vibrations of a single-degree-of-freedom system. A ng1d, Circular, weightless disc of radius r resting on the surface of an elastic half-space was first considered (Fig. 6.7a). This disc was subjected to a vertical oscillating force
ELASTIC HALF-SPACE METHOD
in which
c.
0
Q = Qoeh<>~
221
Gro ( -[2 = --;;;--
!'1 + !'2
r; , 17':.
)
= -;;0
(
- [2
v Gp !'1 + !'2
)
and
(6.8)
(6.13)
The vertical displacement is given by
(6.9) By differentiating Eq. (6.9) with respect to time, one gets
Thus, both c, and k, · are dependent upon a 0 and v. Next Hsieh also considered a rigid cylindrical footing of total weight W placed on the surface of an elastic half-space and excited by a vertical periodic force P (Fig. 6.7b). The equation of motion of such a system is given by 2
dz wQ eiwt dt = G~ (if; - [,)
W d z g dt 2
( 6.10)
0
This leads to
(6.12)
=
P- Q
(6.14)
By substitut1hg Q from Eq. (6.1'i):into Eq. (6.14), one obtains
f 1 wz- f z dz ~
=
Qow Gr
(f'1 +f')e'w' 2
2
W d z dz P iwt --+ e 2 c -+kz=P= g dt ' ' dt 0
0
or =
(6.15)
Equation (6.15), which is known as Hsieh's Analog, illustrates that vertical vibrations of a footing on an elastic half-space can be represented in terms of an equivalent damped spring-mass model with the difference that both the spring constant and damping are frequency dependent.
~~ u; + t~l 0
(6.11)
Lysmer's Analog. Lysmer and Richart (1966) proposed a simplified mass-spring-dashpot analog known as Lysmer's Analog for calculating the response of a rigid circular footing subject to vertical oscillations. They also defined a new displacement function F as
p
Weightless rigid disk
Q
4
Block mass = m
F= - 1 -v
G, v,
p
B,
+z Q =
dz
Czdt + Ia)
kzz (b)
Figu~e 6:7. Parameters in Hsieh's equations. (After Hsieh, 1962. Published by Thomas Telfo d
Publications.)
.
F 1 + tF2
(6.16)
The components of F are practically independent of v, as shown in Fig. 6.8. Lysmer also defise,d a modified mass ratio as
Q G, v, p
f=
r
1-v
1-v m
= -4- b = -4- - , pro
(6.17)
in which B.= modified mass ratio for vertical vibrations. By using the values of F and B, Lysmer and Richart (1966) developed the response curves shown in Fig. 6.9. The effect of frequency ratio on the variation in damping and spring factors was studied. It was observed that frequency-independent constant
223
ELASTIC HALF-SPACE METHOD
values of these quantities could be used in the frequency ranges of practical interest. The spring constant was taken as equal to its static value and is given by
'
t
4Grb k =-z 1- v
112
.;: I
114
""•
(6.18)
which is same as Eq. ( 6. 7b) and damping could be represented by
0
3 .4r; _ r-:::7"
c
(6.19)
c.= (1- v) v pG
0
~
E 0 u
By using the above values of spring and damping constants and the theory of vibrations (Chapter 2), natural frequency "'"' is determined as w
1.0
1.5
Dimensionless frequency, a0
~ii~~::.~-~~ ~riation of modified displacement function with Poisson's ratio. (After Lysmer and
6
nz
=
'V!7S T:i
(6.20)
The responss;; curves between the magnification factor, M, and a 0 , shown by the dotted lines in Fig. 6.9 wef<;'.''obtained. These are close to the exact solutions obtained with the elastic half-space model. The equation of motion for the Lysmer's Analog may thus be written as
3.4r! mi + ( _ v) 1
4Gr
vPGi + (1 _
~) z
=
P
(6.21)
The damping ratio ~' is obtained as 3r-----~r------,-------,------~ - - Half-space theory
0
"'G'""'-~
(v =
- - - - Simplified analog
""::; II
1 3}
2
Bz
=
or
5
::;,"
(6.22) Resonance occurs only when B, "'0.3, and the following approximate formulas for resonanct< condition for frequency-independent constant force excitation were establr~hed: w 2 Dimensionless frequency, a 0
;igure 6·?· ,Response of a rigid circular footing to a vertical force developed by a constant orce exc1tahon. (After Lysmer and Richart, 1966.)
222
nz
V, yB,-0.36 =-
ro
(6.23a)
Bz
and (6.23b)
225 224
FOUNDATIONS FOR RECIPROCATING MACHINES
The amplitude at operating frequency can be obtained by using theory of vibration [Chapter 2, Eq. (2.44b)]: A =
P,
' k,V (1- r
2 2 )
+ (2/j) 2
P, k,{[1- (wlw,,)']' + (2~,w/w,j}ll
2
(6.23c)
ELASTIC HALF-SPACE METHOD
7-8v
-m -
7-8v
w --,
B x = ~--=---,32(1- v) pr; - 32(1- v) 'YJ 0
(6.25)
The dimensionless frequency factor a ox is equal to wr"yprG. The expressions for the equivalent spring and damping factors are as follows: The equivalent spring (6.26)
For a frequency-dependent exciting force, which is normally the case with forces associated with machine operation, the resonant frequency is given by
(6.24a) The maximum vibration amplitude for frequency dependent exciting force is given by A
=
'
_m_,e c::-::~,;;B~,== m 0.85y B,- 0.18
and the equivalent damping _ 18.4(1- v) 2, f::(J ex7-Sv ravPv The damping ratio ~x is given by ~
(6.24b)
(6.27)
ex
0.2875
CC
~
=-=-X
(6.28)
The equation of the analog for sliding is in which me= unbalanced rotating mass and e =eccentricity _of mass "m " from the axis of rotation. e
The mass in the above analog is the total mass vibrating on the surface of the elastic half-space. The shape of the magnification factor M versus a for
(6.29) By comparison with Equations (2.37) and (2.11),
constant force excitation in Fig. 6.9 shows that the peaks at ;esonancea are
relatively flat, and significant damping is associated with the vertical mode of
(6.30)
vibration.
The curves for the magnification factors M x versus nondimensional frequen-
Sliding Vibrations of a Footing
As pointed out in Section 6.2, the sliding and rocking vibrations of a rigid block foundation are coupled and occur simultaneously, but for simplicity it IS necessary to study the cases of pure sliding and rocking vibration first. The information on natural frequencies of pure rocking and sliding is used to compute response of foundations undergoing simultaneous rocking and shdmg. It Will be shown later that the natural frequency in sliding alone is very close to the lower natural frequency of combined rocking and sliding. Arnold et al. (1955) and Bycroft (1956) presented analytical solutions for horizontal translation of a rigid circular disc resting on the surface of an elastic half-space and excited by a horizontal force,,~px = Ptiiwt. Their results were expressed in terms of the dimensionless frequency ratio a and mass ratio b; the solution was valid for all values of v. In a manne; similar to Lysmer's solution, Hall (1967) developed an analog between the elastic
cy factor aox derived through the elastic half-space solution are compared with the curves for the analog solution in Fig. 6.10 (Hall, 1967). The flat peaks on the curves in Fig. 6.10 indicate that the mode of vibration for horizontal sliding is also associated with significant damping as in the case of vertical vibrations. It may be mentioned here that the expressions for sliding along the y axis are also similar. Rocking Vibrations of a Rigid Circular Footing
The problem of pui'b rocking vibrations was analyzed by Arnold et al. (1955) and Bycroft (1956). They assumed that the vertical pressure below the footing varied according to (6.31)
half-space solution and an equivalent, damped-spring-mass system.
Hall's Analog.
Hall (1967) defined the modified mass ratio for sliding as
in which MY is the exciting moment in the ZX plane that causes rotation
227 FOUNDATIONS FOR RECIPROCATING MACHINES
226
ElASTIC HAlF-SPACE METHOD Mysinwt
Bx
=
r-"\
5 - - Exact solution
---Analog solution
- - Exact sqluf1on ---Analog solution
3
2
:£' ;'l"
"' c 0
"'•u
2
~
'2
00
•
" o.sot__L-~L-~-~-~o~.s~-L-~~-L-_L-~-~~k--~~
"''
1 frequency factor for rocking only of a Figure 6.11. Ma~nification f~ctorlvfersus l(~~ns•o~a~~s 1967. © 1968 The University of New rigid circular footmg on elastic ha -space. er ' Mexico Press.) d'
a~
Figure 6.1 0. Response of a rigid circular footing on an elastic half space for pure sliding. (After Hall, 1967. © 1968 The University of New Mexico Press.)
about tbe Y axis, and cf> is the angle of rotation. Figure 6.11 illustrates the geometry of the problem and sbows a plot of the magnification factor, M 1 vs. dimensionless frequency factor ao.P for different values of the inertia ratio B 1 (analogous to mass ratio in case of translation.) The inertia ratio Bq, is defined as
B = 3( 1 - V) _M_mo = ~ _M-"'m"'-o('--1_-_v-'-) 5 " 8 pr o 8 'Y, 5 -r
g
(6.32)
0
in which Mmo is the mass moment of inertia of the foundation and machine about the axis of rotation (in this case they axis, not shown). It may be seen in Fig. 6.11 that the response curves are characterized by relatively sharp peaks compared to the case of vertical vibrations. Hence smaller damping is associated with the rocking mode of vibration (see Fig. 4.40).
·
· . ring-dashpot model that . . . he Hall (1967) proposed an eqmva1ent mass-sp . could be used to evaluate the response of a rockmg ngtd footmg ond t b y surface of an elastic half-space. His analog for rockmg ts represente
(6.33) in which
kq, =spring constant for rocking cq, =damping constant
.,,
The terms k• and
c• c;n be computed as follows: 8Gr~
kq,
= 3(1- v)
(6.34a)
and Cq,
= (1-
v)(1 + B 4J
(6.34b)
229 228
FOUNDATIONS FOR RECIPROCATING MACHINES
ELASTIC HALF-SPACE METHOD
For critical damping, (6.35) and the damping ratio for rocking l;q, is given by
(; = ,S;_ = 1
0.15
(l+B,p)VB:;
c1"
The undamped natural frequency of rocking Wn
wn
(6.36)
is given by
={If.
(6.37)
mo
The analog solution is shown in Fig. 6.11 by dotted lines along with the elastic half-space solution and is in rather close agreement with it. Torsional Vibrations of a Rigid Circular Footing
0
The problem of torsional vibrations (rotation about the Z axis) of a circular footing resting on the surface of an elastic half-space was analyzed by Reissner (1937) and Reissner and Sagoci (1944). The horizontal displacement in the case of a rigid footing varies linearly from the axis of rotation as shown in Fig. 6.12. The inertia ratio, s., for this case may be defined as (6.38) in which Mm, =polar mass moment of inertia of the footing around the vertical axis of rotation. The analog solution for the case of torsional vibrations may be expressed as follows (Richart et al. 1970):
br:fifiro Gof=~
• . 1 f nc factor at resonance for torsional Figure. 6.12. lnert~ahratioHv~rsusd ~:~:·o~~i~ra;~~~eof ~oils and Foundations,"© 1970, P· vibrations (After RIC art, a an I d crff N Jersey) 215. Repr.inted by permission of Prentice-Hall, Englewoo ' s, ew .
The undamped natural frequency
wn.;
of the torsional vibrations is given by (6.41a)
The amplitude of vibrations A'' is given by
(6.39) in which tf; = angular rotation of the footing around the vertical axis of k~, =equivalent soil spring constant for torsional vibrations, and M,e'w' =horizontal exciting moment acting about the Z axis. The spring constant k 1, and the damping constant c 1, are given by (Richart and Whitman, 1967)
(6.41b)
rotation, c"' =damping constant for torsional vibratit?ns,
k,,
=
16
3
3
Gro
1.6r~y'Gp c1, = 1 + B,,,
( 6.40a) {6.40b)
in which the damping ratio (; 1• is given by I; = " (1
0.5
(6.42)
+ 2B,1.)
effective damping in case of As in the case for rocking vibrations, t he torsional vibrations is small (see Fig. 4.40).
230
FOUNDATIONS FOR RECIPROCATING MACHINES
Coupled Rocking and Sliding Vibrations of a Rigid Circular Footing
Equation (6.43) max then be written as
It has already been. stated (Section 6.2) that either rocking or sliding
Rx
alo~e IS
an tdeal condttton. In actuahty, the motion of a footing excited by a honzontal force or a vertical moment involves both rocking and sliding. Figure 6.13 tllustrates the conditions of a rigid circular footing that rests on the surface of an elastic half-space and is excited by a vertical moment, M,(t) = M,e'w' and a horizontal force Px(t) = Pxe'w' acting at its center of gravity. Let the footing's center of gravity lie on the vertical axis which passes through the center of the circular base, at a height L above the surface of the half-space. The motion of the footing may be expressed in terms of the translation of the center of gravity x and the rotation angle q,. The equation of motion may be obtained by considering the limiting eqmhbnum of the exciting and resisting forces and moments in terms of Newton's Second Law. The horizontal resisting forceR at the base is given X by (6.43) in which
X0
=displacement at the base and is given by
xo
=X-
(6.44)
Lcf>
231
ELASTIC HALF-SPACE METHOD
=
cxi + kxx- Lex¢- Lkxcf>
=
0
(6.45)
Similarly, the moment M R, which represents the moment due to the resistance of the elastic half-space, may be written as ( 6.46) The equation of motion for sliding is (6.47) arid rocking ( 6.48) in which M m us the mass momentp.frinertia of the foundation about an axis that passes through the system's center of gravity and perpendicular to the plane of vibrations. By substituting MR and Rx from Eqs. (6.46) and (6.45), respectively, in Eq. (6.48), one obtains
+ •
}i"
+x
~+Px (b)
I
/~
l_ 0 X
I
I
I
llr-
•
(a)
• 1-1
I
7
I I
I I r
11
'I
I!
!
~Fx 1:
(6.50a)
'N
I
I
~
I
r-
+
I I
J(--
and
0
.~
(6.50b)
l '/
-A'r£0 I
(c)
Figure 6.13. Coupled rocking and sliding vibrations of a rigid circular block on an elastic halfMspace. (After Richart and Whitman, 1967.)
Equations (6.47) and (6.49) demonstrate that coupling of the two motions, i.e. sliding and rocking, takes place because the center of gravity of the footing and the point at which the horizontal reactive force R x of the elastic half-space is applied are not the same. If L equals zero, there is no coupling effect, and sliding and rocking are independent. Particular solutions of Eqs. (6.47) aud (6.49) may be obtained by substituting
in wbich A and B are arbitrary constants. When a footing rests on an elastic half-space, the values of both the spring constant and damping coefficients are frequency dependent and must be calculated at any given frequency before the above equations can be
solved. However, if the spring constant and damping are assumed to have a frequency independent constant value as in the case of analog solutions for
232
FOUNDATIONS FOR RECIPROCATING MACHINES
sliding a~d rocking, Eqs. ( 6.47) and ( 6.49) can be easily solved. The natural freq~encws of coupled rocking and sliding are obtained by making the forcmg functiOns Px · e'w' and M/w' in Eqs. (6.47) and Eq. (6.49) e ual to zero. Th1s leads to: q ( 6.5la) and
ELASTIC HALF-SPACE METHOD
233
w, 1 and w, 2 are the two natural frequencies of the soil-foundation system undergoing vibrations due to combined rocking and sliding. The rocking and sliding may be in-phase or out-of-phase depending on the value of operating frequency w and the two natural frequencies w, 1 and w, 2 • This point is discussed in detail in Section 6.7. Damped amplitudes of rocking and sliding occasioned by an exciting moment MY can be cbtained as follows:
A
_ _M Y [(w2nx )2 + (2
x
(6.55a)
(6.51b) By substituting Eqs. (6.50a,b) into Eqs. (6.51a,b) and rearranging the terms, the frequency equat10n IS obtamed as given below (Prakash and Puri 1980, 1981b) ' 4 [ wnd
_
w2
2
w,d (
n>
+4[~xwnxwnd 'Y
+ (I) nx 2
4< <
_
Sx S Wnx (t)n>)
y
y
(w2 -w2 )+ nd
rt
2
2
W nx W n] + ~~""""""
(6.55b) The value of A(w 2 ) is obtained from Eq. (6.56)
2
y
~q,W,q,W,d
( 2-
y
(t)I!X
2 ]2-0
(l)nd)
(6.52) in which
Damped amplitudes of rocking and sliding occasioned by a horizontal force Px are given by Eq. 6.57(a,b) ( 6.53)
w,d = damped natural frequency in coupled rocking and sliding. ~x = damping ratio for sliding vibrations
1;1
=
px Ax=-M m m
damping ratio for rocking vibrations
If ~x
= {;1 = 0, that is, when there is no damping in the system then Eq. (6.52) reduces to ' 2
4-
w,
Wnx
2
w,
(
2
+ Wnq,) + y
2
2
W,xwn1>
y
=
0
(6.54a)
in which w, is the undamped natural frequency o"r the system. Because the effect of damping on natural frequency is small one may calculate the undamped natural frequency for coupled rocking and sliding by usmg Eq. (6.54a). Solving Eq. (6.54a) as a quadratic in w~, we get w2
nJ,2
1
= _
2y
[w 2 + (J)n(l)Nx] /IX
and
{J)/JX
(6.54b)
(6.57b) In case the footing is subjected to the action of a moment and a horizontal force, the resulting amplitudes of sliding and rocking may be obtained by adding the corresponding solutions from Eqs. (6.55) and (6.57). Rigorous solutions for Eqs. (6.47) and (6.49) can be obtained for both elastic half-space and Halls' analog by using numerical techniques on a high-speed digital computer. .For numerical solutions, frequency-dependent stiffness and damping can also be considered.
234
FOUNDATIONS FOR RECIPROCATING MACHINES
235
EFFECT OF FOOTING SHAPE ON VIBRATORY RESPONSE
So far in our discussion, we assumed that the footing has a circular contact area. Mostly the footings are either rectangular or square. We will now consider the effect of footing shape of its dynamic response. 0.16
6.5
EFFECT OF FOOTING SHAPE ON VIBRATORY RESPONSE
0.12
~ I
The natural frequency of a footing is influenced by its shape. Elastic half-space theory was developed for an oscillator with a circular contact area. Its applicability to footings of rectangular and square base areas has also been investigated. The problem involving vertical oscillations of a rigid, rectangular footing on the surface of an elastic half-space was analyzed by Sung (1953b), who used the earlier solution of Lamb (1904). Kobori (1962) and Thomson and Kobori (1963) obtained the displacement functions, [ 1 and / 2 for the displacement at the center of a uniformly loaded rectangular surface area. Elorduy et al. (1967) obtained solutions (in terms of the displacement functions f 1 and / 2 ) for vertically vibrating rectangular (alb= 2) and square (alb= 1) footings resting on the surface of elastic half-space for a typical case of v = ! and compared them with the solutions of Sung (1953a) and Bycroft (1956), who based their calculations on an equivalent circular area. Their solutions are shown in Fig. 6.14 for the cases of square and rectangular footings, respectively. Because these displacement functions are practically the same, it is acceptable to use solutions of a rigid circular base having an area equivalent to a given square or rectangular contact area for alb up to 2.0 for approximate response calculations. Similarly, for rocking or torsional vibrations of rectangular or square footings, one may compute an equivalent radius of a circular footing so that the moment of inertia of the given footing about the axis of rotation is the same as that of an equivalent circular footing about the same axis. Thus, the equivalent radius ro, may be calculated as follows: For translation along the Z, X, or Y axis,
5
..........
0.08
""
-(2
- - - Elorduy et ~1:-~igid rectangular}-._ 1 - - - - Bycroft-R1g1d c1rcular v - 4 --Sung-Rigid circular
0.24 ~------,-------,-------,
'0
=
ba') ( 37T
'
(6.58b)
'\;
:-"\
~ I
0
'Y-;:~-/
0.12
', "
<
'-
-fz
'
a
- ~ 2 b
' 1.0
0.5
+ Rectangle
'- '-' '
0
~12
ea/2 o/2
b/2
~
'\
-~---~
1.5
(b)
in which b = width of the foundation (parallel to the axis of rotation for rocking)
=r =(ab(a'61T+ b') )II'= (21")11' ol)t
" ~"(I"
11
For torsional vibrations about Z axis,
o
·''>
Fi ure 6 .14. Displacement functions for vertical. vibration. of rigid squar~ an~ rectangular fo!tings. (After Richart, Halk and Woods, 11 Vibrat1ons of Sm~s and Foundations, © 1970, P· 212. Reprinted by permission'-of Prentice-Hall, Englewood Chffs, New Jersey.)
For rocking about Y or X axis,
r
1.5
1.0
0.5
( 6.58a)
roo/= ro
..............................
1T
and
(6.58c)
a = length of the foundation (perpendicular to the axis of rotation for rocking).
236
FOUNDATIONS FOR RECIPROCATING MACHINES
For footings with length to width ratios greater than 6, an ideal twodimensional condition can be assumed and the footings treated as strip footings which can be analyzed by applying Quinlan's (1953) method. The above discussion implies that a footing of any shape whether it is a circle or a rectangle, will respond similarly as long as the areas of the footings are the same. A footing having an area of 8.0 m 2 will have an equivalent radius of 1.6 m for translational oscillations regardless of its shape, which may be a circle with a diameter of 3.2 m or a rectangle with length of width ratio of 4.0. However, the actual response of these two footings may not be the same. It has been found that two footings of different shapes will not behave identically, even though the equivalent radius based upon equal areas for translational modes are the same (Chae, 1969). In an accurate analysis, footing shape must be taken into account. Based upon his experimental observations (Chae, 1969) suggested that the concept of equivalent circular areas may be used to predict natural frequencies but that the perimeter characteristics should be taken into account for reasonable predictions of the amplitudes. The problem of vertical vibrations of rectangular footings has also been studied by Dasgupta and Rao (1978), who used a three-dimensional finite element model and three different vertical pressure distributions at the footing soil interface. However, their model is too complex for the design of ordinary machine foundations. Besides, no comparison with actual observations has been reported. It is a common practice to transform area of any shape to an equivalent circle of same area (for translational modes) or equivalent moment of inertia (for rocking or torsional modes) (Richart and Whitman, 1967; Whitman and Richart, 1967). Dobry and Gazetas (1986) and Dobry et a!. (1986) have suggested that this procedure of using the concept of equivalent radius has limitations and the foundation shape defined by the aspect ratio a/ b has a significant influence on dynamic stiffness and damping values, especially in cases of long foundations.
6.6
Vertical Vibrations
Bycroft (1956) considered the problem of a vertically vibrating rigid circular footing resting on an elastic layer. He. assum~d a ngtd~base-type distribution of vertical pressure below the footmg (this assumption IS not strictly correct) and computed values of average static_ displacement for different values of layer thickness ratio Hlr 0 in which HIS the th1ckness of the elastic layer as shown in Fig. 6.15. The values of the ratiO of average static displacement (Z,li for the layered case, to the value of a stahc displacement (Z,)~ when H-'>oo (elastic half-space) and the Hlro ratiO are plotted in Fig. 6.15. The values of the ratio of a spnng coeffiCient (k,), for the layered case, to the value of (k,)" for the elastic half-space, versus the H f r value are also plotted in the same figure. Th1s figure shows clearly the stiff~ning effect that an underlying rigid layer has on the vertlcal m~twn of a footing, which is obvious from the reduced statlc displacement and mc~eased spring coefficient. For large values of H/r 0 (>4), the hehavwr approx1mates that of a footing on an elastic half-space. For lower values of H fro ( <1), the motion is signijicantly affected by the stiffness of the_ underlymg ng1d layer. Bycroft (1956fconsidered the case of.a weightless ng1d Clfcular d1sc (B, :" 0) resting on the surface of a half-space, and Warburton (1957) _obtamed solutions for the case of B, > 0. Both of them noted that amphtudes of motion become infinite at resonance for the case of B 2 equal to zero. Th1s occurs because of the resonance of the layer itself , which acts as a column
VIBRATIONS OF A RIGID CIRCULAR FOOTING SUPPORTED BY AN ELASTIC LAYER \
In elastic half-space theory it is assumed that the medium 'is homogeneous. Actual soil deposits are layered and in certairi" cases, the response of a footing may be affected by the stiffness or rigidity of underlying layers of soil. The problem of a vibrating footing on a layered medium has received the attention of several investigators (Reissner, 1937; Warburton, 1957; Arnold eta!., 1955; Bycroft, 1956). It is assumed that the footing rests on the surface of an isotropic, homogeneous, elastic material of thickness H that extends to infinity horizontally. This layer rests on a semi-infinite body of infinite rigidity.
237
VIBRATIONS OF A RIGID CIRCULAR FOOTING
0,2
(kz)I,;,,I~
Y...... __(k,)oo
(kz}o;, =
'
4Gr0 (I - ,)
-----------OL__L__j2___L__4L_~--~6~-L--~8~_L~~~--~~~ H
'o' Figure 6.15.
Static displacement and spring constant for vertical loading of a rigid circular
footing on an elastic layer (After Rycroft, 1956.)
238
FOUNDATIONS FOR RECIPROCATING MACHINES
of elastic material fixed at the base, free at the top, and restrained against lateral deformations on the side. In this condition, it behaves like a rod of elastic material. The conditions involving the vibrations of such a column of soil have already been discussed in Chapter 3 (Section 3.1). Vibrations in higher modes are possible for such a case. For a footing which has a certain weight in which B, > 0, the amplitudes are finite at resonance, but are amplified by the underlying hard layer. The increase in amplitude at resonance occurs because the underlying hard layer obstructs the transmission of the wave energy away from the vibrating footing and reflects back a part of this energy into the layer. This results in reduced geometrical damping. The energy is finally dissipated by transmission in a horizontal direction. Kuhlemeyer (1969) discussed the transmission of wave energy into a layered medium. An estimate of the amplitude magnification at resonance for a vertically vibrating footing on anelastic layer is given in Table 6.1. The magnification factor, ML in Table 6.1 is defined as ( 6.59) in which (Z,) 1 =dynamic amplitude of the elastic layer and (Z,)oo =static deflection (elastic half-space). The term (Z,)oo is given by (Z) = (1- v)P, s oo 4Gro
239
VIBRATIONS Of A RIGID CIRCULAR FOOTING
It may be noted here tbat a hard rock underlying a relatively thin layer of elastic soil may cause large amplification of the vertical amplitudes because of the reflection of energy back into the soil. Special care must be taken m the design of a machine foundation for such a case.
Torsional Vibrations
The problem of torsional vibrations of a rigid circular footing on an elastic layer overlying a rigid layer was investigated by Re1ssner (1937), Arnold et al. (1955) and Bycroft (1956). The results provided by Bycroft (1956) on B, vs. a0 for different values of H/r are shown in Fig. 6.16. Based upon model tests as well as theoretiCal inve~tigations, Arnold et al. (1955) observed that as the. elastic layer becomes thinner in comparison with the radius of the footmg, effective damping is decreased and is less than that for a torsionally vibrating footing on an elastic half-space. Bycroft (1956) demonstrated that as the H/ro becomes sm}tller, the natural fr$quency of the torsional vibrations approaches the natural frequency of ''rod of radius r o and length H v1bratmg in torsion as a column fixed at the base and free at the top. Cases of torsional vibration of such columns have been discussed in Chapter 3 (Section 3.1).
(6.60)
By using the magnification factors given in Table 6.1 for an appropriate value of b, the amplitude at resonance can be computed by using Eqs. (6.59) and (6.60).
Table 6.1. Magnification Factors for Vertical Vibration of Rigid Circular Footing Supported by an Elastic Layer (v
= l) ML for
H ro
Rigid
b=O
1 2 3
00
4
00
00
1
00 00
b=5
b = 10
b =20
b =30
5.8 8.0 4.7 3.4
11.4 16.1 9.5 5.9
20.5 30.6 23:7 15.6
28.9 40.8 36.0 27.9
1.21
1.60
2.22
2.72
Source: Richart, Hall, and Woods, "Vibrations of Soils and
Foundations," © 1970, p. 234. Reprinted by permission of Prentice-Hall, Englewood Cliffs, New Jersey.
""•
figure 6.16. lntertia ratio vs. dimensionless frequency at resonance for torsional vibrations of a rigid circular footing on an elastic layer. (After Rycroft, 1956.)
240
6.7
FOUNDATIONS FOR RECIPROCATING MACHINES
241
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
t
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
Pz sin wt
P, sin wt
The linear elastic weightless spring method is sometimes used for analysis of machine foundations and utilizes the concept that the displacement of a loaded foundation resting on the surface of a soil can be determined by simulating the soil with a set of independent linear elastic springs that can produce equivalent reactive forces to the displacements developed. This concept commonly known as the elastic subgrade reaction theory has been described by Hayashi (1921), Heteyni (1946), and Terzaghi (1943, 1955). The idea of using elastic springs was extensively developed by Barkan (1962) for the purpose of predicting the dynamic response of machine foundations. His concept is based upon the following simplifying assumptions:
m
m /
Ia I
(b)
Pz sin wt
m
1. The foundation block is infinitely rigid compared with the soil.
2. 3. 4. 5.
The soil underlying the foundation is weightless. The soil can be simulated by linear elastic springs. Damping in the soil beneath a foundation may be neglected. The foundation is resting on the surface of the soil.
These assumptions make it possible to represent the foundation-soil system with an equivalent mass-spring system in which the mass represents the foundation and machine, and the spring represents the elasticity of the soil. The methods for determining the elastic soil springs for different modes of vibration have already been discussed in Chapter 4 (Section 4.4). Vertical Vibrations
For the purpose of computation, consider a situation in which a foundation block rests on the surface of the ground and is excited by the vertical unbalanced force P,(t) generated by the operation of a machine (Fig. 6.17a). Let the unbalanced force be represented by P,(t) = P, sin wt and the idealized equivalent spring-mass system by Fig. 6.17b. If the center of gravity of the foundation and machine and the centroid of the base area of the foundation in contact with the soil lie on f vertical line that coincides with the line of action of the exciting force P,, then the foundation will vibrate vertically only. Because the foundation block is assumed to be rigid, its displacement may be defined by the displacement of its center of gravity, and the vibrating mass may therefore be considered to be a concentrated
point mass. This assumption justifies the model shown in Fig. 6.17c. The problem of a vertically vibrating foundation is thus reduced to the analysis
lei
figure 6.17. Vertical vibrations of a rigid block: (a) actual case, (b) soil replaced by equivalent spring kz, and (c) equivalent model.
of a vibrating centered mass that rests on a spring, and the theory. of vibrations for an undamped single-degree-of-freedom system can be applied (Section 2.3). The equation of motion for the system in forced vtbrat10n ts therefore written as (neglecting damping m Sect10n 2.5) ( 6.61) in which m = mass of the foundation and the machine z = vertical displacement of the foundation with respect to equilibrium
.
position;
~::
k =equivalent spring constant of the soil . for vertical vibrations; w =frequency of operation of the machme. It is shown in Chapter 4 (Section 4.4) that the value of k, can be determined from
k,
=
C,A
( 4.27)
242
FOUNDATIONS FOR RECIPROCATING MACHINES
243
LINEAR ElASTIC WEIGHTlESS SPRING METHOD
in which
Px sin wt
Px sin wt
A = area of contact of the foundation with the soil. Equation (6.61) may thus be written as (6.62) Therefore, the natural frequency of a vertically vibrating system is given by wnz =
[
c A]'/2 ~
rad/sec
(6.63a)
or /,
nz
1 [cmA]'
=-
27T
-"-
lei 1 '
Hz
Figure 6.18.
Block foundation that slides only, and its equivalent model.
(6.63b)
in which Therefore, Eq. (6.65a) may be written as w," =The circular natural frequency (undamped) of the soil foundation
mi +
system in vertical vibration (rad/sec); f"' =Natural frequency of vertical vibrations (Hz)
c·';tf =
PX sin wt
;
(6.65b)
The frequency of sliding vibrations of the system is The amplitude of the vertical vibrations A, is given by A = _P_,'-,;Csi::::n_w~t~ , m( Wnz 2 _ z) W
(CA) m
wnx
=
/,
=-
112
rad/sec
-'-
(6.66a)
(6.64a)
or the maximum amplitude of motion is given hy
or (6.64b)
1 (C,A) 27T m
nx
--
112
Hz
(6.66h)
in which =The circular natural frequency of the sliding vibrations in radians/ sec and =The natural frequency of the sliding vibrations in cycles/sec or Hz. /,nx wnx
Sliding Vibrations
Consider a horizontal unbalanced force, Px(t) = Px sin wt, to act on a block foundation as illustrated in Fig. 6.18. The vibrations of the foundation in this case are analogous to vertical vibrations and may be expressed in terms of Eq. (6.65a):
The amplitude of the sliding vibrations is given by A
(6.65a) in which
x =sliding displacement of the foundation, from its equilibrium position; and A =the base area of the foundation; and
kx =the equivalent spring constant of the soil in sliding; and is given by (4.28b)
x
=
,
px
z
m(w nx - w )
(6.67)
The expressions for frequency and amplitude in sliding are thus similar to the expressions for vertical vibrations. Rocking Vibrations
Consider now only the rocking vibrations induced in a fo~ndati?n bl~ck · · moment M Y (t) • Here , M y (t) = M y sm wt m whtch by an externally excttmg
244
FOUNDATIONS FOR RECIPROCATING MACHINES
245
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
M, denotes the moment acting in the XZ plane (Fig. 6.19a). The footing is symmetrical about the Y axis, and the center of mass of the foundation and the machine and the centroid of the base area lie on a vertical line and in the plane of the moment. The displaced position of the foundation is shown in Fig. 6.19a, and its rotation is cf>. The equation of motion may be obtained by applying Newton's second law of motion as explained below.
Because the displacement angle cf> is small, tan cf> = cf> in radians and the moment
1. Moment M; occasioned by the inertia of the foundation is given by
L = the distance between the center of gravity and the axis of rotation, and W =the weight of the foundation.
M=-M ;;.. 1 mo
(6.68)
in which Mmo is the moment of inertia of the mass of the foundation and machine with respect to the axis of rotation. 2. Moment Mw occasioned by the displaced position of the center of gravity of the foundation is given by
Mw = WL tan cf>
(6.69)
(6.70) in which
3. Moment MR occasioned by the soil reaction. Consider an element dA of the foundation area in contact with the soil and located at a distance l from the axis of rotation (Fig. 6.19b). The soil reaction depends upon the displacement at the point under consideration and varies from zero at the center of rotation to a maximum value at the edge of the fo9ting. The soil reaction over the elementary area dA is given by r,,,/" dR =
(6.71)
c.zq, dA
in which C =the coefficient of elastic nonuniform compression. The reactive " moment dM R occasioned by the soil reaction dR is given by Initial position 2
dMR = ldR = -C,I c/> dA Displaced position
If the foundation does not lose contact with the soil, then the soil reaction will be as shown in Fig. 6.19b, and the total reactive moment MR against the foundation area in contact with the soil is given by MR =-
(a)
I"
r c,/cf>
dA =- c.q,
I I'
dA = - c,)cf>
(6.72)
in which I is the moment of inertia of the foundation area in contact with the soil with respect to the axis of rotation. "~(
4. The exciting moment My(t) = M, sin wt. The equation of motion may therefore be written as
- Mm,;j, (b)
Figure 6.19. Rocking vibrations of a rigid block: (a) Block under excitation due to an applied moment (b) Soil reaction below the base.
+ WLcf>- C"lcf> + M, sin wt =
0
or
(6.73)
246
FOUNDATIONS FOR RECIPROCATING MACHINES
The natural frequency of the system is given by wn> = (
C I- W£)112 "'M rad/sec mo
(6.74a)
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
247
in which h is the height of the point above the base where amplitude is to be determined. It may be further noted that for a footing that rocks about Y axis and has dimensions, a and b along the X and Y axis, respectively, I is given by
or
ba 3 12
1=_ 1 (C1 I-WL)';' fn>- Z1r M Hz mo
(6.74b)
in which w"'" =circular natural frequency of rocking vibrations (rad/sec) fn> =natural frequency (Hz)
In Eq. (6.74a,b), the value of WL is negligible compared to the value of c I and may be neglected. This leads to "' w
n>
=(C•I)II' M mo
(6.74c)
By comparing Eq. (6.73) with Eq. (6.61), and neglecting WL one obtains k1 =
c1 I
(6.79)
The response of the footing to rocking is thus affected by the dimension of the footing perpendicular to the axis of vibration, and this principle may be used to an advantage in proportioning the foundations undergoing rocking vibrations. Rocking vibrations occur mostly in machines that are mounted on high pedestals and have unbalanced horizontal forces and exciting moments. Torsional (Yawing) Vibrations A foundation is excited in torsional vibration when it is acted upon by a horizontal moment, M,(t) = M,.,.~in wt, around the vertical axis that passes through the center of gravity of"ihe foundation, and the position of the foundation at any time may be defined in terms of the angle of rotation
(6.75)
in which
z
k > = Equivalent spring for rocking vibrations.
The amplitude of rocking vibrations A A
= >
1
Mz sin wt
is given by
MY Mmo(w~>- w 2 )
(6.76)
Th~ effect of ro~king is to increase the amplitudes of the vertical and t" honzontal vtbratmns. The maximum amplitude of vertical castoned by rocking is given by mo !On oc-
X
y
(a) lsometeric view
(6.77) in which a. is the dimension of the footing perpendicular to the axis of rotatmn. Stmtbriy, the contribution of rocking, towards the horizontal amphtudes ts given by A x)
(b) Plan
Figure 6.20. Torsional vibrations of rigid block:
(6.78)
(a)
Development of nonuniform shear below the base.
Block subjected to horizontal moment. (b)
248
FOUNDATIONS FOR RECIPROCATING MACHINES
the soil will thus be a nonuniform shear, which may be defined in terms of a coefficient of elastic nonuniform shear C1,. As for rocking vibrations, the equation of motion for torsional vibrations may be expressed as (6.80) in which M m< = mass moment of inertia of the machine and foundation about the
vertical axis of rotation (polar mass moment of inertia) and J" =the polar moment of inertia of the foundation's base area. The natural frequency of the torsional vibrations is given by (6.81a)
249
liNEAR ELASTIC WEIGHTLESS SPRING METHOD
assumed that the center of gravity of the machine and foundation and centroid of the foundation base area are located on a vertical axis. Figure 6.21 shows a foundation that is excited by the following forces and moments , referred to the combined center of gravity of the foundation and the macQ.ine: 1. Vertical force, P"(t) = P" sin wt, 2. Horizontal force, PJt) = P, sin wt, and 3. Moment, My(t) =MY sin wt If the origin of coordinates is located at the center of gravity, 0, the following displacements of the foundation need to be considered (Fig. 6.21): 1. Displacement in the vertical direction z, 2. Displacement in the horizontal direction x 0 at the base and 3. Rotation of the base >
or
!,,<
=
1 (
-2 71"
cM"' J' ) 112 Hz m<
(6.81b)
The equatio~~ of motion referrecf't6 the center of gravity may be written by applying d'Alemberts principle as follows:
By comparing Eq. (6.80) with the equation of motion of a vertically vibrating footing [Eq. (6.61)], one finds that the spring constant for torsional vibrations (non-uniform shear conditions at the base) is given by
-mi + Z,=O
(6.85)
-mi+X,=O
(6.86)
-Mm'¥ -"+M=O 1
(6.87)
(6.82) in which
The amplitude of torsional vibrations, A • is given by
A,,= M
M" ( 2
mz WmJ!- W
2)
( 6.83)
and the horizontal displacement A;, occasioned by torsion by Initial position
Ah
=
rA,1,
(6.84)
Displaced position
in which r = horizontal distance of the point on the foundation from the axis of
yawing
_l__ •
Vibrations Accompanied by Simultaneous Rocking, Sliding, and Vertical Displacement In ?'ost situa~ions, a machine foundation will simultaneously slide, rock, and VIbrate vertically under the action of corresponding exciting forces. It is
Figure 6.21.
Block subjected to the action of simultaneous vertical P"(t), horizontal P,..(t)
forces and moment Mr(t).
250
FOUNDATIONS FOR RECIPROCATING MACHINES
251
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
z
(6.93)
X,= pr?jection of all external forces acting on the foundation, on the X
6. Moment M 2 occasioned by the horizontal soil reaction x 1 is given by
Z, =projection of all external forces acting on the foundation on the axis ' aXIS
M, = sum of all external moments acting on the foundation and lying on the XZ plane, and M m = mass moment of inertia of the machine and foundation about an axis passing through combined center of gravity and perpendicular to the plane of vibrations.
M 2 = C,AL(x
~
L>)
( 6. 94)
7. Moment Mn occasioned by the soil resistance [as in Eq. (6.72)] ( 6. 95)
By substituting the value of the forces and moments in Eqs. (6.85) to (6.87), one obtains
At any time t, the following forces will act on the foundation: l. Weight W of the foundation and machine. Projection on the X axis . equals zero and on the Z axis equals Z 1 = ~ W.
mi + C.Az = P,(t)
z,
mx + C,Ax ~ C,AL> =
2. Soil reaction occasioned by settlement of the foundation under the action of tbe static weight W:
( 6.88)
in which
z" =elastic settlement caused by the weight of the foundation and machine
A = the area of the foundation
3. Soil reaction Z 3 at any time t occasioned by the displacement z of the foundatwn measured from the equilibrium position is given by ( 6.89) 4. Horizontal soil reaction X 1 at the base is
(6.96)
Px(t)
(6.97)
Mm¢ ~ c,ALx + (C•I ~ WL + C,AL )> = My(t)
( 6.98)
2
·;!c·'·:~
Equation (6.96) contains only the terms of z, and in no way depends upon Eqs. (6.97) and (6.98). Hence it follows that the vertical vibrations of the foundation occur independently of any other motion. Equations (6.97) and (6.98) contain both x and q, and are interdependent. Therefore, sliding and rocking are coupled modes. Because the vertical vibrations of a foundation occur independently of any other vibrations, the treatment given earlier in this article for vertical vibrations will hold in the present case also. A solution for simultaneous rocking and sliding vibrations will now be obtained. Natural Frequencies of Coupled Rocking and Sliding. The system that is considered here is a two-degree-of-freedom system. The solutions for natural frequencies are obtained by considering the free vibrations of the system, and, therefore, the forcing functions in Equations (6.97) and (6.98), may be replaced by zero. From this, one obtains
(6.90) in which
X0
( 6.99)
is given by
and X 0 =X~
(6.91)
L>
in which x =horizontal displacement of the center of gravity. By substituting this value of X 0 in Eq. 6.90 one obtains X,=~ C,A(x ~
L¢)
the foundatwn-machme-system is given by
Mm¢ ~ C,ALx + (C4,I ~ WL + C,AL 2 )>
=
0
(6.100)
Particular solutions of these equations may be assumed to be
(6.92)
5. Moment M 1 occasioned by the displacement of the center of gravity of
'ii'J:.
X sin (w t +a)
(6.101)
(6.102)
X=
0
and
252
FOUNDATIONS FOR RECIPROCATING MACHINES
!n ';hich Xd: <1> and " are arbitrary constants whose values depend upon the mttta1 con thons of motion. . Bdy substituting Eqs. (6.101) and (6.102) into eqs. {6.99) and (6.100) and d 1v1 mg by sm (w, t + "), one obtains 2
-mw,X + C,AX- C AL
=
=
W
w'[( C•/-M WL) + C,A(MmM+ mL ""
253 2
4 _
m
m
)
m
C,A ( C.;IMm
+ m
WL)J
=
0
(6.109) By definition, the quantity (Mm + mL 2 ) is the mass moment of inertia of the foundation and machine about an axis that passes through the centroid of the base contact area and is perpendicular to the plane of vibrations. This is denoted by Mmo. Thus,
0
or
X(C,A- mw!)- C,AL
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
0
(6.103)
and
(6.110a) Further, by letting
Mm Mmo
-- =
or
y
where 1 > y > 0
(6.110b)
Equation (6.109) may be rewritten as 2
-C,ALX+
(6.104)
(6.111)
From Eq. (6.103),
X=
C,AL
However, (6.105)
By substituting the value of X from Eq. (6.105) into E ( obtains q. 6.104), ooe
<1>[-c;A'L'+(C.;l-WL+CAL'-M w n2 )(CAr m 7
C,A
C.; I- WL
')]-O -
Mmo
(6.106) 2
4
Th~ term w,: which represents the natural frequency in combined sliding an rockmg, IS the only unknown in Eq. (6.107), which can now be solved ·· · EquatiOn (6.107) may be rewritten as follows: ' z ' +C,A(C,I-WL)-CAM w' -CAL' - C'A'L' , +C,AL 2 T T mwll -(C.J-WL)mw 2 +Mm mw'=O (6.108) /11
'+'
"
/11
2
(6.74a)
= wn
By making these substitutions, Eq. (6.111) may be written as wn-
(6.107)
(6.66a)
and
mwn
For a nhontrivial solution, <1> cannot be zero. Hence the expression within the parent eses must be zero. This leads to
2
----;;;:-- = W nx
(wnx
+ 'Y
2
wll) wn2 +
2
2
wnxwn 'Y
=0
( 6.112)
Equation (6.112) is known as the frequency equation for combined rocking and sliding and is the same as Eq. (6.54a). This equation has two positive roots, w 111 and W 112 , which correspond to two natural frequencies of the system, w, 1 and w,,,''and have the following inter-relationship with the limiting natural frequencies, wnx and W q,. The smaller of the two natural frequencies or lower natural frequency, w112 , is smaller than the smallest of the two limiting frequencies,. and the larger natural frequency is always larger than w,, and w, •. The roots of Eq. (6.112) are: 11
11
By dividing by mMm and rearranging, one obtains
w' nl.2
=
~2 [(w~x +'Y w~•) ±
(6.113)
254
FOUNDATIONS FOR RECIPROCATING MACHINES
255
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
(6.120)
Equation ( 6.113) may be rewritten as or 2
2 CAL + ClWL- Mmw A " ,, A X~ C"AL •
(6.54b) Also
(6.121a)
By substituting for A, from above in Eq. (6.119),
2 2) (C"AL 2 +C 4,J-WL-M"'w )(C"A-mw A -CALA ~P .p r x C"AL
(6.114) 2 2
w,l
X
2
W
11
w,z
2
q, W 11 x
y
(6.115)
and
which gives A,~~
2
2
Wnl- W,z
1 [( W" 2 2 )2 + Wnx y
=-
4 YWnWnx 2 2 )]1/2
-
C"AL (6.116)
2
M [{;,A(C,I- WL) _ w".[TC"A(mL + Mm) m m mM m {. mMm
Amplitudes of Vibration
----::---:c_.::.C~,A:.::L::..__ _ _--: P,
Having determined the natural frequencies of the system, one may now compute the amplitudes of vibration for the following three cases:
Case I. If only the horizontal force Px sin wt is acting, Eqs. (6.97) and (6.98) may be rewritten as follows: mi + C"Ax - C"AL>
~
Px sin wt
2
m Mm [
2
2
2 + w' wnxwn,P- "'- ( wnx n
y
) + w' J
(6.121b)
y
By substituting from Eqs. (6.114) and (6.115) into Eq. (6.12lb), we get
C"AL A > ~ mMm[wnlw,z-w 2 2 2( 2 + Wnl
(6.117)
2 )
w,2
and
+ w 'l
p X
(6.121c)
Mm¢ + >(C"AL + C4J- WL)- C"ALx ~ 0 2
(6.118) Let
Assume that the particular solution to these equations are
(6.122)
x=Axsinwt > ~
Thus,
A 1 sin wt
in which Ax and Aq, are the maximum sliding ·and rocking amplitudes, respectively. By substituting these solutions into the above equations, one obtains
(6.119) and
_ C"AL P A,- t.(w2) x
(6.123)
By substituting for A in Eq. (6.121a), we get (6.124)
256
FOUNDATIONS FOR RECIPROCATING MACHINES
Case II. If only moment MY sin wt is acting. then Eqs. (6.97) and (6.98) may be rewntten as
nVi + C~Ax-
C~AL = 0
(6.125)
and
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
257
forms of vibrations correspond to the frequencies W111 and W112 of the foundation. The vibrations are characterized by a certain relationship between the amplitudes Ax and A"', which depends upon the foundation's size and soil properties but not on the initial conditions of the foundation's movement. Assume that the foundation is subjected only to the exciting moment MY. From Eqs. (6.127) and (6.128), one obtains
(6.126)
(6.131)
By assumi?g solutions as for Eqs. (6.117) and (6.118), it can be sho h the fol!owmg expressions hold: wn t at A _ C~AL x-
b.(w') MY
(6.127)
and
in which the radius vector p is the ratio of amplitudes in sliding and rocking, respectively. It can be seen from Eq. (6.131) that when w is very small, p "' L and the foundation rotates about an axis that passes through the -centroid of the base contact area, and sliding in absent. As w increases up to W112 , (w!x2 ) is greater than zero, and p is therefore greater than L, and Ax and Aq, have the same sign. It means that during vibrations at frequencies w < w112 , when the center of gravity is displaced >m· a result of sliding.Jn,. the positive direction of the x axis, the rotation of the foundation is also •positive and the sliding and rocking are in phase. This form of vibration is shown in Fig. 6.22a. In such a case, the foundation will undergo rocking vibrations with respect to a point at a distance p from the center of gravity of the foundation. The value of p is given by the absolute value of expression after substituting "'nz for win Eq. (6.131). It can be seen that when w = wnx, p ~co and the foundation experiences only sliding vibrations. When the frequencies w > wnx, p becomes negative, and the axis of rotation shifts above the center of gravity. The sliding and rocking then occur out of phase by 180". This form of vibration is shown in Fig. 6.22b. .
w!
( 6.128) Case III. If both the unbalanced force Px and moment M are actin , the amplitudes of motwn are determined as follows: Y g
and A·= (C~AL)Px + (C~A- mw 2 )Mr q, b.(w')
(6.129b)
The total amplitude of the vertical and horizontal vibrations may be computed by using Eqs. ( 6. 77) and (6. 78), respectively. Tbus,
'
' I
(6.130a)
I
r---
and
/ /
/
_.--..-"\
\
\
\ \
_,
\
(6.130b) in which h = height of the top of the foundation above the combined center of gravity.
p
L
Form of Vibrations Associated with Combined Rocking and Sliding In the ca~e of simultaneous rocking and sliding, the soil foundation system is charactenzed by the two natural frequencies wn, and wn,. Well-defined
\
L
\
opposition.
\
v
.... ..--
0 (•}
Figure 6.22.
\
(b)
Simultaneous rocking and sliding (a) in-phase with each other; (b) in-phase
258
FOUNDATIONS FOR RECIPROCATING MACHINES
Effect of Eccentric Distribution of Foundation and Machine Mass on Natural Frequenctes
In the analysis so far, it has been assumed that the center of gravity of the mass of the. foundatwn and machme and the centroid of the base area lie along a vertical axis. An eccentric distribution of machine mass may occur when a machme and a generator or a motor are coupled on the same shaft. Sometimes an. eccentricity in the mass distribution is caused by asymmetry of the foundatwn because of the presence therein of cavities and openings. Such asymmetry can often be ehmmated by adjusting the centroid of the foundatwn area that is in contact with the soil. Sometimes when this cannot be d~ne, one has to take into account the asymmetric distribution of the m~ss m ord~r to compute the foundation's vibrations. In such situations, the soil foundatwn system will behave as a three-degree-of-freedom system and should be analyzed accordingly.
Validity of Assumptions For Computing Response of Block Foundations
While developing the basis for linear weightless spring theory, several Simphfymg assumptions were made. The effect of these assumptions on the computed response of a foundation will now be evaluated. 1. Rigidity of the Foundation. The assumption concerning the rigidity of a foundatwn IS defimtely verified in practice, because a concrete block can be considered infinitely rigid in comparison with soil. 2. Soil U~derlying the Foundation Is Weightless. This assumption is not v~hd m a stnct sense, because a certain mass of soil will oscillate in phase With a v1bratmg foundation. Pauw (1953), Balakrishna (1961), and Hsieh (1962) suggested methods for estimating this soil mass. Barkan (1962) estimated that the sml mass beneath a vertically vibrating footing does not exceed 23% of the foundation's mass and may be accounted for if desired. However, the effect of this additional soil mass if included in the calculatwns will reduce the computed frequencies by about 10% from those denved when the soil mass has been neglected. Also, the spring constant may be so defined that the effect of the soil mass is offset. The correction for the effect. of soil mass on the response of foundations for machines is neglected m all the analyses being followed at the present time (1988). 3. Lmear Elastic. Behavior of Soil. The stress-strain behavior of soils under combmed static and dynamic loads has been discussed in Chapter 4 (Sect~ons 4.2 a~d 4.3). The magnitudes of dynamic loads associated with ma_chme operatton generally do not exceed 10% of the combined static Weights of the found~tion and machine (Prakash and Puri, 1969). Thus, for
the_ order of dynamtc st~esses associated with machine operation, small
residual settlements occaswned by repetitions of the dynamic load may build
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
259
up only for the first few cycles of loading and unloading, and subsequently the soil may be considered to behave elastically. 4. Damping. It cannot be assumed that damping can be neglected because every soil-foundation system that vibrates results in dissipation of energy into the supporting medium. The actual amount of damping varies with the vibration parameters, namely, the frequency and amplitude, the mode of vibration, and the geometry of the foundation, and the nature of the soil. Methods for determining the amount of damping were discussed earlier in Chapter 4 (Section 4.8). Damping affects the computed natural frequency as well as the amplitudes of vibration of a foundation. If the damping gin a system is known, the damped natural frequency can be computed from
(6.132) for g = 10%, wnd = 0.995 wn and g = 3\)%, wnd = 0.953 wn The effect of damping on naturaffrequencies is therefore generally small. It is known from theory of vibrations (Chapter 2) that vibration amplitudes are significantly affected by damping. Even a small amount of damping can considerably reduce the amplitudes of vibration, especially for systems subjected to forced vibrations near their natural frequency. The effect of damping on the amplitude of vibration can be taken into account as discussed earlier in Section 6.4. 5. Embedment of Foundation. A surface footing is only an idealized concept. All footings are founded at some depth. Embedment affects a foundation's response as follows: 1. The weight of overlying soil acts as a surcharge and affects the value of
the dynamic soil parameters at the foundation's base. 2. Additional reactive resistance of the soil is mobilized at the sides of a foundation. 3. When an increased amount of a foundation is in contact, with the soils, a much greater amount of energy is dissipated into the adjoining . soil and the ove4":all damping in the system is increased. 4. The effective soil mass participating in the vibrations may also mcrease and may be more than that for a surface footing. In general, embedment will increase the natural frequency of a foundation-soil system and decrease the amplitude of vibration compared to surface footing for a given exciting load. Methods for computing the
response of embedded foundations are described in Chapter 11.
FOUNDATIONS FOR RECIPROCATING MACHINES
260
6.8
DESIGN PROCEDURE FOR A BLOCK FOUNDATION
So far the methods of analyzing block foundations (Sections 6.4 and 6.7) and design requirements of foundations for reciprocating machines have been considered. Two approaches currently being used for the design of foundations for reciprocating machines are: l. Elastic half-space approach (Richart et al., 1970). 2. Linear weightless spring approach (Barkan, 1962).
A step-by-step procedure for design with either approach is given below. It is essential to procure pertinent machine and soil data before a rational design for a foundation to support a machine can be attempted. The obtaining of the relevant data is the first and most important step and is discussed first, followed by guidelines for selecting trial size of foundation and dynamic soil constants. 1.
Machine Data
The following information should be obtained from the manufacturers of the machine for guidance in designing: a. Layout of the machine and a detailed loading diagram consisting of a plan, elevation, and section showing details of connections and the point of application of all loads on the foundation. b. Height of the axis of the main shaft of the machine above top of the foundation. c. Capacity or rated output of the machine. d. Operating speed of the machine. e. Exciting forces of the machine and short-circuit moment of the motor, if any. f. Position of cavities, open spaces, and anchor bolt locations. g. Permissible amplitudes of vibration. 2.
Soil Data The following information about the subsurface soil should be known: a. Soil profile and data (including soil properties generally for depth equal to twice the width of the proposed foundation or up to hard stratum). b. Soil investigation to ascertain allowable soil pressures and to determine the dynamic properties of the soil. c. The relative position of the water table below ground at different times of the year.
DESIGN PROCEDURES FOR A BLOCK FOUNDATION
261
The minimum distance to any important foundation in the vicinity of the machine foundation should also be ascertained. Trial Size of the Foundation
3.
A suitable size of the foundation should be selected for preliminary analysis. The following guidelines wiJI be useful for this purpose and will be helpful in minimizing the number of trials. Area of Block. The size of a foundation block (in pla?)should be larger than the bed plate of the machine it supports, with a m1mmum all-around clearance of 150 mm. Depth. In all cases, the foundation should be deep enough to rest on . good bearing stratum and to ensure stability against fmlure. Center of Gravity. The combined center of gravity of the machme and the block should be as far below the top of the foundation as possible, but in no case shall it be above the top of the foundation. Eccentricf;Jy. The eccentricit)p,bould not exceed 5 percent of the least dimension in any horizontal direction. . . To simplify computations, it is advisable to select a simple shape m plan. Any grooves, projections, and asymmetry should be avmded except when these are necessary. 4.
Selecting Soil Constant
For a preliminary design, the soil constants can b~ obtained. from. the procedure given in Chapter 4 (Section 4.7). For all Important JObs, 1t IS recommended that dynamic soil properties should be determmed. m the laboratory and in the field for at least three different stram levels: Th1s pomt should be kept in mind when conducting soil explora~wn. A particular value may be selected for an anticipated strain level m a g1ven design problem. A correction for the effective confining pressure and shear stram levels must be . applied before proceeding with the design. Often it may be desirable to select a range of sod constants and to work out limiting values of the natural frequencies and motiOn amphtudes for th1s range of the values of soil constants selected. ,.~~,
5.
Centering the Foundation Area in Contact with Soil and Determining Soil
Pressures
Determine the combined center of gravity for the machine and the foundation in the x, y, and z planes and check to see that the_eccentricit_y alongx or y axis is not more than 5 percent. This is the up~er hmtt f.or this ty_pe of analysis. If eccentricity exceeds 5 percent, the add1t10nal rock1~g occasiOned by vertical eccentric loading must be considered in the analysis.
262
FOUNDATIONS FOR RECIPROCATING MACHINES
The static soil pressure should be checked. It should be less than 80 percent of the allowable soil pressure under static conditions. This condition is met in most machine foundations. 6.
Design Values for Unbalanced Exciting Loads and Moments
The values of the exciting forces and resulting moments may now be determined with respect to the combined center of gravity of the system. If the vertical unbalanced force acts at some eccentricity, it will give rise to a moment. Similarly, if the horizontal unbalanced force acts at a certain distance above the top of a block foundation, the magnitude of the moment occasioned by the horizontal force equals the product of the horizontal force and the distance between the center of gravity of the combined system from its point of application. The nature of the unbalanced forces and moments should give the investigator an idea about the nature of the foundation's vibrations. 7.
[ f
For Torsional Vibrations: I = Polar moment of inertia of the base contact area about the vertical . axis through its center of gravity
r
(6.136)
l
M
m•
(6.137)
I B. Noncircular Foundations f
i
For Rocking Vibrations: I= moment of inertia of the base area about an axis passing through the centroid of the base contact area and perpendicular to the plane of vibrations. It equals ba 3!1?,.-in which 'a' is the dimensio? of the rectangular area in the plarle of vibration, and b the dimension perpendicular to this plane [Eq. (6.79)]. Also
M mo =Mm +mL
A. Circular Foundation For Rocking Vibrations: I. = IY = I, moment of inertia of the base area about an axis passing through centroid of the base contact area and perpendicular to the plane of vibration 7TY
Mm =
'H'Ycfo' 1Tro
4g
I,= J, = ab
M
8.
moment of inertia of the foundation system about an axis passing through the centroid of the base and perpendicular to the plane of vibration. mO
2
I m•
=
(a + b 12
2
)
(6.138)
mass moment of inertia of the machine and the foundation about the vertical axis
M m' M mx and M my are obtained as in Appendix 4.
~Mass
M = 1rr:H (r: + H') g'Yc4 3
(6.110a)
(6.134)
in which 'Yc =unit weight of concrete.
Mmo
I
(6.133)
ro =radius of the base of the foundation. M m = mass moment of inertia the foundation system about an axis passing through its centroid and perpendicular to the plane of vibration
2
in which L is the height of the combined center of gravity above the base. The value of 1' may then be determined as 1' = Mm/Mmo For Torsional Vibrations: Iz = polar moment of inertia of the base area about the vertical axis through the center of gravity
4
I x = yI = I =4-0
=Mass moment of inertia of the foundation about the vertical axis of rotation
f
I
The following moments of inertia and mass moments of inertia need to be determined:
.
I
I
Determining Moments of Inertia and Mass Moments of Inertia
263
DESIGN PROCEDURES FOR A BLOCK FOUNDATION
(6.135)
Natural Frequencies and Amplitudes of Vibration
Steps 1 through 7 give the information which will be used for computing the natural frequencies and amplitudes of vibration. This information _is common and subsequently the dynamic response may be calculated either by Elastic Half-Space approach or by Linear Weightless Spring approach. These steps of calculation for both these approaches are given below:
FOUNDATIONS FOR RECIPROCATING MACHINES
264
265
DESIGN PROCEDURES FOR A BLOCK FOUNDATION
C. Elastic Half-Space Approach
(6.23c)
(i) Equivalent Radius. For noncircular foundations, determine the equivalent radius ro of the foundation contact area by considering the direction of the vibrations. For translation
Torsional Vibrations (6.41a)
(6.58a) For rocking vibrations
and (6.58b)
For torsional vibrations
= (ba(a' + b2))114 = (21,)'14
r o~
6w
w
(6.58c)
(ii) Determination of Mass Ratio, Spring Constants, and Damping Factors. The values of mass or inertia ratio, spring constants, and damping factors can be computed from Table 6.2. Damping ratios can also be determined from Fig. 4.40.
(iv) Coupled modes. The natural frequencies of combi~ed rocking and sliding are obtained from Eqs. (6.52 and 6.54) w1th "'"x and "'"• , .". obtai!led as follows: 1 (6.30)
(6.37) (iii) Natural Frequencies and Amplitudes of Vibration in Uncoupled Modes.
Damped natural frequencies are obtained as the roots of Eq. (6.52)
Vertical Vibrations (6.20)
Table 6.2. Mass or Inertia Ratio 8, Damping Factor {;, and Spring Constant k for Rigid Circular Footing on a Semi~lnfinite Elastic Half~Space
Mode of
Mass (or
Damping
Vibration (1)
inertia) ratio (2)
factor (3)
Vertical
Sliding
B,
=
B x
=
(1- v) _!'!... 3 4 pro (7-8v) m 32(1- v) pr! 3(1- v) Mmo 8 pro
--,
Rocking
s.
Torsional
B,. =M"'"
pr"'
g
=
0.425
=
0.2875
'
g
(4)' k, = 4Gro
B, 0.15
g.=
(1
+ s.JVB:; 0.5
g'1' = 1 +2B
(t)n1,2
=
2 , l 1 . [(wnx Z'to/ + wn¢> ±
V< wn¢> + wnx l' 2
2
1- v _ 32(1- v) G kx- 7-Sv ro
B,
X
Undamped natural frequencies can be obtained by using Eq. (6.54b) Spring Constant
•
k = 8Gr~ • 3(1- v) 16 ' k.;, = 3 Gro
Source: Richart, Hall, and Woods, "Vibrations of Soils and. Foundations," © 1970, p. 382. Reprinted by permission of Prentice-Hall, Englewood Cliffs, New Jersey. "'From analog solution.
, 2 1 4 ywn
Damped amplitudes for motion occasioned by the applied moment, can be obtained from Eqs. (6.55a,b).
A
=
x and
M
_Y
Mm
2 112
2 )2 + (2 < [( wnx sx«>nx ) ]
A(w 2 )
(6.55a)
266
FOUNDATIONS FOR RECIPROCATING MACHINES
22 2 112 [(w~x- w ) + (2gxw""w) ] ~(w')
(6.81a)
(6.55b) and
in which ~(w) 2 is given by Eq. (6.56)
~(w2) =
{(
w' _ w2( (w~i>:
W~x)
_
267
DESIGN PROCEDURES FOR A BLOCK FOUNDATION
(6.83}
4gxg¢~"'W"';) + w~x:~• )' (ii) Combined Rocking and Sliding
+ 4(g
w""w (w' - w2)
x
')'
+ gq,w,i>w 'Y
n
(w2 - w'))'}I/2 nx
(6.56)
Sliding and rocking are coupled modes of vibration. The natural frequencies are determined as follows:
Damped amplitudes for motion occasioned by an applied force P acting at the center of gravity of the foundation may be obtained from Eqs. (6.57a,b)
(6.66a}
_ ~Cq,I-WL M
wn
(6.74a)
mo
and
( 6.57a) The amplitudes of vibration can be computed with the following equations:
and 2 Wnx ( wnx
+ 4"~XW 2)1/2
~(w')
D. Linear Weightless Spring Approach
(6.63a) (6.64b}
2
+ C1 I- WL- Mmw )Px + (C"AL)My ~(w 2 )
Ax= and
(i) Uncoupled modes. Vertical oscillations and torsional vibrations occur independently of any other vibrations. The natural frequencies and corresponding amplitudes can be determined with the help of the following equations: Vertical vibrations
Torsional vibrations
(C"AL 2
(6.57b)
Aq,
=
2 (C"AL)Px + (C"A- mw )My ~(w')
(6.129a)
(6.129b)
in which A = linear horizont<\1. amplitude of the combined center of gravity Ax = the rotational a;;,plitude in radians around the combined center of
•
1 f gravity. · The amplitude of the block should be determined at the beanng 1eve o
the foundation as a
2 A,,
(6.130a)
A,= Ax+ hAq,
(6.130b)
A"= A,+
FOUNDATIONS FOR RECIPROCATING MACHINES
268
269
EXAMPLES
r
in which A h is the horizontal amplitude at bearing level, h is the height of the bearing above the combined center of gravity of the system, and A" is the maximum vertical amplitude. 9.
Check for Adequate Foundation
The natural frequencies and amplitudes of vibration calculated in step 8 should be compared with operating speed and permissible amplitudes, respectively, to check if the foundation size selected is adequate. The natural frequency of the foundation soil system should be at least 30% (preferably 50%) away from the operating speed of the machine. The amplitude of vibration should be smaller than the limiting values of amplitude specified by the manufacturer of the machine. If it has not been possible to procure this information, the permissible amplitude should be fixed with due consideration to stability of the machine and also the effects of vibrations on machines and structures in the neighborhood. From the data available so far, it appears that vibrations in neighboring structures will be negligible if the vibration amplitude of the foundation is less than 0.20mm. For machines vibrating in different modes, the resultant vibration amplitude should not exceed the permissible value. 10.
_j_ 0.15
I 2
~~~05; ~===~-----4.0
m Ia)
y
0.5 m
/:~
' I'
3.0 m
-
+
2.0 m
I
Several Machines on a Common Mat
A number of similar machines erected on individual pedestals may be mounted on a common raft. They should be placed symmetrically along the two axes so that there is no rotation in the raft under dynamic forces. The analyses for these machines should be made as though tbeir foundations were independent of each other by breaking the raft into sections corresponding to separate foundations. The design value for the permissible amplitude of vibrations may be increased by 30 per cent for such cases. In addition to proper design of the machine foundation, its construction aspects need special consideration. The construction aspects of machine foundations are discussed in Chapter 14.
1.5 m
ly
0.5 m
'
L'
0.5 m
X
-L
3.0 m
~
0.5 m
(b)
Figure 6.23.
layout of foundation illustrative examples (6.9.1-6.9.6). (a) Section, (b) plan.
2
Assume that the dynamic shear modulus G = 500 kg/cm , .v = 0.33, 3 the density of the soil y = 1.65 t/m 3 and unit weight of concrete !S 2.4 tim· Determine the following: a. The natural frequ.r.ncy of vertical vibrations and
b. The amplitude of vibration. 6.9
EXAMPLES
EXAMPLE 6.9.1
The concrete block shown in Fig. 6.23 is to be used as a foundation for a reciprocating engine operating at 600 rpm and mounted symmetrically with
respect to the foundation. The weight of the engine is 1.1 t. The vertical unbalanced force due to operation of the engine is given hy P, = 0.2 sin wt.
Solution 1. Machine data Weight of engine= 1.1 t Operating speed of the engine= 600 rpm= 62.83 rad/sec
Vertical unbalanced force= P, = 0.2 t
270
FOUNDATIONS FOR RECIPROCATING MACHINES
2. Soil data G ~ 500 kg/cm 2 ~ 5000 tim' v ~ 0.33, x ~ 1.65 t/m 3
oz
~ (A
v-;
~
W0 ,~
~ ~
(AJd
(6.18)
Damping Ratio, !;, < ~ 0.425
~ \10_
5045 ~ 0.598 or59.8% of critical damping
+ (2
X
0.598
X
62.8/124.2) 2
The foundation block shown in Fig. 6.23 is to be used to support a compressor mounted symmetrically. The weight of the compressor is 2.5 t, and it generates an exciting moment, M, ~ 0.5 sin wt t m. The operating speed of the compressor is 450 rpm and its center of gravity is 0.15 m above the top of the block. By using the elastic half-space approach, determine (a) the natural frequencies and (b) the amplitude of vibration. Assume that G ~ 600 kg/cm 2, -y, ~ 1.65 t/m 3, v ~ 0.33, and the density of the concrete 1', is 2.4 t/m 2• Solution 1. Machine Data Weight of compr.essor ~ 2.5 t Operating speed~ 450 rpm~ 47.12 rad/sec Exciting moment M, ~ 0.5 t m
58340tim
0.425
58340V[I- (62.8/124.2)
EXAMPLE 6.9.2
5000 X 1.9544 tim (1 - 0.33)
,, '-IIi:
~
0.2 2 2 ]
~ 3.6 x 10- 6 m~ 0.0036 mm
( 6.17)
Spring Constant, k,
~
99.55 rad/sec
Amplitude of vibration (d
0.5045
X
WIZZ
(fnJd ~ 15.84 Hz
(1- 0.33) X 37.1 4 X 1.65 X (1.9544) 3
4
~
(6.58a)
!4X3 ~ 1.9544 m
k ~ 4Gr0 , z 1- v
~
(w,,)d ~ wn;l/1- ;;; ~ 124.2\IJ- 0.598 2
\j---;;-
pr!
(6.20)
7r ~ 19.76 Hz 2 Damped natural frequency can also be determined as follows:
fn,
B~1-v.!'!._
4
fif
~ ~ 58340 X 9.81 37.1 ~ 124.2 rad/sec
4. Mass Ratio, Spring Constant, and Damping Mass Ratio, Bz z
271
5. Natural Frequency and Amplitude Natural Frequency, f.,
3. Foundation data (Fig. 6.23) Density of concrete -y, ~ 2.4 t1 m 3 Weight of foundation~ ( 4 X 3 x 0.5 + 3 x 2 x 1.5)2.4 ~ 36.0 t Total weight of foundation and engine ~ 37.1 t Area of the foundation's base~ 4 x 3 ~ 12m' Equivalent radius, roz
r
EXAMPLES
(6.22) 2. Soil data G ~ 600 kg/cm 2 ~ 6000 tim' v ~ 0.33, -y, ~ 1.65 tim 3
272
FOUNDATIONS FOR RECIPROCATING MACHINES
3. Foundation data Density of concrete 1', = 2.4 t/m 3 Weight of the foundation= 36.0 t (as calculated in Example 6.9.1) Total weight of foundation and machine= 36.0 + 2.5 = 38.5 t Area of the foundation= 4.0 x 3.0 = 12.0 m 2
273
EXAMPLES
Mass moment of inertia about Y axis through center of gravity of the base
Mmo
Mm + mLz
=
=4.7051+
4. Selection of soil constants
The values of soil constants, i.e., springs and damping will be obtained using G = 6000 !1m 2 and v = 0.33 5. Centering of the foundation area The height of combined center of gravity L above the base is given by:
Lm;
(6.110a)
38.5 2 2 _ (0.9344) =8.1316tmsec 9 81
Mm Mmo
y = - =0.5786
Moment of inertia of the contact area about y axis
4' 4 I=3 x - =16m 12 Sliding Vibrations
2.5(2.0 + 0.15) + ( 4 X 3 X 0.5 X 2.4)(0.25) + (3 X 2 X 1.5 X 2.4)(1.25) 38.5 =0.9344m
Equivalent radius for sliding, r ox
rox:~
Because of symmetry of the foundation block and the machine about the vertical axis, the eccentricity ex= eY = 0.
=
(6.58a)
~ = 1.9544m
Mass ratio, Bx 6. Unbalanced Exciting Loads and Moments
7-8v W Bx = 32(1- v) - ,
Exciting moment MY = 0.5 t m
(6.25)
'Ysr ox
7. Moment of Inertia and Mass Moments of Inertia
(7- 8 X 0.33)(38.5) = _ 0 6356 3 32(1- 0.33) X 1.65 X 1.9544
Mass moment of inertia about an axis through the combined center of gravity and parallel to they axis, Mm (see Appendix 4)
By replacing r 0 with rox in Eq. (6.26), we get Spring constant, kx
=I[~ (a;+ a;)+ m{(x =
~:l
2
(2.15- 0.9344) +
1 -
3
X0
2 )
k
+ (z 1 - zoJ) 2 ]
2 X ~.;ll~\~ .4
+
3X2X1.5X2.4 2 9.81 (1.25- 0.9344)
+
4 X 3 X 0.5 X 2.4 ( 42 O ,) 9.81 X 12 + .5
(3
2
+ 1.5
= X
2 )
32(1- v) G 7-8V YOX
= 32(l- 0 ·33 ) X 6000 X 1.9544 t/m 7-8 X 0.33 """' 57663 tim Damping coefficient, t;x t = Cx
4 X 3 X 0.5 X 2.4 (O 934 _ O )' 9.81 . 4 .25 2 = 4.7051 t m sec +
0.288 ~
0 288 · = 0.36 or 36% vo.6356
(6.28)
274
FOUNDATIONS FOR RECIPROCATING MACHINES
275
EXAMPLES
Rocking Vibrations (6.37)
Equivalent radius, r00 from (Eq. 6.58b),
ro
=
( b') 3:
11 '
=
~
(3x4')
=
2.1245m
Replacing r0 with ro
B
=
3(1- v)Mmo
8
'Y,
=
~228990 8.1316
=
167.81 radlsec
Combined Rocking and Sliding
5
-rot/>
2
2
2
2
w'("'"x +Y "'""')w'n + w,x"'n.P n 1'
g
3(1- 0.33) X 8.1316 1.65 5 8 X _ X 2.1245 9 81 = 0.2806
2
(121.21 + 167.81 .w,0.5786
2
4
)w'" + 121.210.5786 x 167.81 2
=
0
=
0
(6.54a)
2
or
Spring constant, k> kq,
w~ -7.406 X i'Q4i:,! + 7.1505 X 10 8 = 0
8Gr! 0 3(1- v)
=
(6.34a) _7.406x10'( +I _4x7.1505x10 1 1 42 "'"'·'2 - \1 (7.406 X 10 ) 2
3
I 8 X 6000 X 2.1245 3(1- 0.33) t m rad 228990 t m I rad
=
2 wn1,2
=
Damping constant, {;0 0.15
?;"'
=
(1 + s.)~
(6.36)
=
[, 1
=
)
7 .4° 6 X 104 (1 ± 0 6917) 2 .
w~ 1 = 6.264 "'"' =
0.15 (1 + 0.2806)\1'0.2806
8
X
4 10 , w~ 2 = 11414
250.2 radlsec 39.82 Hz,
"'"' = [, 2
=
106.8
17.0 Hz
0.22 or 22% of critical damping
8. Natural Frequencies and Amplitudes Natural frequency of sliding, "'"x (6.30) = ~ 57663 X =
Natural frequency of rocking
9.81 38.5 121.21 radlsec (6.57)
FOUNDATIONS FOR RECIPROCATING MACHINES
276
2
A(w ) = (47.12) - (47.12) '(167.81 . +121.21 0 5786 _ 4 X 0.36 X 0.22 X 167.81 X 121.21) 0.5786 2
{[
2
167.81 X 121.21 + 0.5786 + 4[
lz
A"ftf
2
4
2 ]'
0.22 X 167.81 X 47.12 ( ,_ ,)] 121 .21 . 47 .12 + 0.5786
2
112
2.0m
}
l
= 6.3558 X 108 0.5 X y/(121.21 2 ) 2 + (2 X 0.36 X 121.21) 2 . . Ax= 4.7051 X 6.3558 X 10 8
oo(ij
__ J 1
- t"Tw'"'" / -t
k~'·
!
0.36 X 121.21 X 47.12 ( , _ ,) 47 .12 167 .81 0.5786
---
\!I
l __
L
~ 0.9334m
---
X a.
2
---J ;rl
=2.0m
= 2.45 x 10- 6 m= 2.45 X 10- 3 mm A
=
•
MY Y(w~x- w ) + (2gxwnxw) Mm A(w 2 ) 2 2
2
(6.55b)
0.5y/(121.21 2 - 47.12 2 ) 2 + (2 X 0.36 X 121.21 X 47.12) 2 4.7051 X 3.1813 X 108 = 2.19 X 10- 6 rad Total horizontal amplitude, A, (Fig. 6.24a)
---
A,=Ax+hA•
= 2.45 X 10- 3 + (2.15- 0.9344) X 2.19 X 10- 6 X 10 3 mm = 5.112 x 10- 3 mm = 0.005112 mm
\
/
I
I
(6.130b)
I
Vertical amplitude, A, a 4 -6 3 A=2A•=2X2.19x10 x10 mm
I I I
3
= 4.38 x 10- mm = 0.00438 mm
/
"'ilg:.
I
I \
EXAMPLE 6.9.3
The block shown in Fig. 6.23 is excited by a horizontal moment, M = 0.5 sin wt m, which acts around the vertical axis through the system's center of gravity. Determine the amplitude of vibration if the frequency of the exciting moment is 10Hz. Use the elastic half space approach. Assume 3 G = 600 kg/em', v = 0.33, Y, = 1.65 t/m and unit weight of concrete y = 3 ' 2.4t/m.
/
/
/
7
"'
' I
I
I
I I I
•
/
-(b)
Figure 6.24.
E 1 6 9 2 and 6.9.5. (b) Examples Illustrations for Problems 6.9.2-6.9.6. (a) xamp es · ·
6.9.3 and 6.9.6. 277
FOUNDATIONS FOR RECIPROCATING MACHINES
278
Solution 1. Machine Data. The weight of the machine is not given in the problem and is therefore negl~cted. Operating speed w =10Hz= 62.83 rad/sec Horizontal unbalanced moment M, = 0.5 sin wt t m
279 EXAMPlES
Inertia ratio, B!/J
Mm, B =•
(6.38)
"Y 5 -roof!
g
5 434- - = 1.0182 65 5 1. X 1.9973 9.81
-:-c::-_.4_-
2. Soil Data
G = 600 kg/cm 2 = 6000 t/m 2 3 v = 0.33, "Y, = 1.65 t/m . 3. Foundation Data (refer to Example 6.9.1) Weight of the foundation = 36 t Area of the foundation = 12 m 2
Spring constant, k,, 16 3 k,=3Groo/
4. Selecting Soil Constants The values of the soil constants, i.e., the soil springs and damping will be obtained using G = 6000 tlm 2 and v = 0.33.
3 = 16 x 6000 x (1.9973) t m/rad 3 = 254965 t m/rad
Damping. coefficient, I;• ~''
o's:
5. Centering of the Foundation The eccentricity ex = eY = 0
!;•
=
8. Natural Frequency and Amplitude of Vibration
7. Determination of Moment of Inertia and Mass Moment of Inertia Moment of inertia of the foundation contact area around the Z axis = 1, from Eq. ( 6.138)
Natural frequency, wn•
_Jk; Wnl/1-
Mass moment of inertia around the Z axis= Mm, (see Appendix 4),
= (4
X
(a;+ a;)+ m[(x,3 X 0.5 X 2.4 9.81 X 12
X0
2 )
+ (y,-
)< 4, + 3,) + ( 3
X
yoJ 2]]
2 X 15 X 2.9 9.81 X 12
(6.41a)
'/Af:
~ ~254965 5.4434 = 216.42 rad/sec
(6.138)
~
(6.42)
1 + 2B,,
0 ·5 =0.165 - 1 + 2 X 1.0182
6. Unbalanced Exciting Loads . M, = 0.5tm
Mm, = [
(6.40a)
fn• = 34.44 Hz
)< 3
2
+ 2,)
Amplitude, A" (6.41b)
= 5.4434 t m/sec2 Mass (Inertia) Ratio, Spring Constant and Damping Equivalent radius, ro.p 2
r m/'
= [
ab(a + b
= [ 4x3
67T
J
)
(4'+3')]1/4 6
112
=
114
2
1T
(6.58c)
254965
[(
1-
( 62 83 )')'
216.42
(
+ 2 X 0.165 X
62.83 )'] 216.42
6
=2.1 x 10- rad =1.9973m
f h f f "E" Figure 6 24b is, Horizontal displacement of the edge o t e oo mg ' .
280
FOUNDATIONS FOR RECIPROCATING MACHINES
A• = rA"'
(6.48)
= Y(2 2 + 1.5 2 )2.1
X
10- 6
= 5.25 x 10- 6 m = 0.00525mm Examples 6.9.1-6.9.3 can be conveniently solved using the computer program giVen m Appendix 1.
EXAMPLES
281
EXAMPLE 6.9.5
The foundation block shown in Fig. 6.23 is to be used to support a compressor mounted symmetrically. The weight of the compressor is 2.5 t, and it transmits a horizontal unbalanced force of Px = 0.2 sin wt, which acts at a height of 0.15 m above the top of the block. The operating speed of the compressor is 450 rpm. Determine (a) the natural frequencies and (b) the amplitude of vibration. Assume that C" = 6.0 kg/ cm 3, the density of the concrete is 2.4 t/m 3, and the center of gravity of the engine to be at 0.15 m above the top of the block. Use the linear weightless spring approach.
EXAMPLE 6.9.4
Compute the response of the block foundation in Example 6.9.1 by the hnear weightless spring approach. Assume that c" =5kg/cm' for !Om'
area.
Solution
1. Machine Data. Same as in Example 6.9.1. 2. Soil Data
Solution 1. Machine Data Weight of compressor = 2.5 t Operating speed = 450 rpm = 47.12 rad I sec ·Horizontal unbalanced force, Px = 0.2 t (along the X axis) Point of application of the horizontal unbalanced force above top of the foundation block= 0.15 m . ;::~-
3
3
3
2
C" = 5 kg/cm = 5 X 10 t/m for 10m area
·;~/if"
2. Soil Data 2
4. 2 Soil Constants. Because t~e area of the foundation's base is more than 10m , the value of C" for 10m will be used 5. Natural Frequency and Amplitude Natural frequency
w n<
=
~CuA m
= ~5000
(6.63a)
12 X 9.81 37.1 = 125.95 rad/sec X
,
Az
=
3. Foundation Data Density of concrete y,·= 2.4 t/m 3 Weight of the foundation= 36.0 t (As calculated in Example 6.9.1) Total weight of foundation and machine= 36.0 + 2.5 = 38.5 t Area of the foundation= 4.0 x 3.0 = 12.0 m 2
4. Selecting the Soil Constants Since the area of the foundation base in contact with the soil is 12m2 > 10m 2, the values of C", C, and C1 as for the 10m2 area will be used for calculation. 5. Centering of the Foundation Area (Refer to Example 6.9.2) L=0.9344m
fu, =20Hz Amplitude of vertical vibration A
2
C" = 6.0 kg/ cm for 10m area 2 C, = 1/2 Cu = 3.0 kg/cm , Cq, = 2 Cu = 12 kg/cm 3
3. Foundation Data. Same as in Example 6.9.1 Total weight of the machine and foundation = 37 .I t Area of the foundation base= 12m 2
'if'J:.
'
P, 2 2 m(wn,- w ) 0.2 2 37.1 2 9.81 ((125.95) - (62.83)
(6.64b)
t'
= 4.4 x 10- 6 m= 4.4 x 10- 3 mm
e=e=O X y
6. Unbalanced Exciting Loads and Moments Horizontal unbalanced force= Px = 0.2 t Vertical distance between line of action of Px and the combined center of gravity= (2.0 + 0.15- 0.9344)m = 1.2156 m. Moment MY about the combined center of gravity occasioned by the horizontal unbalanced force P My = 0.2
X
1.2156 = 0.243 t m
282
7. Moment of Inertia and Mass Moment of Inertia (Refer to Example 6.9.2)
283
EXAMPLES
FOUNDATIONS FOR RECIPROCATING MACHINES
i-
1.!',·.• .·
(6.122) =
Mm = 4.7051 t m sec 2 Mmo = 8.1316 t m sec' 1' = 0.5786 I= 16 m 4
38 5 · X 4.7051(49018.98- 47.12') 9.81 X (7634.91- 47.12
= 4.679 X 10
2 )
9
Amplitude A,
8. Natural Frequencies and Amplitudes Limiting natural frequencies
(C.J- WL
Ax= (6.66a)
=
_ Wnq,-
y/3x10
3
X12x9.81 . = 95.775 rad/sec 38 5
/CI WL "V M
1 [(12 X 10 3 X 16-38.5 X 0.9344 9 4.679 X 10 2 + 3 X 103 X 12 X 0.9344 2 -4.7051 X 47.12 )0.2 5
(6.74a)
= a. Natural frequencies in combined rocking and sliding
(6.113) 1
±
2
5786
2
[(95.775 + 153.646 )
Y(95.775
2
·:.·''c
Amplitude, Aq,
1 A , = --2 [(C"AL)Px+(C"A-mw')M,] Ll.(w )
3
= /12 X 10 X 16-38.5 X 0.9344 y 8.1316 = 153.646 rad/sec
X 0.
(6.129a)
Ll.(w')
.
+ (3 X 103 X 12 X 0.9344)(0.243)] = 1.0849 X 10- m
mo
2
+ C"AL 2 - Mmw')Px + (C"AL)M,
+ 153.6462 ) 2 - 4 X 0.5786(95.775 2 ) X (153.646) 2 ]
(6.129b)
1 [(3 X 103 X 12 X 0.9344)(0.2) 4.679 X 109 38 5 2 + .(3 X 103 X 12- 9.81 · X 47.12 )0.2431
= 2.8549 x 10- 6 rad b. Combined amplitud~s of vibration from Fig. 6.24a. Horizontal
A,= Ax+ hA.; = 49018.98, 7634.91 sec- 2
= 1.0849 X 10- 5 + (2.15- 0.9344) X 2.8549 X 10-
Therefore,
6
= 1.431 x 10- 5 m w., 1 = 221.40 rad/sec,
f., 1 = 35.23 Hz,
w., 2 = 87.37 rad/sec
!,,, = 13.9 Hz
Check
= 0.01431 ..: Vertical
A,= 87.37 < 95.775 < 153.646 < 221.40 Amplitudes of vibration
a
(6.130a)
:z A 0
= ~ (2.8549 X 10- 6 ) = 5.7 X 10=0.0057mm
6
284
FOUNDATIONS FOR RECIPROCATING MACHINES EXAMPLES
285
EXAMPLE 6.9.6
Using the line~r weightless spring approach, compute the dynamic response of ~e found~l!on flock m Ex~mple 6.9.3 in torsional oscillations. Assume C"- 6.0 x 10 tim for a 10m area.
Amplitude, A.p
M, A,1, = M
Solution 1. Machine Data. (Refer to Figs. 6.23 and 6.24 and to Example 6.9.1). The we1ght of the machine is not given in the problem and is therefore neglected. Operating speed = 10Hz
mz Wnof!
=
5.5 x 10- 6 rad
A=rA.;.=Y(2 2 +1.5 2 )x5.5x10
6
(6.84)
6
13.75 x 10- m
=0.01375mm EXAMPLE 6.9.7
• Design a foundation for a reciprocating compressor. The following data are supplied. .,,. 7 1. Machine Data
4. Selecting Soil Constants. Since actual foundation is greater than 10 ' m' the values of Cq, as for 10 m'will be used. 5. Centering of Foundation Area. Not needed for this problem. 6. Design Values of Exciting Loads and Moments Exciting moment M, = 0.5 t m
Frequency of excitation= 10Hz= 62.83 rad/sec
Operating speed of the compressor = 405 rpm Weight of the compressor = 9 t Height of the center of gravity of compressor above its base= 0.5 m Operatiog speed of the motor= 1470 rpm Weight of the motor = 2.0 t Height of the center of gravity of the motor= 0.5 m Bearing level of the compressor above its base= 0.5 m Unbalanced forces and moment occasioned by the operation of the compressor
7. Determination of Moment of Inertia and Mass Moment of Inertia.
From Example 6.9.3
Horizontal primary force =
P; = r; = 0
= P~ = P~ = 0 Vertical primary force= = 165 kg Vertical secondary force = = 40 kg Horizontal primary moment = M; = 185 kg m Horizontal secondar~ moment = M; = 0 Vertical primary moment= M~ = 1750 kg m Vertical secondary moment = M; = 450 kg m Permissible amplitude (peak to peak) = 0.025 mm
Horizontal secondary force
r; r;
2
J,=25m 4 8. Natural Frequency and Amplitude Natural frequency of torsional vibration w
no/1
(6.81a)
=
(6.83)
')
Horizontal displacement of the foundation due to torsion Fig. 6.24b
=
3. Foundation Data (Refer to Example 6.9.1) Weight of the foundation = 36 t Area of the foundation= 12m'
Mm, = 5.4434 t m sec
(t)
05 · 5.4434(143.76 2 - 62.83 2 )
Horizontal unbalanced moment M, = 0.5 sin wt t m 2. Soil Data Cu=6X 103 t/m 3 c, = 0.75 C" = 4.5 x 10 3 tim'
( , _
~4.5 X 10
3
X 25
5.443 4
= 143 rad/sec = 22.88 Hz
2. Solid Data
Allowable soil pressure= 25 t/m 2 Data on dynamic soil properties is the same as in Example 4.9.3. Design by (1) elastic half-space method and (2) linear elastic weightless spring method.
286
FOUNDATIONS FOR RECIPROCATING MACHINES
EXAMPLES
287
Solution 3000-----+~----3000-----1-:
1. Machine Data. The machine data is listed in the problem explicitly and is not repeated here.
-·-.--
2. Soil Data. As listed in problem and Example 4.9.3 (Chapter 4). 3. Trial Dimensions. Trial dimensions of the foundation are shown in Fig. 6.25. In selecting these dimensions, the requirements with regard to geometrical layout of the machine, as required by the manufacturer of the machine, were taken into account. 4. Selection of Soil Constants
Amplitude of vibration=
0
500
·~25
~
= 0.0125 rnrn
0.0125 _ x = 2.1 x 10 6 6 1000 Plot C in Fig. 4.47 gives the values of G versus y, at 2.4 rn depth, and the value of G, corresponding to y0 = 2.1 X 10- 6 is found to be 2 1080 kg/ crn at a mean effective confining pressure of 1.0 kg/ crn 2• The h . I S ear stram eve!
r, =
~.---r----,,-,-
l!LI
500
4000
1----
I
1255~-l+' - - 1 7 4 5 - - - + ! 1
T ~ -f~ I'r--t- . 3: . +Motor
ci.
-r--
z
1106
3oo~\oo
~ "-'-+----:~ ~
375
375
r----;...x z
L__..,L
3oo
f---a5o~t----s5o--j
I
4000
y
2894 Compressor axis
r Motor - - - - . - Motor axis 500 400
5
1300
+ Figure 6.25b.
X
Plan for Design Example 6.9.7. All dimensions are in millimeters. ~~:
mean effective confining pressure at a depth of 6/2 =3m below the foundation is now calculated.
___ (2K 3+ 1) 0
ao- av 5000
Figure 6.2Sa. millimeters.
Section of the foundation for Design Example 6.9.7. All dimensions are in
0'01 = 300 iiv 2 = 4q/
X
2.05 = 615 g/cm'
288
FOUNDATIONS FOR RECIPROCATING MACHINES
289
EXAMPLES
Here
1.13 C, =
q =static stress intensity= 2.948 t/m 2
m=
4
3
3 = 1.33, n = 3 = 1
= 294.8 g/cm 2
(see Step 5)
5. Centering of the Foundation Area and Determination of Soil Pressures
ii, = 615.0 + 224.0 = 839.0 g/cm = 0.839 kg/cm 2
O
Total mass of foundation and machine from Table 6.3, (column 5) = m = 14.4265 t m - l sec 2
I
ii, 2 = 4 X 294.8 x 0.190 = 224.0 g/cm 2
_
2 X 8076(1 + 0.33) 1 I 3 (1 - 0.33 2 ) v'IO t m
= 8614.5 t/m 3
From Fig. 4.46, I= 0.190
uo = .839
X
Area of the foundation= 8 X 6 = 48.0 m' .4 x . 14 265 9 81 Static soil pressure below the foundation = 48 2.948 t/m 2 Coordinates of the combined center of gravity of the foundation block and machine
I I
2
( 2 X 0.5 + 1) = 0.5593 kg/cm 2 3
The value of G for design 0.5593) O.S = 1080 x ( - = 807.6 kg/cm 2 1
c
=
"
= 8076 t/m 1
nHiss of element ':'~ i.:,: x,, y,, z, = coordinates of the center of gravity of the element with reference to the X, Y, Z axis. The details of the computations of the mass of the various elements of the machine and foundation and the products m;X;, miyi, and miz; are shown in Trible 6.3, columns 2-11. - 58.6326 - 43.2718 Y= .4 = 4.0642 m X= .4 =2.9994m 14 265 14 265 14.8583 Z= L= .4 = 1.0299 m 14 265 mi =
2
1.13£ (1- v 2 ) VA
_ 1.13(2G(I + v)) 1 (1- v 2 ) VA
-
Because the area of the foundation is larger than 10m2 the value of C 2 " for a 10 m area will be used in design
I !.
1:
~
rable 6.3. Computations for Example 6.9.7
-----'----'~=:::c_.::.:.::::_--~~------11.··· -~-------::--::-----------~lements
ffsystem
::Ompressor
notor I
2
3 4 5 6(-)
Mass 1 2 (tm- sec )
Coordinates of
Center of Gravity ofElementm
.
Xoi
f
m,.x,.
m,y,
,1.
m,z,
=
(i- i,) (m)
Yo;= (Y- y,) (m)
Zo;
m,(y!, + z~;) (tmsec2 )
(i- z,) (m)
m,(' ') IT ay, + a",
15
16
31.4370 13.3429 0.3831 0.0558 2.0148 -0.0087 47.2249
5.8266 1.0692 3.5955 0.2509 0.9148 0.1143 1.9594 -0.2140 13.5167
m1 ( 2 2 ) 12 a..,,+ aY, (tmsec 2 )
2
2
m,(xot +Yo;) (tmsec 2 )
¥~--------=-----------
m,
x,
y,
z,
5
6
7
8
9
10
11
12
13
14
0.9174 0.2038 5.8715 4.4036 0.8372 0.1669 2.0857 -0.0596 14.4265
1.93 3"805 3.0 3.0 1.93 2.78 3.805 1.93
4.044 4.369 4.0 4.0 4.044 4.044 4.369 4.044
3.55 3.3 0.25 0.80 2.075 1.85 1.95 2.925
1.7705 0.7754 17.6145 13.2108 1.6157 0.4639 7.936 -0.1150 43.2718
3.7099 0.8904 23.486 17.6144 3.3856 0.6749 9.1124 -0.2410 58.6326
3.2567 0.6725 1.4678 3.5228 1.7371 0.3087 4.067 -0.1743 14.8583
1.0694 0.8056 0.0006 0.0006 1.0694 0.2194 0.8056 1.0694
0.0202 0.3048 0.0642 0.0642 0.0202 0.0202 0.3048 0.0202
" 2.5201 2.2701 0.7799 0.2299 1.0451 0.8201 0.9201 1.8951
[
r '
t
K I
I •
17
18
48.9291 22.3849 0.2450 0.0256 2.0148 -0.0111 73.5883
1.04953 0.1512 0.0242 0.01830 0.9577 0.0082 1.9893 -0.0682 4.1302
290
FOUNDATIONS FOR RECIPROCATING MACHINES
EXAMPLES
6. Eccentricity Along the X and Y Axis
passes through its center of gravity of the base and perpendicular to the YZ plane is
(~- 2.9994)100 Eccentricity ex =
. . E ccentnc1ty
eY
=
6/2
291
3
( 4.0642- 4.0)100 _ = 1.605% 40 <5%
Polar moment of inertia,
7. Design Values of Unbalanced Forces and Moments. The values of various exciting forces and moments are explicitly listed in statement of problem under machine data. 8. Determination of Moments of Inertia and Mass Moments of Inertia. The mass moment of inertia Mm about an axis that passes through a common center of gravity of the machine and foundation and is perpendicular to the YZ plane is (see Appendix 4) =
I
(6.138)
9. Computation of Natural Frequencies and Amplitudes of Vibration. In terms of the given data, the foundation is subjected to an unbalanced vertical exciting force, a moment in the YZ plane (coupled rocking and sliding) and a horizontal moment (torsion or yawing). The dynamic motion characteristics corresponding to these exciting forces and mollJs:nts will now be analyzed. The operating speed is 405 rpm or w = 42.41' rad/sec. ,,.c.'
The design of the foundation will first be evaluated by the elastic half space approach. Steps 1 to 7 are common with the solution by linear spring approach as well.
Mm = 47.2249 + 13.5167 = 60.7416 t m sec 2 Mass moment of inertia Mmo
t
(6.110a)
i
Mass Inertia Ratio, Spring Constants and Damping a. Vertical Vibration
t
2
I
=75.0786 t m sec 2 Mm 60.7416 ")' = Mmo = 75.0786 = 0.8090
(6.110b)
Mass moment of inertia m about the Z axis through the common center of gravity (see Appendix 4) 2 M mz = [ m, z z 12 ( a xi+ a zyi ) + mi ( Xoi + YoJ
2
A. Elastic Half-Space Approach
From Table 6.3 (columns 15 and 16),
= 60.7416 + 14.4265(1.0299)
=
f-
~ [ ~~ (a~,+ a;,)+ m,(y;, +z;,)]
M mo =Mm +mL 2
2
ab(a + b ) 12 8 6 2 2 4 J, = ~ (8 + 6 ) = 400 m J '
Because the eccentricity is less than 5%, it may be neglected in calculating the dynamic response of the foundation ..
Mm
3
I = ba = 6(8) = 256 z x 12 12 m
=0.02%
J
From Table 6.3 (columns 17 and 18)
Mm, = 73.5883 + 4.1302 = 77.7185 m sec 2 Moment of inertia of the base contact area, Ix about the axis that
Equivalent radius, r"' from Eq. (6.58a),
I
I
48)112
= ( -;
=3.9088m
(6.58a)
1- v mg -4--,-
( 6.17)
Mass ratio B, 1-v m B=---,z 4 pr oz
'Y_~T oz
= (1- 0.33)(14.4265) (9.81) = 0.1936 2.05 4 X (3.9088) 3 By replacing r0 with r0 , in Eq. (6.7b) we get spring stiffness, k,
292
FOUNDATIONS FOR RECIPROCATING MACHINES
= 4Gr
k
293 3
0,
ro
l- v
z
4
X
=
-x 37T
,,
VB;
~' =
0.425 \1'0.1936 = 0 "965
(6.22)
k
•
X
(6.34a)
8076 4 2493 X · = 2465757.9 t m/rad 3(1- 0.33)
Inertia ratio, B., roy
3(1- v) -Mmo 8 5 pr oq,
Broy=
(A)Iiz = 3.9088 m 7r
.-
3(1- 0.33) 8
(6.58a)
32(1- v)Gr oy 7- 8v
Damping ratio,
(~:~~ )<4.249)
(6.26)
= (7- 8 X 0.33) _ _.::._14:.:._.4:.::2=65_ _ 32(1 - 0.33) = 0 ·235 2.05) 3 ( 9.81 . (3.9088)
d. Torsional Vibrations Equivalent radius, r0 ~, 2
rm~
(ab(a + b . 61T
=
114
2 )
=( 8 x 6 ~7T+ 6
Damping ratio ~Y
2
~ = 0.2875
YVB;
(6.36)
0 15 · = 0.551 (1 + o.o651)v'0.0651
=
(6.25)
5
0.15 (1+ Bq,)VB;,
~. =
Mass ratio BY
7-8v ~ 3 32(1- v) pr oy
= 0.0651
~.
_ 32(1- 0.33)8076 X 3.9088 = 155230.8 tim 7 _ 8 x 0 _33
Y
(6.32) (75.0786)
Spring stiffness kY
B =
=4.249m
8Gr!. = ~---"''-:3(1- v) 8
b. Sliding Vibrations
Y
114
Spring stiffness k., < = 0.425
k =
114
(ab ) 37T
(6 8')
8076 X 3.9088 = 188462.5 tim 1 _ 0 _33
Damping ratio ~'
Equivalent radius
EXAMPLES
Inertia ratio B
(6.58c)
'l)I/4
=3.9947m
~:.
0.2875 = \1'0.235 = 0.587 c. Rocking Vibrations Equivalent radius ro4> by interchanging a and b in Eq. ( 6.58b), we get
(6.38) 77.7185
_ ___:_cc..:..:...::=.:::..._ = 0.3656
G:~~)(3.9947)'
294
FOUNDATIONS FOR RECIPROCATING MACHINES
295
EXAMPLES
By replacing r0 with r0 • in Eq. (6.40a), we get spring stiffness k•
b. Natural Frequency in Pure Sliding, w"'
_ !k; = wny-
=
16
1155230.8
\j "J:i
(6.30)
\j 14.4265
103.73 rad/sec
= 3 x 8076 x (3.9947) 3 = 2745665 t m/rad c. Natural Frequency in Pure Rocking, wn
Damping ratio, I;• 0.5
!;•
= 1 + 2B
,,
(6.42) =
~2465757.9
=
181.22 rad/sec
75.0786
0.5 = 1 + 2 X 0.3747 = 0 ·2888
d. Coupled Rocking and Sliding. Undamped in coupled rocking and sliding are given by
9. Natural Frequencies and Amplitudes of Vibration
natural
frequencies
a. Vertical Vibrations
(6.54a)
Undamped natural frequency,
W0 ,
2
4 _
w0
= ~ = p88462.5 w"'
\1-;:;
'(
w0
w:-
f
f
!\' I
Damped amplitude, (A,)d (6.23c)
I '
i.
((0.165 + 0.04)
=
2.265 x 10- 6 m
=
2.265
X
X
0.965 X 42.41) 114.29
2
10- 3 mm < 0.0125 mm
r: is acting at 2w. However, the final value of amplitude of motion will not be appreciably affected.
+ ( 103.73
2
181.22 0.8090
)
X
2 ))
=O
or
= 114.29 rad/sec
1(
2
(6.20)
14.4265
42.41 )')' (2 (188462 .3 ) \j 1 - ( 114.29 + .
103.73 + 181.22 0.8090
'
2
_
Wol,Z-
=
5.3894
5.3894 X 10 2
4
[
X
+
10 4 w~
+ 4.3678 X 108 = 0
/1- 4 X 4.3678 X 108 ]
1 - \j
(5.3894
X
10 4 ) 2
5.3894 X 104 (1 ± 0.6312] 2
w~ 1 = 4.3955 X 10 4,
W 01
=
w~ 2 = 9938.0,
W
= 99.68 rad/sec,
02
209.65 rad/sec
fo 1 = 33.36 Hz foz
= 15.86 Hz
f
lI
!
i
I
I! I
v
(6.56)
296
FOUNDATIONS FOR RECIPROCATING MACHINES
Ll.(w2) =
{[<42 .41)' _ (42 .41)2( 181.220.8090 + 103.73 2
_ 4 X 0.587
0.551 X 103.73 0.8090 2 + 103.73 X 181.222] 2 0.8090 . X
+ 4 [ 0.587 X 103.73 X 42.41 0.8090
+ 0.551 = 4.5049
(A )
=
M [(w2 '
X
"Y
X
X
(6.41a} =
1
d
_x
. /2745665
'V 77 .7185
2 2 (181.22 - 42.41 )
112 181.22 X 42.41 2 2 ]'} 0.8090 (103.73 -42.41 )
(6.41b}
10 8
+ (2 < w )']'12 'Y
"Y
ny
0.185
(6.55a)
2745665 [( 1- ( 42.41 . )')' + ( 2 X 0.2858 187 95 8 ;,'f" 7.30 X 10- rad ·:;.••r;
o:.yWnyW
(6.55b)
= (3 2 + 42 ) 112
2.2 [(103.73 - 42.41 2}2 + 4 X 0.587 2 X 103.73 2 X 42.41 2] 112 60.7416 4.5049 X 108 = 4.30 x 10- 7 rad
X
7.300
X
10- 8
X
103 mm
= 3.65 x 10- 4 mm <0.0125mm B. Linear Elastic Weightless Spring Approach
Combined amplitudes (damped) Vertical, A,
(A,)= (A,)+~ (A 1 ) X
( 6.84}
(Ah}d = r(A•}d
2
= 2.265
X
2 112 42.41 ) ] . 187 95
Maximum horizontal displacement of an edge of the foundation occasioned by yawing, (Ah)d
Ll.(w 2 )
Mm
10- 3 + 4.530
(6.130a) X
10- 7
X
The design of the foundation will now be evaluated by the linear weightless spring approach. Steps 1-7 are common with the elastic halfspace approach.
103 Natural Frequencies and Amplitudes
= 4.077 x 10- 3 mm < 0.0125 mm (A h)= (Axl
= 187.95 rad/sec
/, = 29.91 Hz
= 1.286 x 10- 8 m= 1.286 x 10- 5 mm M [(w2 - w')2 + 4<2 2 2]112
(A ) =
297
181.22)
MmLI.(w'} 2 2.2 (103.73 + (2 X 0.587 X 103.73) 2] 112 60.7416 4.5049 X 10 8
Y d
EXAMPLES
2
+ h(A 1 )
a. Vertical Vibratio'"ris (6.130b}
5
= 1.286 X 10- + (3.33- 1.0299)4.30 X 10- 7 = 1.548 x 10- 3 mm < 0.0125 mm
X
10 3
Undamped natural frequency, "'"' . = wnz
e. Torsional Vibrations
/8614.5 X 8 X 6 .4 = 169.2993 rad/sec 14 265 fu, = 26.95 Hz =
Natural frequency (undamped) '
w
m{l
[C"A]t/2 m
'V
(6.63a)
FOUNDATIONS FOR RECIPROCATING MACHINES
298
299
EXAMPLES 2
Undamped amplitude, A,
4
-
wn!,2-
= 9.0328 X 10 2
(6.64b)
(0.165 + 0.04) -7 5 29 10 2 2 14.4265((169.2993) - ( 42.41)) = · x m
w!
1
=
7.6773
10
X
4
wn 1
,
= 0.000529 mm
fn!
Check
w ~2 = 13558.23 ,
fnz
A,= 0.0000529 mm < 0.0125 mm
=
~C;,A
C,=
2 C" =
wny
=
1
=
8614.5 3 - =4307.25t/m 2 6
X
]
1 ± 0. 69981
=277.079rad/sec = 44.09 Hz = 116.439 rad/sec = 18.53 Hz
2
(6.66a)
/4307.25 X 8 .4 14 265
[
~(w 2 )= mMm(w~ 1 - w )(w! 2 tr
Y
4
'1
9
4 X 1.0406 X 10 (9.0328 X 10 4 ) 2
Undamped amplitude
b. Limiting natural frequency in sliding along the Y axis, wny wny
Wnz
42.41 0 25 169.2993 = ·
w
I1 -
9.0328 X 10 [1 + 2 -
w
-
2
(6.122)
)
'~;,~o.:r
14.4265
= 7.7267
X
X
60.7416(277.079
10
2
-
2
42.41 )(116.439
2 -
42.41
2 )
11
Maximum exciting moment= Mx
= M~
+ M~
= 119.7126 rad/sec
Mx = (1750 + 450) = 2200 kg m = 2.2 t m
c. Limiting Natural Frequency in Rocking about X axis
_
fC.;I- WL
'V
wncf> -
( 6. 74a)
M
mO
Cq, = 2Cn = 2 X 8614.5 = 17299 tlm _
/177229
Wn
Replacing Ax and MY with AY and M" respectively, in eq. (6.127), we get
X
256
3
(14.4265 X 9.81) 75.0786
Y
1.0299
X
= 4307.25 X 48 X 1.0299 11 7.7267 X 10
= 242.3732 rad/sec d. Coupled Rocking and Sliding. Undamped natural frequencies in coupled rocking and sliding are given by w4- w2 n
n
wny
n
w' _ w2( 119.7126 n
(
2
2
2
)
2
2
+ (J)ncf> + wnywnc/>
=
0
'Y
'}'
0.8090 w~- w~(9.0328
0.8090 X
10
4
)
9
+ 1.0406 X 10 = 0
O
X
10
_7 m
iS.:.
_ [C,A- mw A.,~(w2) (4307.25
2
2.2 = . 6 062
= 6.062 x 10- 4 mm 2 ]
M
(6.112)
+ 242.3732 ) + (119.7126 2 X 242.3732 2) =
X
X
(6.128) Y
2
48- 14.4265( 42.41) ](2.2) 11 7.7267 X 10
= 5.147 x 10- 7 rad
300
FOUNDATIONS FOR RECIPROCATING MACHINES
OVERVIEW
e. Combined Amplitudes of Vibration
axis of rotation and is given by
Vertical amplitude, A"
Ah (6.130a) 10- 4 + 4 X 5.147
5.29
=
25.878 x 10- 4 mm < 0.0125 mm
X
X
f.
=
AY + hA"
=
6.062
=
17.90 X 10- 4 mm <0.0125 mm
X
10- 4 + (3.33 -1.0299)5.147
X
10- 7 X 103 mm
Undamped natural frequency, w""'
(6.81a)
w,~,
X
8614.5 = 6460.8 t/m 3
/6460.8 X 400 = 182.353 rad/sec . 77 7185 29.02 Hz
= 'I
fn~· =
Undamped amplitude, A., A -
M,
~'-M[2
(6.83)
2
m,wn,~-w)]
0.185 77.7185[182.353 2 =
-
rA.,
(6.84)
=
(3 2 + 42 ) 112
X
7.567
X
10- 8
X
103 mm
<0.0125mm
Torsional Vibrations
C~, = 0.75C" = 0.75
=
= 3.7835 x 10- 4 mm
10- 7 X 10 3 mm
=
Horizontal amplitude, Ah (at bearing level of compressor) Replacing Ax witb AY in Eq. (6.130b) we get, Ah
301
42.41 2 ]
7.567 X 10- 8 rad
The maximum horizontal displacement occasioned by yawing occurs at the corner of the foundation farthest away from the vertical
6.10
OVERVIEW
Two methods for the analysis and design of foundations for reciprocating machines have been discussed: the linear elastic weightless spring method and the elastic half-space method. The effect of the soil mass participating in the vibrations and the nonlinearity of the soil have not been included. The effect of embedment is examined in Chapter U; and pile-supported machine foundations ar~,considered in Chap!er_ 12. Damping has' not been considerea'Jn the discussion of tbe linear elastic weightless spring method, but in the elastic half-space method it has been seen that radiation damping affects amplitudes, particularly at resonance. In tbe linear elastic weightless spring method, it has been assumed that the soil can be simulated by elastic springs. The assumption is in keeping with the small amplitudes of motion that are associated with the operation of a machine supported on a well-designed foundation. In the elastic half-space method, analogs have been established as spring-dashpot systems. Both the spring and the damping in the elastic half-space method can be defined in terms of the elastic constants of the material, the geometry of tbe problem, and the mode of vibration of the foundation. Fortunately, for the practicing engineer, these analogs yield satisfactory answers. Solutions for simultaneous rocking and sliding have been based upon the analogs derived from the theory of an elastic half-space; however, if sliding and rocking response frequencies differ by a factor of three or more, the two motions can be analyzed independently and the amplitudes of motion ~uperimposed (McNeill, 1969). The response of the soil--'foundation system can also be computed by using the concept of comJ?liance-impedance function. This method is discussed in Chapter 11 (Sec'ti.on 11.5). The mass of the soil spring has not been included in the foregoing analysis. The mass-spring system in Fig. 2.16 has a spring of weight w per unit length. With regard to the length of the spring, it is shown in Chapter 2 (section 2.8) that the natural frequency is defined by:
w~~-
~
kg W+~wl
(2.70b)
302
FOUNDATIONS FOR RECIPROCATING MACHINES
That is, one-third of the mass of the spring can be assumed to be concentrated at the center of the vibrating mass. Pauw (1953), Balakrishna and Nagraj (1960), and Balakrishna (1961) and Hsieh (1962) attempted to compute the effective in-phase soil mass (or mass moments of inertia of the soil) (Table 6.4). However, Richart et al. (1970) recommended that even tf the "in-phase mass" could be determined satisfactorily, this information would not lead directly to an evaluation of the vibration amplitude. Also, in fixing the quantities of the analog, the in-phase soil mass (or mass moment of inertia) has not been considered. Therefore, an in-phase soil mass may not be considered at this stage in the analysis, but the problem may be realistically analyzed in the future. The effects of the nonlinearity of a soil on a foundation's response have been studied by Novak (1970), Funston and Hall (1967), and Ehlerchritof (1968), but such effects may be insignificant in the design of ordinary machine foundations, because after the first few cycles of operation, the soil behavior essentially approaches that of linear elasticity for the small amplitudes of motion. However, nonlinear effects may be significant for radar installations and missile launching facilities.
Table 6.4. Effective Mass and Mass Moment of Inertia for Soil below a Vibrating Footing
Vertical translation Horizontal translation
Rocking Torsion (about vertical axis)
= 1/4
v = 1/2
l.Opr~ 0.2pr~
2.0pr~ O.lpr~
not
computed 0.3pr~
v 0.5pr~ 0.2pr~ 0.4pr~ 0.3pr~
0.3pr~
(1981) have found, however, that better correlations between computed and observed values of amplitudes are possible if the values of the sot! parameters are selected to be consistent with the effective confining pressures and shear strain amplitudes. Based on the results of small-scale field experiments, Novak (1985) pointed out that the elastic half-space theory grossly overestimates the values of geometrical damping. Further investigations are therefore needed to be able to arrive at methods for predicting realistic values of damping. Also the stiffness as well as damping are significantly influenced by an underlying hard stratum and this fact must be carefully considered while designing the foundations for the machines and also while evaluating its observed performance (Dobry and Gazetas, 1986; Novak, 1985). Based on an evaluation of the performance of a reciprocating compressor foundation Prakash and Puri (1981b) stressed the importance of conducting ' . . suitable in situ dynamic soil investigations and properly mterpretmg the test data to arrive at the design values of soil parameters. Moore (1971) and Prakash and Puri (1981b) observed that there is relatively little field confirmation pf the accuracy of commonly used design methods. Thus, there is a great need to monitor the pe,fformance of prototype machine foundations. Such efforts will be meaningful if necessary geotechmcal mvestigations of the sites are conducted simultaneously. '
REFERENCES
Effective mass or mass moment of inertia of soil Mode of vibration
303
REFERENCES
Source: Hsieh (1962). Published by Thomas Telford Publications
Data on the performance of machine foundations are scant. The procurement of such data will increase confidence levels in the design. Based upon a limited number of observations, Barkan (1962) found good agreement between the computed and the observed natural frequencies of vertically vibrating foundations designed on the basis of the linear elastic weightless spring method. His computed amplitudes, however, were not in agreement with observed amplitudes. Richart and Whitman (1967) compared model footing test results with solutions based on the elastic half-space method.
Their computed amplitudes for the vertical vibrations ranged from 50% below to more than 50% above the observed amplitudes. Prakash et al.
Arnold, R.N., Bycroft, G, N., and Warburton, G. B. (1955). Forced vibrations of a body on an infinite elastic solid. Trans. ASME 77, 391-401. Balakrishna, R. H. A. (1961). The design of machine foundations related to the bulb of pressure. Proc. Int. Conf. Soil Mech. Found. Eng., 5th, Paris Vol. 1, 563-568. Balakrishna, R. H. A., and Nagraj, C. N. (1960). A new method for predicting natural frequency of foundation-soil systems. Struct. Eng., 310-316. Barkan, D. D. (1962). Dynamics of Base and Foundations. McGraw-Hill, New York. Bycroft, G. N. (1956). Forced vibrations of a rigid circular plate on a semi-infinite elastic space and on an elastic stratum. Philos. Trans. R. Soc. London, Ser. A 248, 327-368. Chae, Y. S. (1969). Vibrations of non-circular foundations. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 95 (SM-6), 1411-1430. Dasgupta, S. P., and Rao, N. S. V. K. (1978). Dynamics of rectangular footings by finite elements. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. 104 (GT-5), 621-637. Dobry, R., and Gazetas, G:'~i(1986). Dynamic response of arbitrarily shaped foundations. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. 112 (GT-2), 109-135. Dobry, R., Gazetas, G., and Stoke, K. H. (1986). Dynamic response of arbitrarily shaped footings. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. 112 (GT-2), 136-159. Ehlerchritof, O.M. (1968). Non-linear parameters of vibrating foundations. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 94 (SM-6), 1190-1214. Elorduy, J., Nieto J. A., and Szekley, E. M. (1967). Dynamic response of bases of arbitrary shape subjected to periodic vertical loading. Proc. Int. Symp. Wave Propag. Dyn. Prop. Earth Mater., Albuquerque, NM, 105-121.
jU4
FOUNDATIONS FOR RECIPROCATING MACHINES
Funston, N. E., and Hall, W. J. (1967). Footing vibrations with non~linear subgrade support. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 93 (SM~5), 91-211. Hall, J. R. (1967). Coupled rocking and sliding oscillations of rigid circular footings. Proc. Int. Symp. Wave Propag. Dyn. Prop. Earth Mater. Albuquerque, NM, 139-148. Hayashi, K. (1921). "Theorie des Tragers auf Elastischer Unterlage." SpringerNerlag, Berlin and New York. Heteyni, M. (1946). "Beams on Elastic Foundations." Univ. of Michigan Press, Ann Arbor. Hausner, G.W., and Castellani, A. (1969). Discussion on the paper, Comparison of footing vibration tests with theory by F. E. Richart, Jr. and R. V. Whitman. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 95 (SM-1), 360-364. Hsieh, T. K. (1962). Foundation vibrations. Proc. lnst. Civ. Eng. 22, 211-226. Kobori, T. (1962). Dynamical response of rectangular foundation on an elastic half space. Proc. Jpn. Nat!. Symp. Earthquake Eng., 81-86. Kuhlemeyer, R. L. (1969). Vertical vibrations of footing embedded in layered media. Ph.D. Dissertation, University of .California, Berkeley. Lamb, H. (1904). On the propagation of tremors over the surface of an elastic solid. Philos. Trans. R. Soc. London, Ser. A 203, 1-72. Lysmer, J., and Richart, F. E., Jr. (1966). Dynamic response of footing to vertical loading. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 92 (SM~1), 65-91. McNeill, R. L. (1969). Machine foundations, soil dynamics specialty session. Proc. Int. Conf. Soil. Mech. Found. Eng., 7th, Mexico City, pp. 67-100. Moore, P. J. (1971). Calculated and observed vibration amplitudes. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 97 (SM-1), 14!-158. Novak, M. (1970). Prediction of footing vibrations. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 96, (SM-3), 837--861. Novak, M. (1985). Experiments with shallow and deep foundations. Proc. Symp. Vib. Frob. Geotech. Eng., Am. Soc. Civ. Eng., Annu. Conv., Detroit, 1-26. Pauw, A. (1953). A dynamic analogy for foundation soil systems. ASTM Spec. Tech. Pub[. STP 156, 90-112. Prakash S., and Puri, V. K. (1969). Design of a typical machine foundation by different methods. Bull. Indian Soc. Earthquake Techno!. 6 (3), 109-136. Prakash, S., and Puri, V. K. (1980). "Design of a Compressor Foundation-Observations and Predictions," Mach. Found. Des. Anal. Lect. Notes. University of Missouri-Rolla, Rolla. Prakash, S., and Puri, V. K. (1981a). Dynamic properties of soils from in-situ tests. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. 107 (GT-7), 943-963. . Prakash, S., and Puri, V. K. (1981b). Observed and predicted response of a machine foundation. Proc. Int. Conf. Soil Mech. Found. Eng., lOth, Stockholm, Vol. 3, 269-272. Prakash, S., Puri, V. K., and Horst, W. D. (1981). Some aspects of machine foundation design. Proc. Int. Conf. Soil Mech. Found. Eng., 10th, Stockholm, Vol. 4, 868-871. Quinlan, P. M. (1953). The elastic theory of soil dynamics. ASTM Spec. Tech. Pub!. STP 156, 3-34. Reissner, E. (1936). Stationare Axialymmeterische durch line Schuttelnde Masse Erregte Schwingungen lines Homogenen Elastichen Halbraumes. 'lng. Arch. 7 (6), 381-396. Reissner, E. (1937). Freie und Erzawungene Torsionschwingungen des Elastichen Halbraumes. Ing.-Arch. 8(4), 229-245. Reissner, E., and Sagoci, H. F. (1944). Forced torsional oscillations of an elastic half space. J. App/. Phys. 15, 652-662. Richart, F. E., Jr. (1962). Foundation vibrations. Trans. Am. Soc. Civ. Eng. 127, Part I, 863-898.
305
REFERENCES
· F E J d Wh"tman R. v. (1967). Comparison of footing vibrations tests with Rtchart, . ., r., an 1 , . ( M 6) 143 168 theory. J. Soil Mech. Found. Div., Am. Soc. Clv. Eng. 93 S - , · Richart, F. E., Jr., Hall, J. R., and Wood~, R. D. (1970). "Vibrations of Soils and Foundations." Prentice-Hall, Englewood Cliffs, New Jersey. Sung, T. Y. (1953a). Vibrations in semi~infinite solids due to periodic surface loading. ASTM Spec. Tech. Pub/. STP !56, 35-63. . . . Sung, T. Y. (1953b). Vibrations in semi-infinite solids due to penodtc surface loadmg, S. D. Thesis, Harvard University, Cambridge, Massachusetts. . Terzaghi, K. (1943). "Theoretical Soil Mechanics." Wiley, Ne~ York. . K • (!955) . Evaluation of coefficients of subgrade reaction. Geotechmque, 5, 297-326. 1, . erzag h T Thomson, w. T., and Kobori, T. (1963). Dynamical compliance of rectangular foundattons on an elastic half space. Trans. ASME 30 579-584. Warburton, G. B. (1957). Forced vibrations of a body upon an elastic stratum. Trans. ASME 24, 55-58. 't R v and Richart F. E. Jr. (1967). Design procedures for dynamically loaded Wh1 man, · ., ' ' · ( 6) !69 193 foundations. J. Soil Mech. Found. Div., Am. Soc. Clv. Eng. 93 SM- , ·
!.·
7
307
METHODS OF ANALYSIS
Foundations for Impact Machines
There are several types of machines that produce transient dynamic loads of short duratiOn that may be characterized as impacts, shocks, or pulses. Examples of shock producing machines are the hammers, presses, crushers, and mills. Ha'?mers are most typical of impact machines and are of many types. Accordmg to their functiOns, they may be classified into forging hammers (proper) and drop hamm~rs. Drop hammers are used for a variety of purposes such as forgmg, stampmg, and ore breaking. The hammer-foundation system consists of a frame, a falling weight known as "tup," the anvil and the foundation block as shown in Fig. 7.1. The. fra'?e may be mounted on the foundation block (Fig. 7.1a) or on the anvil (F1g. 7.lb). The anvil is a massive steel block on which material is forged into desired shape or br?ken by the illlpact of repeated blows of the tup. The anvil may be fixed or 1! rna~ move m a manner similar to the tup. An impact machine Ill WhiCh the anvil also moves is known as a counter-blow hammer. Impact machines are rated according to the nominal weight of the tup or the droppmg parts and the height of the drop. The tup's movement is me.asured by the number of blows or impacts it makes per minute. The weight of the tup may vary from 0.25 to 10 tons. The height of the drop may range from 0.3 to 2.0 m or more. Steam or air pressure may be used to lift the tup, which 1s then allowed to fall freely. This class of hammers is known as drop hammers. The forging P?wer of the hammer may be enhanced by
~ncrea_smg the velocity of drop wtth compressed air-·or steam. The hammers
m which steam or air pressure acts on the tup, both during the process of hemg lifted or dropped, are known as double-acting hammers. A part of the Impact energy IS used up in causing plastic deformation of the material being
(a)
(b)
Figure 7.1. Typical arrangement of a hammer foundation resting on soil. (a) Frame mounted on foundation
l
blo_fJ<;
(b) frame mounted on;~,tl:!,e anvil.
operation on the foundation itself, and also on the adjacent structures and machines and people in the vicinity. The foundation for a hammer generally consists of a reinforced concrete block. In case of small hammers, the anvil may be mounted directly on the foundation block (Fig. 7.2a). To reduce the transmission of impact stresses to the concrete block and the frame, an elastic pad consisting of rubber, felt, cork, or timber is generally provided between the anvil and the foundation block (Fig. 7.2b). In case of high capacity hammers, special elements such as coil springs and dampers may be used in place of elastic pads (Fig. 7.2c). The foundation block is mostly designed to rest directly on soil (Figs. 7.2a, b, c). When the soil conditions are poor, the foundation block may be supported on piles. The foundation block may have to be supp?rted on elastic pads or spring absorbers, if necessary, to reduce transmission of · vibrations to adjoining facilities (Figs. 7.2d, e).
7.1
METHODS OF ANXLYSlS
The dynamic response of hammer foundations is computed by modelling them as lumped-mass systems. The number of degreess of freedom that must be considered in the analysis depend on the foundation type and also
forged and conversiOn mto heat, and the remaining energy is transmitted to
on whether the tup moves along the centerline of the system or at an
the foundation and the soil. A proper design of the foundation for the hammer IS therefore essential to avoid any harmful effects due to hammer
eccentricity. When the anvil is rigidly mounted on the foundation block without an absorber pad (Fig. 7.2a) and the impact is central, the system
306
Anvil and /
foundation block
Foundation block
I
resting on soil
Elastic pad below anvil
Anvil and
-tr-
~1
1
foundation block (b)
(a)
Air gap
(a)
Spring
absorbers
~jt:[}i];'11r'"r.;.nrnr~t~
~Tup
llft0Jf~J~€6~f,ht
Anvil Damping in pad
~Tup
I
Spring k3 of pad
I
below anvil
Foundation block resting on soil (d)
(c)
Anvil Damping in absorber
Foundation
I Spring k2 of
block
/
Trough
Foundaf1on
Foundation block
elastic pad
~1
L"""'----,,---r:-oii'amping in the pad
block Spring absorber below foundation
block
Damping in soil (e)
(d) (c)
(c) anvil on spring absorbers; (d) foundation block on elastic pad; (e) foundation block on
. h -foundation-soil systems: (a) single~degree· Some models f?r repre_sentm~re:~~e:ee~oMreedom model (eccentric impact); (c) of~freedom model (central •mpact), (b)l '. I)~ (d) three~degree~of-freedom system (central two-degree-of-freedom system (centra lmpac ,
springs.
impact).
Figure 7.2. Schematic diagram showing different arrangements for supporting anvil and foundation block: (a) anvil resting directly on the foundation block; (b) anvil on the elastic pad;
Figure 7.3.
309 308
311 FOUNDATIONS FOR IMPACT MACHINES
310
will undergo only vertical vibrations and rna y be modeled as a single-degreeof-freedom system, as shown in Fig. 7.3a. When the impact is at an eccentricity, the same system of Fig. 7.2a will undergo not only vertical vibrations, but also rocking (rotation) in a vertical plane and sliding (horizontal translation) and may thus be modeled as shown in Fig. 7.3b. For cases when the anvil rests on an elastic pad and the foundation block rests directly on soil (Fig. 7.2b, c), the foundation soil system may be modelled as shown in Fig. 7.3c. If the impact is central, the system of Fig. 7 .3c can be analyzed as a two degree freedom system undergoing vertical translation. If the impact is at an eccentricity, each of the masses" m 1 and m 2 (Fig. 7.3c) will have three degrees of freedom consisting of vertical translation and coupled rocking and sliding resulting in an overall system with six degrees of freedom. When the foundation block rests on absorber (Fig: 7.2d), the system can be modeled as shown in Fig. 7.3d and will have three degrees of freedom for central impact and, overall, nine degrees of freedom when the impact is at an eccentricity. As the stiffness of trough (Fig. 7.2d) is usually very high compared to tbat of the pad below the foundation block, the trough may be assumed to be rigidly supported on the soil (Novak, 1983) and a two-mass model will be sufficient for all practical purposes. The eccentricity of impact is generally avoided by suitably controlling the geometrical layout of the foundation, and proper alignment of the tup and frequent maintenance, and most practical cases can thus be analyzed by using a two-degree-of-freedom model (shown in Fig. 7.3c). The computations can be further simplified by making the following assumptions: 1. The anvil, foundation block, frame, and tup are rigid bodies. 2. The pad and the soil can be simulated by equivalent weightless, elastic springs. 3. The damping of the elastic pad and soil is neglected. 4. The time of impact is short compared to the period of natural vibrations of the system. 5. Embedment effects are neglected. Validity of the Assumptions
Assumption 1 about the rigidity of the anvil, foundation block, tup and frame is practically correct. The pad material and·' soil cari be considered to behave elastically (assumption 2) for small amplitudes of vibration. As the pad between the anvil and the foundation block becomes older, it may start losing its elasticity and should be replaced after regular intervals of operation. Assumption 3 about neglecting the damping in the pad and the soil is not correct. The foundation block supporting the anvil undergoes vertical vibrations and a significant amount of geometrical as well as material
METHODS OF ANAlYSIS
Foundation block I
L-!:===t====~?ss~o·,ll spr"m-it'~f I
:::wk//ffff~//////,/,1,/I///,?/
Figure 7 .4. Simplified twowdegree-of-freedom model.
· ·ated w1'th this mode of vibration (Section 6.4). Similarly, · f h d d ampmg are assoc1 the elastic pad has a finite damping depending upon the matenal o t e/-a · The duration of the impact (assumption 4) is generally very sma11 an l(s a (N ovak , 1983) · The embedment . effects as· conservative assumptiOn . sumption 5) can be neglected if an air gap or a trench filled w1th sawdust IS constructed around the foundation block, otherwise the embedment may significantly modify the frequencies and amphtudes (Novak, 1970). 'b . These assumptions simplify the process of computatmn o~ VI ratton characteristics of the hammer foundation, which can be ldeahzed as an undamped two-degree-of-freedom system, as shown Ill F1g. 7 .4. Equations of Motion
,.
The s stem shown in Fig': 7.4 is similar to the two-degree-of-freedom system . F' y 2 22 and the equations of motion for free vibrations may therefore be m 1g . . written as
m1
z
1
+ k 1z 1 + k,(z 1 - z 2 )=0 m 2i
in which
2
+ k 2 (z 2 - z 1) =0
(2.95c) (2.95b)
312
FOUNDATIONS FOR IMPACT MACHINES
m 1 =mass of the foundation and includes the mass of the backfill and the frame (if mounted on the foundation block as in Fig. 7.1a) m, =mass of the anvil (and includes the mass of the frame if it is mounted on the anvil as in Fig. 7.1b) k 1 = equivalent soil spring below the foundation block for vertical vibrations k 2 = equivalent spring of the pad below the anvil z 1 = displacement of the foundation z 2 =displacement of the anvil The value of the equivalent soil spring k 1 may be ~htained either by using the elastic half-space method (Richart and Whitman, 1967; Richart et al., 1970) or from the linear spring approach (Barkan, 1962). The value of k 1 from the elastic half-space method is obtained as
= k = 4Gr0
k 1
'
(1- v)
METHODS OF ANALYSIS
313
= the limiting natural frequency of the anvil and foundation resting on soil (assuming the anvil is rigidly attached to the foundation block) w,,, = the limiting natural frequency of the anvil vibrating on the elastic pad or springs, and
w
nl,
The values of wn 11 and wn 12 may be calculated as wnll =
~ m: m 1
(2.99a)
2
and (2.99b)
(6.18) ~'
Similarly the value of k, from linear spring approach may be obtained from Eq. ( 4.27): ( 4.27) in which A 1 = area of the foundation block in contact with the soil. The value of k 2 may be obtained from Eq. (7.1):
''- ,;,.'f'
Amplitudes of Anvil and Foundation Motion The amplitudes of anvil and foundation vibration may be computed by considering the free vibrations of the anvil-foundation soil system as being triggered by an initial velocity imparted to the anvil by the impact of the ram on the metal piece being forged. The particular solutions for amplitudes of vibration may be obtained by expressing z 1 and z 2 as follows:
(7.1)
(7.2a)
in which
(7.2b)
E = Young's modulus for the material of the pad b = thickness of the pad
The initial conditions of vibration may be expressed by using Eq. (7.3). At timet= 0
A, =area of the anvil base in contact with the pad
(7.3a)
In case the anvil is supported on spring absorbers, k 2 should be calculated as the combmed stiffness of all the springs.
(7.3b) ""+:.
Natural Frequencies The two natural frequencies w, 1 and w, 2 of free vibration of the system of Fig. 7.4 may be determmed from the frequency equation given below:
in which Va is the initial velocity of anvil motion. By substituting z 1 and z 2 from Eqs. (7.2) into Eqs. (2.95b and c) and using the initial conditions given by Eq. (7.3), the expressions for z 1 and z 2 are obtained Eq. (7.4): (7.4a)
(2.98) in which
and
314
FOUNDATIONS FOR IMPACT MACHINES
315
METHODS OF ANALYSIS
Initial Velocity of Anvil Motion V,
The contribution of the higher of the two natural frequencies ' wnl and w n2' towards the amplitude of motion is small (Barkan, 1962) and may be neglected for all practical purposes. By neglecting the terms of sin wn 1t (in which wn 1 > w" 2 ) in Eqs. (7.4a and b), the maximum displacements (sin w" 2 t = 1) of the foundation and anvil become (7.5a)
The initial velocity of the anvil just after the tup's impact can be determined by using the law of conservation of momentum. The impact of the tup on the anvil may be central or eccentric (Fig. 7.5). In cases of centrallffipact, the initial velocity V" may be computed by assuming that the impact takes place in a vertical plane through the centroid ~f th~ foundal!on. The centered impact will result only in translatiOnal vtbrauons m the verl!cal direction and only linear momentum need be considered. The momentum ?f the tup before impact is (W)g)VT,. The anvil is initially at rest, and tts mo1Uentum before impact is zero. The momentum of the tup and anvtl after impact is given by
w
w
-"v+--'V g 1 g a
and (7.5b) The values of amplitude of anvil and foundation can thus be determined from Eq. (7.5a, b) by substituting the values of wnl, wnl• Wnz• and V,. The initial velocity of anvil motion can be determined by considering the impact of the tup on the anvil and the velocity of the tup at the time of impact.
in which V is the velocity of the anvil after impact, W2 the weight of the anvil (including the frame if it is mounted on the anvil), and V, the v~locity of the rebol!Ild of the tup after im,pf!Cl. Accordmg to the pnnctpal of tmpact for rigid bodies, the momentum' before and after the impact in a conservative system is constant. Therefore,
w
gTI
Initial Velocity of Tup at the Time of Impact
For a single-acting drop hammer, the initial velocity of the tup VT, at the time of impact after the !up's free fall is (7 .6)
in which h is the drop of the tup in meters, g the acceleration occasioned by gravity, and 17 the efficiency of the drop. For efficiency of drop TJ, one must consider the energy lost in overcoming the friction to the !up's movement and the resistance of the counter pressure of air or steam. For well-adjusted hammers TJ will be close to 1 and a value of 0.65 may generally be used. For double-acting hammers, operated by pneumatic or steam pressure, VT, is given by: (7.7) in which
w
w
_ov.=-"V +-'V gt
ga
(7.8)
Equation (7.8) has two unknowns, V1 and V,. A second equation may be
l'
~
I
I
Tup
I : Anvil
Ch e.G.\
r
•(
Elastic pa d
I
I
Foundation
Wa =the gross weight of the dropping parts, including upper half of the die p = steam or air pressure A P = the net piston area
block
i I
Figure 7.5. Model for calculating initial velocity of anvil for the case of eccentric impact.
316
FOUNDATIONS FOR IMPACT MACHINES
obtained from Newton's law, according to which the coefficient of elastic restitution "e", is defined by
317
METHODS OF ANALYSIS
after eccentric impact, it is V, + r¢,- V1 • By applying Eq. (7.9), we obtain (7.13)
relative velocity after impact e = relative velocity before impact
(7.9) From Eqs. (7.8), (7.12), and (7.13), the values of V, and¢, are obtained as
The coefficient of elastic restitution depends upon the material of the bodies involved in impact. For impact between two perfectly rigid bodies, e is equal to 1. When a rigid body impacts a plastic body, e is zero. The value of e thus lies within the range of 0 and 1 for forge hammers. The value of e depends on factors such as the temperature of the forged piece, the dimensions and form of the grooves in the stamping hammers, and the elastic properties of the tup, die, and anvil. The coefficient of restitution increases with the number of blows as the piece being forged cools. To account for the most unfavorable conditions of operation, a value of e = 0.5 may be adopted for design purposes. Equation (7.9) is the second equation used to solve for V1 and V, and may be rewritten as
(7.10)
follows: (7.14a) and . _ "'"- (1
s(1 + e)r V: . 2 2 + s)(r +e)- r n
(7.14b)
in which k' is equal to Mm 2 g/W0 • When r is zero, i.e., for central impact, Eqs. (7.14a) and (7.14b) yield
This expression for V, is the same as in Eq. (7.1la).
From Eqs. (7.8), and (7.10), the value of V, is
va =
1+e 1+ s
VTi
(7.1la)
in which (7.1lb)
m, =mass of the dropping parts. If the impact of the tup on the anvil is not central, then in addition to the linear motion in the vertical direction, the anvil will rotate around the axis that passes through the center of gravity of the anvil and is perpendicular to the plane of the impact. In addition to the linear momentum angular momentum also needs to be considered in the case of eccentric impact. The moment of momentum of the tup and anvil is given by (Fig. 7.5). (7.12) in which r is the eccentricity, Mm 2 the mass moment of inertia of the anvil and tup around the axis of rotation, and ¢,the initial velocity of rotation of the anvil. The relative velocity of the system before impact is CVn - 0), and
Stress in the Pad Maximum compressive stress in the elastic pad below the anvil depends upon the relative displacements of anvil and the foundation block. The worst case of compression in the pad develops when the anvil moves downward, and at the same instant of time, the foundatiOn block moves upward. The maximum compressive stress in the pad IS thus expressed by u p = k,
z, A+ z,
(7.15)
2
Dynamic Force Transmitted by the Foundation The dynamic force
Fdyn
transmitted to the soil is given by Fdyn =
k,Z,
(7.16)
Stresses in the Soil Stresses transmitted to the soil u through the combined static and dynamic loads are expressed by (7.17)
318
7.2
FOUNDATIONS FOR IMPACT MACHINES
DESIGN CRITERIA
A hammer foundation mu~t be . designed to ensure long-term efficient operatwn ~f the hammer With m1mmum disturbance to the environment This obJective can be achieved if the foundation for the hammer is designed to possess the characteristics given below:
DESIGN PROCEDURE FOR HAMMER FOUNDATIONS
319
of a foundation. Sometimes, the amplitudes of vibration of a foundation block may be within the specified limits from consideration of the operation of the tup but not acceptable for adjacent machines and structures. In such cases, it may become necessary to design a foundation that is supported on springs or vibration absorbers. Restrictions on available area for the machine's foundation may also necessitate the use of absorbers. The design of foundations on absorbers is discussed in Chapter 10.
1. Its natural frequency should be either 30 percent smaller or 21 time 2 s the frequency of the impacts. 2 .. Natural frequency of the foundation should not be a whole number multiple of the operating frequency of the tup . . 3 .. The amplitudes of vibration of the anvil and the foundation should be
With~n the permiSSible values specified by the manufacturer. If these specificatiOns from ~he manufacturer are not available, the values of maximum
7.3
Having studied the analyses of a foundation for an impact machine, a step by step design procedure can now be outlined for this type of foundation.
allowable amplitudes may be obtained from Table 7.1. 4. The maximum stresses in the elastic pad below the anvil should not exceed the permissible values. The allowable stresses in the elastic pad depe_nd upon the material ofthe pad. The stresses for timber pads may be obtamed from codes of practiCe. For material, such as rubber or neoprene the values of the permissible stresses are generally specified by the supplier: in the soil should not exceed the permtsst · "ble values, h.5.h The · 0 stresses · w IC IS .8 times the value for purely static loads. 6. The design of the entire foundation system should be such that the centers of grav.Ity of the anvil, the foundation block, frame, and the centroid of t,he foundatiOn contact area lie on a vertical line defined by center of th tup s descent. e
The criteria mentioned above should ensure the satisfactory performance
Table 7.1. Maximum Allowable Amplitudes for Hammer Foundations
Maximum Amplitude, mm
Weight of the Tup (ton)
Anvil
Foundation Block
(!)
(2)
(3)
<1 2
1.0 2.0
>3
1.2 1.2
4
1.2
Source: N~vak (1983). Reproduced with the permission of the Canadwn Geotechnical Journal.
DESIGN PROCEDURE FOR HAMMER FOUNDATIONS
Design Data
Obtain all the necessary information pertaining to the machine, soil conditions, and layout of the floor pla~; ..9f the shop.
Machine Data. All data pertaining to the size and the weight of various components of the drop hammer, method of supporting the frame, height of movement of the tup, and weight of the upper and lower parts of the die should be obtained. Information on the permissible amplitudes of the anvil motion and the foundation block should also be obtained. If this information is not available, the amplitudes of motion given in Table 7.1 may be considered as limiting values.
Soil Data. For drop hammers of up to 1-ton capacity, soil data should generally be collected to a depth of 6 m. For heavier impact machines, it is preferable to investigate soil conditions to a depth of 12m or to a hard stratum. If piles are to be used, the investigation should be conducted to a suitable depth. Layout Plan. The layout plan of the shop in which the drop hammer is to be installed should be obtained, and the position of any precision machines in the vicinity and adjoining structures that may be adversely affected by the hamm~'s operation should be ascertained so that vibration isolation measures may be incorporated at the design stage, if necessary. 2
Soil Constants
The values of the dynamic soil constants should be determined following the procedure suggested in Chapter 4 in a manner that is consistent with
conditions of confining pressure and anticipated strain levels likely to be induced in the soil by the prototype foundation.
320
FOUNDATIONS FOR IMPACT MACHINES
321
DESIGN PROCEDURE FOR HAMMER FOUNDATIONS
The allowable soil pressure below the foundation should he determined from appropriate field tests.
k,
=
4Gr" 1- v
(6.18)
EA 2
(7.1)
k,= -b3
Trial Dimensions of the Foundation Components
Select the trial dimensions of the anvil, foundation block, and size and thickness of the pad below the anvil. The guidelines given below may be followed in determining the trial dimensions. The area at the base of the foundation block should be such that the safe loading intensity of the soil is never exceeded while the drop hammer is operating. Area.
Depth. The depth of the foundation block should be designed so that the block is safe in punching shear and in bending. The inertial forces that develop should also be included in the calculations. The minimum thickness of the foundation block should be 1.0 m for light hammers and 2.5 m for hammers of medium weight. The weight of the anvil may generally be kept at 20 times the weight of the tup. The weight of the foundation block, Wu generally varies from 60 to 120 times the weight of the tup (Barkan, 1962; Major, 1980). Weight.
in which G = dynamic shear modulus of the soil r o = equivalent radius of the foundation v =Poisson's ratio E =Young's modulus of pad material A = area of the anvil 2 '1 b = thickness of pad below the anv1 ·
Equivalent radius r" is given by: (6.58a) . in which A is the area of the foundation block. The nat~ral frequencies of combined system are computed as follows. (2.98)
Thickness of the Elastic Pad. The thickness of the elastic pad below the anvil and the foundation block (if necessary) depends on the material of the pad and should be ascertained by analysis. 5 Velocity of Tup and Anvil 4
Natural Freqencies of the Hammer- Foundation System
Compute the limiting natural frequencies
and
W 01 [
W 01
'
Compute the velocity Vn of the tup before impact as follows:
as Vn=
11Y2g(W" + pAP)h W
(7.7)
0
(2.99a) in which (2.99b) in which k 1 =equivalent soil spring k 2 = equivalent stiffness of elastic pad below the anvil m 1 =mass of the foundation block m 2 =mass of the anvil (including mass of the frame if attached to it)
Values of k 1 and k 2 are given by
W = the weight of the tup p" = the steam of ai~. pressure A = the area of the piston (net) = the vertical distance of the !up's movement =the efficiency of the fall, which is usually taken as 0.65 11
h
Compute the velocity of the anvil V, after impact as follows:
_l+ev V,- l+s Ti
(7.lla)
322
FOUNDATIONS FOR IMPACT MACHINES
in which e is the coefficient of elastic restitution (the value of e may be taken as 0.5) and s is equal to W,fW0 • 6
323 DESIGN EXAMPlE
7.4
EXAMPLE
It is proposed to install a 2.0 t forging hammer in an industrial complex. The
data pertaining to the hammer is given below:
Motion Amplitudes of the Foundation and Anvil
Compute the maximum foundation and anvil amplitudes as follows: For the foundation,
(7.5a) For the anvil,
(7.5b)
Weight of tup without die = 1550 kg Maximum tup stroke h = 750 mm Weight of the upper half of the die = 450 kg Area of the piston A= 0.14 m' 2 Supply steam pressure p = 70 t1 m Weight of anvil block = 32.5 t Total weight of hammer = 45.5 t (anvil+ frame) . 2 Bearing area of anvil= 2.0 X 2.0 = 4.0 m Permissible vibration amplitude for anvil = 1.5 mm Permissible amplitude for foundation= 1.0 mm Material of pad below the anvil'" ·Pine wood , I 2 Modulus of elasticity of the pad= 5 x 10 t m Allowable compressive stress in pad = 400 t1 m 2 perpendicular to grain ~.,
';'y-
in which wn 2 is the smaller natural frequency. 7
Dynamic Stress in the Pad
Compute the dynamic stress in the pad with
uP=
k 2 (z 1 - (-z 2 )) A
(7.15)
__ .,~
Thickness of timber pad= 0.60 m Efficiency of drop '1 = 0.65 Coefficient of elastic restitution e = 0.5
2
The computed values of the natural frequencies and motion amplitudes should satisfy the criteria for safe operation. Also, the stresses in the pad should be smaller than the permissible values for the pad material.
8
Stresses in the Soil
The details of the suggested foundation are shown in Fig. 7 .6. The depth of the foundation is 2.4 m below the natural ground level. The allowabJe soil pressure at the base of the foundation was determmed to be 25 t1 m . The data on dynamic soil properties is as giVen m Example 4.9.3, (Chapter 4). Assume unit weight of reinforced concrete equal to 2.4 tim and the umt weight of soil equal 2.050 tim'. Check and comment on the adequacy of the foundation shown in Fig. 7 .6.
Compute the stress in the soil with
(7.17) in which u represents the stress in the soil, and A 1 the contact area of the foundation block with soil. The stress in the soil should be less than 0.8 times the allowable stress in soil when static loads alone are supported. A computer program for calculating the dynamic response of a hammer foundation using the above listed design procedure is given in Appendix 2.
Solution Design Data Machine Data.
The data on the hammer is listed in the problem. 2
Sm·z oata. The allowable soil pressure is given as 25 t/m· FThe 4data 47 on dynamic soil properties is given in Example 4.9.3. Plot C m 1g. · · · G vs. )' at 2 • 4-m depth at a mean effective confimng pressure of g~ves
l.Okg/cm
2
0
325
DESIGN EXAMPLE
FOUNDATIONS FOR IMPACT MACHINES
324
4
Floor
Selection of Dynamic Soil Constants
I
Width of the foundation = 6 m
Anvil
Amplitude of vibration= 1.0 mill
}oo Foundation block
. . Shear stram level=
f--- 2000 ----+-\
6
l.O 1 66 10- 4 x 1000 = . x 4
From Plot C in Fig. 4.47 the value of G corresponding to 'Yo= 1.66 x 102 and U = 1.0 kg/cm 2 is found to be 535 kg/cm • The0 mean effective pressure below the foundation is now calculated ~-----------8000--------~..1 (a)
uvl = iiv1 + uv2 2
ii, = yZ = 3 x 2.05 = 6.15 t/m = 0.615 kg/cm
Anvil
T
fc-(J"v2
2400
1
4q[
=
in which 2
2
q =static stress intensity= 6.50 t/m = 0.65 kg/m I= influence factor (from Fig. 4.46) 4 3 m=3=1.33,n=3=1 I= 0.190
1--------oOOO------~ (b)
Figure 7.6. section.
2
1
Layout of the foundation (Example 7.4.1). (a) Longitudinal section and (b) cross
ii, 2 = 4 x 0.650 x 0.19 = 0.494 kg/cm
2
ii, = 0.615 + 0.494 = 1.109 2
Weight of the Foundation and Contact Area
From the dimensions shown in Figs. 7.6a and b, we get:
I
Assume Ko = 0.5 (j
2
Foundation area in contact with soil = 8 x 6 = 48m Weight of the foundation block W1 = [8 x 6 x 2.4- 2 x 2 x 1.0] x 2.4 t = 266.88 t Weight of the anvil and frame= W2 = 45.5 t 266 88 · = 27.2 t sec'im Mass of foundation block m 1 = . 45.5 9.81 Mass of anvd m 2 = . = 4.62 t sec 2 /m 9 81 3
Check on Soil-Bearing Capacity . 266.88 + 45.5 Soil pressure= x = 6.50 tim'< 25 tim' 8 6
= 1.109 (2 X 0j5 + l) = 0.738 kg/em'
0
The value of G for design 2 0.(738) 0.5 2 = 460.0 kg/cm = 4600 t/m = 535 ( -'-··1.0
5
Natural Frequencies of Soil-Foundation System
Limiting natural frequency of the anvil on the pad
w.,,=~
w.,, (2.99b)
326
FOUNDATIONS FOR IMPACT MACHINES
5
10 4 _ 0 60 X
X
4 = 33.3
X
4
33.3 X 10 wnlz = = 7.186 _ 4 62 2
X
w~ 1 = 8.471
10 4 t/m
4
10 /sec
=
X
4
10 /sec
w! 2 = 0.3346
I k, 'Y m I + m 2
(2.99a)
10 4/sec
X
6 Velocity of Dropping Parts
/ (Wo + pAP)h
_ k = 4Gro
r" =
(7.7)
0
V
'1 = 0.65,
148 =3.908m V-:;; Vnc;=
4 X 4600 X 3.908 ( 1 0 _33 ) = 107324 · 17 tim
0.65~2 x 9.81
Velocity of anvil motion
2 107324.17 2 wnl, = 27 _2 + _ = 3370.74/sec 4 62
(
2
2
h=0.75m
and
:?~2x 0 · 14 )
x 0.75 = 6.056 m/sec
v. 1+e
v. =
Ratio of the masses
p = 70 tim'
W0 = 2.0 t,
AP=0.14m
Assume v = 0.33 kl =
w
'1\j2g
VTi-
1-
2
wnz = 57.84/sec
k 1 =equivalent soil spring.
I
2
wn1 = 291.36/sec
2
Limiting natural frequency of the whole system resting on soil wn 1, wnll
327
DESIGN EXAMPLE
(7.11)
1 + s Vn
1
05 + · (6.056) = 0.382 m/sec 45.5 1+-2-
=
m
1-' = ----'- = 0.1705 ml
The two natural frequencies of the system are given by: 7
w~- (1 + J.<)(w! 12 + w!, I )w! + (1+ J.<)wn 12 wn 1l = 0 2
2
Amplitudes of Vibration
Amplitude of vibration of foundation (2.98)
w~- (1 + 0.1705)(7.186 X 10 + 0.337074 X 104 )w! 4
+ (1 + 0.1705)(7.186
w~- 8.8056 _
2
[
wnl.z2
w 111
_ -
X X
8.8056 ±
10 4)(0.337074 4
X
10 w~ + 2.8345
Y(8.8056)
2
8.8056 ± 8.1363 2
J
X
4
10/sec
-2
z
z
Wnlz(wnl
z )
~ Wnz Wnz
V "
(7.5a)
Z 1 = 0.00099 m = 0.99 mm < 1.0 mm ~~:
0
4 X 2.8345]
-
2 [
Z1 =
10 4 ) = 0
X 10 8 =
2 2 )( 2 2 ) (wnl2- Wnz Wnlz- Wnl
2
Z X 10 4/sec 2
2 -
2
wn/2-wnl 2 2)
(wnl- Wnz Wnz
V "
(7.5b)
Z 2 = 0.00104 m = 1.04 mm < 2.0 mm The motion amplitudes of the foundation block and the anvil are smaller than the permissible values. The foundation is therefore adequate.
328
8
FOUNDATIONS FOR IMPACT MACHINES
Stress in the Pad
REFERENCES
329
The stress in the soil is computed by Eq. (7 .17) and is less than (0.8 x 25 t/m 2 ).
models with two or more degrees of freedom are to be used (Novak, 1982, 1983, 1985a; Novak and El Hifnawy, 1983). The uplift of the anvil may be a serious problem in many hammer installations. Uplift takes place because in most systems the mass of the anvil may be too small to eliminate tension in the anvil pad. The occurrence of the phenomenon of uplift was confirmed in a full-scale experiment conducted on a large hammer installation (Novak, 1985b; Harwood and Novak, 1986). The inclusion of the uplift phenomenon in design will necessitate a very involved analysis. The uplift of the anvil adversely affects the performance of the hammer. The uplift of the foundation block is less common.
The above ex~mple was also solved using the computer program given in Appendtx 2. A hstmg of the input data and the results is shown in Appendix
REFERENCES
_ k (2 1 - (-22 )) 2 A
(7.15)
(J"P-
2
=
33.3 x
w• (0.99 + 1.04) ( 1000)( 4 )
9
_ , , -169.16t/m <400t!m
Stress in the soil
2.
7.5
OVERVIEW
A simple two-degree-of-freedom model for predicting undamped vertical ~espouse of an anvil-foundation soil system has been discussed. This model ts an approximate representation of a rather complex system and has been com~only used (Barkan, 1962; Major, 1980; Prakash, 1981). The vibration ~mphtudes have been computed by treating the impact as a pulse of mfimtely ~hort du~ation that dies out before the system starts moving, thus resultmg m free v1bratwns triggered by initial velocity. The duration of the pu.lse m case of hammers varies from 0.01 to 0.02 second and as the piece bemg forged becomes cold, the pulse duration gets reduced to 0.001 to 0.002 sec (Novak, 1982, 1985b; Novak and El Hifnawy, 1983). The assumptiOn of a sho~t duration pulse is thus reasonable. The amplitude response can be esllmated by usmg Fourier analysis. Lysmer and Richart (1966) presented a complicated analysis for predicting response of a footing subjected to transtent loadmg u~mg an assumed time history of the pulse. Unfortunately, the exact l!me h1story of the pulse is generally unknown at the time of design. . The effects of embedment and damping have not been considered in the Slmphfied a~alysis. Embedment of the foundation into the .soil results in mcre~sed sllffness and damping. The methods of estimating stiffness and dampmg for embedded footing will be discussed in Chapter 11. Damping for s~rface footing can be estimated by the elastic half-space method, as already d1scussed Chapter 6 (SectiOn 6.4). If the soil conditions are poor, hammer foundal!on may be supported on piles. The stiffness and damping for p1le-supported foundatiOns 1s d1scussed in Chapter 12. Modal analysis may be used for predicting damped response of hammer foundations when
Barkan, D. D. (1962). "Dynamics of Bases and Foundations." McGraw-Hill, New York. Harwood, M., and Novak, M. (1986). Uplift in hammer foundations·. Soil Dyn. Earthquake Eng. 5(2), \1)2-117. Lysmer, J., and Richart, F. E. (1966). Dynatllic response of footings to vertical loading. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 92(SM-1), 65-91. Major, A. (1980). "Dynamics in Civil Engineering. Analysis and Design," Vol. 2. Akademiai Kiad6, Budapest. Novak, M. (1970). Prediction of footing vibrations. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 96(SM-3), 837-861. Novak, M. (1982). Response of hammer foundations. Int. Conf. Soil Dyn. Earthquake Eng., 1st, Southhampton, Vol. 2, 783-797. Novak, M. (1983). Foundations for shock producing machines. Can. Geotech. J. 20(1), 141-158. Novak, M. (1985a). Analysis of hammer foundations. Int. Conf Soil Dyn. Earthquake Eng., 2nd, Southampton, 4-61 to 4-71. Novak, M. (1985b). Experiments with shallow and deep foundations. Proc. Symp. Vib. Probl. Geotech. Eng., Am. Soc. Civ. Eng., Annu. Conv., Detroit, 1-26. Novak, M., and El Hifnawy, L. (1983). Vibrations of hammer foundations. Int. J. Soil Dyn. Earthquake Eng., 2(1), 43-53. Prakash, S. (1981). "Soil Dynamics." McGraw-Hill, New York. Richart, F. E., Jr. and Whitman, R. V. (1967). "Design" procedures for dynamically loaded foundation. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng., 93(SM-6), 169-193. Richart, F. E., Jr. Hall, J. R., and Woods, R. D. (1970). "Vibrations of Soils and Foundations." Prentice-M1JJJ, Englewood Cliffs, New Jersey.
8
LAYOUT OF A TYPICAL TURBOGENERATOR UNIT
331
/'i.
Top or deck slab
/
/
LD' L171. P. turbine
Foundations for High-Speed Rotary. Machines
Top deck
level
Machines such as gas and steam turbines, generators, rotary compressors, and turboblowers fall in the category of high-speed rotary machines. The operating speeds of these machines are generally 3000 rpm or 3600 rpm and may range up to 10000 rpm. Turbine units operating at 1500 and 1800 rpm are also sometimes used. Turbogenerator (T.G.) units are available in different capacities or power rating ranging from 2 MW to 2000 MW. The capacity of T.G. units in nuclear power plants generally varies from 200 to 1100 MW. Units of 2000 MW capacity are used in superthermal fossil power stations. The turbogenerator unit is one of the most expensive and critical pieces of equipment in a power plant both in terms of its initial cost and the performance of the plant. The cost of a turbogenerator ranges from $25 to 50 million and the cost of lost production due to unscheduled shutdown resulting from malfunctioning of the unit may be as high as $250,000 per day. The long-term satisfactory performance of a T. G. unit requires precise alignment that necessitates stringent limits on permissible motion amplitudes under operating conditions. Auxiliary equipment such as condensers, heat exchangers, pipe lines carrying superheated steam, air vents and ducts for electric wiring are essential features of a turbogenerator installation. Also, the entire unit including the machine foundation and auxiliary equipment should be readily accessible for inspection. To meet these requirements, frame foundations are commonly used for turbogenerator units. An isOmetric view of a frame foundation is shown in Fig. 8.1. A frame foundation is also known as a T.G. pedestal and consists of (1) rigid base slab (or mat), (2) set of columns fixed into the base slab at their lower ends and supporting longitudinal beams at their upper ends, (3) transverse beams and (4) a top slab (or deck) with number of opening, depending upon specific machine requirements (see Fig. 8.1). Beams at intermediate levels may be provided to increase rigidity. 330
Figure 8.1.
Isometric view of a typical turbo-generator frame foundation.
Cross walls adjacent to condenser shells are also sometimes required. The frame foundations offer a better choice for turbogenerators compared to massive block-type foundations, both from consideration of performance and economy. Framed foundations may be constructed with reinforced concrete or steel. Reinforced concrete foundations are more commonly used.
8.1
I r
LAYOUT OF A TYPICAL TURBOGENERATOR UNIT
The layout of a turbogenerator unit complete with all accessories is rather complex. The prime mover is the turbine unit, which may be single stage or multistage. Multistage turbine units may consist of high-pressure (H.P.) intermediate-pressure (I.f.) and low-pressure (L.P.) turbine stages coupled to a common shaft with the generator at one end (Fig. 8.1). The speed of operation of generator and turbine may be the same, permitting a direct coupling. In cases where operational speed of turbine and generator is not same, a gear box is used to couple the turbine and generator shafts. An exciter is installed at the generator end. The turbine-generator unit is supported on the top or deck slab with shaft aligned parallel to the longitudinal beams (Fig. 8.1). The condenser is mounted on pedestals in the space between the deck and the base slab below the turbine (the low-
332
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
pressure stage in case of a multistage turbine). The isometric view in Fig. 8.1 shows the layout for a typical 110-MW T.G. unit supported on reinforced concrete frame foundation.
LOADS ON A TURBOGENERATOR FOUNDATION
333
1000 kg/m 2 to 3000 kg/m 2 (200 psf to 600 psf) are accounted for in design depending upon the size of the T.G. unit. Condenser Loads
8.2
LOADS ON A TURBOGENERATOR FOUNDATION
During the service life of the power plant, a T.G. foundation is subjected to a variety of loads. The design of the turbogenerator and the plant layout determine the nature of the loads imposed on the foundation. In general, the loads acting on a T.G. foundation may be divided into two main categories: 1. Loads due to normal operation of the plant. 2. Loads due to emergency conditions.
A brief discussion about the magnitude and characteristics of these loads is given below. 8.2.1
Condenser loads are transmitted to the foundation due to (a) weight of the condenser and (b) vacuum in the condenser.
Load Due to Weight of the Condenser. The condenser loads acting on the foundation are influenced significantly by the method used for supporting them and the type of connection between the exhaust casing of the turbine and the condenser. One of the following schemes is generally adopted for supporting condensers: 1. The condenser mounted directly on rigid supports on the base mat and
a flexible (expansion) joint provided between the condenser and the turbine exhaust (Fig. 8.2a). 2. Condenser supported on spr}~gs (Fig. 8.2b).
Loads Due to Normal Operation of Plant
Loads associated with operation of the plant include the following: 1. Dead load.
2. 3. 4. 5. 6. 7.
Live load. Condenser loads. Thermal load. Pipe load. Unbalanced loads due to machine. Torque load.
Dead Load The self weight of the foundation components, weight of turbine and generator, bed plate and equipment such as control valve, interceptor valve, and boiler feed pump comprise the dead load. The self weight of foundation components is estimated from section details and unit weight of the material. Information on dead weight of machine and equipment, as well as their point of application, is furnished by the machine manufacturer. Live Load The live loads act on the foundation during installation and repairs necessitated by maintenance operations. Codes of practice generally lay down the values of live loads from floor slabs and galleries and usually live loads of
(a)
Figure 8.2. Schematic arrangement of supporting condensers: (a) condenser on rigid supports; (b) condenser on spring supports.
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
334
When the condenser rests directly on the rigid supports, the entire dead weight of the condenser unit is transferred to the foundation mat. When the springs are provided between the condenser and the base mat, the load is transferred partly to the base mat and partly to the deck slab. The spring stiffness determines the proportion in which the load is shared between the base slab and the deck. The stiffness of the springs may be specified by the manufacturer of the turbine or condenser.
Loads Due to Vacuum in Condenser. The pressure on the turbine casing is atmospheric and the pressure in tbe condenser is below the atmospheric pressure. The differential pressure between the turbine casing and the condenser results in a suction or a vacuum load transferred to the deck slab through turbine base plates. The magnitude of the vacuum load is significantly large and may be several times the weight of the condenser and is considered as a distributed load along edges of supporting members. This results in localized stress concentration and torsion. The area over which this vacuum load acts is the area of the opening (joint) that connects the condenser to the turbine outlet. The condenser vacuum load may be calculated using Eq. (8.1):
LOADS ON A TURBOGENERATOR FOUNDATION
33S
stresses in concrete. Heat buildup in turbine casing and bed plates induces thermal loading on the foundation. The expansion of the casing and bed plate of the machine relative to the concrete deck results in frictional loads on the slab that are internally balanced (resulting in local effects but no net resultant load). It is difficult to estimate precisely the magnitude and direction of thermal loads since they depend on a number of factors, such as the distance between the points where bed plates are held down with anchor bolts, friction between the bed plates and concrete, and the load on the bed plate. An approximate estimate of the thermal load may be made by using Eq. (8.2):
(8.2) where F r = thermal load p. = coefficient of friction between material of bed plate and material of
deck'
•.,..•...
P =sum of loads due to machine, condenser, pipes, and normal torque (8.1) in which
Pu =condenser vacuum load A = area of joint opening between the turbine and condenser p" = atmospheric pressure p c = vacuum pressure
The pressure in the condenser is below the atmospheric pressure by an amount ( p" - p,), which represents depression in the condenser. The information on condenser vacuum load is furnished by the manufacturer of the turbine.
Thermal Loads Tbe heat emitted by pipes carrying superheated steam, circulation of steam or hot gases through the turbines and operation of the machine itself give rise to temperature changes that result in temperature gradients between
foundation components causing additional stresses on them. As the machine heats up, the shaft expands. The shaft is supported on a single thrust bearing permitting its free sliding in the longitudinal direction and as such no loads
Loads due to temperature changes are generally taken into account by assuming differential temperatures between the upper and the lower slabs and between the inner and the outer faces of the deck slab as specified by the manufacturer. The deck slab is considered as a horizontal frame and induced moments due to differential temperature are accounted for. Consideration should also be given to change in direction of thermal loads during start and shut down.
Pipe Loads The term pipe load includes self weight of pipe, dynamic effect of fluids in pipe, and thermal effects. The magnitude of pipe load and its distribution on the foundation depend upon pipe material, size, insulation, and layout details. The magnitude of pipe load is specified by the manufacturer.
Unbalanced Loads DtJ'i;. to Machine The turbine and generator rotors are well-balanced equipment. The unbalance in rotors is checked and corrected or minimized during test runs by mounting the prototype on specially designed test or balancing beds. The residual unbalance is ascertained by monitoring the vibration amplitudes at
of the shaft. The thermal conductivity of
bearing levels. The unbalance is specified as the distance between the axis of
concrete is low compared to that of steel and therefore the change in temperatures of turbine and generator result in local distortion and high
the shaft and mass center of gravity of the rotor, and is known as effective eccentricity. The operation of the machine causes unbalanced forces that
are induced due to
~xpansion
336
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
depend upon speed of rotation. The magnitude of unbalanced forces and moments can be calculated using Eqs. (5.38) and (5.39). The unbalanced pulsating loads are transferred to the foundation through the shaft bearings. The T.G. manufacturers provide information about unbalanced loads that should be used in the design of the foundation under normal operating conditions. In designing a T.G. foundation, most unfavorable combination of unbalanced dynamic loads should be used.
Torque Loads
The torque considered here is different from that due to unbalanced loads (moments due to machine operation). Forces due to steam in each turbine section impose a torque on the stationary turbine casing in a direction opposite to the direction of rotation of the rotor. The normal torque on the generator stator acts in the direction of rotor rotation. The magnitude of the torque depends upon the operational speed and power output capacity of the turbines. The turbine manufacturers provide the information about the magnitude of this torque. This torque is applied to the foundation as a couple acting through the machine bed plates. For a T.G. unit having a multistage turbine Fig. 8.3, the torque may be calculated as follows: 10.48PA TA= N tm
(8.3a)
LOADS ON A TURBOGENERATOR FOUNDATION
337
in which TA =torque due to high-pressure (H.P.) turbine (t m) T 8 =torque due to intermediate-pressure (I.P.) turbine (tm)
Tc =torque due to low-pressure (L.P.) turbine (t m) Tg =torque due to generator (t m) PA, P 8 and Pc=power transferred by couplings A, B, C (Fig. 8.3), respectively in kW and N = operation speed in rpm. The points of actual transfer of the torque will depend upon mounting details of turbine stages and generator.
8.2.2
Loads Due to Emergency Conditions
These loads are not associated with the normal operation of the turbogenerator unit, but are imposed on the foundation under extraordinary conditions apd include the followi11g: -~.~-4
'"
1. Seismic or earthquake load. 2. Loads due to system malfunction.
The nature and magnitude of loads due to earthquake and malfunction of the system will now be briefly discussed. Seismic Load
(8.3b)
(8.3c)
The loads due to earthquake depend upon the seismicity of the area in which the plant is located. The magnitude of the lateral force due to earthquake may be determined from codal provisions (Uniform Building Code, 1985). The lateral force may be calculated [see Eq. (8.4)] (Uniform Building Code, 1985, ASCE 1987): (8.4)
(8.3d) where 'if,J~
F = lateral force on the floor (deck slab) ah = seismic zon~ coefficient I = importance factor {3 = soil foundation factor
---BGenerator
Figure 8.3. Torque due to normal operation of a multistage
C = numerical base shear coefficient S = numerical site structure response coefficient turbine~generator unit.
W = vertical load due to weight of all permanent components
338
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
. Ho_rizontal seismic force is considered both in longitudinal and transverse duecttons. Loads Due to System Malfunction
This includes loads due to abnormal conditions, which may sometimes develop dunng the operatiOn of the T.G. unit, and includes following cases: L Loads due to bending of the rotor. 2. Loads to missing turbine bucket. 3, Short-circuit loads.
DESIGN CRITERIA
The nature and magnitude of loads due to normal operation of the machine and emergency conditions have been discussed above. The manufacturer of the turbogenerator furnishes a loading diagram detailing the magnitude, direction, and point of application of different loads for design of the foundation.
8.3
. Loads Due to Bending of the Rotor. Bending of the rotor results in mcreased ecc~ntricity and larger unbalanced loads than those arising from normal operatwn of the T.G. unit. Bending of the rotor may take place due to differential temper~tures, rotor fixed too tightly at both end bearings preve~ltmg Its free shdmg, mductwn of water and improper operation of the machme. The magmtude of unbalanced loads can be determined from anticipated rotor eccentricity, which is generally supplied by the manufacturer, and using Eqs. (5.38) and (5.39).
Loads Due to Missing Turbine Bucket. One of the buckets or blades of the turbme rotor may break during operation of the unit, resulting in an !~creased unbalance in the rotary system. Such a condition resulf; in a s1gmficant Increase in unbalance force on the foundation depending upon the weight of the bucket, the distance of its centre of gravity from axis of rotation and operational speed.
Short: Circuit Loads. Short circuit induces a severe loading condition on the turbme generator foundation. A fault of this type occurs when any two of three generator phase terminals are shorted. The short-circuit moment results due to magnetic induction between the stator and the rotor. Short circuit results in oscillating torque components, the value of which is gene:ally supplied by the manufacturer. Approximate value of the short cirCuit moment may be determined from Eq. (8.5) (Major, 1980):
339
DESIGN CRITERIA
Long-term satisfactory performance of a T.G. unit can be ensured only by maintaining proper shaft alignment and permitting very small deflections under operating loads. The tolerable limits of vibration amplitudes for these machines are rather small compared to reciprocating or impact-type machines. Permissible vibration amplitudes for normal operating conditions depend upon the speed of the machine. Some typical values of permissible - vibration amplitudes under normal operating conditions are given in Table 8.1. The values ~f permissible amplitudes given in Table 8.1 should be considered as representative only. Values of permissible amplitudes as per manufacturer's specifications should be adopted. The necessity to limit the amplitudes and deflection to such small values arises due to many reasons such as to avoid (a) excessive stresses iu the shaft due to differential deflection between the adjacent bearings (b) overloading and uneven wear and tear of the bearings and (c) excessive deflection of the turbine foundation, which may increase unbalance in the rotor. Each of the above factors leads to alignment problems and excessive stresses in machine and foundation components. For smooth running of the machine, differential deflection of the various structural components of the T.G. foundation should be minimized by limiting the total deflection.
Table 8.1. Typical Values of Permissible Vibration Am-
plitudes for
H~h-Speed
Speed of
M"
=
rW, t m
(8.5)
where M" is the short-circuit moment, W, is the capacity of T.G. unit in megawatts and r is radius of the rotor in m.
Rotary Machines Permissible Vibration Amplitude, microns
Machine, rpm
Vertical
Horizontal
3000 1500
20-30 40-60
40-50 70-90
Source: Barkan (1962). mission of McGraw-Hill.
© 1962
and reproduced with per-
340
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
The criteria given below should be followed in designing a T.G. foundation. 1. Natural frequencies of the foundation should deviate from the normal operating speed of the machine by at least 30 percent. 2. Vibration amplitudes at bearing level should not exceed the values specified by the manufacturer. Permissible amplitudes may be adopted from Table 8.1 if the limiting values have not been furnished by the manufacturer. 3. The stresses due to normal operating load in structural components and sod should not exceed their specified values under the worst combination of loads. 4. For emergency loads, appropriate increase in permissible stresses may be allowed as per codal provisions (Uniform Building Code, 1985). 8.4
DESIGN CONCEPTS
Turbogenerator foundations are generally designed as low-tuned or undertuned foundations of concrete or steel. Low tuning implies that the fundamental natural frequency of the foundation is lower than the operating speed of the machine (Fig. 1.2). When foundations are designed on the low-tuned concept for turbines operating at 1800 rpm, as is the case in some nuclear power plants, it may become necessary to support the machine on springs. T. G. foundations have also been conventionally designed using large supporting columns with cross beams to provide a rigid frame. The lowtuned concept is generally preferred because of additional volume of space below the deck which becomes available when slender columns are used and cross beams are omitted. The additional space results in easier access for maintenance and more flexibility in layout of accessories. In any case, the design is made to ensure favorable vibration characteristics under normal operating conditions. For the case of emergency loads, it is ensured that the foundation will not suffer any permanent damage during the short period of ti':'e . for which the machine operates under abnormal loads by installing tnppmg relays so that the machine is automatically shut off as soon as the emergency conditions, such as a strong-motion earthquake or short circuit, develop. The methods for analysis used in designing frame' foundations will now be described. 8.5
METHODS OF ANALYSIS
The different kinds of loads to which a turbogenerator foundation is subjected during its service span have been described in Section 8.2. The
METHODS OF ANAlYSIS
341
methods of analysis for computing the response of T.G. foundation are discussed in this article. The objective of the analysis is to determine if the proposed foundation will perform satisfactorily under the action of various loads acting according to the design criteria laid down in Section 8.3. It may be mentioned here that the design of a frame foundation is different from the design of massive-block-type foundation in several ways. In case of a block foundation, it is adequate to check the natural frequencies and amplitudes of vibration. A structural design of the foundation block is generally not needed. In the case of a frame foundation, it is necessary to check the frequencies and amplitudes of vibration and also to des1gn the members of frame from structural considerations. The stresses induced in the members of the frame due to the adverse combination of various static and dynamic loads should not exceed their permissible values. The aspects of structural design of the frame fouodation are beyond the scope of this text and only the methods of dynamic analysis for computing the natural frequencies and amplitudes of vibration are considered here. The methods for dynamic analysis of the frame foundations may be d1v1ded mto two categories:
1. Simplified methods. 2. Rigorous methods. In the simplified methods, a number of assumptions are made and the analysis is carried out on frame-by-frame basis. A single-degree-of-freedom or a two-degree-of-freedom model is adopted for computing the natural frequencies and amplitudes. The role of the soil below the base is generally neglected. In rigorous methods, the frame fouodation is modeled as a threedimensional space frame and analyzed as a multidegree freedom system. Rigorous solutions accounting for the three-dimensional nature of the problem are, however, complicated and simplified analysis may be used to obtain practical solutions. Both the simplified and the rigorous methods are described below. 8.5.1
Simplified Methods
In simplified methods of,~~;nalysis, the frame foundation is idealized either as a single-degree-of-freedom system or as a two-degrees-of-freedom system. A description of the underlying assumption and the procedure for computation of response for each of these cases is given below. Single-Degree-of-Freedom System
The natural frequencies and amplitudes of vibration in vertical and horizontal direction are computed by using a single spring-mass system. A singledegree-of-freedom model has been used earlier (Rausch, 1959) to compute
342
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
343
METHODS OF ANALYSIS
the natural frequencies by the method known as the resonance method. In the resonance method, consideration was given only to the natural frequency of the system in relation to the operating speed of the machine, and the amplitudes of vibration were not computed. Vibration characteristics in vertical and horizontal directions by this method are calculated as follows.
Pz sin
t
Vertical Vibration. For analysis of vertical vibrations, each transverse frame that consists of two columns and a beam perpendicular to main shaft of the machine, is considered separately. The stiffness of the equivalent spring is calculated as the combined stiffness of the beam and the columns acting together and the mass is determined by the mass of total loads acting on this cross frame. This analysis is based on the assumptions given below:
~--2a
Assumptions 1 and 2 restrict the vibrations to the vertical direction only. Assumption 3 makes it possible to neglect the effect of the longitudinal beam on vertical vibrations of the transverse frame. Assumption 6 implies that the transverse beam and the column vibrate as one· system in the vertical direction. Assumptions 1, 2, 5, and 6, taken together, justify adoption of a single-degree system for analysis. With assumption 4, the natural frequency of the entire frame in vertical vibrations can be taken as the average of the natural frequency of individual frames. A typical transverse frame is shown in Fig. 8.4a. Different types of loads acting on this frame are also shown. The loads consist of a. Uniformly distributed load due to self weight of cross beam equal to q per unit length. b. Dead load due to machine and bearing equaf to Wm. c. Load transferred to the columns by the longitudinal beams W L. d. Unbalanced vertical force due to machine operation P, sin wt The spring stiffness of the frame kz for vertical vibrations is given by
I
I
Baseslab
ends
11
.u -;.;-
~ ~k a' ~ gj
0!
7//7]/////, (b)
:
L_ _ _ _ _ _ _ _ _ _.: ___ _j
assumed fixed
Figure 8.4.
l!
,- l /~~m"'7 V
Column
(a)
(a) Typical transverse frame; (b) singleMdegree~of-freedom model for vertical
vibrations~
· which w is the total load on the frame and !J.z is the total vertical . deflection at the center of the beam due to bending actwn of bearn and ax!'al compression in column. The total load W is given by
Ill
(8.7) in which l is the effective span. The value of effective span [ depends upon the rigidity of the corner sections of the frame and whether the haunches have beeen provided at the corners (Fig. 8.5). The effective span can be defined in terms of the lengths 1 and [ as shown in Fig. 8.4a, in which /0 ~center-to-center distance 1 between the columns and /1 ~clear distance between the columns.
"""·
,---r--+-----" 1
I
_j
I
I (8.6)
-l
lo
1. Frame columns are fixed at their lower ends into the rigid base slab.
2. The difference between vertical deformations of individual frame columns is negligible. 3. The torsional resistance of longitudinal beam is insignificant compared to the deformation resistance of transverse beams. Therefore, the effect of longitudinal beam on vertical vibrations of transverse frames can be neglected. 4. The natural frequencies of individual cross frames are practically of the same order. 5. The effect of elasticity of the soil is neglected. 6. The connection of transverse beam with columns is rigid.
wt
b-4 figure 8.5.
Values of a and b for a frame with haunches
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
344
For perfectly flexible connection between the beam and columns at their ends, the effective span length I will be the same as the center-to-center distance between the columns 10 . For perfectly rigid end connections, the effective span will be equal to the clear span 11 . For other cases, the effective length is determined depending upon rigidity of corner sections. The rigidity of corner section of a frame can be defined by the ratios b/10 and h 0 /l 0 in which h 0 is the height of the column from the top of the base slab to the center of the frame beam (Fig. 8.4a) and b is one-half of the column width for a frame without haunches (Fig. 8.4a) or the distance as shown in Fig. 8.5 for a frame with haunches. The effective span is calculated as (8.8)
345
METHODS OF ANALYSIS
1959) as follows: (8.10)
in which Ll.z = vertical deflection of beam due to the concentrated load W m 1 Ll.z = vertical deflection of beam due to the distributed load q 2
Ll.z = vertical deflection of beam due to shear 3 Az 4 = axial compression in the column
The magnitude of Ll.z 1 is given by 3
The values of a are given in Fig. 8.6. The effective height is given by
Ll.z, =
(8.9)
Wm/ 2K + 1 96I,E K + 2
(8.11)
in which I = moment of inertia of the beam about the axis of bending
in which a is one-half of the depth of the beam for a frame without haunches (Fig. 8.4a) and the distance shown in Fig. 8.5 for frames with haunches. The values of effective span and height are used to calculate deflections in the frames due to applied loads. The magnitude of deflection Ll.z in Eq. (8.6) may be calculated from the available solutions for rigid frames (Kleinlogel, 1949, 1964; Leontovich,
b
E =Young's modulus for concrete K = relative stiffness factor The value of K is given by
I,, h
K=-I" I
(8.12)
in which I is moment of inertia of the column. The magnitude of Ll.z 2 , Ll.z 3 , and Ll.z, is given by ql 4
5K + 2 Ll.z, = 384EI, K + 2
(8.13)
(8.14) (8.15) 0~---J----~-----L-----L----~~
0
0.04
0.08
0.12
0.16
0.20
bllo
Figure 8.6. Graphical determination of coefficient a. (After Major, 1980.)
in which
A, = cross-sectional area of the beam A c = cross-sectional area of the column
346
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES METHODS OF ANALYSIS
The model for computation of the natural frequency of the frame is shown m Ftg. 8.4b. The natural frequency of vertical vibrations of the frame may then be obtained as in Eq. (2.11) as follows: w nz
=
rr;g \JW
347
Horizontal Vibration. The vibration characteristics in horizontal vibrations can also be calculated by an equivalent single degree of freedom model. Such an analysis is based upon the following assumptions: L Columns are fixed into the rigid base slab at their lower ends.
Average natural frequency of vertical vibrations w
""
is given by
n
(8.16)
~n ~J:ich Wnz~, wnzz are the natural frequencies of vertical vibrations of the md!Vldual frames. The average value of the amplitude of vertical vibration may be computed as (8.17a)
in which A, = amplitude of vertical vibration of the foundation Pv =total vertical unbalance force !; = damping expressed as percent of critical damping
2. The deck slab is rigid in its own (horizontal) plane. 3. Resistance offered by the columns in axial compression is large compared to their resistance in bending. 4. Torsional vibrations of the deck slab are neglected. 5. Elastic resistance of the soil at the base can be neglected. Assumptions 1 and 2 are realized in practice. The thickness of the deck slab and the width and depth the beams in longitudinal and transverse directions are large compared to their spans and deck slab may be assumed to be perfectly rigid in its own plane. Assumption 3 is not fully realized. When columns are of large cross-sectional area, this assumption holds. But wben slendeL,columns are used, tl)is.ilssumption is not realized. Assumption 4 is not strictly valid and is hard to. tealize. Torsional vibrations are excited by the nature of the dynamic loads acting on the foundation and also because the mass center of the deck slab and line of action of superimposed dynamic loads and the resultant horizontal reaction due to bending of the columns do not act along the same straight line. These assumptions make it possible to analyze the frame for horizontal vibrations using a single-degree-of-freedom model, as shown in Fig. 8.7. The
Total spring stiffness k, is given by
Wr
= Total weight of
deck slab and
machine
(8.18)
\-
Px sin
in which k,t> k, 2 are the stiffness of the individual frames. To account for possible buildup of amplitudes during starting and shut down, the absolute maximum amplitude (AJ'""' of vertical vibrations may be obtamed by takmg w"" = w in Eq. (8.17a) above
1 Wr
/4M --,1--1 I I I
i
I I !.,;
(8.17b) In case the difference between the natural frequen~y of vertical vibration and operatmg frequency is smaller than 30%, Eq. (8.17b) should be used to compute A,. The amplitude of vertical vibrations of any individual frame can be calculated by usmg Eq. (8.17) with values of vertical spring stiffness natu~al fr~quency, and vertical unbalanced force for the frame unde; cons1derat10n.
Deck slab wt
I
JI I I I
(.Deflected
f Colum n....-
I I
I I
///. 1'///
shape of column
Kht is combined lateral stiffness of all frames
I /r
/
Figure 8.7. Horizontal vibrations of a deck slab on columns in a single-degree-of-freedom model
349
METHODS OF ANALYSIS FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
348
deck slab undergoes horizontal translation along a direction perpendicular to the mam shaft of the ~achme. The spring stiffness is provided by the columns due to their bendmg action (Fig. 8.7). Natural frequency of horizontal vibratz"ons . The natural frequency of horizontal vibrations w"h is given by
been referred to as amplitude method (Barkan, 1962) and combined method (Major, 1980). In the amplitude method, the vibration characteristics are evaluated in terms of the amplitudes only, whereas in the combined method, both the frequencies and amplitudes are calculated. Some simplifying assumptions are made. These assumptions and the methods of analysis are described below for the case of vertical and horizontal vibrations.
(8.19)
Vertical Vibrations. The natural frequencies and amplitudes of vibration in the vertical direction are calculated based on the following assumptions:
in which k• =combined stiffness of all the transverse frames in bending kx1, kx2' · · · , =lateral stiffness of individual transverse frames Wr =total weight of deck slab and machine.
1- Columns are fixed at their lower ends into the rigid base slab. 2- The effect of longitudinal beams on vertical vibrations of the trans-
verse frames can be neglected. 3. The difference between vertical deformations of individual frame
columns is negligible. 4. The role of soil below the base slab can be neglected.
The lateral stiffness kx of any frame is given by
k
= X
12£1" (6K + 1) h3 3K+2
(8.20)
Amplitude of horizontal vibrations. Amplitude of horizontal vibration A may be calculated as follows: •
A·=~k~.V7~(1~~(w=l~w=".~~~~]~2 =+~(~2=fw=l=w=",~,)2
(8.21a)
These assumptions have been gi.:;cussed already for the case of vertical vibrations using a single-degree-of-freedom system. They enable a frame-byframe analysis to be carried out for the case of vertical vibrations. No assumption has been made about the rigidity of the end connection between the transverse beam and the columns. The vibrations of beams and columns of a transverse frame can therefore be represented by a two-degrees-offreedom system, as shown in Fig. 8.8. In Fig. 8.8a, tbe beam undergoes
in which P• is the total horizontal dynamic force. The maximum horizontal amplitude that may build up during starting or shut down of the machine is given by
tp,
,,
sin wt m)
"
(8.21b)
----
2 \
\
The single-degree-of-fre~dom method used for computation of vertical and honzontal VIbratiOns IS an oversimplification of a complex problem. Only the fundamental f~equency in vertical or horizontal mode is computed. Because of the s1mphfymg assumptions on which this method is based the results are highly approximate. A better estimate of vibration characteristics can be ma?e by using a two-degrees-of-freedom model which is a refinement over the smgle-degree:of-freedom model. The two:degrees-of-freedom system will now be descnbed.
m]
2
\
Direction of
.!'!.....\ 2
\
mz
I
I
rvibmtion columnsof
\ \ \
l
'~!;~
The pr_oblem of vertical and horizontal vibrations is analysed in each case by Ideahzmg the system as a two-degrees-of-freedom system. The method has
0!
I I I
~kj
(o)
Figure 8.8.
5.~kz
~
1!'!. I 2
/
Two-Degrees-of-Freedom System
mz
~ (b)
system. (a) Vertical vibrations of a cross frame as a two-degree-of-freedom
deflected shape; (b) mathematical model.
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
350
forced vertical vibrations due to flexural bending, and columns vibrate along their axial direction. The equivalent spring mass model for this system is shown in Fig. 8.8b. The forces in the columns are developed due to their axial deformation and spring k 1 in Fig. 8.8b, represents the stiffness of the column in loading along the axial direction. The spring stiffness k 2 represents stiffness of the transverse beam in bending. Vertical natural frequency. The vertical natural frequency of the system shown in Fig. 8.8b may be obtained as follows: Mass m 1 acting on the column is given by (Barkan, 1962)
mI
=
WL
+ 0.33W, + 0.25WB g
351
METHODS OF ANALYSIS
_ 13 (1 + 2K)W ~z- 96Elb(2 + K)
+ 31W
(8.25)
8GAb
in which G = shear modulus of the beam material
1= effective span of the beam A = cross-sectional area of the beam b 1 = moment of inertia of the beam b
(8.22)
K is defined by Eqs. (8.12); k 2 is then given by
w
(8.26)
k, = ~z
in which WL =load transferred by the longitudinal beams to the columns including self weight of the longitudinal beams W, =weight of the two columns constituting the transverse frame W8 = weight of the transverse beam
Mass m 2 acting at the center of the cross beam is given by (Barkan, 1962) (8.23)
The equations of motion for the spring-mass system shown in Fig. 8.8b are (8.27a)
m,z, + k,(z,- z
1)
= P, sin wt
(8.27b)
Equations (8.27a, b) are similar to Eq. (2.95a, b). The tw? natura~ frequencies of the system may therefore be obtained by solvmg the requency equation given below: (2.98)
in which W m is the weight of the machine. The stiffness of each column for axial deformation (Fig. 8.8a) is given by k1 2
EA,
h
in which
(8.24a)
(2.99a)
in which (2.99b) E =Young's modulus of the material of the columns A c = cross-sectional area of a column h =effective height of the column
m,
p,=-
m,
Amplitudes of Ve~ical Vibrations. The amplitudes of verti~~ ;ibration are obtained from Eqs. (2.108) and (2.109) by replacmg F, wit '
Total vertical stiffness of both columns is given by (8.24b) Stiffness of spring k 2 • The deflection of the beam (Fig. 8.8a) for a load W at the center is given by
(8.27c)
and
352
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
A
-
z -
''-
Ll.(w
2 )
(l + p,)w~l1 + p,w~r2- W 2
'-
m,LI.(w')
P,
METHODS OF ANALYSIS
bending in the frame columns, which thus offers a lateral resistance to the horizontal motion of the deck slab. The lines of action of the exciting and the resisting forces are not the same. As a result, a horizontal couple is generated which induces torsional vibrations in the deck slab, as shown in Fig. 8.9. Torsional vibrations are also excited due to unsymmetrical distribution of unbalanced loads on the deck, the resultant of which may not be passing through the mass center. The deck slab thus undergoes combined horizontal and torsional vibrations under the effect of horizontal exciting loads. The overall motion of the deck slab may be represented by a line diagram shown in Fig. 8.10 in which the deck slab has been replaced by its center-line A 1B 1 in the initial position. This sketch depicts a typical two-bay frame. The load carried by each transverse frame including its self weight, dead loads, weight of machine, and the load transferred by the longitudinal beams may be represented by a point load; 33 percent of the weight of the columns is also added. Thus the mass m 1 is the equivalent mass representing the load shared by this particular transverse frame including 33 percent of the weight of columns of this frame. Similarly the masses m 2 and m 3 represent th({masses for the otherJ~ansverse frames. Dis the mass center of m 1 , m 2 , m 3 (Fig. 8.10). The lateral stiffness of each transverse frame has been replaced by an equivalent spring. The spring stiffness k, 2 thus represents the lateral stiffness of one transverse frame. The value of lateral stiffness for any of the frames may be calculated by using Eq. (8.20). Point C in Fig. 8.10 represents the stiffness center, that is, the centroid of spring stiffness k, 1 , k, 2 , and kx 3 • It is shown in Fig. 8.10 that the deck undergoes horizontal displacement parallel to itself (from A 1 B 1 to A 2 B 2 ) and also rotates about the vertical axis through the mass center (of gravity) D. The
(8.27d)
is calculated using Eq. (2.104).
Vibrations. d Horizontal ·
The horizontal vibrations of the frame founatiOn. are also analyzed by a two-degrees-of-freedom system based on the followmg assumptiOns. l. 2. 3. 4.
353
The columns are fixed at their lower ends into the rigid base slab. Effect of elasticity of the soil below the base can be neglected. Deck slab ts rigid in its own (horizontal) plane. Elastic resistance of the columns to axial deformations is relatively much larger co~pared to thetr elasttc reststance in bending in the transverse duectwn.
Assumpti~ns 1 and 2 imply that vibrations due to horizontal unbalanced forces occu~ '? the part of the foundation above the base slab. Assumption 3 about. the ngtdtty of the deck slabs implies that the deck slab will undergo only ngtd body motton and that flexural vibrations are insignificant. Because of tts ngtdtty, the deck slab can be replaced by a beam with its centerline along the centerline of the mainshaft of the machine. Assumption 4 implies that the rotatiOn . of the deck slab about a horizontal axis is restrained because of htgh sttffness of the column to axial deformations. The deck slab thus undergoes honzontal translation under the effect of horizontal unbalanced forces, as shown in Fig. 8.9. This motion of the deck slab induces
Initial position of deck slab
dec~
Centerline of slab (initial) ... ~ ,_ _
Final position (after coupled translation and
T
L_7 ___ _ Displacement due to horizontal translation
F•b'gure 8.9 .. Vibrations of the deck slab due to combined translation along X-axis and rotation
a out Z ax1s.
~kx3
2
rotation)
t
t
f r
!
1
A1
C D B ._-----'--i-f=---,+--L-----_. 1 m1
m2
m3
A3.__ _ _ _ _
A•,---~r-~~--~~~~----~,,---eB 2 Due to horizontal displacement
Due to
tra~tion
--
-e 83
and rotation
Figure 8.1 0. Spring-mass model for combined horizontal and rotational vibrations of the deck slab.
354
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
355
METHODS OF ANAlYSIS
(8.30c) final displaced position of the centerline of the deck slab is sbown by line A 3 B 3 (Fig. 8.10). The distances of the different masses from the mass center D are shown as a 1 , a2 , and a3 and the distances of the stiffness center Care shown as b 1 , b 2 , and b 3 , respectively. The procedure for computing the natural frequencies and amplitudes of horizontal vibrations is now described.
Natural Frequency of Horizontal Vibrations. The equations of motion for the system shown in Fig. 8.10 may be written as follows [see also Eq. (2.126)]: .
Total value of R.I for all the masses is given by
2:" Rj =
kh(x
(8.30d)
+ elj;)
j=l
in which
k = sum of lateral stiffness of all the frames = E k xj ; =distance of the stiffness center C from the mass center D (Fig. 8.10).
m
m.X + L Rj = Px sin wt
(8.28a)
j=l
The moment M about the mass center due to a force on the mass mj is I
given by
"
Mm,~ + L Mj = M, sin wt
(8.31a)
(8.28h)
j=I
(8.3lb)
in which
m =total mass and is equal to (m 1 + m 2 + · · ·) Mm, =polar mass moment of inertia of all the masses about the vertical axis through the mass center of gravity (point D in Fig. 8.10) Rj =spring force on any mass mj due to its displacement by xj Mj =Moment of the forces Rj about the vertical axis of rotation, i.e. passing through the mass center P, =Horizontal unbalanced force and, M, = horizontal unbalanced moment 1' = angular displacement due to torsion
in which (8.32) k is the equivalent torsional spring for the frame columns. • Substituting for ER and EM from Eqs. (8.30d) and (8.31b) respectively I I in Eq. (8.28), one obtains,
(8.33) (8.34)
The mass moment of inertia M mx is given by
Mmz For a given frame at a distance force R j is given by
ai
"
= j=l 2: mia~
(8.29)
The set of Eqs. (8.33) and (8.34) are similar to the equations o~ motion discussed in Chapter 6 (Section 6.7) for coupled rockmg and shdmg. The frequency equation in this case may be written as Barkan (1962)
from the mass center, the elastic spring
•
2
Wn- (awnx
2) ,
+ Wno/
Wn
,
, _
+ Wnx{t)n~' 0
(8.35)
-il-o:
(8.30a) The value of k,j may he obtained from Eq. (8.20). The displacement xj is due to horizontal movement plus rotation and is given by (8.30b) in which xis the displacement of the mass center. Therefore, Rj is given by
in which (8.36a)
(8.36b)
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
356
w"x represents limiting natural frequency of horizontal translation when the center of rigidity C coincides with mass center D, that is, e = 0 and M, = 0. There will be no torsional vibrations in this case. The system becomes a single degree of freedom system. Similarly w"'' represents the limiting natural frequency of rotational vibrations about the vertical axis when e = 0 and Px = 0. Equation (8.35) can be solved to obtain two natural frequencies of the system wn 1 and wnz·
Amplitudes of Horizontal Vibrations. given by 2
AX =
z
z
The amplitudes of vibration are
e [ zwnx+wn"'-w r
z] ~-wnxM Px z Mz m
mz
~(w')
(8.37)
357
METHODS OF ANALYSIS
on the basis that significant response of the system takes place in the fundamental mode. The higher modes of the system are neglected. In the case of under-tuned foundations, some of the higher-modes may be near the operating speed and cause instability in the system and may c~use excessive vibrations. The effect of longitudinal beam on vertiCal v1bratwns of transverse frames and the role of soil below the base slab has been neglected. These considerations make the computed vibration response approximate. In a realistic analysis, the structural fr~me should be analyzed as a three-dimensional space frame and the mteracllon of s~1l belo': the base should also be taken iuto consideration. Methods of analys1s for th1s purpose are now discussed. 8.5.2
Rigorous Methods
It is possible to improve upon the simplified methods to incorporate the
and 2
e z --(w Px z -w z) M,-w rz
nx
m
nx
Mmz
~(w2)
A"'=
(8.38)
in which Ax is the amplitude of horizontal translation, A"' is the amplitude in rotation, and 4 ( 2 2 ) 2 2 2 A( W 2) = w - awnx+wn 1, W +wnxwnl/1=0
interaction effects of soil below the base slab. For example, the case of vertical vibrations may be analyzed by the model shown in Fig. 8.11. The mathematical model in Fig. 8.l;l.J1as been obtained by assuming the ba~e slab to be rigid and lower ends tif columns fixed into the base slab. Th1s model is a modification of the single-degree-of-freedom model shown m Flg. 8.4b. For vertical vibrations the effect of vertical stiffness of soil at the base has been replaced by an equivalent soil spring k 1. The damping in soil and in the structure can also be included. This model can be analyzed as a two-degrees-of-freedom system.
(8.39)
where
Pz sin
wt
e2
a=l+r 2
,,
(8.40)
in which r is the radius of gyration defined by Eq. (8.41):
r=
~M,;;,
(8.41)
~
01: ....--
~ ~
The resulant horizontal amplitude A, is given by
kz = Vertical stiffness of frames
(8.42) in which a is the distance of the point at which the amplitude is being calculated from the center of gravity of the system. The methods of computing the natural frequencies and amplitudes of vibration by treating the frame foundation as a single-degree or a twodegrees-of-freedom system have been discussed above. These methods work
01.
~
0!~
ki = Vertical soil spring
01.
w.ifbmh
Figure 8.11. Mathematical model for vertical vibra~ tions incorporating soil interaction effects.
358
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
METHODS OF ANAlYSIS
359
be modeled as a space frame resting on a continuum. Modal analysis technique may be used for analysis of the system. The principle of model analysis has been explained in Chapter 2 (Section 2.14). A lumped mass model may be made by two methods:
Px sin wt Deck slab
1. The frame columns are considered fixed at their lower ends. The kh = Horizontal stiffness of frames
Base slab m)
r~/~~1~------r-----~ ~ k¢
kx = Equivalent soil spring in sliding
Figure 8.12.
= Equivalent soil
spring in rocking
/
Mathematical model for horizontal vibrations incorporating soil interaction
effects.
response of the superstructure is calculated and role of the soil is neglected. 2. The superstructure and the foundation slab are incorporated into a single model and the interaction with the soil is accounted for. The plan and sections of a typical frame are shown in Fig. 8.13a, b, and c. Considering the frame as fixed into the base slab and neglecting the interaction effects of soil below the base, the lumped-mass model can be made, as shown in Fig. 8.13d. A rigorous model that accounts for the interaction effects of soil can be made, as shown in Fig. 8.13e. The effect of soil has been simulated by providing equivalent soil spring. The stiffness of the equivale~t soil springs may b~gbtained by using the concepts of elastic half-space theory (Section 6.4) odinear spring method (Section 6.7). The damping in the structure and the soil can also be accounted for. The damping values for soil may be calculated by following the procedure of Section 4.8. The equations of motion for the models shown in Fig. 8.13d and e can be written in the form given by Eq. (8.43):
The interaction effects of soil on horizontal vibrations can be accounted for by modeling the system as shown in Fig. 8.12. The springs kx and k represent the soil stiffness in sliding and rocking respectively. The model i~ Fig. 8.12 is a modification of single-degree-of-freedom model shown in Fig. 8. 7 to mcorporate effects of soil. The interaction model in Fig. 8.12 is a three-degrees-of-freedom system and can be analyzed following the procedure in Chapter 2 for multidegrees-of-freedom systems. Damping in soil in sliding and rocking modes and also in the structure can be included. Models in Figs. 8.11 and 8.12 can be combined to represent the complete system. The two-degrees-of-freedom models can similarly be modified to include elasticity of the soil. A better estimate of the dynamic reponse of the foundation can be made by including the elasticity and damping effects of the soil. The value of equivalent soil springs may be calculated using either the elastic half-space or linear spring approach as detailed in Sections 6.4 and 6. 7 for block-type foundations. When the degrees of freedom are more than two and damping is included, it becomes ne'Cessary to obtain numerical solutions using modal-analysis techniques.
I
2500'-~t<---- 3600,--~+---2500
4800
.t--+-·t --
2000
2800
Hollow
2000
1000
1200
1200
3100
(e)
Three- Dimensional Analysis
The frame foundation resting on the soil represents a multidegree freedom system and should be analyzed as such. The frame foundation may
Figure 8.13.
A typical frame foundation: (a) plan of the deck slab; (b) longitudinal section; (c)
cross section; (d) lumped-mass model with columns assumed fixed into the base slab; (e) lumpedwmass model including interaction effects of soil.
600 400
miO
400 Line memb represent
spr~~s
1000 4800----
900'r-----
9
DO
1000
mg
5400
1000 8600
(d)
(b)
m,, Hollow
600
400
I I
400
1000
1000
5400
3600
Columns
~============~~~============-IJJOOO 6800 (c)
Figure 8.13. 360
(continued). 361
362
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
DESIGN PROCEDURES
e can be written in the form given by Eq. (8.43): [M]{Z) + [C]{Z} + [K]{Z) ~ {F(t))
(8.43)
in which [M] ~ mass matrix [ K] ~ stiffness matrix [ C] ~ damping matrix { Z} ~displacement vector { Z} ~ velocity vector { Z} ~ acceleration vector { F(t)) ~applied load vector The equations of motion [Eq. (8.43)] may be solved by direct integration to obtain the response of the system. Alternatively, they may be uncoupled into a set of linear equations and the solution obtained by the normal mode method. The normal mode method is generally preferred, since it offers the facility to obain the natural frequencies and mode shapes. The total response can be obtained by modal superposition (Section 2.14), in which case the maximum response at any frequency can be calculated from the displacements resulting from each mode. This makes it possible to calculate the amplitude of vibration not only at the normal operating speed of the machine, but also at all speeds through which the turbine passes during starting and shutdown. Amplitude frequency plot showing the vibration amplitude for different speeds of the turbine can thus be obtained. A typical amplitude-,frequency plot is shown in Fig. 8.14. From the values of tbe
..
363
displacements induced at different points on the frame, the magnitude of the load for structural design of the frame members can be calculated using the principles of structural mechanics (Hurty and Rubinstein, 1964). The frame foundation can also be modeled using a finite element approach. The superstructure frame is modeled as a three-dimensional frame by using beam elements. Shear walls within the superstructure are modeled by a finite-element mesh. The base slab and the soil are also modeled by a finite-element mesh. The effects of nonhomogeneity of soil mass and nonlinearity of its stress strain behavior can thus be included. Nonlinearity effects are especially important when emergency loads such as due to earthquake are being considered for critical structures such as turbine foundations in a nuclear power plant. The details of formulating threedimensional models have been given by Arya and Drweyer (1977), Arya et al. (1979), ASCE (1987), and Shen and Stone (1975). When the frame foundation is modeled as a multidegree freedom system, the response calculations can be made using commercially available computer codes such as STRUDL, which can be used for both linear or nonlinear analysis. Thf, effect of soil resi~tance can be simulated in the form of equivalent springs. When the franie-foundation-soil system is to be analyzed by making a finite element model and nonlinearity effects are to be included, the solution can be obtained by using computer programs such as ADINA. For detailed instruction for using these programs, reference should be made to the respective user's manuals (ADINA System, 1981; ICES STRUDL II, 1979a, b). Since tbe permissible vibration amplitudes for a turbo-generator foundation under normal operating conditions are extremely small, the behavior of the soil foundation system can be safely assumed to be in the elastic range, and linear elastic analysis will be sufficient in most practical problems. problems. A brief description of the capabilities of STRUDL and ADINA is given in Appendix 3.
8.6
DESIGN PROCEDURE
The methods for analysis of frame foundations have been discussed in the previous section. A stef;cby-step design procedure will now be given. The design will consist of two stages: 1. Preliminary design. 2. Detailed design. Operating frequency (rpm)
Figure 8.14.
A typical amplitude-frequency plot.
Before attempting the design, the machine, soil and seismic data and foundation details listed below should be procured.
364
8.6.1
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
Design Data
Machine Data
All data pertaining to the machine should be procured from the manufacturer of the machine. These data should consist of the following: 1. Layout of the machine and auxiliary equipment, including their fixing details. 2. Information on openings, depressions, and projections, including any other requirement for the machine. 3. Weight of machine. 4. Capacity and rated output of the machine. 5. Operating speed of the machine. 6. Weight of rotor and eccentricity. 7. A complete loading diagram showing the magnitude and point of application "of all loads that are to be accounted for in design. The information on various loads should include: Dead loads Live loads Construction loads Thermal loads Load due to condenser weight and due to vacuum (depression) in condenser
Pipe loads Unbalanced loads due to machine operation Torque loads Loads due to short-circuit current Loads due to bending of rotor Loads due to missing bucket 8. Permissible vibration amplitudes.
DESIGN PROCEDURES
365
Seismic Data
Seismic studies of the area should be conducted to select the design earthquake. Alternatively, the seismic coefficients may be adopted based on the seismic zone in which the proposed installation is located.
Proportioning the Frame Foundation
Proportioning of the foundation includes deciding the layout of the foundation and selection of sizes of the different components for preliminary analysis. In deciding the layout of the foundation, the following points should be considered. 1. The layout of the foundation should as far as possible be symmetric with respect to a vertical plane through the longitudinal axis of the machine. 2. The machine bearing should be located directly on the transverse frames. The columns and transverse frames should be exactly in planes perpendicul&'J: to the longitudinaL,a]
Depth of the Deck Slab and Longitudinal and Transverse Beams. The depth of the transverse and longitudinal beams should be one-third to one-quarter of the clear span. The depth of deck slab usually ranges from 0.60 m to 1.5 m. The deck slab should be rigid in its own plane. Column Size. The columns commonly used are sized so that the ratio of height to width generally varies from 2 to 10. Adequate haunches should be
Soil Data
provided at intersection of beams and columns to avoid concentration of stress and to ensure rigidity of the connection.
1. Adequate subsoil exploration should be done to obtain subsoil data including allowable soil pressures up to a depth of. three times the width of the foundation or till hard stratum.
achieve a uniform soil reaction so as to maintain the deck slab in a plane and
2. Dynamic soil properties should be obtained by field or laboratory investigation. 3. Position of ground water level and fluctuations in its level (if any
should be ascertained).
Base Slab.
The
b'ri~e
slab should be sufficiently thick and rigid to
keep the shaft alignment intact. Also the slab thickness should be enough to satisfy the condition of fixity of columns at their lower ends. The thickness of the base slab may be taken as 0.07 L 413, where L is the average of adjacent clear spans in m. A minimum thickness of 1m is generally
preferred for the base slab.
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
366
367
DESIGN PROCEDURES
Dynamic Soil Constants. Dynamic soil constants corresponding to the shear strain amplitudes induced by operation of tbe machine should be selected following the procedure of Section 4.7.
~'
=
1'(1 + 2K)W 3/W 96Eib(2 + K) + 8GAb
(8.25)
in which
Check for Soil-Bearing Capacity. The pressure transferred to the soil due to the combined weight of the machine, foundation, and other superimposed static and dynamic loads should be less than 80 percent of the allowable soil pressures for the case of static loads alone. 8.6.2
Dynamic Analysis
The preliminary analysis may be carried out by using a two-degrees-offreedom model for the case of horizontal and vertical vibrations, by following the steps given below:
W = any load acting at the center of the beam A b = cross-sectional area of the beam 3 I b =moment of inertia of the beam given by bd 112 in which b is the width of the beam and d is the depth l = effective length K =relative stiffness factor (Eq. 8.12). The value of l is given by l
Vertical Vibrations Spring Stiffness k 1 and k 2 • The spring stiffness k 1 (Fig. 8.8) is given by k
= J
2EA, h
(8.24b)
=
10
-
(8.8)
2ab
in which l is the. center-to-center distance between the frame columns and b is one-half the width of the column''{fir a frame without haunches and the distance as shown in Fig. 8.5 for a frame with haunches. The values of a are given in Fig. 8.6. The value of k is given by
Ib h ( l
(8.12)
K=--
in which E =Young's modulus for concrete A, = cross-sectional area of the columns h =effective height of the column given by Eq. (8.9)
h=h 0 -2aa
in which I is the moment of inertia of the column.
'
(8.9).
Natural Frequencies of Vertical Vibrations. vertical vibration are given by
The natural frequencies of (2.98)
in which a =coefficient given in Fig. 8.6 a = one-half of the depth of the beams for frames without haunches and the distance shown in Fig. 8.5 for frames with haunches h 0 = height of the column from top of the base slab up to the center of the beam
(2.99a) (2.99b) and
The spring stiffness k 2 (Fig. 8.8) is given by
k,
=
aw '
in which ~' is the deflection in the beam. ~' is given by
(8.26)
Mass m 1 is given by mJ =
WL
+ 0.33W, + 0.25WB g
(8.22)
368
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
369
DESIGN PROCEDURES
Mass Center of Gravity (Point D in Fig. 8.10). center from A 1 is given by
in which WL ~load transferred by the longitudinal beams on one column W, ~ weight of the two columns constituting the transverse frame
WB ~ weight of the transverse beam
x~
·
Mass m 2 is given by
m ~ Wm + 0.45WB 2 g
(8.23)
L:wx I I L:Hj
(8.44)
in which Xi is the distance of center of gravity of load on any frame j from end A 1 • Polar Mass Moment of Inertia of All Masses about a Vertical Axis through the Mass Center, M mz
in which Wm is the weight of machine and bearing. The vertical amplitude A , 1 and A , 2
Amplitudes of Vertical Vibration. are given by
(8.27c)
(8.29) in which
aj
is the distance between center of mass
mj
and the mass center.
:•, - .-~·'
·[''
Stiffness of a Frame in Transverse Direction Kxi
and A
= ,z
b.(w
The distance X of mass
2
)
z = z
(1 + JL)w!fl + f.LW~/2m,b.(w')
w2
P,
1)
k . ~ 12EI, ( 6K + X] h3 3K + 2
(8.27d)
is calculated from Eq. (2.104).
(8.20)
Total Lateral Stiffness kh
Horizontal Vibrations
Load Carried by Each Frame WI
(8.7)
Distance of the Stiffness Center (Point C in Fig. 8.10) from End A
1
in which Wm ~ weight of machine and bearing transferred to a transverse beam q ~ uniformly distributed load on the transverse beam including its own weight WL ~ load transferred by longitudinal beam on one column
and l ~ effective length
(8.45)
in which "%:. X, ~ distance of point C from A 1 (not shown in the figure) X,i ~ distance of center line of transverse frame j from the end A 1 (not
shown in the figure) Total Mass
Distance e between the Mass Center and the Stiffness Center
L:Hj
m~--
g
e ~x-x,
(8.46)
370
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
EXAMPLES
371
Equivalent Spring k• for Torsional Vibrations The resultant horizontal amplitude A h is (8.32) in which b1 is the distance of lateral stiffness kxf (frame j) from the stiffness
center.
Limiting Natural Frequencies of Horizontal Vibration. frequency of horizontal translation is
w
11~1
Check on Dynamic Response
is (8.36b)
Natural Frequencies in Coupled Horizontal Translation and Torsion
(8.35) in which a is given by e' r
a=l+--, r = radius of gyration =
Amplitude of Horizontal Vibration. tion Ax is
(8.40)
~ M,;;,
in which a is the distance of the point at which the horizontal amplitude is to be calculated from the mass center.
The limiting
(8.36a) The limiting natural frequency of torsional vibrations
(8.42)
(8.41)
The amplitude of horizontal transla-
(8.37)
The natural frequencies in vertical and horizontal modes of vibration as calculated above should be compared with the operating speed. The natural frequency in any mode of vibration should be at least 30 percent away from the operating speed. Similarly, the computed amplitudes of vibration should be compared with their permissible values. Amplitudes of vibration should in no case exceed the limiting values of amplitudes. The amplitude buildup during starting or shutdown should be checked by assuming resonance to take place and the maximum amplitude so calculated should not exc!'ced the permissible Y,a!ues. If the results of preliminary analysis indicate a satisfactory behavior it means that the trial sizes are adequate and more precise values should be ascertained by a detailed dynamic analysis. The detailed dynamic analysis should be performed to ascertain that resonance does not develop under normal operating conditions and the vibration amplitudes both under normal operating conditions and during starting and shut off stages are within the permissible limits. The stresses in frame members under normal operating conditions, as well as under the action of anticipated emergency loads, are calculated. The details of the input parameters depend on the computer package selected and can be decided based upon the computer code used for the design. A reference should be made to the user's manual for this purpose.
8.7
EXAMPLES
EXAMPLE 8.7.1
in which
A reinforced concrete frame is shown in Fig. 8.13 (a, b, c) and carries vertical loads at points s~own in Fig. 8.13a. The details of these loads are: I, (8.39)
The amplitude of rotational vibrations A
is
"'
Points #1 and 2 = 5 t each Points #3, 4, 5, and 6 = 2 t each Assuming unit weight of concrete as 2.240 t/m 3 and Young's modulus for concrete as 3 X 106 t/m 2, calculate the natural frequencies of horizontal
(8.38)
vibrations in the longitudinal direction (a) by hand solution treating the frame as a single-degree-of-freedom system. (b) by treating the frame as a
372
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
373
EXAMPLES
multidegr~e heedom system without considering the effect of soil and also
by mcludmg mteractwn effect of soil. Assume the value dynamic shear 3 modulus G = 9000 tim and v = 0.33. Neglect the effect of haunches.
I,=
From Fig. 8.13b
= 0.03645 m
4
Relative stiffness factor K: K
Solution The dimensions of the frame and the various loads acting on it are shown in F1g. 8.13 (a, b, c).
Unit weight of concrete= 2.24 tim 3 Young's modulus of concrete E = 3 x 106 tim' Total weight of the deck slab W W= 8.6 X 6.8 X 0.6 X 2.24- 3.6 X 2.8 X 0.6 X 2.24 = 65.04 t Applied load= 5 + 5 + 2 + 2 + 2 + 2 = 18 t Total load WT = 65.04 + 18 = 83.04 t Effective span of beam and height of column:
~2°· 9 '
06 ·
=
I, lz_ I, l
K = 0.036 0.03645
X
5.5335 = O 9786 5.589 .
Horizontal stiffness kx =
k,
12EI, h3
k = 12
X
x
(6K + 1) 3K+2 6
3 X 10 X 0.03645 ( 6 X 0.9786 + 1) (5.5335) 3 3 X 0.9786 + 2
(8.8)
''k, = 10780.3 tim
(8.9)
wn, = \j
/2
fnx =
(8.20)
X
····'
10780.3 X 9.81 _ = 50.46 radisec 83 04
50.46 z:;;:= 8.03 Hz
10 =5.7m Multi-degrees-of-Freedom System Neglecting Effect of Soil
h 0 =5.7m a= 0.30.
Assnme the columns to be fixed in the base slab and consider one longitudinal frame. The lumped-mass model is made for plane frame analysis following the specific program manuals. i" The natural frequencies were obtained by using a computer program. The values of the first eight natural frequencies obtained are shown in Table 8.2.
b 0.45 /;, = 5.7 =0.0789 h 5.7 -0= - = 1 10 5.7
From Fig. 8.6, a= 0.185
Multi-degrees-of-Freedom System Considering Effect of Soil
f = 5. 7- 2
X
0.185
X
0.3 = 5.589 m
h = 5.7-2
X
0.185
X
0.45 = 5.5335
Moment of inertia of the beam I 6 : I = 2 h
X
(0.6)' = 0 036 ' 12 . m
Moment of inertia of the column I '.·
The values of the soil springs k"' kx and k were obtained by using the procedure in Section 6. 7•Ior the base area of the foundation. The values of these springs are for the base area of the foundation are: k, = 103044.7 tim
k, = 84874.8 tim
k., = 1569928.4 timirad
t There are several computer codes currently available in the market. The choice among these commercially available codes should be based on the 'specific requirements of the problem (see Appendix 3).
374
FOUNDATIONS FOR HIGH-SPEED ROTARY MACHINES
Table 8.2. Computer Output: Plane Frame Analysis, Soil Interaction Neglected Mode 1 2 3 4 5 6 7 8
Eigenvalue 3 .464487D 1.981161D 2. 779605D 1.202309D 3. 747028D 5.200198D 1.074382D 1.631767D
03 05 05 06 06 06 07 07
Frequency Hz 9 .367850D 7.084030D 8.390967D 1.745133D 3.080803D 3.629363D 5.216747D 6.429091D
00 01 01 02 02 02 02 02
Period sec !.067481D-C1 1.411626D-02 1.191758D-02 5. 730223D-03 3.245907D-03 2.755304D-03 1.916904D-03 1.555430D-03
375
REFERENCES (l b) "En ineering User's Manual. Vol. 2. Additional Design and 1 979 ICES STR~DL .rl. . , S h. I Civil Engineering Massachusetts Institute of Technology, AnalysiS Fac1 1ttes. c oo ' Cambridge Massachusetts. ~~einlogel (1.94~). "Rahmenformeln." Springer~Verlag, Berlin and New York. Kleinlogel (1964). "Rigid Frame Formulae." Frederick U~gar Publ. Co., New York. t . h V (1959) "Frames and Arches." McGraw-Hill, New York. . , L eon ovJc ' · · . _ . A · nd Design " Akademiai Ktado, Major, A. (1980). "Dynamics in Civil Engmeenng: na1ysts a .
of
Budapest. Vol. 3. B k Rausch, E. (1. 959 ). "Maschinen Fundamente und Andere Dynamisch Beanspruchte au on. _. structionen." VDI Verlag, DUsseldorf. N E (1975) Natural frequencies of a turbme foundat1on. Proc. ShenSt~~t~De~~~~:~n;~we~ Pian-ts Fa~ilities Conf., New Orleans, Il-~02 to 11-.311. . . nternational Conference of Buildmg Officmls, Whittier, Uniform Building Code (1985). I California.
Table 8.3. Computer Output: Plane Frame Analysis, Soil Interaction Included Mode 1 2 3
4 5 6 7 8
Eigenvalue 5.455556D 6. 768874D 1.594582D 8.922416D 1.357772D 4.283156D 1.387527D 1. 728597D
03 03 04 04 05 05 06 06
Frequency Hz 1.175547D 1.309419D 2.009759D 4. 754028D 5.864538D 1.041603D 1.874740D 2.092509D
01 01 01 01 01 02 02 02
Period sec 8.506678D-02 7.636974D-02 4 .975722D-02 2.103480D-02 1.705164D-02 9.600583D-03 5 .334072D-03 4. 778952D-03
The computed natural frequencies using a computer program are shown in Table 8.3.
REFERENCES ADINA System (1981). "ADINA User's Manual." ADINA Engineering Inc., Watertown, Massachusetts. American Society of Civil Engineers (1987). "Design of large steam turbine-generator foundatiOns." ASCE task committee on turbine foundations. Arya, S. D., O'Neill, M., and Pincus, G. (1979). "Design of Structures and Foundations for Vibrating Machines." Gulf Publ. Co., Houston, Texas. Arya, S.D., and Drweyer, R. (1977). Mathematical modelling and computer simulation of elevated foundations suppbrting vibrating machinery. Trans. -fMACS 19, No. 4. Barkan, D. D. (1962). "Dynamics of Bases and Foundations." McGraw-Hill, New York. Hurty, W. C., and Rubinstein, M. F. (1964). "Dynamics of Structures." Prentice-Hall, Englewood Cliffs, New Jersey. ICES STRUDL II (1979a). "Engineering User's Manual. Vol. 1. Frame Analysis." School of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts.
-?--•
·:"· ..
9 Foundations for Miscellaneous Types of Machines This ~hapter deals with the design of foundations for the following types f machmes: o
FOUNDATIONS FOR lOW-SPEED ROTARY MACHINES
1. The speed of operation of motor-generator units is much lower (below 1500 rpm) than that for turbogenerators (usually 3000 or 3600 rpm).
2. Motor generators have a larger unbalance compared to turbogenerators, whicb are well-balanced machines. Even at low speeds of operation, the motor generators produce large unbalanced forces and moments.
3. Permissible motion amplitudes for motor generators are higber than for turbogenerators. Tolerance vibration limits are rather stringent (Section 8.3) for turbogenerators. Permissible vibration amplitudes for motor generators operating at low speeds of 300 to 400 rpm are in the range of 0.1 to 0.3mm.
For motor generators operating at low speeds (less than 300 to 450 rpm), it is advisable to design a high-tuned foundation in the form of a rigid concrete block. For motor generators operating at 1200 to 1500 rpm, both massive concrete or frame-type foundations may be provided depending upon soil conditions and the space required for equipment and other accessories.
1. Low-speed rotary machines. 2. Machine tools. 3. Stamping, forging and punch presses.
The problem of machines supported directly on building floors has also been mcluded.
>. ,.rr
t,:
To ensure long-term satisfactory p"erformance of motor generator units, the design criteria given below must be satisfied both for static and dynamic loads.
Design Criteria Static Loads.
9.1
The stability criteria in respect of static load are
FOUNDATIONS FOR LOW-SPEED ROTARY MACHINES
This class includes rotary machines operating at Jess than 1500 rpm M t generators, centrifugal pumps, fans and blowers, crushing mills, and ro~i~~ mills are some examples of low-speed rotary machines For these machines both the massive-block-type and frame-type foundatio~s are used. Principle~ gover~mg the design of foundations for this category of machines are essentially the same as discussed in Chapter 6 for block -type foundations (for reciprocatmg machines) and in Chapter 8 for frame foundations Appropnate design procedure should be followed depending upon type of foundation selected. In some cases, it may be necessary to use a pilesupported foundation. Concepts developed in Chapter 12 may b d · such F d · . e use m _cases. oun atton destgn for some of these thachines wil1 0 b descnbed. . ow e Motor Generators
The design of foundations for motor generator units is considered separately from turbogenerator units for the following reasons: 376
377
1. No shear failure in soil. 2. No excessive settlement.
Dynamic Loads.
The design criteria in respect of dynamic loads are
1. Natural frequencies of the foundation-soil system should be at least 30 percent away from the operating speed of the machine. 2. Amplitudes of vibration should be within the permissible limits specified by machine manufacturer. In case specifications by manufacturers are not available, limiting values of design amplitudes may be adopted as follows: "<:
Operating Speed 750 rpm or less 750 rpm to 1500 rpm
Permissible Amplitude (mm)
0.06 to 0.12 0.06 to 0.04
Data Required. Information listed below should be obtained for the design of a foundation for the motor generator unit.
378
FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES FOUNDATIONS FOR LOW-SPEED ROTARY MACHINES
379
Machine Data
f
frequency of pressure waves in water flowing through the pump N = speed of the pump in rpm n = number of vanes in the impeller
1. Weight of motor and generator.
2. Weight of rotor for (a) motor and (b) generator and the effective eccentricity for each. If information on unbalanced forces and moments is available, data on eccentricity is not needed. 3. Weight of flywheel. 4. Operating speed. 5. Short-circuit moment. 6. Geometric layout of the machine units, anchoring details, and openings required for accessories and inspection. i
I
Soil Data 1. Soil profile and data on soil characteristics up to three times the foundation width or hard stratum. 2. Realistic dynamic soil properties consistent with confining pressures and strain amplitude as discussed in Chapter 4 (Section 4.7). 3. Position of water table.
Design Procedure. The choice of the foundation among massive block type or frame type depends upon operating floor level, layout of the accessory equipment, operating speed of the machine, and subsoil characteristics. Vibration characteristics of the block-type foundations may be computed following the procedure recommended in Section 6.8. For frametype foundations, the procedure suggested in Section 8.6 should be adopted. Foundations for Centrifugal Pumps
These pumps operate at low speeds. The foundation size provided from considerations of geometry of the pumping installation is large enough and vibration problems usually will not arise. There are two main sources of vibration for which the foundation response must be checked: 1. The forces due to unbalance in the rotating unit known as impeller. 2. If the clearance between the impeller and the casing is inadequate, the pressure surges increase and the ensuing waves propagate through the water to the casing and to the foundation. The frequency of such waves is given by (Judd, 1955)
Nn
t= 60 in which
(9.1)
=
Design Criteria
From considerations of dynamic stability of the pumping installation, the criteria to be satisfied are: 1. Natural frequency of the soil foundation system should be at lea;\~0 percent away from the operating frequency and the frequency o e pressure surges. 2. Amplitudes must be within permissible limits.
Machine Data. foundation:
The following machine data is needed for design of its
1. Weighf'of the pump.
2. 3. 4.
5.
..,. ' Operating speed of the pump. Number of impeller vanes. Frequency of pressure surges. Unbalanced forces due to pump operation.
Soil data should be procured as discussed for motor generator sets. . Block-type foundations are used for pumping installations. The v1bral!on characteristics are ascertained following the procedure suggested m Section 6.8 for block foundations.
Foundations for Fans and Blowers
A fan consists of a set of blades attached to a rotor. The angle of blades is set so that the rotation of the fan causes a flow of air in the ax1al duectiOn. In blowers, the air flows both axially and radially. Speed of operatiOn of fa.ns and blowers varies over a wide range from 150 to. 750 rpm. High-capacity units usually have 4J;.ver speed of operation. VIbratiOns result due to unbalance in the rotor; drive mechamsm, and the motor. . Fans and blowers are supported on block foundatiOn. Layout of a typical block foundation for a primary fan is. shown in Fig. 9.1. Because ofthe1rlow speed of operation, the vibrations transmitted to the smlby the foundatiOns of the fan may have a detrimental effect on the ad]ommg structures. A trench is generally provided around the
foundatl~m.
.
.
Soil and machine data necessary for foundatiOn design IS the same as discussed earlier for motor generators.
380
FOUNDATIONS FOR MISCEllANEOUS TYPES OF MACHINES
of discharge of primary ""' air fan unit
FOUNDATIONS FOR lOW-SPEED ROTARY MACHINES
381
Jaw crushers may have different geometrical configurations and may be with or without counterweights. Some typical configurations of these crushers are shown in Table 9 .I, column 2. The unbalanced inertia forces for each case may be calculated by equations summarized in column 3, Table 9.1. In the case of gyratory or cone crushers, the ore is pulverized between the crushing head of the main shaft, which undergoes a rocking motion along a circular path, and the armored jacket of the upper stationary part. Magnitude of the inertia force F 1 , due to angular rotation of the main shaft with the attached crushing cone is given by Eq. (9.2a):
Foundation block
of primary air fan foundation
(9.2a) Retaining wall
in which
(a) Plan
m 1 =mass of the main shaft including mass of crushing cone r 1 = distance between center of gravity of the main shaft and crusher cone .from the axis of the.,s~.usher w = angular velocity of shaft rotation.
~Grout~ Covered~ch
t
16
Lr.= ~·
~. v
+ Figure 9.1.
:P ,,,
rb J
_1J T ur~
/
Reinforcement bars as per design (b) Section XX (concrete not shown)
The camshaft, gears, and accessories, e.g., counterweights attached to it, give rise to a force F2 in a direction opposite to that generated due to rotation of the main shaft. The magnitude of this force is given by Eq. (9.2b): (9.2b)
t
in which
m 2 = mass of camshaft, gears, and counterweights that are rigidly fixed to it r2 =distance between crusher axis and center of gravity of camshaft with accessories attached to it.
Typical foundation for a primary air fan.
Foundations for Crushing Mills
Crushers are used for pulverizing ore. Crushers may be classified in three categories: 1. Jaw crushers. 2. Gyratory crushers. 3. Rotary hammer crushers.
The operation of the crusher results in unbalanced inertia forces that induce forced vibrations of the foundation.
The resultant exciting force F may therefore be obtained from Equation (9.2c):
or (9.2c) This force Facts in the plane of rotation of the crushing cone, which is a horizontal plane and may be resolved into two mutually perpendicular
FOUNDATIONS FOR LOW-SPEED ROTARY MACHINES
383
components Px and PY (referring to the horizontal plane of rotation of the cone as XY) given by Equation (9.3).
Px
II
II
wt
(9.3a)
PY= Fcos wt
(9.3b)
=
Fsin
The forces Px and PY will also give rise to exciting moments Pxh and PYh in which h is the height of point of action of Px and PY above the top of the foundation. These moments act in vertical planes and induce rocking of the foundation. The nature of the unbalanced forces associated with jaw crushers (Table 9.1, column 3) and gyratory crushers [Equation (9.2c)] is similar to those associated with operation of reciprocating machines. Rotary Hammer Crusher
382
Rotary hamnter crushers are commonly used for crushing limestone in cement factories. Auto shredders ar~ another example. A hammer crusher has a rotor on which a number of hammers are attached radially (Fig. 9.2). A series of such hammers form different rows. The rotor and the hammer are housed in a steel casing with an inlet at the top and outlet opening at the bottom. The stones that fall inside the crusher (Fig. 9.2) are hit by the rotating hammers. As a result of impact by the hammers, the stones strike the crusher casing, rebound, and again hit the rotating hammers. The process of rebounding and hitting continues till the stones are crushed to the required size and come out of the crusher. Crushers may be used in series to get crushed stone of the required size. The crusher that gives coarser crushed stone is called a primary crusher. The next lower crushed size of stones is obtained in the secondary crusher. The process of crushing of stones is a continuous one. Unbalanced loads are generated due to hitting of the stones by the hammer and due to rotation of the material inside the crusher. The distribution of material inside the crusher is not uniform. Thus the magnitude and direction of the unbalanced forces and moments is difficult to calculate. The data on unbalanced forces should therefore be obtained from the supplier. The usual speed of if; rotary crusher ranges from 300 to 750 rpm. The capacity of the crusher for a cement plant producing 600 tons of cement per day will be about 200 tons per day. The typical weight of the crusher for such a unit will range from 40 to 50 tons, and the weight of the rotating part will be 10 to 20 tons. If the data on dynamic loads is not furnished by the supplier, a dynamic force equal to twice the weight of rotating parts acting at the center of gravity of the crusher should be used for design of the foundation. Foundations for crushers are usually designed as massive concrete blocks.
384
FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES
385
FOUNDATIONS FOR LOW-SPEED ROTARY MACHINES
Centerline
I
Crusher Inlet
([ of oil drain
Radial hammer Pocket for anchor bolt ([of pulverizer unit
Circular binders
,c.Jop of structural
,·-
Radial bar
steel
Outlet
'.
Reinforcement bars
~~oc:=:==fi=s=,:~:"':~d:~:~c:"u"'sh"',"',""""""""J;;;;;;;: Outlet I
Top of
I
concrete vLL'--~
Centerline
Figure 9.2.
raft
Sand filling
Hammer crusher for a cement factory. Section XX
(concrete not shown)
Figure 9.3. Typical foundation for a crushing mill.
A typical foundation for a crusher pulverizer is shown in Fig. 9.3. Frametype foundal!ons can also be provided for crushers. Data required for design IS as follows.
Machine Data 1. Layout of crusher and motor.
2. Weight of crusher and motor. 3. Operating speed of the main shaft. 4. Unbalanced forces or machine data for their calculation.
5. Anchoring details. 6. Permissible vibration amplitudes.
Soil Data. and 8.6.
Information on subsoil data is as discussed in Sections 6.8
-,..:.
Design Criteria
The criteria for static and dynamic stability should be satisfied as for any machine foundation. The limiting vibration amplitudes should not be exceeded. If information on limiting amplitudes is not available, an upper
limiting value of 0.3 mm may be assumed. The design should be carried out following the procedure suggested in
386
FOUNDATIONS FOR MISCEllANEOUS TYPES OF MACHINES
387
FOUNDATIONS FOR lOW-SPEED ROTARY MACHINES
Section 6.8 if a block foundation is adopted and of Section 8.6 if a frame foundation is to be designed.
C,----(D
Foundations for Rolling Mills
The process of rolling is used to convert molten ingots of steel into structural sections by passing them through a set of rollers. During the rolling operation, variable loads are transmitted to the foundation, resulting in foundation vibrations and transfer of dynamic stresses to both soil and foundation. A rolling mill consists of following components:
c
w
E 0
:< A
B
K H
Time (sec)
I
(a)
J
I. Driving motor.
2. Motor generator (power) unit. 3. Rollers and drive-gear stands. A typical roller unit is shown in Fig. 9.4.
Driving Motor. This is a reversible direct current motor which controls the speed of the rollers and is supported on an independent foundation. The speed of operation of these motors is low, approximately 60 rpm. Torque on the shaft of the rotor varies depending upon the stage of rolling. Different stages in rolling during one pass of the ingot and the corresponding torsional moment on the shaft are described below. I. Speeding up the rollers at no load. The torsional moment during this stage is constant and is shown by AB in Fig. 9.5a.
2. The ingot is gripped by the rollers and forced through them and simultaneously the rollers accelerate. The torsional moment increases as indicated on Fig. 9.5a by line BC. 3. Rolling then proceeds at constant speed. The torque stays constant as shown by line CD. Adjusting screw
Adjusting screw
~ rn1~ +-Rolling-~:tl~'"';'\>jl (b)
Time
Time
Ingot exit
lcl
Figure 9.5. MomenHime history of pulse du~ing r~lling operation. ~a) S~~ematic. diag~am for one passage of ingot, (b) idealized momenH1me d1agram, and (c) s•mphf1ed des1gn d•agram.
4. Towards the end of one rolling pass, the rollers slow down resulting in diminishing torque as shown by D EFG. 5. The ingot then exits, torque diminishes along line GH. 6. The exit of the ingot is marked by sudden unloading of the system as shown by lines HIJ. 7. One pass of the ingot is completed and torque gradually falls to zero along line JK. ,.,
Roll stand
Roller
Figure 9.4.
A typical roll stand unit. (After Major, 1980.)
An ingot passes through the rollers several times before the finished section is obtained. The shape of "torsional moment" vs. "time" diagram for each roller pass is similar. The maximum torsional moment on the shaft and consequently the maximum exciting moment on the foundation are developed at the end of the acceleration stage as shown by C, and this moment varies slightly during the rolling process. It may reasonably be assumed that the absolute change (decrease) in torsional moment during exit
388
FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES
FOUNDATIONS FOR LOW-SPEED ROTARY MACHINES
of the ingot is of the same order as the change in moment (increase) when the ingot is gripped by the rollers. The design diagram for torsional moment variation for one pass of rolling may be simplified to that shown in Fig. 9.5b. Further, the exit of the ingot is followed by a decrease in torsional moment and the stresses in the foundation can exceed those during the process of steady rolling when the torsional moment is maximum. The loading diagram may therefore be further simplified to that shown in Fig. 9 .5c. The torsional moment will induce rocking vibrations of the foundation. The stress induced in the soil will be the sum of the stress due to static load and rotation of the foundation and may be obtained as follows [Eq. (9.4)]: (9.4) in which
Operating and Drive Gear Stands
I I
=maximum value of stress induced to soil W = weight of foundation and equipment thereon A = foundation area in contact with soil C.; =coefficient of elastic nonuniform compression of soil a = foundation width in plane of rotation m"' = maximum angular rotation of the foundation qmax
The value of m" may be obtained by dynamic analysis as for a foundation excited by a moment (Sections 6.4 and 6.7). Alternatively, the value of m"' may be taken as the value of angle of rotation of the foundation if the given torsional moment were applied statically and multiplied by a dynamic loading factor. A dynamic load factor of 2 has been suggested by Barkan (1962). Motor Generator (Power Unit). The power unit supplies power roll motor and consists of one or more direct current generators flywheel mounted on the same shaft. The generators are operated electric motor. The value of torque Mi due to operation of the power given by
to the and a by an unit is
(9.5)
389
The roller stand provides support to the bearing and transmits the forces arising during rolling process to the foundation. The gearbox houses the gears that drive the rollers. A torque equal to the torque on the shaft of the drive motor acts on the gearbox. The foundations for the gearbox should therefore be designed in the same manner as for the driving motor. When the drive-gear and working stands are mounted on a separate foundation, not tied to that under the driving roll motor, the dynamic effects of external loads on the foundations are evaluated separately but similarly. Drive-gear stand, working stand, and driving roll motor may be mounted on a common foundation, the drive-gear stand is then subjected to the action of a torsional moment whose sign is opposite to that of the moment acting on the stator of the driving roll motor. The sum of all the external dynamic loads transmitted to the foundation and soil in this case equals zero, and the foundation will be under the action of internal torsional moments whqse magnitude equal~ t:!Je moment of the shaft of the motor, as well as under the action of the (;quipment weight. These loads should be considered in the stress analysis of the foundation and its components. For designing foundations for a rolling mill, the data as given below should be obtained: Machine Data
Layout of the complete unit. Weight of the rolling-mill equipment. Weight of the driving roll motor. Maximum moment at the motor shaft. Horizontal force transmitted to the footings. 6. Erection loads.
1. 2. 3. 4. 5.
Soil Data. As for other foundations (Sections 6.8 and 8.6). The drive roll motor is usually supported on a separate block foundation. The motor generator may be supported on massive-blocks or a frametype foundation. · ~-q'
Operating and drive gear stands are usually mounted on a common
in which
foundation. Static computations of the foundation are limited to
I= mass moment of inertia of all rotating masses of the power unit N = speed of operation
dNI d.t =rate ,of change of speed, which varies from 3 to 10 cycles (per sec.) (Barkan, 1962). The value of the design moment should be taken as 2Mi to account for the most unfavorable conditions.
1. Stress analysis of separate units of the foundation, such as units weakened by openings, cantilevers, and others.
2. Computation of local stresses under supporting slabs. 3. Analysis of stresses within the foundation. 4. Computation of pressures transmitted to the soil.
390
FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES
The foundation is assumed to be a girder of varying rigidity resting on an elastic base. A dynamic factor of 2 is used in respect to the computations 1 and 2 above. Calculations 3 and 4 are based on actual weights. . A typical f~undation for the roller unit and driving gear for a light steel millis shown m Fig. 9.6.
FOUNDATIONS FOR MACHINE TOOLS
391
Foundations for Grinding Mills
Grinding mills are of two types: (a) drum mills and (b) tube mills. The tube mill has a narrow diameter and larger length. The diameter and length or height of a drum mill are comparable . A tube mill has an outer shell that is rotated by a motor. The tube contains a charge of steel balls that impinges upon the material being ground. Drum mills work on the same principle as the tube mills. The grinding mills may be used both for dry and wet grinding. For design of foundation for grinding mills, the following data should be procured (Major, 1980): 1. Layout details of the mill equipment.
2. 3. 4. 5.
Weight of the mill casing, ball charge, and material to be ground. Characteristics of the driving motor and drive. Direction of rotation of the mill. Distance of the axis of the drumshaft and the top of the foundation.
Dynamic analysis is generally 'n'cit needed for the design of mill foundations. Soil stresses must, however, be checked and ensured to be below the permissible values for the loading conditions given below: 1. Weight of machine. 2. Weight of foundation. 3. The horizontal component of the centrifugal force in the direction of motion of the bottom generator of the tube. The magnitude of the horizontal component should be taken as 10 percent of the weight of the mill (W) for mills with short drums and 20 percent of W for tube mills. Weight W is the weight of the mill without ball charge and material to be ground. Permissible values of the soil stresses are generally reduced to 80 percent of those under static loads. Weight W is considered to be uniform over the two supports of the drum. ±0.00
When the soil conditions are good (permissible soil stresses 3.5 kg/ cm 2 or more), separate foundations may be provided for intake and discharge ends of the mill and also for driving motor and the reduction gear. When soil 2 conditions are not favol\(lble (allowable soil stress <1.5 kg/cm ), the entire mill should be placed on a common foundation.
9.2
FOUNDATIONS FOR MACHINE TOOLS
(b)
Fig~re 9.6. Foundation for a light steel rolling mill. (a) Roller stand, and (b) driving gear. (After
MaJor, 1980.)
Machines such as lathes, milling, drilling, and boring machines are known as machine tools. A dynamic analysis for foundations for machine tools may be
392
FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES
needed only in exceptional cases. A static design is generally adequate. In the case of precision machines, it may be necessary to protect the machine against vibrations from other machines such as hammers, crushing mills, compressors, and railroad and traffic vibrations. In such situations, the foundation may be mounted on suitably designed vibration absorbers (Section 10.3). Vibration isolation can also be achieved by providing a trench around the machine tool (Section 10.5).
9o3
FOUNDATIONS FOR STAMPING, FORGING, AND PUNCH PRESSES
Presses are commonly used in practice for stamping, forging, and punching purposes. A press is assigned a name depending upon the specific operation for which it has been designed. A press consists of a crosshead, a bed plate and columns. The bed plate rests on a base that is anchored to the foundation. A schematic sketch of a press is shown in Fig. 9.7. The bed plate can also be directly anchored to the foundation depending upon the type of the press. The crosshead moves up and down and has a fixed travel known as stroke. The base plate can be adjusted and fixed at any desired elevation. The desired process of forging, stamping, or punching may be performed by attaching the appropriate tool to the crosshead. For stamping purposes, the stamping dye is attached. Likewise for forging operations a forging tool of the required size and shape is attached. The punching operation is performed by fixing a suitable cutting tool to the crosshead. It should be noted that the forging operation in a press is performed by applying a compressive force through a suitable tool to the red hot metal piece and differs from the forging under impact done with the forging hammers.
II
Columns I
I
I I
I
I
I I
I I
I I
I II
:: ! : : I i
i
weight of machine minus the moving crosshead cross-sectional area of all columns height of the columns Youngs' modulus of column material
The dynamic factor F is given by (9.7)
Crosshead
where V is the velocity of impact. The dynamic force is given by (9.8) /
"'%~
Adjustable --bed plate
in which a is the fatigue factor, which may be adopted as 1 ° 1 to 1 ° 3. For pun~h presses, no method for analyzing the forces transmitted to the foundation is available. In the opinion of the authors, the load transferred to the foundation may be obtained from consideration of energy stored in the metal piece being punched before the yield condition develops. The energy
h
~///~/:;~~;>/ _L Base/
W= A = h= E=
~
II I
:
(9.6)
in which
v
I : I
ii
"•·•' Wh us-=· EA
F=--
r
393
The crosshead may be actuated through hydraulic or pneumatic pressure, or an eccentric or a friction drive may be used. Accordingly, the type of press is: (1) hydraulic press, (2) eccentric press, and (3) friction press. A press unit complete with the material being pressed represents an internally balanced system, and no net load is transferred to the foundatmn. Dynamic analysis, therefore, is not needed. A dynamic overload factor of 2 and inclusion of twice the weight of the material being forged is sometimes considered in design. Dynamic overload in stamping presses is caused by the drop of an upper ram on forge piece. In eccentric presses, there is a dynamic torque in the horizontal plane of the foundation. In the case of a large eccentric press, impact moments are also caused. Dynamic effects for hydraulic presses (Fig. 9. 7) can be accounted for by consideration of deformations in the system (Rausch, 1959). Let hand A be the height and cross-sectional area of the anchor columns of the hydraulic press (Fig. 9. 7). The elastic deformation As of the columns is given by
I
I I
I I
FOUNDATIONS FOR STAMPING, FORGING, AND PUNCH PRESSES
stored in the metal piece will be released suddenly as the punch cuts through Figure 9.7. Schematic sketch of a hydraulic
press.
the metal subsequently. This release of energy may result in vibrations of the system.
395
ILLUSTRATIVE EXAMPLE
FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES
394
The data required for designing foundations for presses will now be discussed.
Design Data. The data giVen below should be obtained for designing foundations for a press. Layout of machine installation. Maximum force exerted by the press. Stroke of-the press. Weight of the press. Weight of material to be forged or pressed. Load-time characteristics of the pulse or the dynamic loading effects due to stamping or forging, including dynamic torque for friction presses. 7. Data on soil profile and its static and dynamic characteristics.
Foundation block
1. 2. 3. 4. 5. 6.
The design of the foundation may then be attempted as usual.
9.4
r
Figure 9.8.
I.
Setting a machine directly on floor.
MACHINES SUPPORTED ON FLOORS Washer
Smooth-running machines such as small electric motors and machine tools may sometimes be installed on building floors. It is essential to ascertain that such an installation should not result in excessive vibrations of the floor or the building structure. While installing machines on the building floor, the following points should be considered (Major, 1980): 1. It must be established by analysis that the natural frequency of the floor and the building structure is far from the operating speed of the machine. This will ensure that the building is not exposed to undesirable vibrations. If it becomes necessary to install a machine at an upper floor level, the buildup of amplitudes should be checked by rigorous analysis. 2. Vibration isolation should be provided by inserting vibration absorber pads between the machine and the floor. An arrangement for providing an absorber pad between the machine and the pad is shown schematically in Fig. 9.8. The use of a rubber sleeve is illustrated in Fig. 9.9.
Figure 9.9.
Fixing machine leg on floor.
is felt that the performance of some existing machines resting on floors and vibration response of floors and adjoining structural elements needs to be monitored so that thf~ topic may be given a sound design basis.
3. For a vibration-sensitive environment, the machine ·must be supported
on soft springs of steel or rubber. 4. Concrete floors should be protected against chemical attack due to leakage of oil from the machine. No design procedures have been developed to date for machines supported directly on floors. A modal analysis of the building frame may be performed to ascertain its vibration response and to ensure a safe design. It
9.5
EXAMPLES
Design of foundation for a grinding (tube) mill A grinding (tube) mill is to be installed as a part of the cement plant. The following data on the size and capacity of the mill is given.
FOUNDATIONS FOR MISCElLANEOUS TYPES OF MACHINES
396
2. Weight of motor and equipment= 35 t. 3. Weight of foundation block at inlet end W,
l. Diameter of the tube= 3.0 m.
2. Length of the tube= 12.0 m. 3. Weight of the tube W, = 95 t.
W, = (6
steel balls Wb = 45 t. the material to be pulverized (capacity) W, = 10 t. accessories at discharge end= 5.0 t. motor and gear Wm = 35 t.
4. Weight of 5. Weight of 6. Weight of 7. Weight of
397
ILLUSTRATIVE EXAMPLE
X
6 X 2.5 + 3.6 X 1.8
X
0.5]2.4 t
= 223.78 t 4. Weight of foundation block at discharge end is Wd.
Wd = 2.5(8.8 X 6 X 2.5 + 2(1.8 X 3.6 X 0.5)- 3 X 6 X 0.7]
= 294.91 t A sketch of the proposed foundation is shown in Fig. 9.10. The soil investigations at the site of the mill indicated that the safe allowable soil pressure may be taken as 2.0 kg/ cm 2• Check the suitability of the proposed foundation. Assume that the unit weigbt of concrete is 2400 kg/m3. Solution It is proposed to provide separate foundations for the inlet and discharge
ends. The motor and other machinery will be supported on the same foundation as for the discharge end. l. Weight of machine, balls, and material being pulverized
=
w, + wb + w, = 95 + 45 + 10 = 150 t
5. Allowable soil pressure= 2.0 kg/cm 2 = 20 t/m
2
Check of foundation design at the inlet end Static 150 Weight of machine+ foundation= 2 + 223.78 = 298.78 t Base area'= 6 x 6 =36m 2 .•, .••.• ,· 298.7 Soil pressure due to static load = ~ 2 =8.29 t/m Dynamic Horizontal centrifugal force P for tube mills= 0.2 x W, 95 X =0.2X =9.5t Moment about the base Md = 9.5 x 3.7 = 35.15 t m 2 2 Dynamic stress at base due to moment= 35.15/(k x 6 x 6 ) = 0.97 t/m Stresses on the soil due to combined static and dynamic load 2 2 =8.29 ± 0.97 tim 2 = 9.26 t/m and 7.32 tim The stresses induced are less than the allowable soil pressure. Check of foundation design at discharge end Static 150 Weight of machine and foundation=- + 35 + 294.91 = 404.91 t
z
1T 1 1
I
Plan
1800 ~
6000
l----3000
l
I
3600
' l•l
I
36
]+---
I
.
t-
6000
I
i 3000
30004
~
I
E
1800
1000 1800 (b)
1 l
• I1000 • •I
All dimensions
'm mm
Figure 9.10. Foundation for a grinding mill: (a) inlet end and (b) discharge end. (See Example).
2
Base area= 8.6 X 6 = 51.6 m . 404.91 Stress due to static load = 51:"6 =7.84 tim 2 •~:
2
Dynamic Moment due to horizontal centrifugal force = 35.15 t m 2 35.15 Stress due to moment= 1 2 = 0.81 t/m 6 X 8.6 X (5.5) 2 2 Total stress= 7.84 ± 0.81 t/m 2 = 8.65 t/m and 7.03 tim
398
FOUNDATIONS FOR MISCELLANEOUS TYPES OF MACHINES
The stresses induced in the soil due to combined action of static and dynamic loads are smaller than the allowable pressure both at the inlet and discharge ends. So the foundation proposed is adequate.
REFERENCES Barkan, D. D. (1962). "Dynamics of Bases and Foundations." McGraw-Hill, New York.
Judd, S. (1955). Vibration in hydroelectric power plants. Proc. Am. Soc. Civ. Eng. 81. Major, A. (1980). "Dynamics in Civil Engineering: Analysis and Design," Vol. 2. Akademiai Kiad6, Budapest. Rausch, E. (1959). "Maschinen Fundamente und Andere Dynamisch Beanspruchte Baukonstructionen." VDI Verlag, DUsseldorf.
10 Vibration Absorption and Isolation
Many times the designer is faced with the problem of designing the machine foundations. for which the permissible amplitudes of motion are much smaller thai( those considered n6fmally acceptable for uninterrupted machine operation. It may often be difficult to design the foundation for such limiting amplitudes by proper selection of the mass or the foundation contact area or by increasing the rigidity of the base. A foundation on absorbers may provide an answer to the design problem in such a case. The absorbers used for this purpose may be rubber, cork, felt or neoprene pads, or steel springs. Pneumatic absorbers are also used for this purpose. A foundation supported on spring absorbers is shown in Fig. lO.la. There may be situations when amplitudes of vibration of the machine may be within the acceptable limits so far as the performance of the machine itself is concerned but the resulting vibrations may adversely affect the performance of other machines in the vicinity or may he harmful to adjacent structures. A vibrating footing becomes a source of wave generation in the soil mass (Fig. 3.14). The energy transmitted by the waves may cause vibrations of the structures in the intervening medium. The harmful effects of these waves depend on the operating speed of the machine, the amplitude of motion of the footing, and the nature of the intervening medium. The adverse effects of machine vibrations on surroundings may be avoided either by limiting the amplitudes of vibrations of foundation using absorbers or by providing trench or pit!\' barriers. The trench barriers for screening of waves may be provided either around the source of vibrations (active isolation) or around the precision machine or structure sought to be protected against vibrations (passive isolation). Sensitive eqnipment such as seismographs are usually isolated against vibration due to traffic or machine operation by providing trench barriers around them. In some situations, the depth of the trenches for achieving effective isolation may be very large and may pose construction and maintenance problems. Pile barriers may be effectively used for vibration isolation in such cases. 399
400
VIBRATION ABSORPTION AND ISOLATION
PRINCIPLE OF VIBRATION ABSORPTION
uneven bearing, improper design or construction, and changes in soil conditions such as fluctuations of the water table subsequent to construction of the foundation. Inadequate soil investigation of the site may also lead to such a situation. Depending upon the cause of excessive vibration, certain remedial measures will then be required before the machine can be operated safely. In this chapter the design of machine foundation on absorbers, vibration isolation using trenches and piles, and remedial measures against excessive vibrations of existing machine foundations are described.
Foundation block
10.1
(a)
Foundation block
+ machine Equivalent
I
[f:7~~,~'""' spring kt
ff///bm (b)
401
(c)
PRINCIPLE OF VIBRATION ABSORPTION
A foundation on absorbers is usually made of two parts: a lower slab or a sole plate on which the absorbers are placed and an upper foundation block resting on the absorbers. The machine is anchored to the upper foundation block. For light machines, the base plate of the machine may be anchored directly to the absorbers and an upper foundation block may not be needed. A schematic ~ketch of a machine forlndation on absorbers is shown in Figure 10.1a, and the commonly used model for analyzing this system is shown in Fig. 10.1b. Each of the rigid masses (1) m, due to the foundation and (2) m 2 for the concrete slabs will have six degrees of freedom. The total number of degrees of freedom for the whole system are thus 12. Vibration absorbers are generally used for machines undergoing vertical vibrations and having vertical unbalanced forces. As mentioned in Section 6.2, the vertical vibrations are independent of vibrations in other modesc The problem of machine foundations on absorbers may thus be analyzed by treating the system as a two-degrees-of-freedom system (Fig. 10.1h). The motion characteristics of this system can be obtained as explained previously in Section 2.13. Assuming the masses of the system to be concentrated at their centers of gravity and located on the same vertical line, the differential equation of motion may be written as in Eqs. (2.95a, b):
z + k z + k (z z = 0 m z + k (z z = F sin wt
Figure 10.1. (a) Schematic diagram of a foundation on absorbers (supported type). (b) Equivalent two-spring-mass model of foundation absorber system. (c) Free-body diagram.
m1
1
1
2
2
2
1
2 -
2)
1 -
2
1)
0
(2.95a) (2.95b)
where The process of reducing transmission of vibrations by controlling the amplitude of vibration of the footing by use of absorbers or by providing barriers has been extensively used in the industrial world (Adiar, 1974; Baxa and Ebisch, 1982; Barkan, 1962; Klein and Crockett, 1953). Amplitudes of a machine-foundation system that are within acceptable limits at the time of its commission may increase following a period of its operation. This may be occasioned by an increase in unbalanced loads,
F0 sin wt =exciting force
w =frequency of machine operation rad/sec
z, z 2 =vertical displacements of centers of gravity of masses m 1 and m 2 , respectively k 1 =equivalent stiffness of vertical soil spring below the base and is given by
402
VIBRATION ABSORPTION AND ISOLATION
k = k = 4Gro I z 1- V
(6.18)
and k, is the total equivalent stiffness of all springs in the absorber system. The frequency equation for the system is (Section 2.13)
w;-
(w;
11
+ w; 12 )(1 + !L)w; + (1 + !L)w; 11 w; 1,
=
0
PRINCIPLE OF VIBRATION ABSORPTION
403
in which S is a constant, depending upon machine characteristics such as unbalanced mass and eccentricity. Substituting the expressions for F0 and 2 4 Ll.(w ) in Eq. (2.108) and dividing the numerator and denominator by w , one gets
(2.98)
(10.2)
in which w" 1 , 2 are the natural frequencies of the system, and in which (!)nil
r =-1
is the limiting natural frequency of the entire system resting on soil (when no absorbers are used) and is given by
w '
(10.3)
W"11
w
-
nll-
~ m k+1 1 m,
In case no absorbers are used, the amplitude of vibration of the entire foundation resting on soil is given by
(2.99a)
and w"t 2 is the limiting natural frequency of the mass m 2 resting on absorbers a_nd calculated on the assumption that the system below the springs has large ngidity. w" 12 IS given by
(10.4a)
or (10.4b)
(2.99b) The maximum amplitudes Z 1 and Z 2 are given by 2 wn/2
(2.108)
z1 = m1 Ll.(w2) Fo
z _ (1 + !L)w; 11 + !Lw; 12 2-
m, Ll.(w2)
w2
Fo
(2.109)
in which
(2.104) and w is the operating frequency. The exciting unbalanced force due to a machine is proportional to the square of the frequency of machine operation and, therefore, F0 may be expressed as
It is seen from Eq. (10.2) that the amplitude with absorber will be small only if the ratio r 2 is small. When r 2 , i.e., w" 12 /w is negligible, the amplitude of vibration .Z1 is almost zero and the absorber efficiency is high. The effectiveness of the absorber is thus maximum when r 2 = 0 and decreases as the ratio r 2 increases. For very large values of r 2 (r 2 ~oo), the value of Z 1 approaches the value Z for the no absorber case. From the preceding discussion, it may be concluded that for the absorbers to have a favorable effect on the amplitudes of foundation vibration, the natural frequency of the mass above the absorbers should be as small as possible in comparison with the frequency of machine operation. The required natural frequency of the foundation above the absorbers may be achieved by using absorbers of suitable stiffness and by appropriate selection of mass above the absorbers. For machines operating at high speeds, the required condition bet~een w" 12 and w can be easily satisfied without significant increase in the' weight of the foundation above the absorbers. For machines operating at low frequency, the relationship is usually difficult to satisfy by just decreasing the rigidity of the absorber because this decrease beyond a certain limit is not practicable due to strength requirements. In such a case, massive foundation above the springs is necessary. A proper choice of the type of absorber is very critical in such cases. The absorber
F0
= Sw
2
(10.1)
system may be designed based upon the value of r 2 , which depends upon the required degree of absorption 1) defined by
404
VIBRATION ABSORPTION AND ISOLATION
405
COMMON VIBRATION ABSORBERS
(10.5) The principle of vibration absorber explained above will now be used for developing a procedure for the design of foundations on absorbers. Different types of absorbers commonly used are described first. 10.2
COMMON VIBRATION ABSORBERS
Materials capable of undergoing elastic deformation can be used as vibration absorbers. Commonly used vibration absorbers are l. Steel or metal springs.
2. 3. 4. 5. 6.
Cork pads. Rubber pads. Timber pads. Neoprene pads. Pneumatic absorbers.
10.2.1
Figure 10.3. A multiple spring absorber assembly. (Courtesy Korfund, Inc. 1986).
Steel or Metal Springs
Helical springs made of steel are the most effective elastic supports for reducing amplitudes of vibration in machine foundations. A single spring absorber is shown in Fig. 10.2. This type of spring absorber will be suitable only for very low capacity machines. For machines of medium to high capacity, absorber units having several springs are used (Fig. 10.3). Two arrangements of mounting the spring absorbers are possible for
supporting machine foundations. They are supported-type (Fig. 10.1a) and suspended-type (Fig. 10.4). In a supported-type arrangement, the springs are placed directly under the machine or the foundation (Fig. 10.1a). In a suspended-type absorber system, the springs are located at or close to the floor level, and the main foundation is suspended from the springs. A typical suspended-type absorber is shown in Fig. 10.4.
Spring
Suspended inertia block
Spring
(a)
(b)
Raise foundation for a minimum clearance of 1"
Figure 10.2. Spring absorber having only one spring: (a) without housing; (b) with housing.
Figure 10.4. Typical isolated double-frame hammer foundation-suspended-type absorber.
(Courtesy Korfund, Inc. 1986.)
(Courtesy Korfund, Inc. 1986.)
406
VIBRATION ABSORPTION AND ISOLATION
The choice of any arrangement depends on the balance of the machine and its operational speed. For high-speed machines that are relatively well balanced, a supported-type arrangement is used since in such cases a heavy foundation mass above the springs is not generally necessary. For lowfrequency machines, a heavy mass above the springs becomes necessary and a "suspended-type" absorber arrangement is generally adopted. Suspendedtype arrangement provides easy access to the casings housing the springs. Analysis of the absorber foundation system, irrespeCtive of the supported or suspended type, can be made by treating it as a two-degrees-of-freedom system, as described in Section 10.1. Spring absorbers are commercially available in several sizes and capacities. The information on the load-deflection characteristics is furnished by the manufacturers of these absorbers. Steel springs are affected by the environmental conditions and should be protected against corrosion.
10.2.2
Cork
Natural cork is one of the best vibration and noise absorbing materials. It has a low unit weight, high compressibility, and is impermeable to gases and
Bolting not required
Bolting required
Vibracork
l"
Vibracork (a)
407
COMMON VIBRATION ABSORBERS
liquids. It can undergo large compression. Cork p_ads are placed under the machine or the foundation as necessary. The sllffness of a cork pad to vertical compression without lateral expansion may be calculated by usmg Eq. (7.1)
EA
(7.1)
k=-
t
To prevent lateral deformation, the cork pads may be framed, or the arrangement shown in Fig. 10.5 may be used. Cork pads may need frequent replacement. Oil and water hasten the decay of the cork pad. 10.2.3 Rubber Rubber provides an excellent absorber material because of its resilient properties. Pads and springs made of rubber can be used for th1s purpose. Rubber can be directly vulcanized to metal. Rubber spnngs can be made by vulcanizingc:rubber to metal and, j~ave the advantage of bei?g able. to take compression, shear, or torsion. A typical rubber spnng of th1s type 1s show? in Fig. 10.6a. A rubber spring in the form of a hollow cyhndncal sleeve IS shown in Fig. 10.6b. For preventing buckling under heavy loads, hollow cylindrical rubber springs are bonded witb metal plates (F1g. 10.6c). The stiffness properties of rubber pads are frequency dependent (Snowdon, 1979) and are also influenced by the relal!ve stze of the loaded area to the total area of the pad (especially the area of the lateral surface of the pad). Resilient properties of rubber are strongly influenced by the env1ronme~t, for example, operating temperatures and mdustnal otis. Uneven loadmg also reduces the life of a rubber pad.
(c)
Retaining curn for equipment tending to walk
or concrete
Machine base
Vibracork (b) (a)
Figure 10.5. Typical cork arrangement:
(a)
using channels. to apply load to vibracork for light
(b)
(c)
machinery; (b) continuous layer of Vibracork for heavy machinery and concrete foundations
Figure 10.6. Bonded rubber pads: (a) rubber spring bond~d ~etween two metal plat~s; (b)
above the floor line; (c) Vibracork for isolation of equipment on concrete foundation below
rubber spring in the forin of hollow cylinder; (c) rubber sprang m the form of hollow cyhnders bonded to grooved steel plates. (After Major, 1980.)
floor line. (Courtesy Korfund, Inc. 1986.)
408
VIBRATION ABSORPTION AND ISOLATION
409
COMMON VIHRA TION ABSORBERS
to the piston, and let i5 be the downward movement of the piston. If the area of the piston is A, then the new pressure p 2 is given by PIA. According to Boyle's law (10.6)
Ia) (b)
in which n is an index. Also,
Figure 10.7. Timber pads with multiple layers: (a) two layers; (b) three layers.
v, = 1 0.2.4 Timber Timber pads are commonly used below anvils in hammer foundations. Single or multiple layers of hard wood (such as oak, pine, or beach) beams are used for the purpose. The timber beams are arranged so that the direction of the compressive load is across the grain. Arrangement of timber beams in multiple layers is shown in Fig. 10.7(a, b). The stiffness of a timber pad in compression is given by Eq. (7 .1). The values of E for different grades of timber are available in "Codes of
V1 -
oA
Therefore, (10.7a) or (10.7b)
Practice."
10.2.5
Neo11rene
Differentiating Eq. (10.7b) witb respect to B, we get
Neoprene and neoprene cork pads prove very resistant to industrial environment. These are commercially available in different sizes and with a wide range of load deformation characteristics. 10.2.6
Pneumatic Absorber
A pneumatic absorber uses gas or air as a resilient material. The stiffness of a pneumatic spring may be obtained by using the gas laws governing adiabatic compression. Consider a piston cylinder system having air at a pressure p 1 and occupying a volume V1 (Fig. 10.8). Let a force P be applied
r
~Cylinder
Figure 10.8. Principle of a pneumatic absorber.
If the change in volume is small, then the above expression becomes:
(10.8) The pneumatic springs may be the single- or double-acting type (Harris and Crede, 1976). Pneumatic springs can be made to provide damping as well. The load supporting area may vary with deflection. Pneumatic springs are commercially available in different shapes and capacities. A typical convoluted air spring knowif-as "Airmount" manufactured by Firestone, Inc., is shown in Fig. 10.9. The performance characteristics of the pneumatic springs are supplied by the manufacturers and must be ascertained before planning use of such systems. When the loads are heavy and the required natural frequency of the absorber is low, the static deflection in the usual (spring or pad) type of absorbers will be large. However, in pneumatic absorbers, the static deflection can be controlled by adjusting the air or gas pressure to support the load while maintaining the low stiffness necessary. Baxa and Ebisch (1982)
410
VIBRATION ABSORPTION AND ISOlATION Air i I
411
DESIGN PROCEDURE FOR FOUNDATIONS ON ABSORBERS
Blind nut
2. First Trial. Make a trial design of the foundation without absorber following the procedure of Section 6.8, satisfying the limiting amplitudes. The foundation size may turn out to be too big for the size of the machine or for the space available. 3. Second Trial. Depending upon the requirements of minimum foundation size for the machine and available space, select the area of the foundation in contact with the soil and the weight of the foundation part below the absorber W1 (Fig. 10.1a). 4. Determine the equivalent spring stiffness of the soil k 1 below the base ( 4.27)
or 4Gr0 k 1 = (1- v).
(6.18)
in which
C" = coefficient of elastic unifofm compression A 1 = area of foundation base G = shear modulus of soil r0 = equivalent radius of the foundation contact area v =Poisson's ratio
I
I Figure 10.9. A typical convoluted air spring. (Airmount® isolator by Firestone.)
5. Determine the limiting natural frequency of the whole system resting on soil, wnll" wnli
in which m 1 = W1 /g and m 2
r
=
~
m;m I
(2.99a) 2
= W2 /g. Compute the ratio of masses
f.L
m, m,
f.L = -
have reported the successful use of the pneumatic absorber to control the vibrations of a 4000-hp hammer mill operating at 600 rpm. The total weight isolated was around 550 tons and was supported on 16 Airmounts.
6. Determine the ratio of frequencies r 1
(10.3) 10.3
DESIGN PROCEDURE FOR FOUNDATIONS--ON ABSORBERS
Design procedure for a foundation on absorbers and supporting a reciprocating machine having its main unbalanced force component in vertical
where
is the operating speed of machine. 7. Compute the amplitude Z for the system resting on soil (no absorber w
provided).
direction is described below:
(10.4a) 1. Design Data. Procure all design data about the machine and soil listed in Section 6.8 and the limiting amplitudes.
in which F 0 is the unbalanced vertical force.
412
VIBRATION ABSORPTION AND ISOLATION
8. Calculate the degree of absorption 71
z ]:,
T/=-
in Which Zl is the amplitude of foundati.On Wit . h a bsorber or permissible amplitude of foundation vibrations.
PRINCIPLES OF VIBRATION ISOLATION WITH WAVE BARRIERS
413
15. Check the safety of the spring. From consideration of stresses in spring P, < P. But the absorber system works within narrow ranges. So P, = P will be more reasonable. The process of designing an absorber is illustrated with an example in Section 10. 7. We shall now discuss the process of vibration isolation using wave barriers.
9. Determine the frequency ratio r2 , i.e., wn 12 fw from Eq. (10.5)
71
=
.!:_
zl
=
10.4
[1- (1 + ~t)(ri + r;- r;r;)] ri[(1+~t)(ri-1)]
(10.5)
in which wn 12 is the limiting natura I fr equency o f foundation above the absorber and is given by
(2.99b) 10. Determine
w~ 12
,z = 2
2 £t) 11{2
w
2
or
11. Determine the total vertical stiffness of absorber k k 2--
2
mzwnf2
2
(2.99b)
12. Select the typ~ of absorber. An absorber having total stiffness k, may now be chosen. This selectiOn can be easily made from the information gtven m ~anufacturers catalog about load us deformation characteristics of commercial absorbers.
PRINCIPLES OF VIBRATION ISOLATION WITH WAVE BARRIERS
A vibrating footing results in transmission of energy through the surrounding soil in the form of waves (Fig. 3.14). A major part of this energy is carried by Rayleigh waves, which travel near the ground surface and may adversely effect nearby structures, precision machines, and people. One of the possible ways to protect a structure from the harmful effect of Rayleigh waves is to locate it far away from the source of vibrations. The amplitude of wave motion decreases with increasing distance from the source because of the atte!luation of wave energY,due to geometrical and material damping. This method of isolation with dis.tance is, however, of limited significance since its advantage cannot be practically taken without sacrificing effective space utilization. Effective protection from harmful effects of Rayleigh waves may be obtained by using concepts of vibration screening which is made possible by proper interception, scattering and diffraction of surface waves with wave barriers (Woods, 1968). The wave barriers may consist of open trenches, trenches filled with bentonite slurry, sawdust, or sand, sheet piles, and piles. Screening problems may be classified into two groups as follows:
Active Isolation. The isolation is provided at the source of vibration. A wave barrier is provided close to or surrounding the source of disturbance as shown schematically in Fig. 10.10, in which a circular trench of radius R 0
I
Oscillating
13. Find the amplitude of vibration Z 2 of the system above the absorbers.
Z - (1 + f.t)W~ll + f.LW~/2- w' ,-
m A(w 2 ) 2
Fo
(2.109)
force
in which Circular, open trench of
radius Ro
(2.104)
and depth H.
14. Actual load per spring P, Figure 10.10. Schematic of vibration isolation using a circular trench surrounding the source of
(10.9)
vibrations~active
isolation. (After Woods, 1968.)
VIBRATION ABSORPTION AND ISOlATION
414
AMPLITUDE OF SURFACE
INCOMING
RAYLEIGH
WAVE
SENSITIVE INSTRUMENT OR
/TOOL
Figure 10.11. Schematic of vibration isolation using a straight trenchMpassive isolation. (After Woods, 1968.)
and depth H surrounds the foundation for the machine (the source of vibration). Passive Isolation. The isolation is provided near the location of the structure sought to be protected from th\, incoming waves. The wave barriers are thus provided remote from the source of vibration but near the site where reduction of vibration amplitudes is required. Figure 10.11 shows an example of passive isolation in which an open trench of length L and depth H is used to protect a sensitive instrument from the harmful effect of waves. The criteria for design of trench and pile barriers for effective vibration isolation is discussed in this article. The wave barriers are considered effective in reducing vibrations if the amplitude reduction factor (ARF) is 0.25 or less. The ARF is defined as (Woods, 1968): ARF"'
10.4.1
amplitude of vertical vibration with trench amplitude of vertical vibration without trench
(10.10)
Trench Barriers
There have been several successful and unsuccessful applications of the trench barriers for vibration isolation in the past (Barkan, 1962; McNeill et at., 1965). The problem of designing trench barriers for effective vibration screening has been the subject of several experimental and analytical investigations. Based upon the results of a series" of field tests, Barkan (1962) pointed out that a reduction in vibration amplitudes as a result of a trench or sheet pile barrier, is achieved only when the trench dimensions are sufficiently large compared with the wave-length of the surface waves generated by the source of disturbance. Dolling (1966) studied the effect of size and shape of the trench on its ability to screen vibrations. Woods and Richart (1967) and Woods (1968) conducted a comprehensive
415
PRINCIPLES OF VIBRATION ISOLATION WITH WAVE BARRIERS
series of field tests to evaluate the screening effect of trenches. The cases of active as well as passive isolation were investigated. The effect of parameters such as the trench dimensions (length and depth), ItS distance from the source and the frequency of vibrations, on their effectiveness as wav~ barriers was assessed. The tests were conducted at a prepared site. The soil conditions at the test site consisted of uniform silty, fine sand (SM) up to 4ft - 104lb/ft3, w = ·7% ' e = 0 ·61 ' and. Vc =940ft/sec3 at surface, _ m d eep w1'th 'Ydand sandy silt (ML) from 4 to 14ft deep with 'Yd = 91lb/ft, w- 23-ro, e = 0.68, and Vc = 1750 ft/sec at upper boundary. The water table was below 14 ft in depth. A small vibration exciter was used as the sourc~ of vertical vibrations and was set up at the center of the test site. Usmg velocity transducers the amplitudes of vertical ground motion were measured at selected points throughout the test site before installation of the trench (no trench condition) and after installation of the trench (after trench condition). A comparison was made of the amplitudes of motio? for the "n? trench" and "after trench" conditions and was used m evaluatmg the effectiveness of the barrier. Amplitude reduction factor (ARF) defined in Eq. (10.10) was used to give a quantitative evaluation of the effectiveness of the trench barrier at the point of measurement. . The critical dimensions of the trenches used in all tests were normah~ed with respect to the Rayleigh wave length ( AR) for the frequency of vibratiOn used in a particular test when comparing r~sults from two or more tests at different frequencies. The velocity of Rayleigh waves VR and the wavelength An were determined by steady-stat~ vibration tests (Section 4.6). Thevalues of v and A at different frequenCies used m the test program are giVen m R R Table 10.1 (Woods, 1968). . In the active isolation tests, the depth of trenches was vaned from 0.5 to 2ft the radius R of the annular trench (Fig. 10.10) varied from 0.5 to l.O,ft, and the an;ular dimension e was varied from 90' to 360' around the source of vibrations. Frequencies of 200 to 350Hz were used m tests (Table 10.1). The values of scaled depth H!An thus varied from 0.222 to 1.82 ~nd those of scaled distances R 0 / An varied from 0.222 to 0.910. Some typical results of the investigation by Woods (1968) are shown m Fig. 10.12 m the
Table 10.1. Wavelength and Wave Velocity for the Rayleigh Wave at the Test Site Frequency, Hz
AR ft
VR, ft/sec
200
2.25
450
250 300 350
1.68 1.38 1.10
420 415 385
Source: Woods (1968).
416
VIBRATION ABSORPTION AND ISOLATION
417
PRINCIPLES OF VIBRATION ISOLATION WITH WAVE BARRIERS
4. Partial circle trenches having angular length 0 < 90", did not provide an effectively screened zone. 5. Trench width is not an important parameter. 6. Amplification of vibratory energy occurred in the direction of "open side" of the trench.
Woods (1968) also conducted passive isolation tests using open rectangular trenches and investigated the effect of trench length L, width B, depth H, and the distance from the source R 0 . A typical layout for these tests consisting of two vibration exciters (operated one at a time), 75 transducer locations, and a trench is shown schematically in Fig. 10.13. The trenches ranging in size from 1.0 ft deep by 1.0 ft long by 0.33 ft wide to 4.0 ft deep by 8.0 ft long by 10ft wide were used in the tests. Frequencies of excitation
01.25-0.50
!:::::: l 0.50-0.25
D
H!An
RofAR
1.452
0.726
32,
> 1.25
D
1.25-o.5o
l::::::j
0.50-0.25
Hf>...n
RofAR
0.596
0.596
I
N
o.25-0.125
0.25-0.125
tw~mllld <
24'
':i'. :.<-'
I
0
I
7:0
0
o.l25
8'
0
20 ft (b)
Figure 10.12. Amplitude reduction factor (ARF) contour diagrams for active isolation: (a) full
0
0 0
0'
/~ 0
0
0
0 0
0
/
0
0
0
0
form of amplitude reduction factor contour diagrams. It may be seen from this figure that ARF of 0.25 or less was achieved for the trench dimensions used in these tests. By comparing the results of different tests in which ARF of 0.25 or less was achieved with those in which ARF of 0.25 could not be achieved, the following conclusions were drawn regarding the use of trenches for active isolation.
I
0
0
circle trench; (b) partial circle trench. (After Woods, 19&8.)
0
0
0
Ia I
16'
0
0
~·75pick"P benches
1. For full circle trenches (0 ~ 360"), a minimum value for HIAR of 0.6 is required to achieve ARF equal to or less than 0.25. 2. The zone screened by a full circle trench extended to a distance of at least 10 wavelengths (10AR) from the source of excitation. 3. For partial circle trenches (90" < 0 < 360"), the screened zone was defined as the area outside the trench extending to at least 10 wavelengths (10AR) from the source and bounded on the sides by radial lines from the center of the source through points 45" from ends of the trench (Fig. 10.12b). A minimum value for HIAR of 0.6 is required for the trench to be effective.
Figure 10.13. Schematic of the test site for passive isolation tests. (Woods, 1968.)
418
VIBRATION ABSORPTION AND ISOLATION
from 200 to 350Hz were used in the tests. The values of HI An varied from 0.444 to 3.64 and that of R 0 1An from 2.22 to 9.20. It was assumed in these tests that the zones screened by the trench would be symmetrical about the 0" line. It was defined that for effective isolation, the values of ARF should be less than or equal to 0.25 in a semicircular zone of radius of ! L behind the trench. Typical amplitude reduction contour diagrams for one of these · . tests are shown in Fig. 10.14.
················· .. .. ...... .... .. .... ... . .. .... . ....············ .... .. .. .... .... .. . .. .. .... ············· .················· . .. .. .. . .. . .. .. . .... .. .. .. .. .... .. .... ...... .. . .. ...... .... .. ...... .... .... .. .... . . . ... . .. .. .. . ···············
PRINCIPLES OF VIBRATION ISOLATION WITH WAVE BARRIERS
Significant results of this study were as follows: 1. For effective passive isolation (R 0 = 2An to ?An), the depth of the
trench H should be at least 1.33A.. 2. Larger trenches were required at greater distances from the source. To maintain the same ARF, the scaled area of the trench (HI An x LIAR= HLIA~) should be increased with increasing scaled distance R 0 1An. The least area of the trench in the vertical direction should be 2.5A~ at R 0 = 2An and 6A~ at R 0 = 7. 3. Trench width had practically no influence on effectiveness of open trench (for BIAR = 0.13 to 0.91). 4. Amplification of vertical motion occurred in zones in front of trenches and to the sides of the trenches. Sridharan et al. (1981) conducted an experimental investigation to deter· mine the effectiveness of open trenches and trenches backfilled with sawdust and sand as isolation barriers. Surface and embedded square footing 45 em x 45 em x 7.5 em and 45 em•?< 45 em x 180 em respectively were excited into vertical vibrations with a mechanical oscillator and served as the source of disturbance. Several combinations of static and dynamic loads were used in the investigations and trenches 30, 60, and 120 em in depth located at 15 em from the footing were tried. Their observations revealed that the open unfilled trenches are the most effective and the performance is better with sawdust as compared with sand. Haupt (1981) conducted model tests to study the effectiveness of open trenches, concrete core walls and void cylindrical holes in reducing vibration amplitudes. The problem of effectiveness of trench barriers has also been the subject of several analytical studies aimed at investigating the behavior of Rayleigh waves in a homogeneous medium with discontinuities. Finite element methods, finite difference methods, and special numerical techniques have been employed for this purpose. Lysmer (1970) and Lysmer and Wass (1972) used a lumped-mass model to study the propagation of Rayleigh and SH waves in homogeneous layered medium. The method was also extended to determine the effect of a trench in a homogeneous layer resting on a rigid base and at the surface of which a harmonic load induced horizontal shear wave motion. They ob~rved that the reduction in vibration amplitudes was a function of trench depth and was achieved for several frequencies. Haupt (1977) used the finite element method to study the effectiveness of concrete core walls as an isolation barrier and found that the isolation capacity depended on the cross section of the wall and not on its geometry. Segol et al. (1978), and May and Bolt (1982) developed special finite element methods to study the reduction of amplitudes of surface waves under plane strain conditions. Segol et al. (1978) observed that: l,'.i
~
c:::::::J
>
1.25
1.25-0.50 0.50-0.25
0.25-0.125 .·::·-::·~~:-:·:
<
H/).R
L/).R
Roi>.R·
1.19
1.79
5.96
0.125
Figure 10.14. Amplitude reduction factor (ARF) contour diagrams for passive isolation. (After Woods, 1968.)
419
•i{.. .•)J?'
VIBRATION ABSORPTION AND ISOLATION
420
PRINCIPLES Of VIBRATION ISOLATION WITH WAVE BARRIERS
421
1. The effectiveness of the trench as wave barrier is primarily a function of the ratio HI An and significant reduction of amplitudes occurs when H!An is larger than 0.6. 2. The trench location and shape of its cross section have only a marginal influence. 3. Open trenches are more effective in reducing vibration amplitudes.
Beskos et a!. (1985) used the boundary element method to evaluate effectiveness of open and infilled trenches as wave barriers. The cases of active as well as passive isolation were investigated. They observed that the trench barriers are effective in screening vibrations when HI An 2 0.6 for open trenches and BH!A~ 21.5 for concrete-filled trenches (B =width of trench). It may thus be noted that the analytical studies generally support the qualitative conclusions of Woods (1968). The open (unfilled) trenches are more effective as wave barriers but may present instability problems necessitating trenches backfilled with bentonite slurry, sawdust, concrete, or sand. In some other situations, it may not be possible to use trench barriers and the designer may prefer to use pile barriers. The use of pile barriers is now discussed. 10.4.2
Pile Barriers
When vibrations are occasioned by a source operating at a very low frequency, the Rayleigh wavelength will be long afid may range up to 50 m or more. For a trench to be effective in such a case, its depth will range from 30m (0.6An) for active isolation to 66.5 m (1.33An) for passive isolation. When the Rayleigh wavelength is long, the trench depth often limits the application of trenches and open or slurry-filled trenches are impracticable. Piles may be used as barriers in such cases as they can be installed to any depth. This alternative of using rows of piles as passive isolation barriers has been investigated by Woods et a!. (1974). They used the principle of holography and observed vibrations in a model half-space to evaluate the effect of void cylindrical obstacles on reduction of vibration amplitudes. Fine sand medium in a box 100 em X 100 em X 30 em constituted the model half-space. A schematic sketch of the test set showing the geometry of the problem is given in Fig. 10.15. In Fig. 10.15, Dis the diameter of the void cylindrical obstacle and sn is the net space between two consecutive void holes through which energy can pass through the barrier. The effectiveness of the barrier was evaluated numerically by obtaining an average effect (ARFs) from several lines beyond the barrier in a sector ±15° on both sides of an axis through the source and perpendicular to the barrier (that is 8 = 30°, Fig. 10.15). The values of HI An and LIAR were kept 1.4 and 2.5, respectively, in all tests. The isolation effectiveness for these tests was defined as Effectiveness = 1 - ARF
(10.11)
Figure 10.15. Definition of parameters for:~,~y~indrical hole barriers. (After Woods et al., 1974.)
The data from different tests was plotted in the form of nondimensional plots of effectiveness versus Sn!AR as shown in Fig. 10.16. From these results, Woods eta!. (1974) concluded that void cylindrical holes may act as isolation barriers if
D
1
->-
AR- 6
(10.12a)
0.8 ~
"'"' I ~
o. 6 H-\--''r--t--''<:"'---'
0.10
0.15
figure 1 0.16. Isolation effectiveness as a function of hole diameter and spacing. (Woods et al., 1974.)
423
DESIGN PROCEDURE FOR WAVE BARRIERS
VIBRATION ABSORPTION AND ISOLATION
422
Soil type
and
't
(10.12b)
Barrier materials
I nfinitely
,; gid pile
Two rows of void obstacles were found to be more effective and could be used if a single row of cylindrical void obstacles spaced closely enough is not possible or when depth is not optimal. Slurry-filled barriers were found to be effective provided solidification of the slurry is prevented. The solid wave barriers were also effective, but their behavior was different and needed further investigation. Liao and Sangrey (1978) used a two-dimensional acoustic P-wave model to simulate in an approximate manner the passive isolation of foundations from Rayleigh waves. Experiments in multiple acoustic scattering were conducted as an extension of the work of Woods et al. (1974) and the effect of diameter, spacing, and material properties of the soil-pile system on isolation effectiveness was investigated. Their results indicated that the earlier conclusions of Woods et al. (1974) as given in Eq. (10.12) are generally Valid and Sn = 0.4,\R may be the Upper limit for the pile barrier to have any effectiveness. The effectiveness of the barrier was found to be significantly affected by the material of the pile and void holes and acoustically soft piles were more efficient than acoustically hard piles. The relative hardness or softness was defined in terms of impedance ratio (IR) as follows: IR =
stee I 10 7 -
Concrete
E
10
'-
Gravel
Timber
Dense sand Hard c\ay
10 5 -
.
Silt Loose sand Very soft clay
.,..,. 10
..
~;;. ;:~
Plastic foam
o3 -
,J
·Void borehole
10
Rayleigh wave impedance of the pile Rayleigh wave impedance of the soil medium
Figure 1 0.17. Estimated values oi Rayleigh wave impedance for various soils and pile materials. (After Liao and Sangrey, 1978.)
(10.13) 10.5 in which Pp = density of pile material p, = density of soil medium VR(p) =Rayleigh wave velocity in the pile material VR(•) =Rayleigh wave velocity in the soil medium.
The piles are considered soft if IR < 1 and hard if IR > 1. The values of Rayleigh wave impedance (pVR) for various materials are given in Fig. 10.17. Liao and Sangrey (1978) also observed that the two row barriers can be more effective than single row barriers.
A step-by-step procedure will now be given for design of trench and pile isolation barriers.
DESIGN PROCEDURE FOR WAVE BARRIERS
The information listed below should be procured before attempting the design of any type of vibration isolation barriers: Data Required "' 1. Source Data. The information on operating frequency f of the source of vibration should be obtained. 2. Soil Data. The soil profile, unit weights, and water c?ntent of the soil at the site and information on its dynamic properties should be obtained. The velocity of Rayleigh waves VR may be measured at the site with steady-state vibration test. Alternatively, the shear wave velocity may be determined by cross bore hole method. If no
VIBRATION ABSORPTION AND ISOLATION
424
information is available, the value of dynamic shear modulus may be determined by using Eq. (4.9). G
~ 1230(0CR)ke·~7! ~e)' (0'0 ) 0 5
(4.9)
425
DESIGN PROCEDURE FOR WAVE BARRIERS
10. Location of the Trench. The trench should preferably be located between 2.0AR to 7 AR from the source. 11. Length of the Trench L. The length of the trench may be determined
from Eq. (10.17). Ar~LH
The shear wave velocity V, may then be calculated as
v, ~ v; {G
or
Ar H
(3.30b)
Assume VR = v;. 3. The distance between the source of disturbance and the structure to be protected. 4. The size of the area over which effective vibration isolation is desired.
(10.17)
L~
The value of area of the vertical projection of the trench Ar is given by (Woods, 1968)
Ar ~ 2.5A~
for R 0 ~ 2AR
(10.18a)
and
Design of Isolation Barriers
(10.18b)
5. Calculate the wavelength of the Rayleigh waves through the soil by Eq. ( 4.33a),
(4.33a)
For other values of R 0 between 2AR and 7 AR, the value of Ar may be determined by interpolation. The minimum length of the trench should not be smaller than the value given by Eq. (10.17) or the length of the structure being protected.
Active Isolation- Trench Barrier Passive Isolation- Pile Barriers
6. Calculate the minimum trench depth H as H~0.6AR
(10.14)
12. Depth of Pile H. The minimum depth of the pile H should be 1.33AR [Eq. (10.16)]. If the bedrock is shallow, piles may be placed on the
bedrock. 7. Location of the Trench. The trench should be located as close to the source as possible. The distance R 0 between the centerlines of the source and the trench should be less than AR, especially when afull circle trench is used. 8. The length of the partial circle trench 8° (Fig. 10.12b) should be obtained as follows: Determine the angle a subtended by the outer boundaries of the protected area at the center of the source (Fig. 10.12b). The value of angular length 8 is then given by
13. Diameter of the Cylindrical Void (Pile) D. The diameter of the pile D
may be calculated from Eq. (10.12a): (10.12a) 14. Spacing of Piles Sn. The clear spacing between the piles Sn is given by
sn
1
-
(10.15)
(10.12b)
15. Material of the Pile. Calculate the Rayleigh wave impedance of the soil p,VR(.>)' Using Fig. 10.17, select the pile material so that the
Passive Isolation- Trench Barrier
impedance ratio IR is <1.
9. Calculate the minimum trench depth H for passive isolation as (10.16)
(10.13)
426
VIBRATION ABSORPTION AND ISOLATION
16. Length of the Pile Barrier L. The length of the pile barrier L may be determined by joining the center of the source "with the outer boundary of the area to be protected and then placing the pile barrier perpendicular to the axis through the source as shown in Fig. 10.15. 17. If one row ?f piles is inadequate, a second row of piles may be provided to mcrease the effectiveness of isolation.
10.6 METHODS OF REDUCING VIBRATION AMPLITUDES IN EXISTING MACHINE FOUNDATIONS Excessi~e foundation vibrations may sometimes develop soon after the mstallatwn of the machine or sometimes thereafter due to an increase in the unbalanced loads arising out of wear and tear of the machine, change in the subsoil conditions, defective design or construction" It may be possible to reduce or limit these vibrations by appropriate selection of the following remedml measures" It must be emphasized that before any remedial measures are considered, the cause of excessive vibrations ffiust be established by proper investigation, which will also help in choosing the most effective measure. Improper selection of the remedial measures may further worsen the situation rather than improve it. The methods used to reduce vibrations in existing machine foundations are: 1. 2. 3. 4.
Counterbalancing the unbalanced exciting loads. Chemical soil stabilization. Structural measures" Providing vibration dampers.
Each of these methods will be discussed now. Counterbalancing of Unbalanced Loads Due to Machine Operation
Counterbalancing of unbalanced loads in a machine will result in smaller ~nbalanced exciting loads and hence result in reduced amplitudes of vibration. The unbalanced forces for rotating machinery such as centrifugal pumps, turbogenerators and turbines are given by F0
in which
=
mew 2 sin wt
(5.37)
METHODS OF REDUCING VIBRATION AMPLITUDES
427
Reduction in Fa can be achieved by attaching additional mass (masses) on the rotor in such a manner that the effective eccentricity of the rotating mass decreases. In rotating components such as flywheels, the eccentricity is reduced by cutting grooves on the flywheel at predetermined locations. Unbalance of the rotors in high-speed rotary machines is checked as a part of normal maintenance operation. Primary inertia forces in reciprocating machines can also be balanced by means of counterweights in two ways: 1. The component of the force in a direction perpendicular to piston motion may be completely counterbalanced and the component along the direction of piston motion may be only partly balanced. 2. It is also possible to counterbalance completely the first harmonic of the exciting force in the direction of piston motion by using counterweights at appropriate distances from the axes of rotation" The component of unbalanced force in the perpendicular direction of piston motion will then increase.
The use elf a particular method>~nd its efficiency in counterbalancing the exciting forces induced by an engine for the purpose of decreasing foundation vibrations depends on the type of engine and on special features of the foundation" In a horizontal reciprocating engine, the most dangerous foundation vibrations are those occasioned by simultaneous rocking and sliding. In this case, a decrease in the vibrations of the foundation may be achieved by counterbalancing the inertia forces of the engine by the second method, even if it leads to some increase in vertical vibrations. Therefore, if an engine was counterbalanced by the first method but impermissible horizontal vibrations were observed after the construction of the foundation, then counterbalancing by the second method (i"e., by changing the character of counterbalancing) may be used as one of the simplest measures to decrease these vibrations. Cases in which vertical vibrations of an impermissible amplitude are present in systems with horizontal motors; the use of second method is unsuitable, and the first method should be applied. Similarly, for vertical motors, the method of counterbalancing selected will depend on the type of foundation vibrations-vertical, horizontal, or rocking. •( The main advantage in using this method for reducing vibrations is that installation of counterweights for balancing a motor does not require dismantling or prolonged interruption of operation" The shutdown is only for th~ time needed to attach the counterweight to the sides of the crank.
Fa = unbalanced force m = mass of the rotor
Chemical Soil Stabilization
e = eccentricity of the mass w = speed of operation
Foundation vibrations may sometimes be reduced by chemical stabilization of the soil that results in an increase in the rigidity of the base and,
428
VIBRATION ABSORPTION AND ISOLATION
consequently, in an increase in the natural frequencies of the foundation. This method is effective only when the natural frequencies of the foundation before stabilization of the soil are higher than the operational frequency of the machine. An increase in rigidity of the soil, in such cases, will increase still further the difference between the frequency of natural vibrations and the operating frequency of the engine, resulting in a decrease in the amplitudes of foundation vibrations. If the foundation has natural frequencies that are smaller than the operational frequency of the machine, then soil stabilization may cause an increase in the amplitudes of vibration, because the natural frequencies of the foundation after stabilization may come closer to the operating speed. Chemical and cement stabilization of soils is economical as its costs are low in comparison with other methods of reducing vibrations. Another advantage of this method is that it can be applied without any prolonged interruption of the machine. The extent of soil stabilization below the footing depends on the nature of its vibrations. For a foundation undergoing only rocking vibrations, stabilization may be necessary only near the foundation edges (perpendicular to the plane of vibrations). If the foundation vibrations are vertical, stabilization of soil below the entire area of the footing will be necessary. The depth of the stabilized zone should not be less than the width of the footing and it should extend by 30 em beyond the periphery of the foundation. Barkan (1962) reported a case where vibration amplitudes of foundation for a horizontal compressor decreased by 50 percent as a result of soil stabilization achieved by injection of silicates. Structural Measures
The structural measures are used with the object of changing the natural frequencies of the soil-foundation system in such a way as to achieve the largest possible difference between these natural frequencies and the frequency of operation of the machine. These consist in increasing the rigidity of the foundation by providing piles below the foundation by method of underpinning; by increasing contact area of the foundation with the soil; by increasing foundation mass and by attaching suitably designed slabs to the existing foundation. Appropriate choice of structural measure depends on the nature of the vibrations and the relationships between the natural frequencies and the operating speed. It is also possible to. increase the foundation mass without inducing changes in the frequency of foundation vibrations, resulting in a decrease in the amplitudes of vertical vibrations. For undertuned machine foundations, an increase in rigidity of the foundation will produce undersirable effects. In such a case, it may be better to decrease further the natural frequency of the foundation by increasing the foundation mass without increasing its area in
429
METHODS OF REDUCING VIBRATIONS AMPLITUDES
I
P:r: sin wt
/
Foundation block
~
Attached slab
/ I
I
Foundation block
.
I
L rr
--:c-
ofareaA1
~
';?.mmmmmmmmmt!%''(, "' ~
Figure 10.18. Use of special slabs in reducing vibrations.
contact with the soil. Local condition may govern the choice in some cases. If a vibrating footing lies close to another foundation, it may be attached to the latter. Use. of special slabs (Fig.,.,;[0.18) attached to the main foundation may also prove effective in reduCing vibrations in some situations. The weight of the slab and its area in contact with the soil must be analyzed to ensure the effectiveness of this measure. Vibration Dampers
Amplitudes of vibration of a foundation undergoing vertical vibrations may be decreased by attaching two auxiliary masses as shown in Fig. 10.19 by means of elastic tie rods. The attached mass and the tie rod constitute the vibration damper. The system consisting of the foundation and the damper can be represented by Fig. 2.22a in which m 1 is mass of the foundation and machine, m 2 is the attached damper mass, k 1 is the equivalent soil spring, and k 2 is the stiffness of the tie rod or the spring that attaches m, with the foundation. The principle of vibration damper has been explained in Sectmn 2.13, where it has been established that if the damper mass and stiffness are chosen so that the natural frequency of the damper wn 12 is equal to the operating frequency of the machine w [Eq. (2.106)], then the amplitude of vibration of the foundation becomes zero. The amplitude of vibration of the attached damper mass ~then given by (2.107) Amplitude Z 2 thus equals the static deflection of mass m 2 produced by a force of magnitude equal to the maximum value of the exciting force Fa. It may be noted from Eqs. (2.106) and (2.107) that neither the frequencies nor the amplitudes of vibration of the damper depend upon the properties of the
430
VIBRATION ABSORPTION AND ISOLATION
t
P, sin
wl
EXAMPLES
431
only for some optimum value of damping. Dampers for foundations undergoing rocking or sliding vibrations can be designed similarly.
10.7
EXAMPLES
Foundation block EXAMPLE 10.7.1
Ia I
Design the foundation for a reciprocating machine operating at a speed of 750 rpm. The weight of the machine is 2.0 t and it produces a sinusoidally varying unbalanced force of 0.5 t in the vertical direction. Due to limited available space, the area of the foundation should not exceed 3m X 2m. Due to presence of precision machines in the vicinity, the vibration amplitude should be less than 0.025 mm. Assume the dynamic shear modulus of the soil G = 1950 tim 2 and v = 0.305. Unit weight of concrete y, may be taken as 2.4 tim'. Solution 1. Design, Data
~I ..
L
Figure 10.19. (a) Vibration dampers attached to the foundation; (b) damper system with dashpot.
soil below the base or the mass of the foundation. By satisfying the condition wn 12 = w, it is theoretically possible to damp the vibrations of a foundation and dampers with even small masses will be effective. A smaller mass of damper necessitates a smaller value of the stiffness of the spring k 2 , which will result in large amplitudes of vibration of the damper mass. The stiffness of the damper is thus governed by its strength requirements. The damper will work efficiently if w, the Operational speed of the machine, is constant. If the operational speed of the machine fluctuates and comes closer to wn 12 , large amplitudes will build up. If the fluctuations in the operational speed are large, it is difficult to design a vibration damper. The working range of the damper may be increased by introducing damping into the damper system, as shown in Fig. 10.19b, which results in a decrease in amplitudes of vibration as w approaches w" 12 • These effects will be possible
Weight of the machine= 2.0 t Operating speed= 750 rpm= 78.53 radisec Vertical unbalanced force P 2 = 0.5 t Dynamic shear modulus G = 1950 t/m 2 Poisson's ratio for the soil= 0.305 Permissible amplitude of vibration= 0.025 mm
2. First Trial The limiting amplitude of foundation vibrations is only 0.025 mm. Considering the limitation of 3 m x 2m on the foundation area, it will not be possible to design a simple block foundation satisfying the criteria for adequate design. The amplitude of vertical vibrations is 0.2024 mm as computed subsequently. A foundation resting on absorbers must be designed. 3. Second Trial Adopt a foundation \'rea of 3m x 2m. Let the size of the foundation below the absorber (F~: 10.1a) be 3m x 2m x 0.3 m and the size of the foundation block above the absorber be 3m X 2m X 1m. Weight of foundation block below the absorber= W 1 = 3 X 2 X 0.3 X 2.4 = 4.32 t 4.32 2 Mass m 1 = . = 0.4403 t sec im 9 81 Weight of foundation block above the absorber W2 = 3 x 2 x 1 X 2.4 = 14.4 t
432
VIBRATION ABSORPTION AND ISOLATION
Total weight above the absorber= 14.4 + 2 = 16.4 t M 16.4 2 ass m 2 = = 1.6177 t sec /m 9 81 . f · m 1.6717 R atto o masses t-t = - 2 = ~~ = 3 796 m1 0.4403 ·
EXAMPLES
or
k, = k,
=
J"!
1- (1+ t-t)r;
r 22 -
(1 + t-t)(11-1)(r; -1)
-
=
r2
4. Stiffness of Soil Spring below the Base k 1 Equivalent radius r 0 ro =
433
2
2
1- (1 + 3.796)(1.0906) (1 + 3.796)( -10- 1)(1.09062 -1)
10. Determination of w~ 12
= 1.382m
4 X 1950 X 1.382 ( _ _ = 15510 tim 1 0 305 )
2 2 Wntz Tz = - 2 -
w
(6.18)
or w ~ 12 = (0.4706)(78.53) /sec 2
5. Limiting Natural Frequency of the Whole System Resting on Soil W
n/1-
wnll
=
~
~ m k+I m I . 2
= 0.4706
wn
11
2
or (2.98a)
15510 0 .443 +1. 6717 =85.69rad/sec
wn 12
= 53.87 rad/sec
11. Stiffnes~. pf the Absorber k,_ .• ·i':''''~'c
.,
(2.98b)
6. Frequency Ratio r 1
2
= (1.6717)(0.4706)(78.53) = 4851 tim
85.69 rl = ~ = 78.53 = 1.0911 Wnll
'
7. Amplitud¢ of the System Resting on Soil (No Absorber Case)
Z=
Fo (m 1 + m 2 )(w~ 11 - w 2 )
k 2 = 4851 tim is the total stiffness of the absorber system. Use eight absorber units, each having a stiffness of 600 tim. The actual value of wn 12 = "1/(8 X 600) /1.6717-53.58 rad/sec. Step 12 is omitted since pertinent data is not supplied.
(10.4a) 13. Amplitude of Vibration of the System above the Absorber Z 2
0.5 (0.4407 + 1.6716)(85.692 -78.53 2 )
2
(1 + t-t)w~ 11 + f-'W~ 12 - w z, = m,A(w') F0
= 0.0002024 m = 0.2024 mm
A(w
2
8. Degree of Absorption
z
11=-
zl
4
= w - (1 + t-t)(w~ 11 + w~ 12 )w + (1 + t-t)w~I1W~ 12
2
A(w 2 ) = 2.200
(10.5)
2
z,
0.2024 11 = O.D25 = 8.096 Adopt 17 = -10 for the design
)
(2.109)
X
(2.104)
108 2
2
= (1 + 3.796)(85.65) + 3.796(53.58) -78.53 (0.5) 8
0.000053 m = 0.054 mm
Dynamic load on each absorber= 0.000053 x 600 = 0.0324 t 9. Frequency Ratio r 2 , i.e., wn 12 /w
_ [1- (1 + t-t)(r; + r~- r;r~)] 1)r;[(l+t-t)(r;-1)]
(10.5)
Note that the amplitude Z 2 is calculated to check the stresses in the absorber material. The amplitude Z 1 has been restricted to a value less than the specified value of 0. 025 mm. The value of Z 1 may be calculated using Eq. (2.108).
434
VIBRATION ABSORPTION AND ISOLATION
435
EXAMPLES
2 Wn/2
zl ~ mlb.(w') Fo
(2.108)
(53.58) 2 (0.4403 )( 2 .20 X 10') (0.5) ~ 0.0000148 m
Solution 1. Source Data Operating speed of the compressor ~ 1200 rpm 1200 Operating frequency f ~ 6o ~ 20Hz
~0.0148mm
2. Soil Data
<0.025mm
Shear wave velocity v; ~150m/sec Rayleigh wave velocity VR = v; ~150m/sec
EXAMPLE 10.7.2
A compressor having an operating speed of 1200 rpm was installed in an industrial unit. Two precision machines were added to the plant later on. "It was felt necessary to protect these precision machines from any harmful vibrations due to operation of the compressor. The locations of the compressor ( C1 ) and precision machines (P1 and P2 ) are shown in Fig. 10.20. Design an open trench barrier to provide effective vibration isolation for the cases of (a) active and (b) passive isolation. The velocity of shear waves was determined at the site by cross bore hole method and its values was found to be 150m/sec.
3 and 4. The location of the compressor C1 (source of disturbance) and the precision machines P 1 and P2 (object or structure to be protected) are shown in Fig. 10.20. The distance between the center of the source and the center of the structure is 60.0 m. 5. Wavelength of Rayleigh wave AR A ~,.!::'"':~ 150 ~7 5m R
r
2o
.
(4.33a)
Active isolation f--20.0
m----1
~ II I I I I I Ct
PI
I', I
I
I
1
I I
I
Precision \ machines \
Trench
I
I I
\
6. Depth of the Trench H for active isolation
1•
1
60 .0 m
45oj /::::45' >< >/ 50~/ a
1/
CI
lal
I
I I I
I I I I
I I I
I
I
. 45.0 m
I I I
I I I I Jl II
~c, (b)
Figure 10.20. Layout of compressor and precision machines in the industrial building (Example 10.7.2). (a) Active isolation; (b) passive isolation.
~0.6
x 7.5 ~ 4.6m
7. The trench may be located 5 m from the center of the source, as shown in Fig. 10.20a. 8. It is planned to use a partial circle trench. Angle a is determined as (Fig. 10.20a):
30 m--+,-->j
\
(10.14)
H~0.6AR
15.0 m
\ \ ! --+--,----,-
I I
Trench
)
I
I
I I
1!:!1
I
I I I I I
Compressor \
p2 }
~
I
I I
m----1
f--20.0
1!:!1
a~ 2 tan- 1 G~) ~ 18.92°
The angular length of the trench 0 is given by: 0 ~ a0
Passive isolation
+ 45° + 45° ~ 18.92 + 90 ~ 108.92° ~ 110° ""-
9. Depth of the trench H for passive isolation. H
~
1.33AR
~
1.33 x 7.5
~
9.975 m
(10.16)
10. It is planned to locate the trench at a distance of 15m from the precision machines as shown in Fig. 10.20b. This distance is within 2AR to 7 AR from the source.
436
VIBRATION ABSORPTION AND ISOLATION
REFERENCES
437
11. Length of the Trench L Dolling, H. J. (1966). "Efficiency of trenches in isolating structures against vibrations," Proc. Symp. Vib. Civ. Eng. Butterworth, London. . "Firestone Airmount Isolators" (1986). Firestone Industrial Products Company, Nobesvtlle,
Scaled distance= ;,~ = 6AR Scaled depth= 1.33,\R Minimum scaled area A r required [by interpolation of Eqs. (10.18a, b)]= 5.3A~ A L = ___I H
(10.17) 2
5.3 X 7.5 O ( ) 8 9 .975 =29. 8m= 30. m say which is also more than the distance between P 1 and P • 2
10.8
FINAL COMMENTS
The principles of vibration absorption and isolation have been discussed in this chapter. The absorber system is designed on the assumption that the operating speed is constant. Fluctuations in the operating speed will adversely affect the efficiency of the absorber system. The damping in the absorber system has not been included. Damping has a favorable effect on the performance of the absorber and takes care of the influence of minor fluctuations in the speed of the machine. Performance characteristics of commercial absorbers are supplied by their manufacturers and are helpful in selecting the appropriate absorber system for given operating conditions. Isolation procedures using trench barriers have been investigated both experimentally and analytically. ~ased on the available information (1988), design procedures have been developed as described in this chapter. Standardized procedures are not currently available for design of pile barriers. Based on present recommendations, a procedure has been suggested for the design of pile barriers which may need modification as more information becomes available. REFERENCES Adiar, A. (1974). The design and application of pneumatic vibration isolators. Sound Vib. 8(8), 24-27. Barkan, D. D. (1962). "Dynamics of Bases and Foundations," McGraw-Hill, New York. Baxa, E., and Ebisch, R. (1982). Controlling automobile shredder vibration through pneumatic isolation. Foundations for equipment and machinery. Publ. Am. Caner. Inst. SP~78, 33-46. Beskos, D. E., Dasgupta, B., and Vardoulakis, I. G. (1985). Vibration isolation of machine foundations. Proc. Symp. Vib. Frob. Geotech. Eng. Am. Soc. Civ. Eng., Annu, Conv., Detroit, 138-151.
Indiana. · c . M . an d Harns,
.,
cre de , c . E . (1976) · "Shock and Vibration Handbook,
G 2nd ed. Me raw-
Hill, New York. ·z M h Haupt, w. A. (1977). Isolation of vibrations by concrete core walls. Proc. Int. Conf. Soz ec · Found. Eng., 9th, Tokyo, Vol. 2, 251-256. 'l M h Haupt, w. A (1981). Model tests on screening of surface waves. Proc. Int. Conf. Sot ec · Found. Eng., 10th Stockholm, Vol. 3, 215-222. . . . · A. M., an d c roc ke tt ' J . H . A . . (1953) . Design and construction of a fully vtbratton24(5) 421-444 Klem, controlled forging hammer foundatiOn. J. Am. Caner. Inst. , · "Korfund Vibro Isolators" (1986). -Korfund Dynamics Corporation, Westbury, New York .. · Ltao, S., an d sangrey, D . A . (1978) . Use of piles as isolation barriers. J. Geotech. Eng. DlV., Am. Soc. Civ. Eng. 104(GT9), 1139-1152. . Lysmer, J. (1970). Lumped mass method for Rayleigh waves. Bull. Setsmol. Soc. Am. 60(1), "J E M h 89-107. . Lys.mer, J., an d Wass, G . ( 1972) . "Shear Waves in Plane Infimte Structures. . ng. ec · Div. Am. Soc. Civ. Eng. 98(EM1), 85-105. . '' Major, A. (1980f:,'"Dynamics in Civil Engih~~ring: Ana Iysis. an d D estgn, Vol · 2 · Akademiai ,_. . d · · · t'on Kiad6 Budapest. May, T . W'., an d B oIt , B . A . (1982) . The effectiveness of trenches m re ucmg setsmtc mo l • Earthquake Eng. Struct. Dyn. 10, 195-210. . . . McNeill R L Marganson B. E., and Babcock, F. M. (1965). ''The Role of So~l Dy~amtcs. m De;ign. of. Stable Test Pacts." Guidance and Control Conference, ~inneapohs., Mmnesota. reductton of surface waves by ( M3) 621 641 Segol, G., Lee, P· c · y ·• an d Abel ' J · F. (1978). Amplitude trenches. J. Eng. Mech. Div., Am. Soc. Civ. Eng. 104 E , · · t'ton, " NBS .Handb · No · J c (1979) "Vibration Isolation: Use and Charactenza d of Commerce, National Bureau of D.C. Sridharan A Nagendra, M. v., and Parthasarathy, T. (1981). Isolatton of mach.me foun· datio~s b; barriers. Int. Conf. Recent Adv. Geotech. Earthquake Eng., St. Lows, Vol. 1, 279-282. Woods, R. D. (1968). Screening of surface waves in soils. J. Soil Mech. Found. Div., Proc. Am. Soc. Civ. Eng. 94(SM·4), 951-979. . 1 art , F · E ·• Jr · (1967) · Screening of elastic waves by trenches. Proc. Int. Woods, R .D. ,an d R tci 4 Symp. Wave Propag. Dyn. Prop. Earth Mater., Albuquerque, NM, 275-28 . . W00 d R 0 Barnett N. E. and Sagessor, R. (1974). Holography-A new tool for sml Eng.' Div., Am. Soc. Civ. Eng. lOO(GT·ll), 1231-1247.
Sno~2~~·u.S. Depart~ent
d~~a~ics.''1 . Geote~h.
Standar~s, Washmgt~n,
11
439
ELASTIC HALF-SPACE METHOD
350
1-------Wr--1----+---j--
Dynamic Response of Embedded Block Foundations In the discussion of the analysis and design of machine foundations in Chapter 6, it is assumed that the foundation rests on the surface of the ground. The real foundations, however, are founded below the ground surface. For an embedded foundation, the soil resistance is mobilized both below the base and on the sides. The additional soil reaction that comes into play on the sides of an embedded footing may have a significant influence on its dynamic response. The results of the field investigations of embedded foundations by Novak (1970, 1985}, Beredugo (1971}, Beredugo and Novak (1972}, Fry (1963), Stokoe (1972}, Stokoe and Richart (1974) and of laboratory experiments by Chae (1971), Gupta (1972}, and Vijayvergiya (1981) show that as a result of embedment, the natural frequency of the foundation-soil system increases and the amplitude of vibration decreases as compared to the response of a surface footing under otherwise identical conditions. These conditions are illustr~ted in Fig. 11.1 by typical response curves for a concrete block of diam'eter d, having different depths of embedment h, and subjected to an unbalanced horizontal force. It is observed from this figure that the natural frequency of this system increases and the horizontal vibration amplitude A, decreases with an increase in embedment ratio hid. The determination of the response of an embedded foundation supporting a machine is thus of great practical importance and has been the subject of several theoretical and experimental studies. The available analytical procedures for computing the response ·of an einbedded footing may be classified into the following categories: 1. Approximate methods, which consider the effect of the soil on the sides of the footing separately and include extension of the elastic half-space method for the surface footing (Anandakrishna and Krishnaswamy, 1973a; Baranov, 1967; Beredugo and Novak, 1972; Novak and Beredugo, 1971, 438
3000
3500
--~-,pm
Fi
11 1
Response curves of horizontal vibrations of concrete foundation with different
h~~~~s of ~ontact with surrounding undisturbed soil at the same excitation intensity. (After Novak, 1970.)
1972} and extension of the linear-elastic-weightless spring approach of Barkan (1962} byPrakash and Puri (1971, 1972). 2. Rigorous methods, which include th~ finite ele~ent methods with o~ without special energy absorbing boundanes (Chnstmn and Carner, 1978, Dasgupta and Rao, 1978; Day, 1977; Johnson et a!., 1975; Kausel and 1979·, Lysmer , 1980·, Tassoulas , 1981·'· Wass, 1972}, and the ... U sh lJima, boundary integral approach (Dominquez and Roesset, 1978). In the design of machine foundations the elastic half-space method and the linear elastic weightless spring method are commonly used and the design of embedded block foundation~ using these. two m~thods only ts considered in this chaf~ter. Pile foundatwns Will be dtscussed m Chapter 12.
11.1
ELASTIC HALF-SPACE METHOD
The earlier solution for computing the dynamic response of embedded foundations was obtained by Baranov (1967} and has been extended by Novak and Beredugo (1971, 1972), Beredugo (1971, 1976), and Novak and Sachs (1973). The following assumptions are used as a basts for the
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
440
solutions: (1) the footing is rigid, (2) the footing is cylindrical, (3) the base of the footing rests on the surface of a semi-infinite elastic half-space, and the soil reactions at the base are independent of the depth of embedment, (4) the .soil reactions on the side are produced by an independent elastic layer lying above the level of the footing's base, and (5) there is a perfect bond between the sides of the footing and the soil. Assumption 1 is valid in all practical cases. Assumption 2 may not hold in all cases. Assumptions 3 and 4 enable the reactions on the base and side to be independent of each other. The contact between the soil and the sides of the foundation (assumption 5) may not be perfect. The nature and extent of the contact between soil and the sides of the foundation are affected by the nature of the soil, the method of placement and compaction, the amplitude of vibration under operating conditions, and temperature variations, depending upon the specific use of the machine. Assumptions 1-5 considerably simplify the evaluation of the soil's resistance, and the elastic half-space approach can be conveniently extended to account for embedment effects. The vibrations of embedded block foundations in different modes of vibration are discussed below.
11.1. 1 Vertical Vibrations The solution to the problem of vertical vibrations of embedded foundations was obtained by Novak and Beredugo (1972) in the following manner: Consider a cylindrical block foundation of radius r o and height H embedded to a depth h, as shown in Fig. 11.2, and subjected to a vertical exciting force
///////
I )
Side layer
//////T
P,(t). The equation of motion may be written as mi(t)
=
P,(t)- R,(t)- N,(t)
(11.1)
in which m is the mass of the footing, z the vertical displacement of the footing, R,(t) the dynamic vertical reaction at the base, a?d N,(t) th~ dynamic vertical reaction along the side surface of the footmg. The sml reaction at the base is obtained from the elaslic half-space approach as follows: (11.2) R,(t) = Gr (C + iC,)z(t) 1
0
in which (11.3)
and
The displacement functions/, and/2 (Reissner and Sagoci, 1944; Sung, 1953; Bycroft, 1956; Luco and Wes~n, 1971) depend upo? (1). the dimensionless frequency ratio, a0 = wrWfJ7G, (2) the Pmsson s ratiO v, and (3) the stress distribution below the base. G is the dynamic shear modulus of the medium. The dynamic soil reaction N,(t) on the side is obtained by using
N,(t)
=
f
s(z, t) dz
(11.4)
in which s(z, t) is the time-dependent soil reaction per unit length on the vertical side of the footing and is a functiOn of the shear mo~ulus of the side layer c,, its mass density p,, depth of embedment h,d!menswnles~ frequency ratio a , and the quality of contact between the soil and th~ footmg. If the 0 soil reaction is considered to be independent of the depth, sIS equal to s(t), and by Baranov's (1967) approach, its value is given by
s(t)
= G,(S 1 + iS2 )z(t)
(11.5)
in which S 1 and S2 are given by
+e.G.
----=:N:c-,-(t)1-b-'----,-----I-[-N,_It)::-==G-,,-p,__ z(t)
-
441
HASTIC HALF-SPACE METHOD
.. J 1(a 0 )J0 (a 0 ) + Y 1(ao)Y0 (ao) S, "& 21Tao J~(ao) + Y~(ao)
lh
(11.6)
and
_ _ ...J._ _ _ _ _ - - - - _J....---
1---,o-
G,p
s, =
R,lt)
Elastic half-space
Figure 11.2. Vertical vibrations of an embedded foundation according to the elastic half-space approach. (After Novak and Beredugo, 1972.)
4
J 02 (a 0 ) + Y '0( a0 )
(11.7)
in which J (a 0 ) and J1 (a 0 ) are Bessel functions of the first kind of order zero 0
442
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS ELASTIC HALF-SPACE METHOD
and one,. respectively, and Y0 (a 0 ) and Y 1 (a 0 ) are Bessel functions of the second kmd of order zero and one, respectively. The value of the soil reaction on the side is thus given by
f
N,(t) =
G,(S1
+ iS2 )z(t) dz
G,h(S 1 + iS2 )z(t)
=
443
The natural frequency from Eq. (11.13) can be determined by trial and error, because both C1 and S 1 are functions of the frequency ratio a0 • Calculation of the dynamic response can be simplified considerably if the stiffness parameters are C 1 and S1 taken as frequency independent, and the damping parameters, C2 and S2 , as proportional to frequency. Therefore, by assuming that
(11.8) (11.16)
By substituting the values of R,(t) and N,(t) into Eq. (11.1), one obtains .. () G [ . G, h ] mz t + r0 C 1 + •C2 + G ro (S1 + iS2 ) z(t) = P,(t) =
P,(e'w')
=
P,(cos wt + i sin wt)
(11.9)
in which C, C2 , S, and S2 are the constant values of the corresponding parameters and are given in Table 11.1, wherein the frequency-dependent values are also shown. The value of the frequency-independent spring constant can be obtained by substituting C1 = C1 and S1 = S1 into Ijq. \ .. 1 (11.11) thus: "~ ,;v- n-0r<;-· L cvv-'t;r.J- ·.,.&.~ l V' '>I
in which P, is the real force amplitude. The steady-state response is given by z(t) = ze'w'
(11.10)
kz =
r./a +..:a 'oh St·]
Gro~
J
(11.17a)
The value of the frequency-indepen'j:fent damping is !liven by
in which z is the real response from Eq. (11.9). The stiffness k, and dampmg c, are thus given by [compare Eq. (11.9) with Eq. (2.37)]
(11.18a) (11.18b)
(11.11) and
and
( + -G cz = Gr -w° C - ' -h Sz) z Gr
(11.12)
0
t;,
1 = -2-vrlo_b_o
(
c, + s,~v (p,ipJ(a,laJ)
-y7c"",=+~(~G=,/=G=)""(h=/=ro""s,=)-
(11.18c)
~:e undamped natural frequency of the embedded footing w"" is then given
{t)nze
=
fk:. v(Gr v-;::; m m
0 ()
=
The damped amplitude of vibrations A
cl +
"
G h G 'o
_!..
sl )
(1l.l3)
is
(11.14) in which the damping ratio is
t;
=
z
cX 2mwnze
(11.15)
in which b0 is the mass ratio m/pri. A comparison of the response, for which the frequency-dependent values of the stiffness and damping parameters and corresponding constant values of these parameters are used, is shown in Fig. 11.3, in which it may be observed that the two solutions are close to each other in the range of practical interest (Novali;and Beredugo, 1972).
11.1.2 Sliding Vibrations Consider a cylindrical block of radius r0 , height H, and embedded to a depth h, as shown in Fig. 11.4. This block is acted upon by a horizontal exciting force, Px(tl = Pxe'w'. It is assumed that only ~lidi!!K.Y!!?!
445
ELASTIC HALF-SPACE METHOD
8 VI 0
"'
VI 0
---Constant parameters
0
oo
"'"' MM II
-Variable parameters
c-i VI ,f VI
II
rc...i"'rt..JN
00 NO
v-i v-i
II II r\.,)""n;•.J'l
RR
-------
N~
II
II
IC'),..II:.-:JN
Dimensionless frequency a 0
Figure 11.3. Gomparison of vertical re.~90se curves computed with variable and constant parameters (hi r = 0.5, b0 :;:: 8.1, piPs= 0':75, and G.fG = 0.5.) (After Novak and Beredugo, 0
1972.)
be written as (Novak and Beredugo, 1971; Beredugo and Novak, 1972) (11.19) The reaction Rx(t) is tbe same as for a footing resting on an elastic half-space and is given by ~0
"'0
+
I
<---<> <'10 v-i v-i II
II
u"'v('l Side layer 0
..,
I
,.o
GS> Ps
0
I I I I I I
I I I
'f(
i I
I
\
I I
H - h
. /,1'/// 10'/
1
~N,(t)
I I I I
I I I
h
J
x(t) ~
I
\--,o--1
Rx(l)
Elastic half-space
G, P
Figure 11.4. Sliding vibrations of an embedded foundation according to the elastic half-space approach (After Beredugo and Novak, 1972.)
444
446
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
Table 11.2. Stiffness and Damping for Half-Space and Side Layers for Sliding Vibrations
(11.20) in which G is the dynamic shear modulus. The parameters C, and C, 1 depend upon the displacement functions /, 1 and fx 2 for sliding as well as2 upon the dimensionless frequency a0 = wr0VPfG, the Poisson's ratio v, the frequency of excitation w, and the mass density p of the half-space (Bycroft, 1956). The horizontal reaction N .. (t).,_~~ layer is give\!__12Y_{J':Iovak and Beredugo, 1971; Beredugo and Novak, 1972) '' ····--
Poisson's
HalfRSpace Functions
ratio v
C,,
in which Q, isthe dynamic shearlllodulus of thesid.~-l~y.er and the functions S,1 and s~, depen(ftipoii the Poiss6ii;s ratio and-dimensionless frequency of the side~_s;L.J'he values of C, 1 , C, 2 , S 1 , and S,, are gi~abiel1.2. By substituting the values of R,(t) and NJt) from Eqs. (11.20) and (11.21),
0:S:a 0 <2.0
10.39a
0 c,, = 5.333- 1.584a, + a,+ 6.552
0.5
respectively, into eq. (11.19), one obtains
C, 2
= 2.923a 0 -
mi(t) + Gr0 (C, 1 + iC,,]x(t) + G,h(S, 1 + iS,,)x(t)
= P,(t)
Parameters
89.09a
0.1345a 0 C, 2 = 2. 536a, - a, _ 1. 923
(11.21)
Constant
Range
0 = 4.571- 4.653a 0 + a,+ 19 .14
0.0
-~-------~------
Validity
0.1741a 0 a,_ 1.927
Q:s:a 0 <2.0
Side Layer F~_ns;ions
(11.22a)
or 0.0
3.609a 0
S,,
= 0. 2328a, + a, + 0.06159
S, 1
= 150.3a 0 -
3630a~
0.2 s a0 :s: 1.5
s,, = 3.60
+ 3948a;
- 1934a~ + 348Sai
0.8652a,
S,, = 7.334a, + a,+ 0.00874
By comparing Eq. (11.22b) with Eq. (11.9) for the case of vertical vibrations and Eq. 2.37 one obtains
4.320a~
Sxl
= 2.474 + 4.119a 0 3
S, 1
= 1.468vU; + 5.662ya;; '
4
+ 2.057 a0 - 0.362a 0
(11.23) 0.25
and
f (11.24) Both k, and c, are frequency dependent. Beredugo and Novak (1972) established that the values of k, and s_may be approximat.)C.1n;q]lency_!.ndepen4wsa1!Jes_J2!.Jlll practical purposes by the substitutions
----------------
(11.25) (11.26)
0.2!5a 0 o52.0
s,, = 4.00
41.59a,
S,, = 0.83a, + 3.90 + a,
ft
I'I
sxl = 2.824 + 4.7fl:t)a 3
0 -
5.539a~ 4
+ 2.445a 0 - 0.394a 0
0.4
s"' = -1.796vU; + 6.539V'a;; 56.55a,
sx2 = 0.96ao + 4.68 + ao
0.2:::::.;: a0 :s: 2.0 _
s" = 4.10
0,;; a0 ,;; 1.5 S, 2 = 10.60
Source: Beredugo and Novak (1972). 447
448
DYNAMIC RESPONSE OF EMBEDDED BlOCK FOUNDATIONS
ELASTIC HALF-SPACE METHOD
449
The values of C,p cx2• sxl> and sx2 are also given in Table 11.2. Therefore, the frequency-independent value of k, is given by
.---..... My(t)
(11.27)
H! h f//////
+ lL~ r
(11.28) and (11.29)
I I
The undamped natural frequency of the sliding vibration is given by
I I I
1'
and the frequency-independent damping is
'
I
I
I I
ioltl
l '//uu
Nq,(t)
~I
)
Side layer Gs, Ps
I
'o----i
~ R(t)
~ G,
p
Elastic half-space
(11.30)
Figure 11.5.
R:~king vibrations of an e.l~ded foundation according to the elastic half-space
approach. (After Beredugo and Novak, 1972.)
and the damped amplitude of the sliding vibrations A" is
.
(11.31)
h' h C
and C
are dependent upon the displacement functions for circular foundation on an elastic 1956) and are shown in Table 11.3. The parameters, c.,l an ¢2• or k' d nd only upon the dimensionless frequency a,. roc ~g e:fs~in moment N (t) about the center of gravity ?f the system by fhe soil on the sides of the founda!ion m response to rocking is
~~c~i~; vib:~tions ol~
oc~si~~ed
11.1.3 Rocking Vibrations Figure 11.5 shows a cylindrical block being excited in rocking vibrations by a moment My(t) = M/'w'. The base of the block rests on the surface of an elastic half-space with a dynamic shear modulus G, mass density p, and Poisson's ratio v. The side layer has a shear modulus G, and mass density p,. The rotation of the block is defined by cf>(t), and the resisting moments occasioned by the soil reaction at the base and side are R> (t) and N., (t), respectively. The equation of motion of the rocking vibrations is
Table 11.3. Stiffness and Damping for Half-Space and Side Layers for rocking v
Half-Space Functions
c _ 2.654 + 0.1962a 0
1. 729a; + 1.485a; - 0.4881a~ + 0.03498ai c. = 0.000025a0 + 0.01583a; + 0.20~5a,3 ' + 1.202a~ 1.448a: + 0.4491a, ••
(11.32)
(11.33)
reacti~m
(11.34)
0.0
in which M mo is the mass moment inertia of the block about the axis of rotation. The resisting moment about the center of gravity of the system occasioned by the resistance of the soil at the base is (Beredugo and Novak, 1972)
half-spac~ ~ycr~ft,
c•. = 2.50
c.,= 0.43
Side Layer Functions
Any value
Source:
S = 3.142- 0.421a 0 - 4.209a; + 7.165a; •• - 4.667a '0 + 1.093 a,5 S = 0.0144a 0 + 5.263a~- 4.177a~ ., 4 4 5 + 1.643a 0 - 0.25 2a,
Beredugo and Novak (1972).
s•. = 2.50 s., 1.80 =
450
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
The values Sft• and So/ 2 are independent of v, are functions of the dimensionless frequency a0 , and are shown in Table 11.3. The values for Sxt and Sx, are shown in Table 11.2. The equation of motion then is (11.35)
451
ELASTIC HALF-SPACE METHOD
11.1.4 'Coupled Sliding and Rocking Vibrations Consider the coupled sliding and rocking vibrations of the embedded footing shown in Fig. 11.6. The footing is acted upon by a honzontal force, e'w'' and a moment ' Y M (t) = My e'w', ahout its center of gravity.d Px (t) = p x The forces and moments occasioned by the soil's reaction at the base an side are shown in Fig. 11.6. The equations of motion for sliding and rocking are (11.42a)
The frequency-dependent spring kq, is given by and (11.36) The frequency-dependent damping factor is (11.37)
(11.42h) in which M is the mass moment of inertia about an axis that passes through the system~ center of gravity and is perpendicular to th~ plane of the vibrations. The values of the soil's reaction, RxCt), and reststmg moment on the base, Rf(t), are (Beredugo ati
Beredugo and Novak (1972) found that the frequency-independent values of kf and c> may be u~ed for all e_ractical pu_!poses. These ~!lay be obtained by SUbStitutin~ eft= eft• Sf!= Sf!• Sxt = Sxt and eo/ 2 = eo/ 2 a0 , S> 2 = S~ 2 a 0 , and Sx 2 = Sx 2 a0 • The values of Cft• St• Co/ 2 , and So/ 2 are given in Table 11.3. This gives the frequency independent k> as 2
G - )} kf = Gro'{ eft+ a' ( -h ) (Sft + -h2 Sxt
'o
3r 0
(11.43) and
R,p(t) = Gr~(et + ie> 2 )>(t)- Gr 0 (ext + iex 2 )[x(t)L- L >(t}] (11.44) 2
(11.38a)
and frequency-independent damping co/ as
Cq.
, cr; •{ -
= v puro
c>2
G, h (-
+ G ro
so/2
2
1 h + 3 r~
- )} sx2
(11.38b)
The damping ratio is (11.39) The natural undamped frequency is (11.40) and the damped amplitude in rocking is
Elastic half-space G, p
(11.41)
Figure 11.6 Coupled rocking. and sliding of an embedded foundation according to the elastic half-space approach. (After Beredugo and Novak, 1972.)
452
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
The values of horizontal soil reaction, Nx(t), and resistance moment, Nq,(t), of the side layer are (Beredugo and Novak, 1972)
453
ElASTIC HALF-SPACE METHOD
By substituting for x(t), x(t), (t), and ¢(t) from Eqs. (11.49a,b) into Eqs. (11.47) and (11.48) and rearranging, one obtains (11.50)
(11.45) and
and
Nq,(t)
=
G,r~( !!._ ){[
(!!___- L)(s 'o Zro 'o
+ _!_
2 2)
+( h2
3'o
-
h~ + L:)(sx, + iSxzl](t) ro
ro
in which the frequency-dependent stiffness constants are
+iS )x(t)}
xt
(11.51)
(11.46)
x2
The parameters ex,, Cx 2 , Sx,, and Sxz have been defined for the case of pure sliding vibrations and are given in Table 11.2, and the parameters Cq, 1 , C
mx(t) + ro[ G(Cx!
Gro( Cx, + ~ ~ Sx,) '{ (L)' G., (h) (G')(h) Gro c.,+ ro ex,+ G Yo s., + G Yo kx
+ iCX,) + G, : (Sxl +iSX,) ]x(t)
kq,
=
2
X
=
2
h L hL] } [-+--- S 3r~
(11.52)
r~
~/'!fa
(11.53)
xl
and
0
(11.54) =P
eiwt
(11.47)
X
and the frequency-dependent damping constants are
and
Mm¢(t)
+ r~[o,!!._ ( 2h
'o
'o
(11.55) -
!:_)(sx, + iSx2 ) - G !:_ (Cx 1 + iCxzl]x(t) 'o
+ r~{ G(Cq, 1 + iCq, 2 ) + G,:,
'o
(L)' G,)( roh )s"'' ;:;; ex,+ (G h+L hL] ) G)(h)[ 3r~ + (G r~ - r~ Sxz
or: ( Cq, =--;;;-- c.,+
[
2
2
+ ( -h 2
3 'o
-
hL
-,
'o
+ 2L ') (Sx, + iSxzl ] + G 2L
'o
2
'o
Yo
( Cx 1 + iCxz) } (t)
(11.48) Particular solutions of above equations can be found by substituting (11.49a) and (11.49b) in which Ax and A are complex displacement amplitudes.
2
(11.56)
and (11.57) The terms kxq, and ex> represent cross co~pling stiffness ~nd damping terms respectively. Undamped natural frequencies may be obtamed by makmg ex, c and C P (t) and M (t) equal to zero in Eqs. (11.50) and (11.51). This ' x y leads to (11.58)
454
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
and
ELASTIC HALF-SPACE METHOD
455
and (11.59)
(11.70)
(11.60)
Beredugo and Novak (1972) suggested that the parameters CxP Cxz• Sx 2 , c>1> c.,, sl• and s., may be replaced by their frequency-independent values _cxl' cxz, §xl' sxz, ~4>1' co/2' ~!fld s4>2' by ~ubstituting ~xl = cxl' c,t = 9>t' Sxt = Sxl' St = St' Cxz = Cxzao, Cct>z = Cq,zao, Sxz = Sxzao, and s., ~ Sq, 2a0 , respectively as discussed earlier for the cases of pure sliding and rocking. With these substitutions we get frequency-independent springs
For a nontrivial solution
or (k x -mw 2 )(k o/ -Mm w 2 )-k2xo/ ~o •
(11.61) (11.71)
On solving for w, one obtains (11.62) in which w ~ 1 and w ~ 2 are the two natural frequencies of the system in coupled rocking and sliding. Real amplitudes of vibration are obtained (Beredugo and Novak, 1972) as follows:
kq, =
'{ ( L )'Gro Cq,r + '• ex!+
G G(h);:;
Sq,r
+
G G(h) ;:; (11.72)
and (11.73)
(11.63) Frequency-independent damping is given by
and
(11.74)
(11.64)
•{-c.,+ (L)'L - -, hL)- ]} - Cxz + (h) - \1rp;a;[_ ~ -r.:G s., + (h' -3 + -, 2
in which
c = VPGro
a1
~
2 (MY) k
(k > - M m w -
px¢
~
~
p
2
ro
ro
~
sx2
(11.75)
(11.65)
X
(11.66) 2.
px
{3 1 ~ k x -mw - M kxc/>
(11.67)
and ex>=
(h)rp;a;( h)-] -v"--=z[p(iro LCX, + ro \1 '{;' G L- 2 sx2
(11.76)
y
(11.68) E1
=
mMmw'- [mk 1 + Mmkx + CxCq,- c~ 1 ]w + [kxk>- k~
(11.69)
The frequency-independent values ex~> ex,,_ SxP _and ~xz are gi~en in Table 11.2. The frequency-independent values of C.;~> C1 ,, S.;~> and S.; 2 are given in Table 11.3. The use of frequency-independent values for stiffness and damping makes it possible to calculate natural frequencies and amplitudes by hand.
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
457
ELASTIC HALF-SPACE METHOD
11.1.5 Torsional Vibrations
and
Consider a cylindrical foundation of radius r 0 and height H that is excited in torsional vibrations by a horizontal moment M,(t) around the vertical axis through the foundation's center of gravity as shown in Fig. 11.7. It is assumed that the base of the foundation rests on the elastic half-space defined by G and p and that the footing is embedded in an elastic layer characterized by G, and p,. The equation of motion of the torsional vibrations may be written as
c., and c., are given by -!~"
c.,=!'!frt +f'!fr2 ,
c.,=!''''t +f'l/lz
in which f.; 1 and f.;z are the displacement functions (Reissner and Sagoci, 1944; Bycroft, 1956). The side reaction N.p(t) is given by (11.80)
(11.77)
in which s and s are parameters for the side layer and are functions of 1 the dimen~lonless f~equency ratio. By substituting the values R,;(t) and N,;(t) from Eqs. (11.78) and (11.80) into Eq. (11.77), one obtams
in which
.P = angle of rotation R.p(t) =moment of the resistive force at the base of the footing about z axis N.p(t) =moment of the resistive force on the sides of the foundations about z axis M m• = polar mass moment of inertia of the foundation about z axis (vertical axis about which the torsional vibrations take place). The value of
R~,(t)
(11.79)
MmJ;(t) +
Gr~[ C.; + ~ ( 1
:Js.,
+ ;(
C~, 2 + ~ ( ~)s,,,,] i/J(t) (11.81)
= M,(cos wt + i sin wt)
is given by (Novak and Sachs, 1973)
The frequency-dependent stiffness coefficient for torsional vibrations is thus given by
(11.78)
(11.82) and the frequency-dependent damping constant is given by
z Mz(t)
(11.83) H- h
Y.
The natural frequency (undamped) of the torsional vibrations is given by
/
w,., =
H
Jl""·~ Gs,
{li-
(11.84)
m•
Ps
The amplitude of damp"ia torsional vibrations A.;, is given by
.
~'O Figure 11.7.
approach.
(11.85)
G, p R;(t)
Torsional vibrations of embedded foundations according to the elastic half-space
I r
I
in which
458
DYNAMIC RESPONSE OF EMBEDDED BlOCK FOUNDATIONS
LINEAR ElASTIC WEIGHTLESS SPRING METHOD
459
(11.86) {11.90) Novak and Sachs (1973) suggested that the frequency-independent constant values of the spring and d~mping may be used. The frequency-independent values may be obtamed With the following substitutions:
in which
B "' (11.87)
c,,,
s,,
The values of C"" S"" and are given in Table 11.4 (Novak and Sachs, _1973). The value of the frequency-independent stiffness coefficient is then giVen by
(11.88) and the damping constant is given by
(11.89) The damping coefficient is given by
Table 11.4.
Stiffness and Damping Parameters for Torsional Vibrations
Stiffness and Damping Parameters
Constant Parameters
Validity Range
Half-space
c,, C"' 2
= 0.486a~ Side layer
- 2. 76a: + 0.495a; s>JI2=9.04a~ 3 .72ao S !/12 -7 .Sao - 0.455 + ao
Source: Novak and Sachs (1973).
s., ~ 12.4
s., ~ 10.2 s., ~·2.0 s"'2 =5.4
· · ratio · mertta
(6.38)
We will now discuss the extension of linear weightless spring method for calculation of dynamic response of embedded footings. 11.2
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
The linear elastic weightless spring approach of Barkan (1962) was extended by Prakash and Puri (1971, 1972) for the purpose of computing the response of ~.mbedded foundatio_!}§."and is based upon the following sim... ·• · plifying assumptions:
1. 2. 3. 4. 5.
The foundation block is rigid. The soil mass participating in the vibrations may be neglected. The soil behaves as a linear elastic material. The damping in the system may be neglected. The reactions on the base of the footing are the same as those for a surface footing resting at that depth. 6. The soil reactions on the sides of the foundation are provided by an independent elastic layer of soil lying above the base of the footing, and 7. The soil throughout remains in contact with the sides of the footing.
Assumptions 1-4 are the same as for the surface footing (Section 6.7). Assumptions 5 and 6 make it possible to separate the soil reactions on the base and sides and determine them separately. Assumption 7 is the same as assumption 5 in Section 11.1 and its relevance is discussed there. On the basis of the,preceding assumptions, solutions for vertical and horizontal vibrations, c~upled rocking and sliding, and torsional vibrations are presented below.
5.333 + 0.032a 0 1.368a: + 0.743a:- 0.1414a:
s,, 12.58 1.01a0 - 5.912a~ s,, ~ 12.59- 1.855a0 - 3.349a: + 5.335a;
Mmz pro
= -5 =
0.2s a0 :s: 2.0 0 ::5 a0 s 0.2 0.2 :5 a0 ::s: 2.0
11.2.1
Vertical Vibrations
Consider a foundation block of length a, width b, and height H, which is embedded to a depth h, as shown in Fig. ll.Sa. Let the block be subjected to a sinusoidally varying unbalance force P,(t) = P, sin wt, which acts along
460
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
t
t
PAt)
H
L
_ _ _ _ _a_ _ _ _ _
llc,J
(11.91b)
%
t
C,ah,
in which A equals ab, the area of the base of the foundation block in contact with the soil. The vertical soil reaction on the faces of the block occasioned by sliding between the bloc]< and the soil F, is
Crsbhz
_JT
CuAz z(t)
(a)
461
or
P,it)
+T
~
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
(11.92) in which C" is the coefficient of elastic uniform shear of the soil on the sides of the block. The equation of motion may be written as
(b)
t
P,lt)
mi!
+ C"abz + C"(2a + 2b )hz = P,(t)
(11.93a)
mi!
+ [ C.ab + Cit('{;a + 2b )h ]z = P,(t)
(11.93b)
or
The equation for free vibration is therefore .
1c.,
mi! + [C"ab + 2C,,(a+ b)h]z =0
<'////,
(11.94)
A,
and may be rewritten as
(')
Figure 11.8. Vertical vibrations of embedded foundations (linear elastic spring approach). (a)
(11.95)
Actual problem. (b) Soil reactions. (c) Equivalent problem.
the vertical axis through the block's center of gravity. Under the action of the dynamic load, dynamic soil reactions develop on the base and sides of the block, as shown in Figure 11.8b. . The equation of motion of the foundation may be obtained by considering the equilibrium of the actuating and resisting forces and applying Newton's second law. The various forces acting on the foundation are l. The applied vertical force P,(t).
2. The inertial force mi!, in which m is the mass of the foundation and z is the displacement at any time. 3. The vertical soil reaction at the base Rb, and 4. The vertical soil reaction (frictional force) on the sides F,.
in which c., is the equivalent value of the coefficient of elastic uniform compression given by c.ab
cue= =
+ 2C"(a + b )h ab
c" + 2c"(~ + ~)
(11.96a) (11.96b)
The embedded foundation then will be analyzed as a footing resting on the surface and using c.,~iven by Eq. (11.96b) in place of c. (Section 6.7). When his equal to zero in Eq. (11.96), c., is equal to c., as for a surface footing. The natural frequency of the vertical vibrations w., of the embedded footing is given by
The magnitude of the vertical soil reaction on the base Rb is given by
(11.97a)
(11.91a) or
DYNAMIC RESPONSE OF EMBEDDED BlOCK FOUNDATIONS
4&2
(11.97b) The amplitude of vibration of the embedded foundation A, is given by
P,
A, = m ((J)nzez
w
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
the applied external force, Px(t), the inertia of the block, mx, the horizontal soil reaction at the base R x, and the horizontal reaction on all the four sides of the block, R,. The magnitude of horizontal soil reaction at the base is given by (11.101)
(11.98)
z)
in which w is the frequency of the exciting force. If instead of a rectangular or a square block, the block is circular with a radius r and embedded to a depth h, the equation of motion for free vibrations is given by (11.99)
in which C, is .the coefficient of elastic uniform shear at the base of the foundation. The horizontal soil resistance on the vertical faces of the block R, consists of a force occasioned by the uniform compression of the soil against the faces of the block parallel to the y axis and the shearing force on the two faces parallel to the x axis that result from the uniform shear between the soil and the block. Thus,
The value of C"' is thus given by
(11.102) (11.100)
From Eqs. (11.96) and (11.100) for h > 0, C"' is greater than C", and the overall spring stiffness of soil is more than that for a surface footing. 11.2.2
4&3
in which C , the coefficient of elastic uniform compression for the side layer, repre~ib'nts the net force oci:l\'gioned by uniform compression and the decrease in compression on faces of block perpendicular to the direction of vibration. The equation of motion is, therefore, (11.103a) or
Sliding Vibrations
Consider the sliding vibrations of a rigid block that are caused by a sinusoidally varying unbalanced force, Px(t) = Px sin wt, shown in Fig. 11.9. When the block slides by an amount x along the x axis, it is acted upon by
(11.103b) in which C, denotes the value of equivalent coefficient of elastic uniform shear and is given by (11.104a)
zt
Px(t)
(11.104b)
----.---
l
lJ- h
I
"
I I I
h
1
The natural frequency ~f the sliding vibrations of embedded foundation is given by ·
I C.,.s a hx
Gus bhx
(11.105a)
I J -:;._X a
or Cr A x
Figure 11.9. Sliding vibrations of an embedded foundation according to the linear elastic spring approach (a) Actual case, (b) Soil reactions on sides and base.
(11.105b)
464
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
465
LINEAR' ElASTIC WEIGHTLESS SPRING METHOD
The amplitude of vibration is Axe= m ( (J)nxe2 2) w
(11.106)
(Cq,zrf>) X
bdz
For a circular block embedded to a depth h, the soil reaction against the vertical face may be replaced by a net force of uniform compression on the projected area of the face on a vertical plane, and the equation of motion is mJi
+ C, 1rr 2x + Cu,2rhx = 0
(11.107)
and (11.108a) ~-------------a------------~
Ia)
or C,
= (
C, + ; C"'
~)
(11.108b)
It will be noted from Eqs. (11.104) and (11.108) that when h > 0, C, > C,
and hence the overall stiffness of the equivalent soil spring for an embedded foundation undergoing sliding is higher than that for a surface footing. 11.2.3
Rocking Vibrations
Consider now the rocking vibrations of an embedded rectangular block (Fig. 11.10) excited by a moment, My(t) =MY sin wt, acting at the center of gravity of the foundation. Rocking of the block occurs in the plane of the figure, i.e., about they axis. Let the position of the foundation at any time be denoted by the angle of rotation c/>, as shown in Fig. 11.10a. The various actuating and resisting moments on the foundation are
T+ u
(b)
r'r;r
I
---
/
~
I
I I
H
dz
Fore e on elementary area d ue to nonuniform shear = C.;,(z¢>)(adz)
lf---z¢
~
-/¢
I 0
!
Figure 11.1 o. Rocking vibrations of an embedded foundation according to linear elastic spring approach: (a) block under excitation; (b) non-uniform compression on faces parallel to Y-axis; (c) non-uniform shear on faces perpendicular to Y-!lxis.
5. The moment occasioned by the nonuniform compression of the soil against the faces of the block parallel to the axis of rotation, i.e., they axis, M, (Fig. 11.10b),
1. The applied external moment MY sin wt, 2. The moment due to inertia of the block M
(6.68) M,_~
3. The moment occasioned by the displacement of the center of gravity Mw:
(6.69) 4. The moment occasioned by- the soil's reaction at the base M R:
(6.72)
l
h
0
(Cq,,zc/>)(b dz z) = c.,,cf>
bh 3
3
(11.109)
in which Cq,s is the coefficient of elastic nonuniform compression for the soil layer on the sides of the foundation and represents the net effect of the soil's reaction on the faces of the block parallel to the axis of rocking, and 6. The moment occasioned by the nonuniform shearing resistance between the faces of the block perpendicular to the axis of rotation and
466
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
in which c," is the equivalent coefficient of elastic nonunif~rm ~om pression for the embedded foundatiOn, and I the moment of mertia of the base contact area of the foundation around the y axis, which is ba 3112. For h equal to zero, c,, is equal to C•, and for h greater than zero, c,", is greater than c.,.
the soil, MN. The shearing displacement of any height z above the base equal to z. The shearing resistance of an elementary area of the face of the block is c,,(z> )adz. The moment about 0 occasioned by this resistance is given by
Therefore the total moment due to nonuniform shear for the two faces is given by
=2C
M Ft/J
if's
(ah') 3
The natural frequency of the rocking vibrations, w""', may be computed by using, wnct>e
(11.110)
7. The moment M F• occasioned by the shear resistance of soil along the faces of the block parallel to axis of rocking Fig. (11.10a). The shear displacement is (a/2)>, and the shearing resistance of soil is Fs = (a/2)>Cu(bh). Thus,
467
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
= ~(C.,,I-
WL)
M mO
(11.114a)
or
fnq,,
= J__ ~( Cq,J2-rr
WL)
M mO
(11.114b)
The amplitude of the rocking vibrations A"'" is given by
MY
The equation of motion is
.. bh 3 ah 3 Mmo + Cq,l>- WL> + Cq,, 3 > + 2C,, 3 > 2
. + Cn'P_,_ -a bh - =MY sm wt
(11.112a)
2
For circular foundations, the moment occasioned by the soil's resistance on the sides is due to the nonuniform compression of the soil on the sides and the moment occasioned by the shear F,. The moments are given by
and the equation of motion for free vibrations is
Mmo.. + [ Cq,I +
c,, 3bh
3
M,
+ 2C1, (ah') 3 + C"' (a'bh) - 2 - - WL ] >
=0
(11.115)
=
J: c.,,(2r
=
rh') ( 23 c,,
dz z>z)
(11.116a) (11.116b)
(11.112b) and
or (11.112c) in which
M F< The value of
(11.113a) bh
3
ah
3
(11.117)
"%:
c., therefore, is (11.118)
a bh
=C+-----~----=--
I
C"'(r )(2rh )2r = 4C,,r'h>
2
c,, 3 + 2C1, 3 + C,., -2•
=
(11.113b)
The natural frequency and amplitude of the rocking for a circular footing can then be obtained using Eqs. (11.114) and (11.115).
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
468
469
LINEAR ELASTIC WEIGHTLESS SPRING METHOD
11.2.4 Coupled Rocking and Sliding Vibrations
in which
The response of an embedded foundation to coupled rocking and sliding (Fig. 11.11) may be obtained by treating it as a surface footing and using the equivalent values of the coefficient of elastic uniform shear C,. and of the coefficient of elastic nonuniform compression c., from Eqs. (11.104) and (11.113), respectively, and following the procedure described in Section 6.7. The frequency response of embedded foundations may be computed by using Eq. (6.112) in Section 6.7 by replacing the limiting natural frequencies, wnx and wn, with wnxe and wn.Pe, respectively. The characteristic frequency equation for coupled rocking and sliding may thus be written as
2 2 .6.(w 2) = mMm ( wne1w 2)( Wne2w 2)
(11.122)
The horizontal force Px sin wt acting alone: (Cq,,l- WL + C,.AL A"= A(w 2 )
2
-
Mmw')Px
(11.123)
and (11.124)
(11.119)
Both the force Px sin wt and the moment MY sin wt acting:
Equation (11.119) is a quadratic in w~, and will yield two roots, (w,,) 2 and 2 (w"' 2 ) , which are the two natural frequencies of the embedded foundation. The amplitudes of vibration may similarly be obtained as follows:
2
[C.,I- WL + C,.AL 2 - Mmw ]Px + C,,ALMY Ax,= A(w2)
(11.125)
·:.:.,->·~
Moment MY sin wt acting alone:
and (11.120)
2
Aq,, =
and
C,.ALPx + (C,.AL- mw )My A(w 2 )
(11.126)
(11.121)
11.2.5
z
Torsional Vibrations
Consider a cylindrical foundation block of radius r and heig_ht H that is embedded to a depth h in the soil (Fig. 11.12). The block 1s torswnally vibrated by a horizontal moment, M,(t) = M, sin wt, about the vertical axis. The displacement of the block may be defined by angle 1/J. Moments of the various forces acting on the block may be obtained as follows: 1. The applied moment M,(t) is equal to M, sin wt. 2. The moment M 1 occasioned by inertia of the block is expressed by
'
(11.127)
h
3. Moment M n;, oc~isioned by elastic nonuniform shear at the base is expressed by (11.128)
1---------a--------1 Figure 11.11. Coupled rocking and sliding of an embedded foundation according to the linear elastic spring approach.
I
I
in which C is the coefficient of elastic nonuniform shear at the base . and I is the" polar moment of inertia "of the base area about th e z ax1s ' 4 and, is 1rr 12.
470
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
z
<::.b
in whicb c.,, is the equivalent coefficient of elastic nonuniform shear of embedded foundation. The natural frequency w""'' of the torsional vibrations of embedded foundation is expressed by
Mz(t) "" Mz sin wt
I
~! l~ _/JG
L L
(11.132a) and
I Ms.;,
c_p I
(11.132b)
'----M-:;;::-+--;¢=---,~_j~x
The undamped amplitude
4. The moment M, occasioned by elastic uniform shear at the vertical face of the block is expressed by =
C"(hrh )(n/l)r
=
(27rC"hr 3 )1f;
(11.129)
(11.130a) or (11.130b) in which Mm is the mass moment of inertia of the foundation about the · of rotation. axts (11.131a)
'TrY
h
-;:
t.;'. ')
(J)
mpe -
(11.133)
(t)
The response of a non circular Jo;undation to torsional vibrations can be computed by using Eq. (6.58c) to determine the equivalent radius and then following the procedure just outlined.
Thus far, the method for analyzing the response of an embedded block foundation in different modes of vibration have been discussed with regard to the following two methods: 1. Elastic half-space method (Novak and Beredugo, 1971, 1972; Novak and Sachs, 1973); and the 2. Linear elastic weightless spring method (Barkan, 1962), which is extended to include embedment effects (Prakash and Puri, 1971, 1972) A step-by-step design procedure witb both approaches is given below. The information required for making the design and other steps common to both the methods a!'¢ 1. Machine data
(11.131b)
2
= C~~ +4Crs
of the torsional vibrations is given by
11.3 DESIGN PROCEDURE FOR AN EMBEDDED BLOCK FOUNDATION
The equation of motion of the torsional vibrations is expressed by
= c, + (2"'c")h''
A.,,
A~,= M mz (
11.12. hTorsional vibrations of an embedded foundation according to the linear elast'oc SFpig~re rmg approac .
M,,
471
DESIGN PROCEDURE FOR AN EMBEDDED BLOCK FOUNDATION
(11.131c)
2. Soil data 3. Trial size of the foundation 4. Centering the foundation area in contact with the soil and deter" mination soil pressures 5. Design values of unbalanced exciting forces and moments
DYNAMIC RESPONSE OF EMBEDDED BlOCK FOUNDATIONS
472
473
DESIGN PROCEDURE FOR AN EMBEDDED BlOCK FOUNDATION
6. Determination of the moment of inertia and mass moment of inertia. These steps have been discussed in detail in Chapter 6 (Section 6.8) for surface footings and are not repeated here. 7. Natural frequencies and amplitudes.
G,] -p 7J
- h ~p, [ c., + s,,, -;:;;
1
g=-
yc"'
" Zy'"B,
(11.90)
1 + (G,/G)(hlr 0 )S,,
in which Mmz = mert1a . , l ratio . B = -5 "' pro
A. Elastic Half-Space Approach
1. Selecting Values of G, G, p, p,. The values of G for the soil at the base should be determined by following the procedure given in Chapter 4. Because the soil on the sides is usually backfilled and compacted, the value of G, may be taken as 0.5 to 0.75 of G below the base. The density of the soil at the base and on the sides should also be determined. The value of v for the soil below the base should be determined or selected depending upon the type of soil. The appropriate values of equivalent radius r0 for a given vibration mode should be calculated as detailed in Section 6.5. 2. Uncoupled Modes.
The undamped natj!ral frequency wn,., is (11.84) The damped amplitude of torsional vibrations
A,., is
M,
(11.85)
For vertical vibrations, the spring constant k, is (11.17a)
The damping ratio g, is
3. Coupled Modes. The response occasioned by coupled sliding and rocking may be obtained as follows: The stiffness constants kx, ko~, and kxq, are -
kx = Gro [ cxl
(11.18c) 3
-
-
-
ko~
-
in which b 0 is the mass ratio, m/pr 0 . The values of C" C2 , S" 5 2 should be· obtained from Table 11.1 (column 3). The undamped natural frequency wnu is wnze =
(6.38)
fk: = \1-;:; m
h-)
0 (-C, + GGs - St ~Gr m r0
(11.13)
=
G, h
+G
~
S ]
(11.71)
xl
'{ c.,+ _ (L)'(G·)(!!_) -;:; cx1 +GG., (h)s Yo '" + G ro
Gro
2
2
X
h L hL]- } [ 2 + -, - - 2 Sx1 -
3r 0
r0
(11.72)
'o
and (11.73)
The damped amplitude of vibrations A;, is given by
A'"=z=
~[1-(~)']' +[zg,c(--
---~)
(11.14)
The frequency-independ3il!t damping is given by
..---,..-
ex= VPGro'(-ex,+ (h) ;:;; 'J/P,p G,S G
Wnze
For torsional vibrations, the spring stiffness k"' is k~, =
3
Gr 0 (C.p 1 +
(G,h)-) G -;: s,., 0
The damping ratio
g,, is
Cq, =
(11.88)
x2
)
(11.74)
(h)
( L )'fP:G, ypor~ { Cq,z + ~ cx2 + ~ \) =--:: G
L- , + -, [ - + (h' 2
Sq,z
3r 0
r0
hL)-z Sxz ]} r0
(11.75)
474
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
and
The_ valu~s of _ex\' cx2,}xl, and sx2 are given in Table 11.2, and the values of c.,, C> 2 , s.,, and S• 2 are given in Table 11.3. The undamped natural . f requenctes, wn2 1 an d wn2 2 , are w'
nl.2
~!2 [(kx + !.±..) ± ~! (kx m M 4
m
m
_
DESIGN PROCEDURE FOR AN EMBEDDED BLOCK FOUNDATION
described in Chapter 4. The values of C, for the soil on the sides may be similarly determined from suitable tests. The value of C., for the side layers will depend upon the type of soil and the method of placement and compaction. If no other information is available C"' may be taken as 0.5 to 0. 75 Cu. 2. Uncoupled Modes. For vertical vibrations the coefficient of elastic uniform compression c"' for the embedded foundation is given by
!.±..)' _IS__] M + M m
m
475
(11.96b)
m
(11.62)
The natural frequency of vertical vibrations "'"" is
The damped amplitudes of vibration, Ax and A•, are (11.97a) (11.63)
or
and (11.97b) (11.64)
The undamped amplitude of vibration A., is
P,
in which
(11.98)
A,= mwnze-w ( 2 ')
(11.65)
For torsional or yawning vibrations the value of coefficient of elastic nonuniform shear c,, for the embedded foundation is
(11.66) (11.67)
(11.131c) The natural frequency of the torsional vibrations "'""' is given by
(11.68)
(11.132a) or
(11.69) €z ~ -[mc
(11.132b)
(11.70)
',
B. Linear Elastic Weightless Spring Appro/ch 1. Selecting Soil Constants. Soil constants such as C C C and C , . ' U' r' ¢' .p for the sot! below the base may be selected by following the procedure
The maximum undamped amplitude of torsional vibrations A"' is given by ,
A.,,= M i
M,
(
2
mz Wn!fre
_
W
')
(11.133)
476
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
EXAMPLES
co The response. of a noncircular foundation to torsional vibrations can be mp~ted by usmg Eq. (6.58) to determine the equivalent radius and then followmg the procedure outlined above.
477
The resultant horizontal and vertical amplitudes are given by
~· Coupled Modes. Sliding and rocking are coupled modes. The followmg steps are followed: The values of C" and c., are calculated as below:
A he = Axe + hA
and (6.134b)
(11.104b) and 9.
The design of the foundation will be satisfactory if the computed natural frequencies and amplitudes of vibration are within their prescribed values, as discussed earlier in Chapter 6.
(11.113b) The values of limiting frequencies in pure sliding of w and pur k" w are nxe e roc mg nq>" w
nxe
VC"A m
=
11.4 .EXAMPLES EXAMPLE 11.4.1
(11.105a)
and
_ v(Cq,J
wn
r
WL)
(11.114a)
Mrno
The natural frequencies in coupled sliding and rocking given by . ' W4
_
ne
(
wznxe
+ wzm/>e ) I'
2
W
11
'
+
2 2 WnxeWnq.e
I'
=
wne1
an
0
d
wne2'
Check for Adequate Design
'
A foundation block 6 X 4 X 4 m high and made of M-150 concrete is embedded in soil to a depth of 3.0 m, as shown in Fig. 11.13. The block is subjected to a vertical sinusoidal force of P, = 7.6 sin wt tons by an engine. The operating speed of the engine is 600 rpm. The dynamic shear modulus G for the soil below the base is 6300 tim 2 and Poisson's ratio is 0.25. Assume G, = 0.75, the density of the soil below the base, y = 1.8 tim 3 and for the side soil r, = 1.6 tim 3• Determine the natural frequency and am-
are
(11.119)
The amplitudes occasioned by coupled sliding and rocking are "////
3.0 m
and
A
=
C"ALPx + (C"jL- mw')My
"''
The term d(w
2
)
is given by
~(w')
(11.126)
_L'-----------'
! - - - - - - - - 6 . 0 m ---------1 1 1
Figure 11.13. Embedded block (Examples 11.4.1 and 11.4.2).
479
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
478
EXAMPLES
r =
plitude of vibration of the block using the elastic half-space approach. The density of concrete y, may be taken as 2.4 tim 3•
0
=
Solution 1. Machine data The weight of the machine is not given in the problem and has not been included in the calculation. Operating speed of the machine is 600 rpm. w
= 62.83 radisec
2. Soil data For soil below the base =
is
k,
G, h Gro [ cl + G ;:;; sl
=
1
=5.2,
X
3 2 _764
X
] 2.7 tim
The frequency-independent damping ratio ~' is
G,)
3 and 4. Foundation data The data on the foundation is given in the problem. Length of block a = 6.0 m Width of the block b = 4.0 m Height of the block H = 4.0 m Embedded depth h = 3.0 m Weight of the foundation block = ( 6 x 4 x 4)2.4 = 230.4 t Base area of the block= 6 x 4 =24m 2 5 and 6. Unbalanced exciting forces The unbalanced vertical exciting force
- h ~p, ( C' +S2 ro p G
1
~' = 2Vbo V(C1 + (G,iG)(hir 0 )S 1 )
Since the exciting force is in the vertical direction only, calculation of moment of inertia and mass moment of inertia is not relevant to this problem.
Natural frequencies and amplitudes of vibration a. Selecting soil constants The values of soil constants given above are adopted for this calculation. (Follow procedure of Chapter 4 if adequate soil data is available.)
(11.18c)
m W b0 = - = - , pr~ 'Y'o 230.4 = 6.06 3 1.8(2.764) From Table 11.1, column 3,
c, =5.0
P, = 7.6 tons
Vertical vibrations The equivalent radius r 0 is
(11.17a)
= 130734.8 tim
0.75G = 4725 tim 2 ')', = 1.6 tim3
b.
J
sl =2.70
4725 k, = (6300)(2.764) [ 5.2 + 6000
=
7.
= 2.764 m
The frequency-independent spring stiffness k, of the embedded foundation
.c
For soil on the sides G,
(24
\13.14
From Table 11.1, column 3, for v = 0.25,
6300 tim' 3 y= 1.8tim v= 0.25
G
(6.58a)
(A \1-;
sl = 2.1
s,
=
6.7
3 /"'1-c.6~'4'"'72"'5 5.0 + 6 ·7 X 2.764 \11.8 X 6300 ~=------~~~~==~ 112 ' ( 4725 3 ) 2 7 2(v'6.Q6) 5 ·2 + 6300 X 2.764 X · ...,
= 0.816 The undamped natural frequency of the embedded footing w"' is
4BO
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
w.,=~
EXAMPLES
4B1
b. Vertical vibrations . The equivalent value for the coefficient of elastic uniform compression C"' IS
/130734.8 X 9.81 = 74.60 radisec 230 .4
c"' = c" + 2c,,( ~ +
wn, = y
fu, = 11.87 Hz
~)
= 6000 + 2(2250)( ~
The damped amplitude of vibration (A,)d of the embedded footing is
+
(11.96b)
D
= 11625 tim
3
The natural frequency of the vertical vibration, w""' is P,
(A, ) d =
,
112
(11.14)
_ wnze-
k, {[ 1- (wiwn,,)'] + (2g,wlw.,,)'}
~ C",A m
(11.97a)
=~----,--,~~=
- I 11625 X 24 x 9.81 - y
7.6 = (130734.8){[1- (62.83/74.6) 2) 2 + [2 X 0.816(62.83/74.6)] 2 } 112
= 108.99 radisec
= 108.99 = 17.35 Hz
fnze
= 3.88 x 10- 5 m= 0.0388 mm
230.4
21T
The amplitude of vibration \undamped) A" is
.,
EXAMPLE 11.4.2
ze
f<.·~
(11.98)
2Z ..· 2 ,
m(wnze- w )
_ _ _7:...:,.6::___ __ A,= m
Solution 1. Machine data Same as in Example 11.4.1
2.
.
A=
Solve Problem 11.4.1 by using the linear elastic weightless spring approach. 3 Assume that C" =6.0kgicm (for a 10m 2 area) for the soil at the base and that C"' = 0. 75 C".
(
230 .4 )(108.99 2 - 62.83 2 ) 9.81
= 4.08 x 10- 5 m= 0.0408 mm
Soil data 2
C" for the base (for a 10m area)=6.0kgicm 3 =6000tim 3 C"' for the side layer (for a 10m 2 area)= 0.75 C" = 4500 tim'
It should be interesting to compare the natural frequency and amplitud~ of vibration of the foundation if it were resting on the surface of the soil with c = 6000 tim' and the embedment effects were neglected. The natural frquen~y wn, as surface footing would be
3, 4, 5, and 6 are the same as in Example 11.4.1
/6000 x 24 X 9.81 = 78 .3 radisec 230.4
_
wn,- Y 7.
Natural frequencies and amplitudes
a.
Soil constants
t,
nz
=
78 30 · = 12.46 Hz 21T
The amplitude of vibration (undamped), A, as surface footing would be Because the base2 area of the block and the embedded area of each side are more than 10m , the values of cu, cus' and crs for the 10m 2 area will be used. Adopt C" = 6000 tim 3 C"·' = 4500 tim
3
C" = li2C"' = 2250 tim 3
7.6
<0:..
A
=
230 ( .4 )(78.302 9.81
'
m -
62.83 2 )
= 1.482 x 10- 4 m= 0.1482 mm It should he observed that Wnu
wn,
= 108.99 = 1.391 78.30
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
482
that is, the natural frequency of the embedded foundation in this case is 39.1 percent greater than that of the surface footing. The ratio of undamped amplitudes is
483
COMPLIANCE-IMPEDANCE FUNCTION APPROACH
associated rigid massless footing as shown in Fig. 11.14b. For the case of pure vertical vibrations, the vertical impedance K, is then defined as (11.134)
A,_ 0.0408 A, = 0.1482 = 0 ·275
that is, the amplitude in the embedded case is only 27.5 percent the amplitude if the footing were treated as a surface footing.
11.5
COMPLIANCE-IMPEDANCE FUNCTION APPROACH
The response of a vibrating footing may also be computed by using impedance-compliance functions for the footing soil system (Gazetas, 1983). For the case of harmonic excitation, the dynamic impedance is defined as the ratio of the steady-state force (or moment) to the resulting displacement (or rotation) at the base of a massless rigid foundation. Dynamic compliance functions are similarly defined as the ratio of the dynamic displacement (or rotation) to the dynamic resistive forces (or moments) at the base of the foundation. Dynamic compliance functions may also be termed as dynamic displacement or flexibility functions. The impedance and compliance functions will be different for different modes of vibration. Consider the footing shown in Fig. 11.14a to be replaced by an
±
Actual foundation
in which R (t) is the total vertical soil reaction against the foundation and includes th~ resistive forces at the base and on the sides for the case of embedded foundations and, z(t) is the uniform harmonic settlement of the soil-foundation interface. R,(t) will be equal to the vertical exciting force and may be expressed as (11.135a) Similarly, z (t) = ze
iwi
(11.135b)
The dynamic compliance function F, for the case of purely vertical vibrations may be expressed by Eq.-,(~~1.136)
F,
=
z(t) R,(t)
(11.136)
The torsional impedance K"', the horizontal impedance Kx (or Ky) and rocking impedance K can be defined by equations similar to Eq. (11.134). For the vibration coridition involving coupled rocking and sliding, crosscoupled impedance terms such as K,x (or K,y) may also be defined. . The dynamic force and displacement are generally out of phase (Section 2.5). The dynamic displacement can be resolved into two components: a component in phase and another component 90° out of phase with the applied harmonic load. The impedance function [Eq. (11.134)] may be expressed using complex notation in the form (11.137) in which the subscript a represents z, x (or y), >, x (or y) and .P depending upon the form of vibrations. Both the real and imaginary components of impedance are func.tj.ons of frequency w. The significance of real and imaginary components· of dynamic impedance can be eastly understood_ by considering a single-degree-of freedom system as for the case of vertical vibrations. The equation of motion for damped vertical vibrations may be written as mi'(t) + c,i(t) + k,z(t)
(a)
Figure 11.14.
(a) Rigid massive foundation. (b) Associated rigid massless foundation.
• Lettmg z ( t)
=
=
P,(t)
ze iwi an d p , ( t) = p ,eiwi and substituting in the equation of
motion above, we obtain
484
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS 2
(
p (t)
•
k,- mw ) + tc,w
=
z(t)
(11.144) (11.138)
Comparing Eq. (11.138) with Eq. (1.134), we obtain K, = (k,- mw
2
)
+ ic,w
(11.139)
Equation 11.139 shows that the dynamic impedance is a complex number. Further, comparing Eqs. (11.137) and (11.139) Kzl =
kz- mwz
Kz2 = CzW
485
COMPLIANCE-IMPEDANCE FUNCTION APPROACH
(11.140a) (11.140b)
in which f3 is the hysteretic damping ratio. . . . As for the case of dynamic impedance, the dynamic compliance functmns defined by Eq. (11.136) may be expressed in terms of real and imaginary parts as follows: (11.145) Subscript a has already been explained earlier for the case of dynamic impedance. . . . . For uncoupled modes as in the case of vertical or torsw~al vJbratmns, the compliance functions are the inverse of impedance functmns, for example 1 F =" K"
It is observed from Eq. (11.140) that the frequency-dependent real
component of dynamic impedance represents the stiffness and inertia of the soil-foundation-system and the frequency-dependant imaginary component represents the damping (energy loss) in the system. Equation (11.139) may be rewritten as
(11.146)
a= z, .p For coupled ~ode,s, such as rock;';;)iand sliding vibrations, the compliances may be obtained by inverting the impedance matrix:
(11.141a)
K X>
]-t
K,
or K,
=
k,{k + iwc,}
(11.141b)
in which
Equation (11.141b) shows that the dynamic impedance can be expressed as a product of static spring constant k, times a complex number (k + ic,w), that encompasses dynamic characteristics, namely, inertia and damping. Using the dimensionless frequency factor a0 = wr)V,, Eq. (11.141b) can be
For calculating the dynamic response of a massive machine foundation, the pertinent dynamic impedance functions at the anticipated frequency (or range of frequencies) of machine operation are evalua:ed first. Any method used to calculate these impedances must take mto cons1deratmn the shape of the footing, type of the soil profile, the depth of embedment ~n? ?ature of soil contact on the sides of the foundation, and the flexural ngJdJty of the foundation. This can be done by using discrete or continuum type formulations. A detailed discussion of these approaches is given by Hadjian et a!. (1974), Jakub and Rosse!, (1977), and. Gazetas (1983). Using the impedance functions, the equations of motmn for different modes of ' vibration of a massive foundation (Fig. 11.15) may be written as follows:
written as
K, = k,(k + ia 0 C)
(11.142)
in which
v c=c_§._ 'r"
(11.143)
Equation (11.142) may be modified to account for both the radiation and material damping and may be written in the form
mi(t) + R,(t) = P,(t)
(11.147)
Mm,lj,(t) + T(t) = M,(t)
(11.148)
m.X(t) + Rx(t) = P,(t)
(11.149)
Mmy4>(t) + T,(t)- Rx(t) = My(t)
(11.150)
"4~.
in which R T R and T. are vertical, torsional, horizontal, and rocking, Z' o/' X ~ , reactions of the soil acting at the center of the foundatiOn base, and other
486
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
487
COMPLIANCE-IMPEDANCE FUNCTION APPROACH
iw< z () t = ze , Mz{t)
x () t = xe
iw< ,
z=z 1 +iz 2
(11.152a)
= x 1 + ix2
(11.152b)
x
cJ>(t) = cJ>e'w' ,
. > = > 1 + i>2
(11.152c)
= 1/11 + ii/12
(11.152d)
iJ!(t) = 1/Jeiw< ,
1/1
in which z, x, q,, and 1/J are frequency-dependent displacement and rotation . amplitudes at the center of gravity. The soil reaction also will be harmonic in nature and may be wntten as
TI
11
R,(t) = R,eiw<
(11.153a)
RJt) = Rxeiw<
(11.153b)
T,(t)
=
Tq,eiw<
(11.153c)
T,(t)
=
T,e'w'
(11.153d)
The complex soil reaction amp~jtudes in Eqs. (11.153) are related to the displacement and rotation amplitudes through the correspondmg Jmped.ances as given in Eqs. (11.154). Original position
(11.154a) I I I
-i I
X
Rx
=
Kx(X- L>) + Kx>>
(11.154b)
T,
=
K + Kx(x- L>)
(11.154c)
--.._)
T,
f------
Figure 11.15. Definition of variables for vibrations of a massive foundation. (a) Section. {b) Plan.
z =
P,eli(w<+a,)]
Px(t) =
Pxe[i(wt+ax)]
=
Mye[i(w<+a•)]
M,(t)
= M,e[i(w<+••)]
(11.155a)
eio:o/1
"'= K>(w)-' Mm,w
(11.151a) (11.151b)
"- = 'P
2
(11.155b)
K*x,Py M e0"• 1}N
(11.155c)
{K*X M y e0 "• 1 - K*X PX eu"" 1}N
(11.155d)
'*'= {K*x P e0"x 1 >..
My(t)
P ,e Ia~ 2 K,(w)- mw M
quantities are as previously defined in this chapter. For harmonic excitation, =
(11.154d)
K•iJ!
Substituting Eqs. (11.151), (11.152), (11.153), and (11.154) into the Eqs. (11.147) to (11.150), we obtain
(b)
P,(t)
=
(11.151c) (11.151d)
in which a is the phase angle of excitation and its subscript indicates the form of excitation. The steady-state motion may be represented by
in which K~ = Kx(w)- mw
K:.;
= Kx
2
Kx(w)L
(11.156a) (11.156b)
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
488
K;
=
K,(w)- Mm,w' + Kx(w)L
2
N = (K*x K*K*x
-
2Kx
(11.156c) (11.156d)
This approach can be us~d for surface as well as embedded machine foundations by suitably defining the impedance functions. If the forcing function is nonharmonic, it may be decomposed into a number of sinusoids by using Fourier analysis.
11.6
OVERVIEW
The problem of analysis and design of embedded rigid block type foundations subjected to sinusoidally varying unb'i.lanced forces has been discussed on the basis of the elastic half-space approach and linear elastic weightless spring approach. Both solutions are extensions of the solutions for surface footings that have been obtained by extending the method developed by Baranov (1967), who assumed that the soil underlying a footing is an elastic half-space and that the soil on the sides is an independent elastic layer composed of a series of infinitesimally thin independent elastic layers. The compatibility condition between the elastic half-space and the overlying elastic layers is thus satisfied only at the body and far from it. The approach yields reasonable results in the closed form and is very versatile and easily applied to any mode of vibration (Novak and Beredugo, 1971, 1972; Novak and Sachs, 1973). The response of embedded foundations has been solved with the elastic half-space approach by using expressions for the overall spring stiffness and overall damping ratios and accounting for the properties of soil on the base and sides of the footing and the embedment ratio hI r 0 • In these solutions, the stiffness and damping are frequency dependent. Constant values of these parameters have also been given for which the predicted response compares well with exact solutions in the range of practical interest. For the linear elastic weightless spring approach, expressions have been derived for computing the equivalent soil constants c"'' c", c.,. and c.,, in terms of (1) the values of C", C", C
489
OVERVIEW
and Dobry and Gazetas ( 1985) have suggested a different approach for calculating soil stiffness and damping factors for embedded foundatmns of arbitrary shape for the case of vertical vibrations. The soil is assumed to be a homogeneous and elastic half-space. The arbitrary base area Ab of the footing is circumscribed with a rectangle of length L and wtdth B as shown in Fig. 11.16. A dimensionless shape paramete: A.ILB i~ used to acco~nt for the effect of shape of the area on its dampmg and sl!ffness properl!es. The static stiffness is first calculated for a surface footing case and then modified for embedment effects. The static stiffness of the embedded footing is further modified to calculate the values of dynamic stiffness. The method appears promising but more work is needed before it can be adopted by a practicing engineer. . . The response of a footing embedded in a homogeneous u~tform sot! can be computed easily using the same elastic constants and denstty for the sml on the sides as for the soil at the base. The effect of embedment in a stratum can be easily accounted for in the elastic half-space approach by using the proper displacement functions [ 1,2 to compute the stiffness and the dampJ9g parameters c1 ,2 for the base layer. For the case of a stratum, increasing the embedment reduces dependence on the stratum's thickness considerably. The effect of the nonlinear behavior of a soil has not been accounted for. Experimental data on the response of embedded foundations shows goo.d qualitative agreement with theoretical predictions based on the elasttc half-space approach (Novak and Beredugo, 1971). It has been observed that backfilling considerably reduces the effects of embedn:'ent and that nonlinearity causes an inevitable scatter when the expenmental results are compared with linear spring theory. Rigorous analytical solutions of embedded footings have been attempted by Lysmer and Kuhlemeyer (1969), Kaldijan (1971), Krizek et a!. (1972), Wass and Lysmer (1972), and Urlich and Kuhlemeyer (1973). Johnson eta!. (1975) proposed static stiffness coefficients for rigid circular, and stnp
footing can be analyzed in the same manner as a surface footing.
As in the case of surface footing, the effect of soil mass participating in the vibrations has not been considered. The effect of damping has been neglected in the case of the linear elastic weightless spring approach. The elastic half-space solutions were developed for circular footings. The concept of equivalent circular area may be used for footings of other shapes, as already discussed in Chapter 6 (Section 6.5). Gazetas and Dobry (1985)
Figure 11.16.
Rectangle circumscribing an arbitrary area.
490
DYNAMIC RESPONSE OF EMBEDDED BLOCK FOUNDATIONS
footings embedded into an elastic stratum as well as for layered strata. These coefficients can be used for dynamic analysis involving soil-structure interaction. Anandakrishnan and Krishnaswamy (1973b) accounted for the effects of embedment on vertical vibrations by considering an increase in effective damping force occasioned by a skin friction mobilized between the vertical surface of the footing and the surrounding soil as a Coulomb damper and obtained solutions for natural frequency. It will be realized that embedment. significantly affects the dynamic response of foundations. The effect of embedment essentially depends upon the nature of the soil on the sides of the footing and the method of placement and compaction. The nature of the contact between the soil on the sides and the foundation is an important consideration (Novak, 1985). While considering the effects of embedment on the dynamic response of a foundation, the behavior of the soil on the sides under continuous vibration over long periods must be given due consideration along with effects· brought about by the specific operating conditions of the machine. Although embedment reduces the amplitude of vibration of the foundation, more energy is transmitted into the surrounding soil as more surface area of the vibrating footing is in contact with the soil. The energy transmitted in this manner may have adverse effects on adjoining structures. No studies are available on the effects of vibrations of embedded foundations on surrounding structures. The designer must carefully consider the favorable effect of embedment on response amplitudes to any unfavorable effect on surrounding structures. The compliance impedance function approach may be used to analyze the vibration response of a footing. The method is promising as it better defines the contribution of the soil on the vibratory response of the footing. However the process is too complicated for design of an ordinary machine foundation and has generally been used only in soil structure interaction studies under seismic loads.
REFERENCES Anandakrishnan, M., and Krishnaswamy, N. R. (1973a). Response of embedded footings to vertical vibrations. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 99, 863-883. Anandakrishna, M., and Krishnaswamy, N. R. (1973b). Vibrations of embedded footings. Proc. Int. Conf. Soil Mech. Found. Eng., Moscow, Book Nd: 4.3, SpeC. Sess. Soil Dyn. 428-429. Baranov, V. A. (1967). On the calculation of excited vibrations of an embedded foundation (in Russian). Vopr. Dyn. Prochn. 14, 195-209. Barkan, D. D. (1962). "Dynamics of Bases and Foundations." McGraw-Hill, New York. Beredugo, Y. 0. (1971). Vibrations of embedded symmetric footings, Ph.D. Thesis, University of Western Ontario, London, Canada.
REFERENCES
491
Beredugo, Y. 0. (1976). Modal analysis of coupled motion of horizontally excited embedded footings. Int. J. Earthquake Eng. Struct. Dyn. 4, Q · 3-410. Beredugo, Y. 0., and Novak, M. (1972). Coupled horizontal and rocking vibration of embedded footings. Can. Geotech. J. 9(4), 477-497. Bycroft, G. N. (1956). Forced vibrations of a rigid circular plate on a semi-infinite elastic space and an elastic stratum. Philos. Trans. R. Soc. London, Ser. A 248, 327-368. Chae, Y. S. (1971). Dynamic behaviour of embedded foundation-soil systems. Highw. Res. Rec. 323, 49-59. Christian, J. T., and Carrier, W. D. (1978). Janbu, Bjerrum, and Kjaernsli's chart reinterpreted. Can. Geotech. J. 15, 123-128. Dasgupta, s. P., and Rao, K. N. S. V. (1978). Dynamics of rectangular footings by finite elements. ], Geotech. Eng. Div., Am. Soc. Civ. Eng. 104 (GT5), 621-637. Day, S. M. (1977). Finite-element analysis of seismic scattering problems. Ph.D. Thesis University of California, San Diego. Dobry, R., and Gazetas, G. (1985). Dynamic stiffness and damping of foundations by simple methods. Proc. Symp. Vib. Probl. Geotech. Eng., Am. Soc. Civ. Eng., Annu. Conv., Detroit, 75-107. Dominguez, J., and Roesset, J. M. (1978). "Dynamic Stiffness of Rectangular ~oundations," Res. Rep. R78-20. Department of Civil Engineering 1 Massachusetts Instttute of Technology, Cambridge, Massachusetts. ~:..-':t· Fry, z. B. (1973). "Development and Eva1~ation of Soil Bearing Capacity, Foundation of Structures," Waterways Exp. Sta. Tech. Rep. No. 3-632. Gazetas, G. (1983). Analysis of machine foundation vibrations. State of the art. Soil Dyn. Earthquake Eng. 2(1), 2-42. Gazetas, G., and Dobry, R. (1985). Vertical response of arbitrarily shaped embedded foundations. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. 111 (GT6) 750-771. Gupta, B. N. (1972). Effect of ~foundation embedment on the dynamic behaviour of the foundation-soil system. Geotechnique 22 (1), 129-137. Hadjian, A. A., Luco, J. E., and Tsai, N.C. (1974). Soil-structure interaction: Continuum or finite element? Nucl. Eng. Design 31, 151-617. Jakob, M., and Rosset, J. M. (1977). "Dynamic Stiffness of Foundations: 2-D vs. 3-D Solutions," Res. Rep. R77-36. Massachusetts Institute of Technology, Cambridge, Massachusetts. Johnson, G. R., Christiano, P., and Epstein, H. I. (1975). Stiffness coefficients for embedded footings. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. 101 (GT-8), 789-800. Kaldijan, M. J. (1971). Torsional stiffness of embedded footings. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 97 (SM-7), 967-980. ,._ Kausel, E., and Ushijima, R. (1979). "Vertical and Torsional Stiffness of Cylindrical Footings," Res. Rep. R76-6. Massachusetts Institute of Technology, Cambridge, Massachusetts. Krizek, R. J., Gupta, D. ~.,and Parmelee, R. A. (1972). Coupled sliding and rocking vibrations of embedded foundations. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 98 (SM-12), 1347-1358. Luco, Y. E., and Westman, R. A. (1971). Dynamic response of circular footings. J. Eng. Mech. Div., Am. Soc. Civ. Eng. 97 (EM-5), 1381-1395. Lysmer, J. (1980). Foundation vibrations with soil damping. Civ. Eng. Nucl. Power, Am. Soc. Civ. Eng. 2 (Pap. 10-4), 1-18. Lysmer, J., and Kuhlemeyer, R. L. (1969). Finite dynamic model for infinite media. J. Eng. Mech. Div., Am. Soc. Civ. Eng. 95 (EM-4), 859-877.
492
12
DYNAMIC RESPONSE OF EMBEDDED BlOCK FOUNDATIONS
Novak, M. (1970). Prediction of footing vibrations. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 96 (SM-3), 836-861. Novak, M. (1985). Experiments with shallow and deep foundations. Proc. Symp. Vib. Probl. Geotech. Eng. Am. Soc. Civ. Eng., Annu. Conv., 1-26. Novak, M., and Beredugo, Y. 0. (1971). Effect of embedment on footing vibration. Proc. Can. Conf. Earthquake Eng., 1st, Vancouver, 111-125. Novak, M., and Beredugo, Y. 0. (1972). Vertical vibration of embedded footings. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 98 (SM-12), 1291-1310. Novak, M., and Sachs, K. (1973). Torsional and coupled vibrations of embedded footings. Int. J. Earthquake Eng. Struct. Dyn. 2(1), 11-33. Prakash, S., and Puri, V. K. (1971). Dynamic response of embedded foundation in vertical and torsional modes (unpublished report). University of Roorkee, Roorkee, India. Prakash, S., and Puri, V. K. (1972). Coupled rocking and sliding vibrations of embedded foundations (unpublished report). University of Roorkee, Roorkee, India. Reissner, E., and Sagoci, H. F. (1944). F~rced torsional oscillations of an elastic half space. J. Appl. Phys. 10, 652-662. Stokoe, K. H., II (1972). Dynamic response of embedded foundations. Ph.D. thesis presented to the University of Michigan, at Ann Arbor, Michigan. Stokoe, K. H., II, and Richart, F. E., Jr. (1974). Dynamic response of embedded machine foundation. J. Geotech. Eng. Div., Am, Soc. Civ. Eng. 100 (GT-4), 427-447. Sung, T. Y. (1953). Vibrations in semi-infinite solids due to periodic surface loadings. Symposium on dynamic testing of soils. ASTM Spec. Tech. Pub/. STP 156, 35-64. Tassoulas, J. L. (1981). "Elements for the Numerical Analysis of Wave Motion in Layered Media," Res. Rep. R81-2. Massachusetts Institute of Technology, Cambridge, Massachusetts. Urlich, C. M., and Kuhlemeyer, R. L. {1973). Coupled rocking and lateral vibrations of embedded footings. Can. Geotech. J. 10(2) 145-160. Vijayvergiya, R. C. (1981). Response of embedded foundations. Ph.D. Thesis, University of Roorkee, Roorkee, India. Wass, G. (1972). Analysis method for footing vibrations through layered media. Ph.D. Thesis, University of California, Berkeley. Wass, G., and Lysmer, J. (1972). Vibrations of footings embedded in layered media. Proc. Symp. Appl. Finite Elem. Methods Geotech. Eng. Vicksburg, MS, 581-604.
Machine Foundations on Piles
,
'
I
In foundations subjected to vibrations and shocks the use of piles may be necessary ,jn the following situ!lti}lns: . •
~)>:"''"',
1. The pressure due to the combined action of static and dynamic loads exceeds the permissible values of soil pressure and an adequate shallow foundation is not feasible. 2. Soil and water table conditions are such that machine vibrations may result in a significant loss of soil strength and/or buildup of large deformations, thus necessitating that the foundation loads (static and dynamic) be transferred to deeper soil layers. 3. It is necessary to increase the natural frequency of the foundation and decrease its amplitude of vibration, or, 4. Seismic considerations and the value or sensitivity of machines necessitate a foundation resting on piles. If a pile foundation is used because the total pressure on the soil is larger than its bearing capacity, then conventional methods of design are applied. In such cases the practical procedure of pile-foundation design consists in the determination of the number of piles needed from the known value of the bearing capacity l?f a single pile. This value may be determined by load tests or any other "!.'tandard procedure. The length of the piles is best selected on the basis of test pile driving. When a pile foundatioJ1_ !§.. l!.~eded to increase the natural frequency of vibraiTOnanddecr-ease-ihe amplitudeoFVlfirailoii 'orifie fouiiclation, lheii
tlie'practical-Cfesignproceoiirerequfreni'pecrarconslcferallons,'wh1i::'h-will be
examined in this chapter. The natural frequency and amplitude of a soil-pile system for any mode of vibration can be calculated from a knowledge of its stiffness and damping properties by using the theory of vibrations (Chapter 493
494
IJ
MACHINE FOUNDATIONS ON PilES
~~~c!~s::ethods of d':~ermining !~e~~(l~l:P.!.~e~~~fi?:'.'~~~r@.~i~g2~ow If a cyclic pile test under vertical loads is performed as on a plate (Section 4.6), then the elastic settlement of the pile is proportional to the magnitude of the load. Thus, denoting by P the load acting on the pile, and by z 1 the elastic settlement, we obtain
p
=
c,z,
(12.1)
in which C, is a coefficient of proportionality, i.e., the coefficient of elastic resistance of the pile and represents the load required to induce a unit elastic settlement of the pile. The coefficient of elastic resistance of the pile depends on soil properties, pile characteristics (e.g., length), and the length of time the pile has been in the soil. For example, the elastic resistance of a pile may have different values during driving and some time later. The natural frequency of the pile in vertical vibration is then given by
495
ANAlYSIS OF PilES UNDER VERTICAl VIBRATIONS
12.1
ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
Two cases of pile supports will be considered: end bearing piles and friction piles.
12.1.1
End-Bearing Piles
If piles are driven in soft soil and are embedded in smmd .mck-rn;..a hard
stratum at their ti we rna c · ~r that this stratum is rigid. Deformations o p1 e tlp do not occur when dynamic loads are transferred to the pile. The pile can then be considered as an elastic rod fixed at its tip (base) and free at the top, with a mass (m) resting on the top (Fig. 12.1). If no mass rests on top, we then have a solid resonant column with the fixed-free condition, which has a resonant frequency wn given by (2n -1)1rV, 21
wn =
forn=1,
(3.22)
-.:;-;;,.;.:,:·
''7rv,
wn=-zr
(12.2) in which m is the mass of the pile and static load on pile. Based upon the above simple concepts, Barkan (1962) has described test data and typical values of elastic constants of piles and pile groups under both vertical and horizontal vibrations. This analysis does not consider damping in the system and the dynamics of the problem, for which practical solutions are described in the following sections. Prakash (1981) has discussed the solutions of piles under earthquake-type loading, in which case the deflections are relatively large. In machine foundation problems, the permissible amplitudes of vibration are quite small. Hence the values of the elastic constants ( C,) determined from a lateral deflection of the order of 3.4 mm in Barkan's test are not applicable to machine foundation problems. Soli-pile constants to be used in machine foundation problems are to be determined essentially at very small deflections, of the order of a few thousandths of an inch. The soil property, e.g., shear modulus, must essentially be at a low strain value, e.g., around 10~ although there is no
'Oirecr relaUonshtp available between strains in the soil along a pile, particularly in horizontal vibrations and soil deformations around the pile. Puri et a!. (1977) have suggested a procedure for determination of amplitudedependent soil-pile stiffness from dynamic and cyclic pile-load tests. Solutions for vertical vibrations will be presented first, and will be
followed by cases of horizontal and torsional vibrations.
fn
=
V, 1 [E 4/ = 4/ \j
p
(12.3a)
or (12.3b) in which E =Young's modulus of elasticity of the pile, p(=ylg) =mass density of the pile material, and I= length of the pile
V, =compression wave velocity in pile (V,).
I Figure 12.1. Fixed-free rod pile with mass attaciled to free end.
496
MACHINE FOUNDATIONS ON PILES
ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
For the case in which the weight of the pile is negligible as compared to the supported mass, the natural frequency may be obtained as follows: We apply the end condition U = 0, at x = 0, to Eq. (3.15) and find that c = 0 and tbe displacement amplitude becomes
497
1.6 1.4
1.2 •
W X
U= Dsm-"-
(3.23)
V,
1.0
~I" •""
From Eq. (3.13), we get
au ax
=
0.8
N
au ax (A cos wnt + B sin wnt)
0.6
(12.4)
0.4
and
0.2
a2 u at
,
- , = -wnU(A
cos wnt + B sin w t) n
(12.5) IV
For longitudinal excitation of the rod in Fig. 12.1, displacement is zero at the fixed end. At the free end, a force equal to the inertia force of the concentrated mass is exerted on the rod. Therefore,
au
a'u at'
F=-AE=-mJx
(12.6)
Substituting Eqs. (12.4) and (12.5) into Eq. (12.6), we get
au ax
AE- =mw 2 U
(J)n r
wn[
y
r
Cliffs, NJ.)
resonant frequencies of unloaded steel, concrete, and wooden piles, as computed from Eq. (12.8). As the axial load is increased on a pile of given length, the resonant frequency is reduced and can be determined from Fig. 12.3.
12.1.2
=
".
(12.7)
n
Finally, substituting U from Eq. (3.23) into Eq. (12.7), we get AE V cos
Figure 12.2. Graphical solution for Eq. (12.8). (After Richart, Hall, Woods, "Vibrations of Soils and FoundationS," © 1970, page no236.·,,R~rinted by permission of Prenctice-Hall, Englewood
Friction
Pile~
Analysis of floating piles under vertical vibrations is quite different from that for end-bearing piles in which no load transfer from the shaft to the soil occurs. There are at least four methods that could be employed to examine the response of floating piles to vertical loads.
2 W I w nm sin -"-
~
which can be reduced to 1. A three-dimensional analysis (e.g., using the finite element method) in
(12.8) in which Al-y is the weight of the rod and W is the weight of added mass. The solution of Eq. (12.8) is plotted in Fig. 12.2, from which the natural frequency in vertical vibrations, fn may be determined. In order to illustrate the influence of axial loading on the resonant frequency of end-bearing piles on rock, Richart (1962) prepared a diagram th~t.l.~!?}uded t~':_Par!!';'."t.~:~.!. .ii."i.~J2.!!£L!'i!Jjle material (f:\g. 12."3). Tile three curves in the upper part of the diagram illustrate the
which the propagation of waves through the pile and soil is considered. 2. Solution of the,.pne-dimensional wave equation, for example, in a manner similar to the solution of this equation to analyze the piledriving process.
3. An analysis of the response of a lumped-mass-spring-dashpot system representing the pile and soil. 4. An approximate elastic analysis in which the problem is simplified to one of plane strain and it is assumed that the elastic waves propagate only horizontally.
498
MACHINE FOUNDATIONS ON PILES Material Steel
Concrete Wood
E(lb/in. 2)
'}'{lb/ft 3)
29.4 X 1Q6 3.0 X 1Q6
480 150 40
1.2
X
106
ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
499
Barkan (1962) and Maxwell et al. (1969). They have used a single-degreeof-freedom model, while Madhav and Rao (1971} used a two-degree-offreedom model. The single-degree-of-freedom model is simple and useful for design purposes and will be described in detail in this chapter. The fourth approach has been used by Novak (1974, 1977) and Sheta and Novak (1982) to obtain an approximate solution for pile response to vertical loading. It has been assumed that the soil is composed of a set of independent infinitesimally-thin horizontal layers of infinite extent. This model could be thought of as a generalized Winkler material that possesses inertia and dissipates energy. By applying small harmonic excitations, Novak derived solutions for the equivalent stiffness and damping constants of the pile-soil system. This model is regarded as superior to that of Maxwell et al. (1969) and will also be described in detail.
Lumped-Mass-Spring-Oashpot Model
60
80 100
150
200
Pile length, {ft)
F.ig.ure 12.3. Resonant frequency of vertical oscillation for a point~bearing pile resting on a ngrd stratum and carrying a static load W. (After Richart, 1962.)
A three-dimension~! analysis is too expensive and involved for practical des1gn. Such an analys1s may be necessary for pile-supported turbogenerator foun~atwns In nucl~ar p~wer plants, where toleranc.e limits are very critical. Soluti~ns of one-dtmensmnal wave equations, involving extension of the numencal method of analysis used for pile driving, does not appear to have been used for s~lving problems of pile response under vertical vibrations (Poulos and Dav1s, 1980}. The use of a lumped'mass spring-dashpot system h~s been succes~fully apphed to shallow foundations (Richart et al., 1970; R1chart and Wh1tman, 1967) and has been applied to pile foundations by
The single-degree-of-freedom lumped-parameter model for the actual soil pile system Fig. 12.4a is shown in Fig. 12.4b. The foundation response can be determined" from elementary theory of mechanical vibrations (Chapter 2) if appropriate values of the mass,' damping, and spnng constant can be selected for the system. It should be noted that the model m Fig. 12.4b is the same as in Fig. 2.9a for which complete solutions have been obtained (Se~tion-:h-5~ring £On§tant, C, has been used in this model in place of •. /·'kin Chapter 2. This solution differs from Barkan's solution described earlier !' in this chapter in only one respect. Maxwell et al. (1969) considered ···~ ~ 'i damping in the system. V In their model, M_illnyel!~L~LL!2.69) have considered the equivalent mass m to be only above the giund i.e., mass of the oscillator, pile cap, and the static load. They report tests on steel H-piles and concrete-filled pipe piles, in silty sand and cl overlying sand, to determine the relationship between frequency and isplacement. Fro!l1_.1h,.~te_st_Ees\llt_s, v.~l\les of equiv~lentstiffness. C,and da ping ratio g were back-calculated. . At nisoiiince, the dynamiC vafue-oCc;was '£ound to-be 'greater than the statiC sllffness for comparable p1le~ But:J.!._was suggested that use of tlie ~!at[s.§Jiffw:ss w_guld be a
o£ · · eac11·· c;rfflese i>aranieters\voura·n:orTeaa"!oan
ac;cufl'.t1f.:.I1X<>:9i£JJ2ii:§f'Pile'response·a£-atnre
1111\..
>UU
MACHINE fOUNDATIONS ON PILES ANALYSIS Of PILES UNDER VERTICAL VIBRATIONS
501,
Static load
'r:=--:-::-r
Oscillator
2.0 0
Pile cap
eiJ
lA a
1.s
6
"'•
"'fu"~
~§ 111
I. 0
~
t
89i
0. 5 Pile
0
n8~o o" o
0
0.5
1.5
1.0
2.0
2.5
3. 0
Q = Qo sin 2'11" ft 0.25
m
'-r----r-'-r z
=
Aq, sin(2'1!'ft - q,)
8
0
~
0.15
~
E 0.1 0
/l
a
BpB 0.0 5 /
0
c = damping coefficient Co = effective spring constant m = equivalent mass of system
= periodic displacement Aq, = amplitude of displacement q, = phase angle between Qo and z f = frequency (Hz)
0
Legend Static load (tons)
Q = periodic exciting force
z
0
oll_ 0 00%~t
~
c ·c_
Qo = magnitude of exciting force t = time
0~/
I·•
0.20
Figure 12.4. Analytical model pile. (a) Pile and soil system. (b) model. (After Maxwell et Copyright ASTM. Reprinted
for floating Mechanical al., .1969. with per-
mission.)
Maxwell et al. also carried out tests to determine the effeot_()fJ>il~c;~ thJ;.J:-e~<>-OL.pile fou.o>l~s. One test was performed with the cap in' contact with the soil and another test after excavating beneath the cap. Typical test results showed that the dynamic displacements of the pile cap were approximately 0.0385 in and 0.145 in respectively with a constant force excitation of 4T for pile in contact with the soil and pile cap not in contact with the soil.
0
0 0
-- A 0 .5
~ 1.0
B a ~ "'"'"'
a
a
1.5
2.0
2.5
3. 0
f
Frequency ratio, fn
50 100 200
_,._ figure 12.5. Stiffness and damp!ng rati~ vs. fre~u~ncy ratio, pipe pile D-1. (After Maxwell et ' al., 1969. Copyr_ight ASTM. Reprmted w1th perm1ss1on.)
It should be noted ;~at because the stiffness of a pile foundation is generally greater than that of a corresponding surface foundatiOn, the natural frequency of the foundation-soil system Will be mcreased by the use
of piles. '1 b The values of soil-pile-constant C, (spring constant) must necessan y e determined from a pile test aliiJtlie value of the daommg ratio {; must be estimated based on 3!!.eerin_gj~~~l_lt. There may be several questtons 'ontlle3ccuracy-of"-·these values and hence on the predicted response.
502
MACHINE FOUNDATIONS ON PILES
Novak's Model
ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
503
vibration w(z, t) such that (Novak, 1977)
;r~~ main assumptions in development of this model (Novak, 1974, 1977) I. The pile is vertical, elastic, and of circular cross section
2. It is a floating pile. 3. It is perfectly connected to the soil.
'
. The soil abo~e . the tip is modeled as an elastic layer composed of mfimtesi.mally thm mdependent layers. This assumption is equivalent to the assu'_llptlon of plane strain and leads to very reasonable results for endbeanng piles and embedded footings. It actually means that the elastic waves propagate only honzontally. The soil reaction acting on the tip is assumed to b~ eq_ual to that of an elastic half-space. It is further assumed that t?e motmn IS small and the excitation is harmonic. The latter assumptiOn yields the Impedance functions and the equivalent stiffness and dampmg constants of the soil-pile system that can be used in structural analysis. In Fig. 12.6, an elastic vertical pile is shown undergoing complex vertical
w(z, t) = w(z)e'""
(12.9)
in which w(z) is the complex amplitude at depth z, w is circular frequency, and t is time. The motion ~UJ!~e- is re~sted _l?y~ distributed ..".~~~~o_~_'.'.~.s.
p(z, t) dz
=
G(Swl + iSw 2 )w(z, t) dz
(12.10)
in which G is the shear modulus of the soil surrounding the pile and •··
Sw
'
=
21Tao
J,(aoV6(a 0 ) + Y,(a 0 )Y0 (a 0 ) ·2 2 J 0 (a 0 ) + Y 0 (a 0 )
(11.6) (11.7)
in which J0 (a 0 ), ! 1 (a 0 ) are Bessel functions of the first kind of order zero and one, respectively, and Y 0 (a 0 ), Y1 (a 0 ) are Bessel functions of tbe second kind of order zero and one, respectively. Parameters Sw 1 (=S 1 ) and Sw 2 (=S2 ) are functions of the dimensionless frequency, a0 = r 0 w/V,, in which r 0 is the pile radius, V, is VGTP; and pis tbe mass density of soil. Equation (12.10) is similar to Eq. (11.5). Parameters S are shown in Fig. 12.7 as function of dimensionless frequency a0 • With the soil reactions defined by Eq. (12.10), the differential equation of damped axial vibration of the pile is m1
a'w(z, t)
at'
+c
aw(z, t)
at
2
-
E A a w(z, t)
az'
P
+ G(Sw, + iSw,)w(z, t) =
(12.11)
0
~~-
in which m,' is the mass of the pile per unit length; cis the coefficient of pile internal damping, E is Young's modulus of the pile, and A is the area of the pile cross section.~-----..----~---------.... .._---"-----------With harmonic motion described by Eq. (12.9), Eq. (12.11) reduces to an ordinary differential equation Figure 12.6.
Vertical pile and notations.
w(z)[ -m, w2 + icw + G(Sw +iS )]- E A t
w2
P
dzw~z) dz
=
0
(12.12)
504
MACHINE FOUNDATIONS ON PILES
ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
505
the mass density of the pile. Let (12.18)
and 2
r=V(a + b
2
b
tan.p=a
),
(12.19)
Then the frequency parameter A is more conveniently written as (12.20)
in which Dimensionless frequency
Figure 12.7.
ao
Parameters 5"' 1 , 5"' 2 , C,. 11 and C.., 2 • (After Novak, 1977).
A1 =vrcos
The solution to this equation is z
z
w(z) = B cos A y + C sin A y
(12.13)
Note
A2 = vrsin
~
w(O) = 1
(12.23)
.• in which
which, for a pile of circular cross section, is
o
r
0
_ _!_
V,
0
c
. p E ao - r V ao p
G, =shear modulus of the soil below the tip
(12.16)
w(l) = the comple'l1 amplitude of the tip cw1'
and
cwz =;=,.dimensionless parameters depending on the dimensionless frequency, a 0 = r0 w/Vb, and Poisson's ratio v.
(12.17)
in which Vc =
(12.22)
At the tip, the motion of the pile generates a concentrated reaction, R(t) of the soil lying below the level of the tip. This can be described approximately as the reaction of an elastic half-space to the vertical motion of a rigid circular disk. This reaction can be written as R(t) = Re'w', the amplitude of which is
(12.15)
A = _!_ ~ Pr G
(12.21)
The integration constants B and I:Hl;'te given by the boundary conditions. At the head of the pile, harmonic motion with a unit amplitude is assumed w(O, t) = 1e'w' since this form of excitation defines the stiffness and damping of the soil-pile system at the pile head. Therefore, the first boundary condition is
in which B, C are integration constants, I is pile length, and the complex frequency parameter
(12.14)
YEP/pP is the longitudinal wave velocity in the pile, and Pp is
The shear wave velocity of the soil below the tip V, =yG,Ip,, and G,, and p, are the shear modulus and mass density of the soil respectively.~ G,--> oo, Jl!~ mo,!i2!!_Qf the tip vanishes corresponding to an end-bearing -pile. ----- --··-~-··--··---·~-·-y;·---·~·---·--With G,--> G, the pile becomes floating. Tne uJstributed soil reaction,
~
506
MACHINE FOUNDATIONS ON PILES
507
ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
p(z, t), contributes to the total stiffness and damping of the system in both
cases, but to different degrees. Using Bycroft's (1956) solution, the polynomial expressions for the parameters Cw are for v = 0.25:
Cw 1 = 5.37 + 0.364a 0 - 1.41a~
(12.31) With the integration constants established, tbe amplitude of the pile displacement becomes
(12.24a) w(z) = 1 cos A}+ C(A) sin A}= w1 + iw 2
(12.24b)
in which C(A) is obtained from Eq. (12.29). The unit appearing in Eqs. (12.28), (12.29), and (12.32) is actually the amplitude of the head and thus has a dimension of length. The real amplitude of motion is
and for v = 0.5 Cw, = 8.00 + 2.18a 0
-
12.63a~ + 20. 73ai - 16.47a:+ 4.458a~
(12.25a)
Cw, = 7.414a 0
-
2.98a~ + 4.324ai -1.782a:
(12.25b)
w(z) =vw; + w;
z)
(12.26)
The end force of the pile must be equal to the soil reaction given by Eq. (12.23). Thus, the boundary condition for the tip, z = l, is A EPA[ (-BsinA+ CcosA)= -Ghr0 (Cw, +iCw,)(BcosA+ CsinA).
(12.27) Equations (12.22) and (12.23) yield B= 1
(12.28)
The second integration constant from Eq. (12.27) is K'AsinA-(C +iC )cosA C(A)w, wz - K'A cos A+ (Cw, + iCw,) sin A x 1
(12.29)
(12.33)
and the phase angle is
The parameters Cw described by Eqs. (12.24) and (12.25) have been plotted against dimensionless frequency in Fig. 12.7. The axial force in the pile, positive for tension, is dw(z) A ( . z ( ) _-EPA-d-=EANz z p l -BsmA-+CcosAl l .
w,
cf>(z) = a tan ,~-'~'.
(12.34)
wt
Novak (1977) has plotted variations of the amplitude and phase with relative depth z/1, slenderness ratio llr 0 , wave velocity ratio V,IV,, and frequency ratio a 0 for v = 0.5 and density ratio p/pP = 0.7, which is typical of reinforced concrete piles, and shear wave velocity ratios Vb!V, = 1 and 10,000 that characterizes floating and end-bearing piles, respectively. Internal dam~ o1 the pile has been neglected. These plots indicate that the tip condition is particularly important in weak soils (small V,IV,) in which even a very long pile can vibrate almost as a rigid body. Conversely, in stiff soils it is o_n.Jythe upper part of a pile that undergoes significant displacement. The increase in the phase shift, where VISible, is indicative of increased damping. To design ri!":~Ul'IJ
in which Cw, and Cw, are calculated for frequency a 0 = r 0 w/Vb and
K' =EPA Gblr 0
(12.35) in which (12.30)
Fw(A) = - AC(A) = Fw(A),
1
For a circular pile, K becomes
(12.32)
and C(A) is given by Eq. (12.29).
+ iFw(A),
(12.36)
508
MACHINE FOUNDATIONS ON PILES ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
Subscript 1 denotes the real part of Fw, which defines the real stiffness, and subscnpt 2 md~cates the imaginary (out of phase) part of F , which relates to the dampmg. w The stiffness constant k~ of one pile can be rewritten as (12.37) in which
(12.38)
The constant of equivalent viscous damping of one pile is E AF (A) 1 (lw ), which can be written as
P
w
2
(12.39) in which
(12.40)
The stiffness and damping of piles vary with frequency Such variations are shown in Fig. 12.8, in which parameter fw , characterizlng stiffness and ··-------·-··--·······--·~-·-+··--·>~--~-L---------..----·~-----·-------·----~
v = 0.5 p
Vs!Vc
O.D3
E • E ro
• O.Q2
0.
O.Ql
' f.~w1_ -------::-100
-
.............
40
p:
=
0.7
v = 0.5
--
-..... ---.::.. 100 «:---
----
''
-- -- -----40
V8 1Vc "" 0.03
0.03
\
..!!
J
=
p;',!_c_hli_~C>lc_l_e~~~ill:!l..2.~.1?~1'.i!!g, __~I~. JJ!sml'9..19J. Jel".. typi~~l__::,~es . This figure shows that both for slender as well as rigid piles, theAYilmnic stiffness of the soil-pile system varies_()l!!Y~!!!9.ds:rately_w.ith..fr!'.qlll'!!.CY. The 1lampu1g-cfecreases-rapldlfwltli-liicreasing frequency but levels off in the range of moderate frequencies. Since stiffness and damping do not depend much on frequency, it is possible to pre~ent .l'."f.".!!!_etes!'.lw, an_(l::r,;:Joraes1gn ®~®J!l'~-''~!!Ellii2~JiS!~!lL2:ULequeiJ.cy. Figure 12.9 shows the variation of the stiffness and damping parameters of the pile with the shear wave velocity ratio, Vb!V,, of the soil below and above the pile tip. It is seen from this figure that with increasing stiffness of ~i!_ below the tip,the8t!ffi1e880TillepileTiicrea8eSI)titffi.eoarn:piiig decreases. With increasi~oerrgt.ll.,__ t]l~~t~f_f_~"~-':'f end.:_lJear~_eiles decreases while the stiffness of floati)Jg_pile_s_in.9Iffises. Damping increases with pile length in most cases. -In Fig 12 1Q,_s_tiffness and damping h":~~-be~lVooden piles with r 0 = 5tr!(T30 mm) and length either 25r 0 or 50r0 , EP = 1.728 X 10 8 psf (8.28 X 10 6 kN /m 2 ), p/p6 = 2.0, and (V,IVcJ = 0.02. The excitation was quadratic.
a0 = 0.3
Pp = 0.7 N
509
100
40 109
---------
--
40
- - VbfVs = 1 - - - VbiV8 = 5
0 0 Dimensionless frequency a0 = r w/V8
0
Figure 12.8.
Novak, 1977.)
Variations of stiffness and damping parameters of pile with frequency (After
·
Shear wave velocity ratio Vb /V8
Figure 12.9. Variations of stiffness and damping parameters of pile with ratio of shear wave velocities of soil below and above the tip. (After Novak, 1977 .)
~
(
510
MACHINE FOUNDATIONS ON PILES
\
\
511
For design of both e_nd bearing and friction piles, Novak (1974, 1977) ,!!'!.<:!,
Floating pile
- - - End-bearing pile
Rroposed_.."~.E~~!()E••~~il ..':'!_ofi~~~~~ ...~.~!'-:. !~."~--~~~!}•..4.".P.th... N. o.vak and. ,.!2Sllaii1ouby (1983 1 extended t ese so utions ror soil shear modulus decreasilfg·upward'accofl:lilfgl6 a qiiifi:lriitic"parilool'aofor bearing oas'weii as tfoating piles (Figs. 12.12 and 12. 13, respectively). - - - - The geometric damping ratio for a single pile supporting a structure can be computed from the damping constant by using Eq. ( 12.41 ).
1
~
\ \
\
ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
\ fw2 (damping}
\
\
end
---
(12.41)
.'W /'.· \.'\./
\
Pile slenderness lfro
Fig~re 12,1 0. Comparison of floating piles with end-bearing ao- 0.3, V.IV., = 0.03). (After Novak, 1977.) piles
(pi Pp
= 0.7,
v
= 0.5,
.,The calculated response curves for both floating as well as point b . pies are shown m Fig. 12.11. Parameters f. . eanng cy dependent. ' wl,w2 were considered as frequenIt can be seen that the relaxation of the f
d
!~6u:~~~ ~~l~s~mplitude. This reduction can ~~ ::r~c;:a~:~7c~~a:t~~~~a:~ Ia) b
a
0.10
!4
• 0
•I i'
•
~
~e
~~
0.06
-a;Sl Eu
~'3
0
0
T
0.08
..•E
damping
b
u
~
stiff~ess
fw1 fw2 -
0~ ·~
4
0.04
0
0.02
2
v
\
'~Wi ,-
~li ~~ ··~
•
<(
Epile/Gsoil
.;;,
=
250
500
500
250
f--
)00'"
F
Gsoil
1000
~2500(
-+-
10,000
0 0 •
Frequency w (rad/sec)
F•gure 12.11. Vertical response of foot'
piles (/I ro =50, 25, v = 0.5 p/ ,
PP
= 2 ;~i, s~pported by: (a) floating piles; (b) point bearing '
.•
c-
0.02. (After Novak, 1977.)
(b)
20
40
'
60
Pile slenderness
80
100
.!:..
'o
Figure 12.12. ~~a~!! .. ~~':!!e!!!gJ.~£tors for fi~ed-tiJ?~~~~!l~~!!L~~~!!l!.li)KPJ!es. (a) homogeneous soil (b) parabolic soil profile. (After Novak and EI-Sharnouby, 1983.)
0.1 2
I 0.10
,
! i
!'-\'. . . .
Epile/Gsoil
I',
I
~o
I r\rW! f' lf:'-
0
-"
0.04
1/
0.02
~ 0
---
t---- --1000
500
--
~
Gsoil
I
0.0 2
damping
~
0
"""-
0
10,000 40
60
80
100
(b)
t--_
'"
1---
2500
I>
hiooo
v--- -- --
20
--
80
60
Pile slenderness
Gsoil
_l.0.Q9{2_
2500
40
500
1000
2501--
500
-~
10,000
2500
Figure 12.13. Stiffness and damping parameters for vertical response of floating piles. (a) homogeneous soil.
512
~/
0.04
r - - - t-25001000
- ~
250
/ . / I--
20
0.06
1- 500
·~ v
0 (a)
-- --- --.......
I I
"'"" "' E a.
2
! llfw2' i' ...... __ ---1 --- .....·... ...... I I - --'"---. I -fw1-- --
0.08
Epile/Gpile ~ 250
I
~ g 0 . 06 Q) c: Ea>
,"'- ~',·,
250
.Lfw2 I
!!?~
~
- - 1 - - - ·t- --==
0.08 ~-~
llfw -- stiff~ess
0.10
10,000 100
lro
figure 12.13. (Continued). (b) parabolic soil profile. (After Novak and EI-Sharnouby, 1983.)
513
514
MACHINE FOUNDATIONS ON PILES
ANALYSIS OF PILES UNDER VERTICAL VIBRATIONS
where m, is the mass of the cap plus machinery or the portion of the structure vibrating in phase with the cap.
If the pile cap is not in COf!l_a<;t with the ground, Eqs.
c~~)is~ii!.llrectlyto'compute~theresponse o(ihe
f' '\
are increased.
(12.42)
· RG,hS,J
in which n is the number of piles and a, is the axial displacement interaction factor for a typical reference pile in the group relative to itself and to all other piles in the group, assuming the reference pile and all other piles carry the same load. The factor a" can be evaluated from Fig. 12.14. The equivalent geometric damping ratio for the group is given by
Also, from Eq. (11.18b), neglecting the effect of base, C2 w
---------
c~ = hr0 S2VG:P,
0.8
l
"'
0.6 .9 u .1'
I I
0
0
~ 0.4
--- '
oo
I ~
2
3
·-." (-. S/2r \ 0
Figure 12.14.
I
0
4
~
(12.44b)
~---------r
t
,~
0, we get (c,) or
In these equations, h is the depth of embedment o the cap, r 9 is the and total mass equivalent radius of the cap, G, and P,.!':_r_e the shear o d~sity of the backfill, and S(imd S2 are con n s 'II: 1 able 11 v, is the Pmsson 's ratw of the backfill sml. "'-["
k! + K~
/
(12.45a)
= c' + c1 / ---~--~::.w---' .
(12.45b)
Total (k!) ,~.....
=
(12.44a)
cf
f 1.0
_
1
(12.43)
0.2
a
~ In practice it--should be assumed that embedment is effective only in the development of side friction between ilfu ca and oil and only whende~~e nu ar ac fill is used. The sm beneath the base of the cap iSlikely to be oTJ,-;;-;;;-guaHt£:!inCi _ci~£~y_t!J"-~-'Y-~Y.fr_()l1l__tl1".£~· Similarly conesivellaCklill mayshrink away from the sides and become ineffective. Novak and Beredugo (1972) have develo e essions for calculating stiffness and geometric damping cons s for the embedded fpotings (pile cap in this case, see Section 11.1.1 . These are added to the stiffness and damping values obtained in Eqs. (12.42) and (12.43). The sum of the two stiffness and damping values give the total system stiffness and damping for a group of piles. _ In Eq. (11.17a), neglecting the effect of the base (i.e., C1 = 0) and using . get ___::.:__-~~,;.kwf for k , , we .,.,.:"'~------'--;,_-'--'___
Most piles are installed in groups. The group stiffness as well as damping will not, in general, be the simple sum of the stillness anC~
"E•
(12.42)an~(12_,43)
pile group_in _verlical
v'ibrations. Emb;;d,;ient of theplie--cap:-however, has favorable effect on : the--response of the group: the stiffness and damping values of the pile group '
Pile Croups
~
515
=
[ Total(c')
Novak (1974) computed the vertical response of a machine and its foundations (Fig. 12.15). The foundation consisted of a rectangular block of concrete (16ft long X 10ft wide X 8ft high). It was considered both (1) embedded 2ft into the soil and (2) having no embedment. It was supported on 35-ft-long end beaqpg timber piles in a medium stiff clay. The machine weight was 10 t. The reiponse of the pile foundation with varying frequency is shown in Fig. 12.16. It can be seen from this figure that in pile-supported structures, 1. Damping is very low compared to soil supported footings and 2. The use of piling increases the resonant frequency and, in this case,
5
0.2
0.1
·2r0 1S
a., as a flulction of pile length and spacing. (After Poulos, 1968.)
increased displacement amplitude at resonance. Damping can be increased by embedding the pile cap. )Yiaterial diimpmg was not 'CO'nstdered m thts particular _analysis.
)o'
1 1
ANALYSIS OF PILES UNDER TRANSLATION AND ROCKING
It must, however, be noted that if the operating frequency of the machine is less than 60 rad/sec the amplitude of vertical vibrations is reduced by use of piles. If the operating frequency is less than 40 rad/sec, the amplitudes are reduced to less than one-half of their corresponding values without piles. Sheta and Novak (1982) presented an approximate theory for vertical vibrations of pile groups. This theory accounts for: (1) dynamic interaction of piles in a group; (2) weakening of soil around the pile because of high strain; (3) soil layering; and ( 4) arbitrary tip conditions. Effect of pile interaction on damping and stiffness of pile groups, distribution of internal forces in the piles, and response of pile-supported foundations to harmonic excitation have been studied. [ It was found that dynamic group effects differ considerably from static gtou effects and ella! dynamtc stiffness and dam ing of pile groups are' much _pore requen~en ent t an t ose of single piles. It hada'!Sci been concludeCilliat the dynalil'iCbehav'!Orf a pile group is very complel\._ It is therefore recommended that the approximate methoa outlined above be used in practice until better methods become available.
T. 8'
1
12.2
Figure 12.15.
517
ANALYSIS OF PILES UNDER TRANSLATION AND ROCKING
The response of a single pile subjected to a time-dependent horizontal force and moment has been studied by several methods. Some oTtliesememoas are:
Dimensions of pile foundations.
1. The pile is considered to be an equivalent cantilever and the effect of soil is neglected. The resonant frequency and the amplitude of vibration of the cantilever may then be determined by methods applicable to beams. However, no information can be obtained on the moments, stresses, and displacements along the length of the pile 'for dynamic loads. 2. The pile is considered as a beam on an elastic foundation subjected to time-dependent loading and analyzed by finite differences. Moments, stresses, and displacements along the length of the pile may be analyzed, and impact loads as well as harmonic loads can be considered (Tucker, 1964). 3. Prakash (1981) presented an analysis for determination of natural frequencies and mod~ shapes of fully embedded piles under lateral free vibrations. This solut~n can be extended for determination of pile response
a
to horizontal excitations.
4. The approximate analytical derives stiffness and dam in Frequency w {rad/sec)
Figure 12.16. Vertical response of (a) pile foundation, (b) embedded pile foundation (c) shallow foundation, and (d) embedded shallow foundation (b = 1974.}
516
m!pr: =5.81). (After N;vak z
,
Novak (1974) had derived lateral stiffness and damping constants for
I
"-!-,
II
n
~~ .-\ ,~tl] ~~ II
~r--tf' II 0 ~
II
U'l
3
(JQ
I'Jl
.-+
-·
Vl
,g
.....
=:~:
0
c:r
-·
g.
,_.
<<...::'!g,o _, c =.: '"0 o.U'l(ll=.: o o
~
N
r.ll
;·
..
!!,. ..... "t::
:..'\....
'<
'$-
=:
...~
-e.
II
s· '"c:::l
0g
0
g;
~
:.:
""l oN
o,
1
(!)
I
.., "1::1t"l ~ "1::J"""-!
II
~
--.
Pt.
-·
-e.
::l
'-"
Q
I~~ 't:l\"'j
't:l
~
--
-$.
N
5'
("')
::;
Cl'l
0
C1tl
s-
ga
c ....
r.ll
-e......
........
II
II
"'!
"1::1
1oo.,; "":)
~
,-....
N '-'
0
I~~
. , It"l ~
"1::1V·j
0
't;j
o,
tv
.....,
~
~
---
-..
....., '-'
~
,......
........
N '--"
~
N'
r.ll
.....
""t 0
'-'
~
g:,.
2. Z
n· ~
a· .....
J-j
........
a(t)<~o(r ~ ~ ~ 0" Iii " o;· § g :1. 8.. ~ 0o' c.. ~ g. Vl
(D
•
;:::;.; -
c m----
Cll
~
1:' 'I" N 0 ~;;.,c.n'-':; ::r'>-t Vl~os § 8. g s- 0
,....
0.. :::r' ;:::
U'l
§
c.;J;""or=.o..
g ;: :
~ ~ 8" se; _._ ...... ~U'l 'E. 9 - g 8
"
(l)
II
't:l
'
0
a =
::I
II
fa ~
3
r.ll
S r':l
~ ~ · ::(~"'·0 =~
g, '"" c. nSe;gg::l
n
Vl
-e......
It"l o,~
g
'"
S' (D
~
~
U'l
-·
§
r
c..
~
o
('"')
§
a. §
~
Vl
.....
"
0
~
0"
0>
::t
-·
s·
,....
= .....
(D
~
3~
.., ~~
0>
::s
::l 0..
'"0
a~
Vl
.., ~~
a·
(JQ ......
r..ll
S
tl)
(i"
~""
"' 8::I
("')
i-1'>
......
(D Vl
-·
~"'"~~(§"O.c c;l -E.(!) (D
S"
3
'i:l
::o ~~
a·
;!!,
~
(D
(D
~ ;:::;.; 2
o.
'""I
::o ~~
~.8 ;?_
::£
;;· :::::r
§;:c~Oo ~ = c 3 § ~ 8" " 00. g &1' 3 ~ ~ s· - = ~ < so 0 '\:I
~
n a -
s· ~
o""" ,.,""'!"'
I~t"l
,g
"C
0Q.
(")
;:::
Q
't:l
,_.
675 "~ "tJ t:j 0 U'l
lJ.)" ..._
a~~o..??' e:.. ::Ia---..-!<
-
;-;.,
V)
-·
=~(lln:::r'
(tlCilQ.'-"::::::;,:
.._.-
.._,
(/j 0..00.. ~tr1 ..... ..§(l) ..0 :::r' -"t:: Q. t:: (tl (tl ,....
=
,_J~r5C..?' '"""] ..... '""I
0 """" ~ooo ...... co,....~ (tlVl=~(tl -.. -
1--' 1--' .....
::t. 0 (")
!VN§::Io
!'J Ul
!--' '-'
!--'
!'J
!--'
!--'
N
N
+>--
!--'
!'J
+>--
+>--
!--'
+>--
Ul 0
\0
00
-.J
0\
'-'
'-'
'-'
'-'
'-'
-- -· >-t <:1--'::::l~ft
~!'-JQ..;:::~
0 z
=
"'1::1
Ut
o.=
(tl!--' 'VJ ' - ' (tl
~J-->
;!!_-, '
'-'
~-~-~~~~-~.~~--~~·
TABLE 12.1. Stiffness and Damping Parameters of Horizontal Response for Piles with I I r0 > 25 for Homogeneous Soil Profile and I I r, > 30 for Parabolic Soil Profile Damping Parameters
Stiffness Parameters £pile/
v (1)
ff>~~: (3}
Gsoil
(2)
f~~~~ (4Y
!"'" (5}.
~~I
f.,.
(6)
(7)
(&Y
0.1577 0.2152 0.2598 0.2953 0.3299 0.1634 0.2224 0.2677 0.3034 0.3377
-0.0333 -0.0646 -0.0985 -0.1337 -0.1786 -0.0358 -0.0692 -0.1052 -0.1425 -0.1896
0.1450 0.2025 0.2499 0.2910 0.3361 0.1508 0.2101 0.2589 . 0.3009 0.3468
c-0.0252 -0.0484 -0.0737 -0.1008 -0.1370 -0.0271 -0.0519
!~>~,
fx;"
(9")'
!~2 (10)
(a) Homogeneous Soil Profile 0.25
0.40
10,000 2.500 . 1,000 500 250 10.000 2,500 1,000 500 250
.~
0.2135 0.2998 0.3741 0.4411 0.5186 0.2207 0.3097 0.3860 0.4547 0.5336
-0.0217 -0.0429 -0.0668 -0.0929 -0.1281 -0.0232 -0.0459 -0.0714 -0.0991 -0.1365
0.0042 0.0119 0.0236 0.0395 0.0659 0.0047 0.0132 0.02q1 0.0436 0.0726
0.0021 0.0061 0.0123 0.0210 0.0358 0.0024 • •;": 0.0068 , ., 0.0136 0.0231 0.0394
0.0107 0.0297
0.0579"'; 0.0953 0.1556 0.0119 0.0329 0.0641 0.1054 0.1717
0.0054 0.0154 0.0306 0.0514 0.0864 0.0060 0.0171 0.033,9
0.0510 0.0957
(b) Parabolic Soil Profile 0.25
0.40
::: <.c
10.000 2,500 1.000 500 250 10,000 2,500 1,000 500 250
0.1800 0.2452 0.3000 0.3489 0.4049 0.1857 0.2529 0.3094 0.3596 0.4170
-0.0144 -0.0267 -0.0400 -0.0543 -0.0734 -0.0153 -0.0284 -0.0426 -0.0577 -0.0780
0.0019 0.0047 0.0086 0.0136 0.0215 0.0020 0.0051 0.0094 0.0149 0.0236
0.0008 0.0020 0.0037 0.0059 0.0094 0.0009 0.0022 0.0041 0.0065 0.0103
Source: Novak and El-Sharnouby (1983). /~ 1 and /~ 2 are parameters for pinned end.
~0.0790
-0.1079 -0.1461
0 >
:::! 0
VI
Vl
!'J
~
; z ITI 0"'" c: z
1--' ~ Vl ~ .~ ::::::":0'....,._:;0.: ;!; o. CD
y·o
_-., !--'
=:: >
0.0060 0.0159 ,o.03o3: •'1l0491 0.0793 0.0067 0.0177 0.0336 0.0544 0.0880
0.0028 0.0076 0.0147 0.0241 0.0398 0.0031 0.0084 0.0163 0.0269 0.0443
z
i= 1T1 Vl
520
MACHINE FOUNDATIONS ON PILES
ANALYSIS OF PILES UNDER TORSION
It will be seen from the comparison of corresponding numbers for two profiles that the parabolic soil profile shows much lower ·stiffness and damping than the homogeneous soil. Also in Table 12.1, coefficients have been included for both pin-headed and fixed-translating headed piles. For pin-headed pile, 1;1 gives translation stiffness and fx 1 = 0 (i.e., k~ = O): The stiffness and damping of pin-headed piles are much less than for fixed (translating) head piles.
521
o.s r-=:o-r-.......c::---r--r111 1 --r-~~~--r-
~r-..., 0.71----+--f""";;::-
1
1
Values of
8 2
'o-
"'N~
0.6~-"""-::::-::::=1~---t--t--14
It was found, as in case of vertical vibrations, that the frequency dependence of stiffness and damping can generally be ignored, and that the important parameters are (1) the ratio of shear wave velocities in the pile • and soil and (2) the slenderness ratio l/r0 • As for vertical vibrations, the stiffness and damping for a group of piles is ----·-----·-···--·--··----··-····........................ . given by
r-·. . .
'
. . . . . t'.....
0.51---_!_-1--+:-=,~,............. -=+........_--55-t-----J
'~ (f~~d o. 4 f--~-t- Direction of load
O:::::;IJ
head)
(12.52)
---~-~-- r--
'
· r- _
'0
0.31--.==_c+ _-'-'---+-"'~-+--t---j
---- :-....._
n
2: c; c;= -~~--
_.!_ ""'
25 0 2~ 2ra;;_-.;~'0·
(12.53)
(faL)
·
-~ ~-1--
·
v =
I
I - ......... ......_
·o.5
-.,..-.:...2,1--_
.
- - KR = 10 (stiff pile)
(EJ)pile
c I I---+ O.lt- (" _!'--+--t ,I .
in which aL is the displacement factor for lateral translation and may be adopted from Fig. 12.17. Again, as for vertical vibrations, the spring constant and damping due to pile cap translation are, respectively,
""':'- ...... Kn
,\
L
=
KR =
10-5 (flexible.
1
1pile)!
.
(l
~
(EI.,;, l
4
pile length)
~-L_~1~0~~20~~30~~40~~5~0~6~0~7~0~8:,0~90 Departure angle (3 (degrees)
(12.55) in which
h = depth of embedment
•
Figure 12.17.
(12.54)
and
r 0 = equivalent radius of the cap G, and P, = the shear modulus and total mass density of the backfill Sx 1 and Sx, =constants in Table 11.2. --~---------~
Equation (12.54) is obtained by letting<£, =O)n Eq. (11.27), and Eq. (12.55) IS obtamed by lettmg Cx 2 = 0 m Eq. \I128). The total stiffness and total damping values are sums of Eqs. (12.52) and (12.54) and Eqs. (12.53) and (12.55), respectively. Total (kg) = kg+ ktX X X Total (cg) X
= cgX
+ cf X
Graphical solution for at. (After Poulos, 1971.)
For rocking vibrations, the effect of pile groups_ and the pile cap is accounted as for sliding and equations have been wntten m SectiOn 12.4. The use of these equations has been illustrated in example 12.5. The soils very near the surface control the load deformation propertie~ of the pile. Also, a gap may often form behind a later~ loaded_ll!.!!'c Therefore, the value of Q.QL.!',JQ be used f~teral Pil~lli!.IYJ.!~j~ __sm~!ler tfiiiiitllevafueusecl !9£verJiS!/c!J?lL~l'.!'_~lr~.\~ This is true for static as well as dynamic analysis. . . The effect of stat.i,E load was investigated and was found to be stgmficant only with extremely il!competent soils. Most stiffness and dampmg parameters were reduced by the presence of axial load, but the dampmg caused by rotation was increased.
12.3
ANALYSIS OF PILES UNDER TORSION
(12.54a) (12.54b)
Novak and Howell (1977) proposed an analysis for torsional vibrations of a pile. The mai~, ·assumptions in this analysis are
522
MACHINE FOUNDATIONS ON PILES
1. The pile is vertical, elastic, end bearing, circular in cross section, and IS perfectly connected to the soil.
2. The soil is modeled by a linear viscoelastic medium with frequencymdeptndent matenal dampmg of the hysteretic type. 3. The local soil reaction of this medium acting per unit length of the pile IS assu~ed to be equal to that denved for plane strain conditions, i.e., for umform rotatwn of an infinitely long pile. 4. The excitation is harmonic and the motion is small. In Fig. 12.18, the vertical pile undergoes a complex harmonic rotation Eq. (12.56), about its vertical axis. '
.P(z, t) = ,P(z)e'w'
(12.56)
in which
.P(z, t) =complex amplitude of the pile rotation at deptb z w =circular frequency of excitation t =time
'
. The motion of th~ pile is resisted by a torsional soil reaction acting on p1le element dz, wh1ch can be wntten as (Novak and Sachs, 1973, Novak and Howell, 1977)
Gr~(S1, 1 + iS.,)(
ANALYSIS OF PILES UNDER TORSION
in wbich the stiffness parameter (12.58) and the damping parameter 4
s,,,,(ao)
= J'
+ y'
l
l
(12.59)
in which
a0 = r0 w/V8 r 0 = the pile radius V, = -,fG!p = the shear wave velocity G = the shear modulus of soil p = the soil density .T0 (a 0 ), 11(!z0 ) =Bessel functions of the first kind of order zero and one, · respectively Y 0 (a 0 ), Y1 (a 0 ) =Bessel functions of the second kind of order zero and one, respectively . For noncircular piles, r 0 should be the equivalent radius of the possible slip circle around the pile and not its cross-sectional area. The material damping can be included by the addition of an out of phase component to the soil shear modulus, which then becomes
(12.57)
---..M
523
G*=G 1 (1+itan8)
(12.60)
in which
..........
tan a= G 2 /G 1 GP G 2 =real and imaginary parts, respectively, of the complex soil shear modulus G* 8 = loss angle
'
G,p Op
'l'}?;;~~~??,!?;;~~~m/
Figure 12.18.
Vertical pile and notation for torsion.
Thus, G* replaces O.,jn Eq. (12.57) and enters Eqs. (12.58) and (12.59) ·· through a0 • The effect of hysteretic material damping is to significantly increase the damping parameter, s,, by an almost constant amount, equal to 47T tan 8 at low frequencies, and to reduce the stiffness parameters.,, slightly at higher frequencies. Experiments have shown that material damping may be neglected for other vibration modes but is essential for torsion (Novak and Howell, 1977). They have fur!her shown that the displacement of slender piles quickly
524
MACHINE FOUNDATIONS ON PILES
diminishes with increasing depth and varies, to a lesser degree, with frequency. The effect of the tip conditions is less significant for the more slender pile, in which case the tip is fixed by the soil. The degree of this fixity, of course, depends on pile slenderness and the stiffness of soil (wave velocity ratio, V;JVP). Stiffness and damping constants and c~ for fixed tip single piles are given by
Timber
_
(12.61)
I
0
and
0.3
(12.62)
fT, 2 (tan 0
=
0)
--- fr, 2 {tan 5 ~ 0.1) III II II - fTl {tan 5 ~ 0.1) I I , 1ll \ .-f, ......_ _ _ _ l 11\' - - - Vs \ T,I - > 20-= II 'o Vp I I \
k:,
1 GPJ k o/ =-t r T,t
525
ANALYSIS OF PILES UNDER TORSION
0.1
I \ I h,l ' . . . . . __ _...-
II I
.;: ~ E
• ro
in which
Q.
0.2
GP = shear modulus of pile material J = polar moment of inertia of the pile cross section r 0 = effective radius of one pile and V, = shear wave velocity of soil
;T,l} =parameters r.z
which have been plotted for dimensionless input parameters in Fig. 12.19 for timber piles and Fig. 12.20 for reinforced concrete piles. In these figures, VP is shear wave velocity of pile \}GP/pP
It can be seen from these figures that damping parameter fr 2 varies with ' frequency much more than the stiffness parameter fr. 1 • The marked effect of material damping may be seen from the broken lines in Figs. 12.19 and 12.20, ·which were calculated with tan o = 0.1, a representative value for soils. The effect of the material damping of the soil is to increase significantly the total torsional damping of the pile, particularly at low frequencies, and to make the equivalent viscous damping constant somewhat less frequency dependent than it is with tan o = 0 (for higher frequencies). The effect of material damping on the torsional stiffness of the pile is negligible. Stiffness and Damping Constants of Footing
The torsional stiffness and damping constants of a pile have been obtained in the above analysis as moments that correspond to unit rotational displacement and velocity respectively, For a pile located beyond the reference
~1-~-L-L~o~.5~_j~L-~l~.o~_j~--~l.~ a0 = wr0
JPiG
Figure 12.19. Torsional stiffness and damping parameters of timber piles (p/pP = 2). (After Novak and Howell, 1977.)
point, these moments are composed ?f (1)a part that twists the pile and (2) a part that induces translation ?f 1t. ~~t~ reference to F1g. 12.15, the torsional stiffness constant of a p!le-footmg 1s ~~"
(12.63) and the torsional damping constant is
c• = L; [c~ + c;(x; + y~)]
(12.64)
The summation is extended over all the piles. In Eqs. (12.63) and (12.64),
MACHINE FOUNDATIONS ON PilES
torsion diminishes quickly with the ratio Rlr0 • Therefore, the torsion of the piles will be more important for footings supported by a small number of large diameter piles than for footings supported by a large number of slender piles spread far from the reference point. The maximum effect of twisting will occur if the foundation is a caisson, which may behave as one pile. In case the position of the centroid of the footing coincides with the elastic center of the piles in plan, the excitation moment, M, cos wt, produces pure torsional response of the footing, A.,, given by
Concrete - - fT,2 (tan 0 = OJ - - - fT.2 (tan 0 = 0.1) - - fr,l {tan 0 = 0.1)
527
ANAlYSIS OF PilES UNDER TORSION
0.12 0.11
0.10 0.09
0.08 0.07 0.06
(12.65) N
"',•
in which M m• is the mass moment of inertia of the footing about the vertical
2
axis .
E ro
0.05
"-
A reasonably accurate estimate of the frequency-independent constant k~ may be obtained by using parameters from Figs. 12.19 and 12.20. In order tiJ express the respoifs~ in a dimensionless form, because of an excitation of an unbalanced mass whose eccentricity is em and whose horizontal distance from the footing centroid is r", the dimensionless amplitude of rotation is a.,= A1>Mm,/(m,emre). Novak and Howell (1977) have computed the torsional response of the footing, shown in Fig. 12.15. Total weight of the machine is 20,000 lb (9 072 kg). Torsional excitation is caused by rotation of an unbalanced mass m,. Values of m,, em and r" are not required, as the results have been given in dimensionless form. The footing is of reinforced concrete with a density of 150 pcf (2 650 kg/ m 3 ) with plan dimensions as shown in Fig. 12.15. Depth of the footing is 8ft (2.4 m) with no soil contact. The equivalent radius for torsion, determined from the polar moment of inertia, r0 = 7.41 ft (2.26 m). Soil data are as follows:
me,
0.02 0
0
0.5
ao = wr0
1.0
0 1.5
../PiG
Figure 12.20. Torsional stiffness and damping parameters of reinforced concrete piles (p/pP =0.7). (After Novak and Howell, 1977.)
k~ .and c!, are stiffness and damping constants, respectively, of a pile
e
subjected to torswn and are given in Eqs. (12.61) and (12.62), and and c are. stiffness and damping constants, respectively, of a pile subje~ted t~ ~onzontal transla!!on and are given by Eqs. (12.46) and (12.47), respectively. It is obvious from Eqs. (12.63) and (12.64) that the contribution of the translation components increases with the square of the distance from the ~eference point, ~ = + Therefore, in a practical situation, the Importance of torswn of each pile depends on the ratio of the torsional stiffness to the stiffness caused by horizontal translation. Novak and Howell (1977) have shown that the contribution of the pile
Yx; y;.
1
Bulk density= 100 pcf (1 767 kg/m 3 ) Shear wave velocity V, = 220 fps (67 m/sec) Poisson's ratio v = 0,25.
....
The backfill has mass density p, = 0.75p and shear modulus G, = O.SG.
Timber Piles
Computations were done with eight timber piles with the following specifications:
528
MACHINE FOUNDATIONS ON PILES
DESIGN PROCEDURE FOR A PILE-SUPPORTED MACHINE FOUNDATION
529
3
Density= 48 pcf (848 kg/m ); Pile length= 35ft (10.7 m); Effective radius r 0 = 5 in (127 mm); Young's modulus EP = 172.8 x 10 6 psf (8.28 x 10' kN/m') Shear modulus GP =54 x 10 6 psf (2.59 x 10 6 kN/m 2 ) and P1le shear wave velocity VP = 6,020 fps (1835 m/sec); With these parameters, total mass m = 9 813 kg sec'lm (6 584 slugs) and total mass moment of inertia about the vertical axis M = 24 433 k 2· 2 (176 893slugft }, the response curves of the foo;ingm'on timber ~~:e~s pl~tted m F1g. 12.21. This figure also shows the effect of ignoring pi! ' tw1stmg. e Concrete Piles
Computations were done with six reinforced concrete piles with the following specifications: Density= 150 pcf (2650 kgim') Pile length= 35ft (10. 7 m) Effective radius r0 = 12 in (305 mm) Young's modulus EP = 552 x 106 psf (26.4 x 10' kN/m') Shear modulus GP = 200 x 10 6 psf (9.58 x 106 kN/m') anil ~lie she".r wave velocity VP = 6 525ft/sec (1 989 m/sec); four of the piles ave R-6.5ft (1.98m); the remaining four have R=2.5ft (0.76m).
The response of footing with concrete piles is also plotted in Fig. 12.21 with and without twisting of piles. For a group of piles, the effects of pile twisting and soil material damping are not very pronounced. With single piles, such as piers and caissons, these effects are essential (Novak and Howell, 1977}. The analysis presented above may require some corrections based on experiment because of 1. The dependence of the transmission of the torque into the soil by
shear. 2. The possibility of slippage. 3. The variation of shear modulus of soil with depth, and 4. Other factors.
12.4 DESIGN PROCEDURE FOR A PILE-SUPPORTED MACHINE FOUNDATION Based on the analysis presentect"fJi? the previous sections, a design procedure of piles under (1) vertical vibrations, (2} horizontal vibrations, and (3) torsion will now be described. The following soil and pile properties and dimensions must be determined. Soil Properties
Ignoring twisting
8 Timber piles 10 in diameter Including twisting
Shear modulus G, and G6 , Poisson's ratio v, and unit weight y, for the soil both around the pile and below its tip respectively. Pile Properties and Geometry
Pile length, cross section, and spacing in tbe group, unit weight y of pile and pile cap and Young's modulus of pile material. Based upon the above information, (1) v; and VP shear wave velocity in soil and pile respectively and (2} V, compression wave velocity in pile are computed. Vertical Vibrations 6 Concrete piles 24 in diameter Including twisting Ignoring twisting
.,.
1. Compute spring stiffness and damping of single pile (12.37}
Frequency (Hz) diameter
Figure 12.21. Torsional response of piled fooling. (After Novak and Howell, 1977.)
(12.39)
530
MACHINE FOUNDATIONS ON PILES
DESIGN PROCEDURE FOR A PILE-SUPPORTED MACHINE FOUNDATION
The values of functions fw 1 and fwz are obtained from Figs. 12.12 and 12.13 for fixed tip and floating piles, respectively. 2. Compute spring stiffness and damping of pile group k! (piles only)
531
(12.53)
n
k'
=
2: k~
_1_ _.
in which "Lis taken from Fig. 12.17. 3. Compute stiffness and damping due to pile cap.
(12.42)
n
w
2: "· 1
fkxG,hSx 1
and
(12.43)
in which "• is taken from Fig. 12.14. 3. Determine spring stiffness and damping due to side friction
e
w
(12.44a) (12.44b)
Values of sx1 and Sxz are listed in Table 11.2. 4. Total stiffness and total damping are then sum of stiffness and damping values computed in steps 2 and 3, respectively. '
Rocking 1. Compute stiffness and damping of a single pile in both rocking as well as for coupled motion.
Values of S1 and S2 are listed in Table 11.1. 4. Compute total spring stiffness and total damping
1 EPIP k"' = -r- fq,1
(12.45a)
Total (c') = c'w + cfw w
(12.45b)
w
1
1. Compute stiffness and damping of a single pile
EPIP -,ro
fx1
(12.46)
EPIP 1 ex= r2V fxz
(12.47)
kx
=
0 '
x
1 EPIP kx = -,- fx1 ro
(12.50)
1 EPIP . ex= r v fx2
(12.51)
1 g = ..S [k k
1 1 2 1 + kw x'r + k x Zc - 2Zcxc[> k ]
(12.66)
1
2: k1 n
(12.49)
Values off parameters are listed in Table 12.1. 2. Compute stiffness and damping of pile group (piles only )(Novak, 1974) ..,,
n
_1_ _ x
'
t.,
0 '
in which fx 1 and fxz are given in Table 12.1. 2. Compute stiffness and damping of the pile group (of piles only)
kg=
EPIP
c•= V
Translation
1
(12.48)
0
e
Total (kg)= k'w + w
(12.54)
(12.67)
(12.52) in which
532
MACHINE FOUNDATIONS ON PILES
x, =horizontal distance of pile from C. G. (Fig. 12.15) spacing of piles Z, = height of the center of gravity of the pile cap above its base (Fig. 12.15) and o=hlr 0
i
and
h
o=-
I
'o
and
I l
A;\
8•
Co
Do
Eo
F•
Go
H•
I•
J.
K•
L•
M•
No
0•
P•
a•
R•
S•
T•
! 11m
6m
3. Compute stiffness and damping of pile cap. Letting
c
~--------:sm--------~
zc=L,
~
in Eqs. (11.72) and (11.75) \\le get
1.5 m
figure 12.22a. Arrangement of 5 x 4 pile group for Example 12.5.1.
(12.68)
(12.69) 4. Total stiffness and total damping are then the sum of stiffness and damping values computed in steps 2 and 3, respectively.
12.5
t4m 1m
I
T
1_1_
D
+
4m
EXAMPLES
EXAMPLE 12.5.1
Estimate the (a) stiffness and (b) damping in vertical vibrations of' a 20 pile • group in sand (Fig. 12.22a). The concrete piles are 45 em in diameter and 20m long. The following soil and pile properties may be assumed
m
j_ D D Im
0.6
2m
_i_
D
I m
T
.
Structural column vertical
static load gQO tons
~1.3
Soil Properties G, = 4000 t/m 2 throughout
50cmx50cm concrete piles
Also
Medium stiff lean
30m
clay
y,, = 1.59 g/ cm 3 Pile l'p =2.4g/cm
EP = 2.5
X
3
10 6 t/m 2
Figure 12.22b.
Pile foundation Example 12.5.2. 533
534
MACHINE FOUNDATIONS ON PILES
Solution Pile
'o =
.~
A
B
c
22.5 em
D
S= 1.50m
E
'Yp = 2.4 gi cm 3
EP = 2.5 X 106 t/m 2 A=
IP = V
=
F G H I 1
1r
4 (0.45 m) 2 = 0.159 m 2 1r
64
4
(0.45) = 2.013
{!!_
' Y-p
=
X
10- 3 m 4
6
/2.5 X 10 _ Y2 .4 i 9 .81 -3196.7misec
v, = yI 1. 40oo i .
K L
M N 0
_
59 9 81 -157 misec EP 2.5 X 10 6 G, = 4000 = 625
I ro
=
p Q
20 0.225 = 88.89 = 90
R
s
T
Vertical Vibrations
2: "·
fw, = 0.037 (Assume floating piles) (Fig. 12.13) fw 2 = 0.068 (Fig. 12.13)
e =EPA r0
6
V,
A
B
c
F
G
H
1.0 0.60 0.45 0.38 0.33
0.60 1.0 0.60 0.45 0.38
0.45 0.60 1.0 0.60 0.45
0.60 0.53 0.43 0.38 0.30
0.53 0.60 0.53 0.43 0.38
0.43 0.53 0.60 0.53 0.43
0.60 0.53 0.43 0.38 0.30
0.53 0.60 0.53 0.43 0.38
0.43 0.53 0.60 0.53 0.43
1.0 0.60 0.45 0.38 0.33
0.60 1.0 0.60 0.45 0.38
0.45 0.60 1.0 0.60 0.45
0.45 0.43 0.40 0.33 0.29
0.43 0.45 0.43 0.40 0.33
0.40 0.43 0.45 0.43 0.40
0.60 0.53 0.43 0.38 0.30
0.53 0.60 0.53 0.43 0.38
0.43 0.53 0.60 0.53 0.43
0.38 0.38 0.33 0.30 0.29
0.38 0.38 0.38 0.33 0.30
0.33 0.38 0.38 0.38 0.33
0.45 0.43 0.40 0.33 0.29
0.43 0.45 0.43 0.40 0.33
0.40 0.43 0.45 0.43 0.40
8.58
9.31
9.53
9.14
10.01
10.25
Average value offa, = 9.47.
6
- (2.5 X 10 t/m 2 )(0.159 m 2 ) fw,(0.225 m) (0.037)
c' = EPA f. = (2.5 X 10 tim 2 )(0.159 m 2 ) x 0.068 w
535
Ref Pile 1=20m
w
EXAMPLES
157
w,
=
65,384.4 tim
i/
· = 172.16 t secim
~~~: 1is~~d ~ \~!h:o~~~:goia~: i;:;r:~i~;;a~~";s1 ;I,:o~ :~.the piles have 1
From Table 11.1
s, =2.7, k~
= G,hS, = 4000(2)(2. 7) = 21,600 tim
fcw-hrOS2
vc:;;, GsPs
r 0 = (8x6r --:;;- =3.9m
c~ = (2)(3.9)(6.7)V4000
Use l: "• = 9.47.
k' = w
c' = w
2: k~- 20(65,384.4) L "• -
L c~ _ L "· -
9 .47
s, = 6.7
X
1.59 i9.81- 1330.64 ti(misec)
Total Stiffness and Damo,!ng Values = 138,087.4!/m
20[172.16] = 363.59 ti(misec) 9 .47
Assume a pile cap 8 m x 6 m x 3 m thick is embedded 2m into the soil.
Total k~ = 138,087.4 tim+ 21,600 tim= 159,687.4 tim Total c~ = 363.97 + 1330.64 = 1694.63 ti(misec) EXAMPLE 12.5.2
Determine the damping and stiffness constants in vertical and horizontal direction for the pile group shown in Fig. 12.22b. Assume Novak's frequency-independent solution.
536
MACHINE FOUNDATIONS ON PILES
537
EXAMPLES
The data that has been made available for soils and piles is as follows:
m,= ( 40 X 1000 + 4 X 4 x
Soil Properties
\~~0106 x 2.5) /981
4
= ( 4 X 10 4 + 10.4 X 10 ) /981 2 = 147 kg sec /cm
Lean clay G, = 300 kg/ em' v=0.4 y, = 1.8 g/ cm 3 Backfill G, = 400 kg/em' y, = 2 g/cm 3
r0 = (
50 X 50)
0 5 '
7T
= 28.22 em
Mm =mas moment of inertia of m, about C. G. of block 2 2 Mm = [4 X 104 X (300) 2 + 10.4 X 10 4 X ( 400 + 260 ) /12] /981 2 = 5.68 x 106 kg em sec
Piles and Pile Cap
(assuming the 40-t mass to have a radius of gyration equal to 3m)
Rigid cap 'Yc = 2.5 g/cm' Piles EP = 2 x 10' kg/ em' 3 'Yp = 2.5 g/ cm
[ 30 X 100 ro = 28.22 = 106.3 From Fig. 12.13a, for Epilo/G,oil = 667 and llr0 = 106.3
Other dimensions of the pile group are shown in Fig 12 22b The ff f vertical load of the static column load vibrating with th~ piie capem:cy tbve assumed to be 40 t. e
fw, = 0.03(\,,?·;'0:
Vertical vibrations 1.
Assume floating piles
k~ =
E;:
fw 2 = 0.065
(12.37)
fw, 5
=(2x10 x50x50) 36 o.o 28.22
Solution
= 6.378 x 105 kg/ em
Vertical vibrations
Shear wave velocity in the soil
v
1
cw=
ro;
=
1000 X 981 1.8 = 127.8 m/sec X
Compression wave velocity through the pile
v, =VE;TP;, =
.J2
X
10
5
1000 X 981 2.5
X
= 280 X 10 3 em/ sec = 2800m/sec
v; 127.8 V, = 2800 = 0.0456
(12.39)
'
2 X 105 X 50 X 50 127.8 X 100
' 'J-;: P,
= ~300
vEPA fw2 X
0 065 ·
= 2541 kg sec/em 2. To consider the group effect, assume any pile in the group as reference pile. With r0 = 28.22 em, the value of Sl2r 0 for adjacent piles is 3.54 (=200/56.44) and for the diagonally opposite pile is 5.01 (=283/56.44). From Fig. 12.14, ll2r0 = 3000/56.44 = ~.15 aa = 1 for reference pile a,= 0.57 for adjacent corner piles a,= 0.51 for the opposite corner pile :S a,= 1 + 2 X 0.57 + 0.51 = 2.65
Combined stiffness of piles
538
MACHINE FOUNDATIONS ON PILES n
Referring to Table 12.1, for v, ters are
.2: k~
F=-1,_
(12.42)
n
w
539
EXAMPLES
fxl
L aa 4 X 6.378 X 105 _ 2 65
=
=
Lc~ _I_
= {2 X 105 X 5 X 10 /((28.22) = 0.77 x 10 3 kg sec/em 2. Letting the departure angle (3 aL
(12.44a)
= =
!:> w
=
=
c•
3.83 X J0 3 + 8.6 X 10 3 1.243 x 10 4 kg sec/em
5
(4 X 0.105 X 106 ) + 2.2 = 1.91 X 10 kg/em
2: c!
=
(4 X 0.77 X 10 3 )/2.2 1.40 x 103 kg sec/em
3. For pile cap fkxG,hSxl = 240 X 200 X 4.1 = 1.97 X 10 5 kg/em
c~nl) k~(mJ4) = 2541 /2(6.378 x 10 x 147 /4) 0 5 = 0.26 5
V,
=
5
V,IV" = 0.03271,
(12.54)
c[=hr0 (YG,y,lg)Sx 2 (12.55) 3 0 ""'- = 200 X 225.6(240 X 2/981 X 10 ) ·'(10.6) 4 = 1.06 X 10 x kg sec/em 4. Total stiffness
Translation ~et the reduced values of G, be 240 kg/ cm 2 for the backfill and 154 kg/ em for the lean clay = 1298 154 /154 X 1000 X 981 'I LS
(12.53)
(.2: aL)
w
. The pile cap is seen to produce more damping than the piles. For a single p1le ·
2 X 10
(12.52)
()(.L
=
=
0 and from Fig. 12.17,
1.00 for reference pile and assuming flexible pile,
X
= 1.243 X 104/2(11.79 X 10 5 X 147)0 · 5 = 0.472
EP/G,oil
X 9.16 X 10 ]}(0,0563)
.2: k! 2:
=
=
3
F = --
= c~l2(k• m )II' w w- c
~~
=
2
= 0.45 for adjacent corner piles (S/2r0 = 3.54) aL = 0.30 for op65',iite corner pile (S/2r 0 = 5.01) L aL = 1.0 + 2 X 0.45 + 0.30 = 2.2
200 x 225.67 (/400X2) 'V 1000 X 981 6,7 = 0.86 x 104 kg sec/em Total (k!) = 9.63 X 105 + 2.16 X 105 5 = 11.79 x 10 kg/em Total (c!)
X 0,0237
aL
(12.44b)
4.
]
(12.47) 5
k~ = G,hS1 = 400 X 200 X 2.7 = 2.16 X 105 kg/em r 0 (cap) = ( 400 x 400/7r) 0 ' 5 = 225.67 em
=
(2 X 105 X 5 X 10 /(28.22)
3
c! = (EPIP/r~V,)fxz
(12.43)
n I
3,
(12.46) 5
6
4 X 2541/(2.65) 3.83 X 103 kg sec/em
=
0.0563
= 0.105 x 10 kg/em
L aa =
=
k! = (EPIP!ri)fxl
9,63 x 105 kg/em
n
w
fxz
0.0237 ,
=
the stiffness and damping parame-
1. For a single pile
I
cc=
= 0.4,
f
(k~)
= 1.91 X 105 + 1.97 X 105 = 3.88 X 10 5 kg/em
Total damping
= 9.16 x 10 3 em/sec
(c!) = 0.140 X 10 4 + 1.06 X 10
pJpp = 1.8/2.5 = 0.72
=
J
1.20 x 10 4 kg sec/ em
4
540
MACHINE FOUNDATIONS ON PILES
Damping factor (; may be computed as 8
(;
c 2\jk'fmc X
=
Sx 1 = 4.1 ,
+ c1
f
= 1.2 X 10 4 7 2(3.88 X 10 5 X 147) 0 5 = 0.794 The pile cap contributes a very large share of the damping in the whole system. Rocking motion
1. Stiffness and geometric damping constants for single piles.
k~ = (EP9ro)f.;,
(12.48)
5
S.; 2 = 1.8
5
= [2 X 10 x 5 X 10 /127.8
X
(12.49) 6 100](0.26) = 2.06 x 10 kg em sec/rad
X
2
2
z
-
(12.68)
2 2 cq,f = 8r 0··~ y G,-y,/g{Sq, 2 + [8 /3 + (Z,Ir 0 ) - 8(Z)r 0 )]Sx 2 } 05 4 = 0.8865 X (225.6) (240 X 2/981 X 1000) (1.8 + 0.88) = 1.36 x 10 8 kg em sec/rad
(12.69)
Total stiffness k! = (469.1 + 115.60)10 = 584.7 x 10 8 kg cm/rad 8
k~.; = (EPJP/r~)fxq,, 5
z-
kq, = G,r 0 hS 1 + G,[Q.WB /3) + (Z)r 0 ) - 8(Z)r0 )]Sx 1 2 = 400(225.6) 2 X 200(2.5 + ((0.8865) 2/3 + (130/225.6) - 0.8865(130/225.6)](4.1)} = 40.7 x 108 (2.5 + 0.34) = 115.6 x 108 kg cm/rad
8
c! = (EPIP!V,)f.;2 5 10
Sq, 1 = 2.5 ,
4. Total stiffness and damping values are:
5
= [2 x 10 x 5 x 10 /28.22](0.37) = 13.4 x 10 8 kg cm/rad
X
Sx 2 = 10.6,
Also, the height of the center of gravity of the dynamic force above base of pile cap: Z, = 130 em and coordinate x, = 100 em
X
= (2
541
COMPARISON OF MEASURED AND PREDICTED PILE RESPONSE
5
5 X 10 /(28.22) 2](-0.068) = -8.6 X
(12.50) 10 6 kg/rad
c!.; = (EPIPiroV,)fx.;, 5
(12.51) 5
= (2 X 10 X 5 X 10 /(28.22 = -2.76 x 10 4 kg sec/rad
X
127.8
X
Overall damping factor (;
!;! = c!i2V k!Mm ~--------~ 6 8 8
100))( -0.099)
Stiffness and damping parameters were obtained from Table 12.1,
[q,, = 0.37 fx
Total damping'c'f= (1.90 + 1.36)10 = 3.26 x 10 8 kg em sec/rad
= 3.26 X 10 12'1) (584.7 = 0.287
X
10
X
5.68 X 10
(12. 70)
)
The stiffness, damping, and masses have been established in the preceding computations. The response of the pile group may now be determined from principles of vibrations described in Chapter 2.
[.; 2 = 0.26 fx.;z = -0.099
2. Stiffness and damping due to pile group (only). k'
2: [k'1> + k' X w
8
2 '
+ kx1 Z c2 - 2Z,kxJ ' 5
= 4(13.4 X 10 + 6.378 X 10 X (100) 2 + 1.05 X 105 + 2 X 130 X 8.6 X 10 6] = 469.1 x 108 kg cm/rad Cg-
2:[ c1 + cwx, 1 2 1z2 +ex ' -
X
(12.66) (130) 2
1
.; 2Z,cxl (12.67) 6 2 = 4[2.06 X 10 + 2541 X (lOW+ 770 x (130) + 2 x 130 x 2.76 X
10 4)
= 1.90 x 108 kg em sec/rad 3. Stiffness and damping due to pile cap. 8 = hlr0 = 200/225.6 = 0.8865
Fr~m Tables 11.2 and 11.3, frequency-independent constants for the side resistance on the embedded pile cap are:
12.6 COMPARISON OF MEASURED AND PREDICTED PILE RESPONSE
Several dynamic tests on single as well as groups of piles have been -performed on small scale as well as full-sized piles to check if the predicted response tallied with the measured response (Gle, 1981; Novak and ElSharnouby, 1984; No~ak and Griggs, 1976; Woods, 1984, and Ting, 1987). Fifty-five steady-state lateral vibration tests were performed on 11 pipe piles 14 in in outside diameter witb wall thickness of 0.188 .or 0.375 in at three sites in Southeastern Michigan (Woods, 1984). The piles were end bearing and 50 to 160ft in length. Figure 12.23 shows response curves for the pile GP 13-7, 157ft long in soft clay. All piles were excited in steady-state oscillation using an eccentric weight vibrator (Lazan oscillator) attached to the head of the pile, and their response was monitored by two velocity transducers and recorded on a strip
542
MACHINE FOUNDATIONS ON PILES
COMPARISON OF MEASURED AND PREDICTED PILE RESPONSE
Response curves Site: Belle River
Pile: GP 13-7
w-6
Lazan(fl): 2.5-15 deg
Comment: W/K-KRETE ~
....E' -1l
"'.1' w-s
~-·-~v = =
9
-
'b
-
Frequency (Hz)
Figur; 12.2_3. R~sponse curves show a decrease in resonant frequency with increasing a~pl~tudes m horrzontal vibrations. (From R. D. Woods, Lateral interaction between soil and prle, m E;, Beskos, Theodore Kranthammer, and I. Vardoulakis, eds., "Dynamic Soil-Structure
'?·
Interaction,
1984, A. A. Balkema, Rotterdam.)
chart recorder. At the conclusion of the first steady-state test, the eccen. tnc1ty of the Lazan oscillation was increased to increase the oscillating force and the test was repeated. To cover the range of lateral displacements covered by most machine foundation, four or five increasing eccentricities were used. It was observed that the frequency of maximum response decreased as the force level increased, indicating nonlinear response. Woods (1984) used PILAY computer program to determine stiffness and damping elements (Novak and About-Ella, 1977). PILAY is a continuum model accommodating a multilayered soil based on the elastic -side layer approach of B~rano: (1967). However, PILAY as~umes that the soil surrounding the pile m a giVen layer Is the same at all distances from the pile. A dy_nam1c response curve with PILAY solutions is shown in Fig. 12.24 along w1th the field data. The poor correlations between predicted curve and measured response is obvious. In all tests, computed response based on stiffness and damping from PILAY and measured response showed that the amplitudes of motion were greater than predicted and the frequency of mrunmum response was lower than predicted.
In an attempt to match the measured response with the computed response, two approaches were adopted.
8 %8
= ;:::
4 10-5
~ ~5--~~I0~~-1~5~--,2~0,---~2~5----,3~0----~35~---4~0~--~4~5~--~5h0----5J.5
S
200F solution
-
rrr~
_ 9
f?
3
0.
Dynalic respLe
/
~ v
15
:: --
Field data 0
----. X
~
-
l, 9 b
~
r::::::- I--
Dynamic response predicted with PI LAY solution
,/. / 10
543
=
:
-
20
25
30
35
40
45
50
55
Frequency (Hz)
Figure 12.24. 'Typical response curves -p,r.edicted by PI LAY superimposed on measured pile response. (After Gle and Woods, 1984.) ·
1. Only a fraction of the rocking and translation stiffness computed by PILAY had been used in predicting the response. It was seen that even with wide variation in rocking stiffness, the observed amplitudes in the frequency range just above the horizontal translation peak is still higher than predicted. The observed increase is more likely due to change in soil parameters caused by pile driving. A better handle on the disturbed soil zone had been obtained by replacing the original soil with sand fill around the pile up to 4ft in depth. 2. Because of the poor correlation achieved in the initial attempt, a second correlation with the analytical procedure, PILAY 2, was attempted. PILAY 2 permits an inclusion of a "softened" or "weakened" zone surrounding the pile, simulating the disturbance to the soil caused by pile installation (Novak et al. 1981). A good match of the measured and predicted response could be obtained by a considerably reduced soil modulus in the softer zone (one-tenth to two-tenths of the oftginal value) and the extent of the softened zone (one-half to one times the pile radius). A loss of contact of the soil with pile for a short length close to the ground surface also improved the predicted response. El-Sharnouby and Novak (1984) performed dynamic tests on a 102 steel pipe piles group. The length of the piles was 106 em, with outside diameter of 26.7 mm and inside diameter of 20.93 mm. The slenderness ratio of piles was greater than 40 and the pile spacing about three diameters. The pile
544
MACHINE FOUNDATIONS ON PILES
group was placed in a hole made for its placement in the ground and then backfilled with a specially prepared soil mixture. The pile cap was 6 em above the ground level. The pile group was excited by a Lazan oscillator at frequencies of 6-60Hz in the vertical and horizontal directions and in torsional mode. Free vibration tests and static tests had also been performed. The measured response curves were very linear for small amplitudes and indicated relatively small nonlinearity at amplitudes of 0.2 mm. The test results of Gle (1981) and Woods (1984) show definitely nonlinear behavior of insitu piles. Novak and El-Sharnouby (1984) analyzed the data presented earlier by the following methods: I. Using static interaction factors by Poulos (1971, 1975, 1979) and
Poulos and Davis (1980). 2. Concept of equivalent piers. 3. Using dynamic interaction factors by Kaynia and Kausel (1982). 4. Direct dynamic analysis of Waas and Hartmann (1981). Analysis with Static Interaction Factors
Using Poulos' charts the theoretical response curves based on static interaction factors are shown, together with the experimental curve, in Fig. 12.25. Three theoretical curves are plotted against the experimental one: Curve A represents the group response without any interaction effect, curve B was calculated using a static interaction factor for both stiffness and damping,
COMPARISON OF MEASURED AND PREDICTED PILE RESPONSE
545
(~ a. = 23) while the interaction effect has been considered for the stiffness only in curve C. It is seen that the full value of static interaction coefficient cannot be applied for damping. A much lower value of interaction factor for damping is needed in order to obtain a better estimate of response of the group. A theoretical response curve with interaction factors of 40 and 2.4 for stiffness and damping constants respectively is shown in Fig. 12.26. However, Fig. 12.26 shows that even if the match between the theoretical and experimental curves is achieved near the peak, the experimental dimensionless response curve at higher frequencies approaches about 0.5 instead of unity. This may indicate that the apparent vibrating mass may differ from the true mass of the foundation. The apparent vibrating mass determined from the experimental response curve using the technique described by Novak (1971) was found to be about 2.3 times the foundation mass and about 1.3 times the total mass, which comprises the mass of the foundation and the mass of the piles with the soil enclosed between them. With the apparent vibrating mass used instead of the foundation mass, better fit was, obtained (Fig. 12.2].)"A still better fit had been obtained with interaction factors of 16 and 2.4 'Ior stiffness and damping, respectively. Since arbitrary correction factors are applied to stiffness and damping constants, the static interaction factors may not be used for dynamic analysis.
Equivalent Pier Method
The theoretical analysis of the test foundation indicated that the vertical motion of pile tips is almost the same as the motion of the heads. (The
Group of 102 piles .20 plates Vertical excitation
B Symbol 0
N-M 0.1932
...
0.0966
N-M 0.1932 ... 0.0966
Symbol 0
E
~eetle~e
r ill~~~~~~c~~~::~~J O~ ~ 10
20
30
40
50
A 60
Frequency (Hz)
Figure 12.25. Experimental response curves and theoretical curves calculated with static interaction factors: A, no interactions; B, interaction factors applied to both stiffness and
damping; C, interaction factors applied to stiffness only; and f, experimental data (After Novak and EI-Sharnouby, 1984.)
Frequency (Hz)
Figure 12.26. Experimental reSponse curve and theoretical curve with interaction factors of 40 and 2.4 for stiffness and damping, respectively. (After Novak and EI~Sharnouby, 1984).
546
MACHINE FOUNDATIONS ON PILES
547
FINAL COMMENTS
Group of 102 piles .20 plates Vertical excitation
-1l
..§
c_ E
N-M 0.1932 • 0.0966
Symbol
•~
0
.'!' 0 0
__ .... ____ _
·~
0~~~~~~~
E
i5
o_~~~~~~lllluy~~~~~~Ull~ 0
10
20
30
40
o~~~~~~Ullllll~~~llllilli~~~~
50
0
10
Frequency (Hz)
30
40
50
60
Frequency (Hz)
Figure 12.27. Experimental response curve and theoretical curves calculated with apparent mass: A, with Poulos static interaction factors for stiffness and interaction factor of 2 for damping; and B, with interaction factors of 16 and 2.4 for stiffness and damping, respectively (After Novak and EI-Sharnouby, 1984.)
20
,
difference is less than 2 percent suggesting a pier action.) The added mass effect, indicated by the factor of 1.3, appears to result from the high rigidity of the piles and may not be needed for more compressible piles, especially when they are end-bearing. The equivalent pier properties for the· foundation test model were established using the material characteristics of both the steel piles and the soil enclosed. The behavior of the layers below the pier tip was considered up to a depth of 3 times the pier length. The stiffness and damping constants of the equivalent pier were calculated using the same concept used for the single pile and the computer program PILAY2. The dimensionless vertical response curve based on the equivalent pier approach is plotted in Fig. 12.28. It can be seen that both damping and stiffness were moderately overestimated (curve A). A better match of the theoretical curve with the experimental one was achieved when both stiffness and damping were modified by factors of 0.5 and 0.6, respectively (Fig. 12.28, curve B). The concept of equivalent pier may be applicable only to closely spaced piles. Novak and El-Sharnouby (1984) have also described comparison of the theoretical and measured vertical response by the dynamic methods of Kaynia and Kausel (1982) and Wass and Hartmann (1981). Also, a comparison of the theoretical and measured response both in horizontal and torsioual modes by several methods have been presented by the authors. The preceding discussion points to the fact that dynamic interaction is very complicated, and further theoretical and experimental research is needed in the dynamic behavior of piles and pile groups.
Figure 12.28. Experimental response curve and theoretical curve based on equivalent pier concept: A, true stiffness and damping: and B, modified stiffness and damping. (After Novak and EI-Sharno~.by, 1984).
12.7
FINAL COMMENTS
Machine foundation (block) may be supported on piles especially if the bearing capacity of soils at shallow depths is poor. Depending upon the nature of exciting forces, the pile-foundation may be subjected to (1) vertical oscillations, (2) horizontal translation and rocking, and (3) torsion. Simple solutions for single piles in all the preceding modes of vibrations have been included in this chapter. Also the effect of group action on the behavior of the total system as compared to that of the single pile has been included. An effort has been made to present as complete an analysis as possible, but there are certain definite gaps in the present (1988) understanding of "single pile" and "pile group" action under vibrations. Initial analyses by Barkan (1962) and Maxwell et a!. (1969) have been shown to have only limited application. For vertically vibrtting piles, Novak's (1977) analysis for single piles is reasonable and uses rational soil and pile properties. However, in the case of groups, recourse has been taken to group effect, as in the case of static loads (Novak and Grigg, 1976). Sheta and Novak (1982) developed an approximate theory for vertical vibrations of pile groups. This has been discussed in Section 12.1.2. Nogami (1983) and Nogami and Liang (1983) have also obtained solutions for pile groups in vertical vibrations and have shown that the concept of the Winkler soil model could be applicable to pile groups problems for frequency range higher than the fundamental natural frequency of the soil deposit. It was further found that:
548
MACHINE FOUNDATIONS ON PILES
1. A dynamic group effect can be strongly frequency dependent, and depends upon the ratio between the pile spacing and the wavelength propagating in the soil. This is due to the phase shift between the directly induced pile motion and the transmitted motion. Thus, the frequency-dependent behavior of pile groups is controlled by the type of predominant waves induced in the soil, frequency, and pile spacing. The effect of material damping of the soil is primarily a reduction of the amplitude of the motion. 2. A dynamic group effect is more pronounced in pile groups with stiffer piles and with a larger number of piles. 3. Under the dynamic load, the group effect may increase or decrease the values of the stiffness and damping parameters per pile in a group compared to the values for a single pile, though it always decreases the stiffness value under a static load.
On the basis of comparison of predicted and measured response of 102 pile group in vertical vibrations, Novak and El-Sharnouby (1984) concluded · that: 1. Correction for the apparent mass may be necessary, particularly for rigid floating piles. 2. The static interaction factor provided quite a good estimate of the group stiffness but the group damping could not be predicted. 3. The equivalent pier concept provided a reasonable agreement with the experimental data if the theoretical damping constant was reduced by about 50 percent.
Novak and El-Sharnouby (1983) presented solutions for soil modulus variation with depth and effect of frequency on the group action. The solutions are by no means simple in their present form. Therefore, more research is needed to solve the problem completely and put it in a form which can be easily used by the practicing engineer. Horizontal vibrations of piles have been investigated by considering the piles as an equivalent cantilever, a beam on elastic foundation (Tucker, 1964; Prakash, 1981), and also by an approximate method developed by Novak (1974). Equivalent cantilever method does not consider realistic soil-pile behavior. The solutions of beam on elastic foundations need be developed further to put them in readily usable forms. Novak's solution for single pile and that for pile groups for horizontal vibrations is subject to the same limitations as for vertical vibrations. Also, the static group effect differed considerably from dynamic group effect in horizontal vibrations in tests of Novak and El-Sharnouby (1984 ). However, the equivalent pier concept predicted the stiffness well, but not the damping in that particular case.
FINAL COMMENTS
549
For single piles also, Woods (1984) found that softened zone around the pile alters the behavior and needs to be considered in a realistic analysis. For torsional vibrations of vertical piles, Novak and Howell's (1977) solution is a good tool. The dynamic stiffness and damping in torsion depend on soil-pile interaction in terms of dimensionless parameters (1) shear wave velocity ratio (ratio of soil shear wave velocity to pile shear wave velocity), (2) slenderness ratio (ratio of pile length to effective radius), (3) mass ratio (ratio of specific mass of the soil to specific mass of the pile), ( 4) dimensionless frequency, and (5) material damping ratio. For a group of piles, the contribution from torsion to the total stiffness and damping decreases with the relative distance of the pile from the centroid of the footing. Pile foundations can have smaller natural frequencies in torsion than shallow footings, but the increased damping generated yields lower resonance amplitudes. This contrasts with other vibration modes. Comparison with experiments is desirable since pile slippage, together with other effects such as method of installing the piles are not accounted fo,r in any theory, an~,Way affect the comput~d values. The approach throughout in thts:chapter has been to dtscuss sttffness and damping constants in terms of basic soil and pile properties and geometry of the system. Assuming the mass of the pile cap and the superstructure (machine) and knowing the unbalanced forces, natural frequencies and amplitudes of motion are determined from principles of vibration analysis given in Chapter 2. The soil has been considered to be isotropic, homogeneous, and elastic: Nogami (1980) has considered pile vibrations in nonhomogeneous soils. Layered soil has also been considered by Nogami (1983). Soil modulus variation as a quadratic parabola has been considered by Novak and El-Sharnouby (1983) for vertical and lateral vibrations. The interaction of pile cap with soil affects the dynamic response of the system. This can be accounted for in all modes of vibrations on the basis of principles defined in Chapter 11. Based on the approximate solutions presented in this chapter, a step-bystep design procedure has been listed. Solved problems have been included to illustrate the listed design procedure. The soil properties used in defining the stiffness and damping parameters are (1) shear wave velocity V, or shear modulus G and (2) Poisson's ratio. Their values are dete\'?ined from principles and procedures described in ·· Chapter 4. Another solution technique for pile groups has been reported by Aubry and Postel (1985) who considered the soil-pile system as a fiber reinforced composite material. The technique of homogenization of composite materials was used to compute equivalent modulus which was used to compute the seismic response of the equivalent foundation at the soil surface. The method has been shown to be particularly useful for very large numbers of piles beneath a foundation. This method may be regarded as a complimentary solution to Novak's equivalent pier concept.
ted on the pile cap. Pile response was measured through inductance type accelerometers and ink writing recorder. The oscillator could generate a force of 93 kg at 10 cps at full eccentricity. Tests were conducted at two different eccentricity values up to a maximum speed of 35 cps. Typical representative displacement amplitude versus excitation frequency curves for a single pile are plotted in Figs. 12.29a and b. Estimate the design parameters for use in Novak's analysis and list them properly. 2. Estimate the (a) stiffness, (b) damping, (c) natural frequency, and (d) amplitude of motion of an 8-pile group in vertical, horizontal, and torsional vibrations.
Gazetas and Dobry (1984) proposed a method to compute response of single fixed head pile under horizontal excitation at its head. The method consists of estimation of (1) deflections of the pile under static lateral load, (2) dashpots attached to the pile at selected elevations, (3) dashpot at its head, and ( 4) variation of spring coefficient and damping ratio with frequency. The applicability of the proposed method has been illustrated in three linearly hysteretic soil deposits: (a) homogeneous deposit with modulus constant with depth, (b) inhomogeneous deposit with modulus increasing linearly with depths, and (c) layered deposit. The philosophy and methods described in this chapter will need a change as the understanding of the dynamic pile behavior improves. There is an urgent need for more theoretical and experimental research.
10
PRACTICE PROBLEMS 1.
For design of a group of piles to support a turbogenerator, the following in situ tests were performed: (a) (b) (c) (d)
'E
Pile Resonance Tests. In all, four piles under two turbogenerator foundations were tested by exciting them into steady-state forced vibrations using a mechanical oscillator. The motor-oscillator assembly was moun-
8
.s 0 0
Cross Hole Wave Propagation test, Free horizontal vibration tests on piles, Vertical pile resonance tests, and Horizontal pile resonance tests.
The piles were 45 em diameter 20m long bored cast-in-situ R.C.C. piles. The soil at site consisted of clayey to fine sand with traces of silt and ground water table was at 2.0 m depth. The soil above water table had a moist density, y, of 1.70t/m3 with a saturated density, 'Y,., of 1.91 t/m 3• Poisson's ratio v of the soil may be taken as 0.35. The dynamic elastic modulus of pile material was obtained as 2.5 x 106 t/m 2 with a density of 2.4 t/m 3 The shear wave velocity had been measured as 125m/sec at a depth of 3 m below ground level in the cross bore hole test. The piles tested in the free vibration tests as well as the resonance tests were provided with a pile cap 0.7 m square and 0.5 m deep made of the same grade of concrete as the piles. For free vibration tests, one pile was pulled against another horizontally using a 10 t capacity chain-pulley block. Also a specially designed sudden-release clutch and a load cell were attached in series with the pulling device tomeasure the pull. Two piles tested under free vibrations yielded identical natural frequencies. From the test data, the natural frequency of pile with cap under horizontal vibration is obtained as 12.5 cps and the damping factor of 7 percent.
551
PRACTICE PROBLEMS
MACHINE FOUNDATIONS ON PILES
550
~ D
.,.
6
'l5
•
4
u
:E
" E
""
2
0 30
20
10
0
Frequency (cps) (a)
0.6
'
'E
.s 0
0
~ D
.,.
0.5 0.4 0.3
'l5
•
u
0
:E
0.2
""
0.1
"E 0 0
10
20 Frequency (cps) (b)
30
40
Figure 12.29. Resonance tests on 45-cm diameter piles: (a) horizontal 1 (b) vertical.
552
MACHINE FOUNDATIONS ON PILES
The exciting forces are vertical 8 sin 1007Tf (t) horizontal 4 sin 1007Tf (t) torsional 2 sin 1007Tf ( tm)
3.
The superimposed load is 50 !I pile. Assume suitable values of data needed but not supplied,· Check the stiffness and damping parameters of the pile group in Example 12.5.2 if the piles were end bearing.
REFERENCES Aubry, D., and Postel, M. (1985). Dynamic response of a large number of piles by homogenization. Proc. 2nd. Int. Conf. Soil Dyn. Earthquake Eng., Queen Elizabeth II, 4-105 to 4-119. Barkan, D. D. (1962). "Dynamics of Bases and Foundations." McGraw-Hill, New York. Baranov, V. A. (1967). On the calculation of excited vibrations of an embedded foundation (in Russian). Vopr. Dyn. Prochn. 14, 195~209. Beredugo, Y. 0., and Novak, M. (1972). Coupled horizontal and rocking yibration of embedded footings. Can. Geotech, J. 9(4), 477-497.
Bycroft, G N. (1956). Forced vibration of a rigid circular plate on a semi-infinite elastic half space on elastic stratum. Philos. Trans. R. Soc. London, Ser. A 248, 327-368. El~Sharnouby, B., and Novak, M. (1984). Dynamic experiments with groups of pile. I. Geotech. Eng. Div., Am. Soc. Civ. Eng. 110(GT~6), 719-737. Gazetas, G., and Dobry, R. (1984). Horizontal response of piles in layered soils. I. Geotech. Eng. Div., Am. Soc. Civ. Eng. UO(No. GT-1), 20-40. · Gle, D. R. (1981). The dynamic lateral response of deep foundations, Ph.D. Dissertation, University of Michigan, Ann Arbor. Gle, D. R., and Woods, R. D. (1984). Predicted versus observed dynamic lateral response of pipe piles. Pap. presented to World Conf. Earthquake Eng., 8th, San Francisco. Kaynia, A. M., and Kausel, F. (1982). Dynamic behavior of pile groups. Int. Conf. Numer. Methods Offshore Piling, Austin, TX, pp. 509-532. Madhav, M. R., and Rao, N. S. V. K. (1971). Model for machine pile foundation soil system. J. Soil Mech. Found. Eng. Div. Am. Soc. Civ. Eng. 97(SM-1), 295-299. Maxwell, A. A., Fry, Z. B., and Poplin, J. K. (1969). Vibratory loading of pile foundations. ASTM Spec. Tech. Pub!. STP 444, 338-361. Nogami, T. {1980). Dynamic stiffness and damping of pile groups in inhomogeneous soil. Proc. Dyn. Response Pile Found Anal. Aspects, Am. Soc. Civ. Eng., Hollywood FL 1980, 31-52. Nogami, T. {1983). Dynamic group effect in axial responses of gro.uped piles. J. Geotech. Eng., Am. Soc. Civ. Eng. 109(No. GT-2), 220-223. Nogami, T., and Liang, H. (1983). Behavior of pile groups subjected to dynamic loads. Proc. Can. Conf. Eq. Eng., 4th, Vancouver B.C. 414-420. Novak, M. {1971). Data reduction from non~linear response curves. J. Eng. Mech. Div., Am. Soc. Civ. Eng. 97(EM~4),_ 1187-1204.
Novak, M. (1974). Dynamic stiffness and damping of piles. Can. Geotech. J. 11(4), 574-598. Novak, M. (1977). Vertical vibration of floating piles. I. Eng. Mech. Div., Am. Soc. Civ. Eng. 103(EM-1), 153-168.
REFERENCES
553
Novak, M., and Aboul~Ella, F. (1977). "PILAY-A Computer Program for Calculation of Stiffness and Damping of Piles in Layered Media," Rep. No. SACDA 77~30. University of Western Ontario, London, Ontario, Canada. Novak, M., and Beredugo, Y. 0. (1972). Vertical vibration of embedded footings. J. Soil Mech. Found. Div., Am. Soc. Civ. Eng. 98(SM~12), 1291-1310. Novak, M., and El-Sharnouby, B. (1983). Stiffness and damping constants of single piles. J. Geo{ech. Eng. Div., Am. Soc. Civ. Eng. 109(GT~7), 961-974. Novak, M., and El-Sharnouby, B. (1984). Evaluation of dynamic experiments on pile group. I. Geotech. Eng. Div., Am. Soc. Civ. Eng. 110 {GT-6), 738-756. Novak, M., and Grigg, R. F. (1976). Dynamic experiments with small pile foundation. Can. Geotech. J. 13(4), 372-395. Novak, M., and Howell, J. F. (1977). Torsional vibrations of pile foundations. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. 103(GT-4), 271-285. Novak, M., and Sachs, K. (1973). Torsional and coupled vibrations of embedded footings. Int. I. Earthquake Eng. Struct. Dyn. 2(1), 11-33. Novak, M, Aboula~Ella, F. and M. Sheta (1981). "PILAY 2-A Computer Program for Calculation of Stiffness and Damping of Piles in Layered Media," Rep. No. SACDA 81-100. University of Western Ontario, London, Ontario, Canada. PoulOs, H. G. (1968). Analysis of the settlement of the pile groups. Geotechnique 18(4), 449-471. ,,,,,,~ Poulos, H. G. (1971). Behavior of·laterally 1~aded piles. II Pile groups. J. Soil Mech. Div., Am. Soc. Civ. Eng. 97(SM-5), 733-751. Poulos, H. G. (1975). Lateral load deflection prediction for pile groups. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. lOl(GT-1), 19-34. Poulos, H. G. (1979). Group factors for pile~deflection estimation. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. lOS(GT-12), 1489-1509. Poulos, H. G., and Davis, E. H. (1980). "Pile Foundation Analysis and Design," Wiley, New York. Prakash, S. (1981). "Soil Dynamics." McGraw~Hill, New York. Puri, V. K., Bhargava, S., Nandakumaran, P., and Arya, A. S. (1977). Evaluation of dynamic soil-pile constants from in-situ tests. Int. Symp. Soil-Struct. Interact., ·Roorkee, India, 349-354. Richart, F. E., Jr. (1962). Foundation vibrations. Trans. Am. Soc. Civ. Eng. 127, Pt. 1, 863-898. Richart, F. E., and Whitman, R. V. (1967). Comparison of footing vibration tests with theory. J. Soil Mech. Div., Am. Soc. Civ. Eng. 93(SM-6), 143-168. Richart, F. E., Hall, J. R., and Woods, R. D. (1970). "Vibrations of Soils and Foundations." Prentice-Hall, Englewood Cliffs, New Jersey. Sheta, M., and Novak, M. (1982). Vertical vibrations of pile groups. I. Geotech. Eng. Div., Am. Soc. Civ. Eng. lOS(GT-4), 570-590. Ting, J. M. (1987). Full scale'YJ\rnamic lateral pile response. J. Geotech. Eng. Div., Am. Soc. Civ. Eng. 113(1), 30-45. Tucker, R. L. (1964). Lateral analysis of piles with dynamic behaviour. Proc. Conf. Deep Found. Mexico City, Vol. 1., 157-171. Waas, G., and Hartmann, H. G. (1981). Pile foundations subjected to dynamic horizontal loads. Eur. Simul. Meet., Model. Simul. Large Scale Struct. Syst., Capri, Italy, p. 17; also Conf. Struct. Mech. ReactOr Tech. SMIRT, Paris. Woods, R. D. (1984). Lateral interaction between soil and pile. Proc. Int. Symp. Dyn. Soil Struct. Interact., Minneapolis, 47-54.
13 Case Histories
Long-term satisfactory performance of a machine foundation depends upon the interaction of the machine, foundation, and the supporting soil system. It is of utmost importance that a machine should give trouble-free service for a long time. There have been many occasions when a plant had to be operated at less than its rated capacity because performance ofits machine foundations was not up to the required standards. It should be stressed here that the poor performance of a machine may be due to: 1. Machine-related parameters, such as faulty design, installation and
commissioning, and inadequate information about machine unbalances, 2. Foundation-related parameters, such as faulty design or construction, and 3. Soil-related parameters, such as inadequate geotechnical information that leads to unrealistic soil data used in estimating foundation response. Malfunctioning of any installation that is due to foundation-related problems can mostly be avoided by conducting well-planned geotechnical site investigations and using realistic soil properties in the design of the foundation. The analytical theories for computing the response of different types of foundations for reciprocating machines, hammers, and turbogenerators have been discussed in chapters 6-8, and embedded and pilesupported foundations were discussed in Chapters 11 and 12, respectively. These methods should be expected to yield reasonable results only if all relevant machine data has been procured and soil data has been rationally interpreted for determination of dynamic soil properties. It is a matter of common observation that once machines are commis-
sioned, not much attention is given to the performance of their foundations 554
CASE HISTORIES
555
unless some unsatisfactory behavior is noted or there is some distress to the foundations. The authors are of the opinion that a periodic check on performance will not only help one to obtain a better understanding of foundation behavior, but will also provide an insight into the source of potential trouble. Because analyses of machine foundations are based on simplifying assumptions, an evaluation of a machine foundation's performance will enable its designer to identify and correct the inadequacies of design which may subsequently result in the development of more rational design procedures. The opportunities for comparing performance of prototype machine foundations with predicted performance are rare. Such observations are helpful in more than one respect, as shown in the following examples. Richart and Woods (1982) made vibration measurements on a pilesupported auto shredder foundation under operating conditions and observed vertical vibrations at a frequency of 48Hz. According to the data supplied by the shredder manufacturer, the expected vibration should have been in the rocking mode at about 12Hz. This shows a need to improve the methods of 111easuring or estimatfp~ impact forces. Madshus et al. (1985) monitored the performance of two undertuned compressor foundations during test runs and observed unexpected and unsatisfactory behavior. Careful observations revealed that the design assumptions had not been realized in construction of the foundation. Improper clearances and joints that had got filled with concrete had resulted in undesirable mechanical contacts between different units of the foundation. Corrective measures were taken to establish the assumed design conditions and subsequent observations indicated a qualitatively good agreement between the design and performance of the foundations. The observations highlighted the fact that the construction aspects play a very important role in the performance of the machine foundation. Prakash and Kumar (1984) and Kumar et al. (1985) reported the results of field observations on block-type compressor foundations located in poor subsoil conditions. During the first two years following the construction, the block foundations settled up to 60 mm, showed excessive tilt, and posed serious misalignment problems for the compressors. The stability of the foundation and even the strength of the soil was in doubt. A series of tests under static and dynamic loading conditions were planned to assess the future performance of t)Je foundation and to provide remedial measures if necessary. The results rndicated that the settlements had practically stabilized and only 4.0 mm of settlement had occured in the last 12 months. This was further confirmed by preloading tests on one of the foundation blocks which, on being preloaded to 44 percent over the normal working loads, showed a negligible settlement of 0.4 mm. Test runs were then obtained with the compressors running at "no load" and at "full load." A tilt of 1 in 6000 was noted during the first few runs at no load. However, subsequent runs with "no load" and "full load" operating conditions did not result in
556
CASE HISTORIES
further tilt or settlement. The vibration amplitudes were well below the permissible values as specified by the machine manufacturers. These observations on performance of the foundation provided answers to a very crucial question on the long-term stability of the machine foundation in question. The preceding cases clearly bring out the importance of monitoring the performance of a machine foundation. In general, the investigations for studying the response of a machine foundation can be divided into two types: 1. Postdesign, in which the amplitudes of vibration under normal operat-
ing conditions are observed to exceed permissible values thereby necessitating a critical evaluation and possible alteration or redesign of the foundation, and 2. Postmonitoring, in which the vibration response of the foundation is monitored after commissioning, but before it is put into operation so as to ensure uninterrupted, satisfactory, long-time performance.
557
CASE HISTORY OF A COMPRESSOR FOUNDATION
in strain levels for these two conditions had a significant effect on the relevant soil properties. The methods used to analyze the above two conditions were the elastic half-space and the linear spring method. For each case the computed amplitudes and frequencies of the foundation were compared with the observed amplitudes and frequencies. . The purpose of the evaluation was to determme whether remedtal measures were to be taken to reduce the foundation's vibration amplitudes to permissible levels or to redesign the foundation completely. Machine and Foundation Data
The plan of the foundation is shown in Fig. 13.1a, and its cross section in Fig. 13.1b. The reference axes are also indicated in these figures.
N
Two cases histories that illustrate these two types are presented below. The first concerns the study of a reciprocating compressor foundation (Prakash and Puri, 1981, 1984), which, in vibrating beyond its permissible limits, endangered the stability of an entire system. The second involves the vibration response of a hammer foundation (Prakash and Gupta, 1970).
=
location of obse!Vation points ;';
y
lk. of compressor
c 13.1
CASE HISTORY OF A COMPRESSOR FOUNDATION
General
A four-stage, reciprocating air compressor, which had been installed in an industrial plant, was supported on a concrete block foundation that had been constructed to conform to dimensions suggested by the supplier of the machine. When the machine was placed in operation, the foundation vibrated excessively. The amplitudes of vibration at the operating speed were measured, and the natural frequency was monitored by conducting free vibration tests. The dynamic properties of the supporting soil were determined using in situ tests. The design of the foundation was checked to determine one of two possible conditions; 1. Postdesign. The foundation was designed before· the monitored performance of the machine was known, and 2. Postmonitoring. It was designed with knowledge of the machine's monitored performance.
The two sets of computation proved to be similar except that the shear strain levels in the soil for the two conditions were different. The difference
23~ !50 12 490
+
490
230
23~ 0 ~c~~~ 50
~ ~ 230
50
85o
85o
'b I • •I
150
230
(a)
Figure 13.1.
Layout details of the compressor foundation: (a) plan; (b) section.
558
CASE HISTORIES
z
y
CASE HISTORY OF A COMPRESSOR FOUNDATION
559
Soil Data
~
of Motor pulley
In situ tests consisting of block vibration, cyclic plate load, and standard penetration tests were conducted at the site (Prakash and Puri, 1984). The block vibration and cyclic plate load tests were conducted at a depth of 4.0 m below ground level. The standard penetration tests were conducted up to a depth of 14.0 m below ground level. A detailed discussion about the test data and its interpretation is given in Example 4.9.3 (Chapter 4) and is not repeated here. The variation of dynamic shear modulus vs. shear strain at the base of the foundation at a depth of 2.4 m is shown in Fig. 4.47, plot C. The value of G in Fig. 4.47 corresponds to a mean effective confining pressure 170 of 1.0 kg/ cm 2 Observations
Accelerometers were used to measure the vertical and horizontal amplitudes of vibration at 14 points on the foundation (Fig. 13.1a). It was found that the maximum amplitude of vibration in the Z direction was 0.1085 mm. The maximum amplitude of horizontal ·¥i~ation at the top of the block in the Y direction was 0.3156 mm. The foundation was excited in free vibrations in the X direction. The natural frequency of the free vibrations was observed to be 17.5 Hz Lean concrete (b)
Figure 13.1.
(Continued).
Machine
Operating speed = 405 rpm Weight of compressor and motor= 11.0 t Horizontal unbalanced force =0 Vertical unbalanced force, P, = 0.205 t Horizontal moment, M, · = 0.185 t m Vertical moment, Mx = 2.2 t m Permissible vibration amplitude= 0.025 mm (peak-to-peak) Foundation
Area A =7.103m 2 Weight W = 49.79 t Depth D = 2.4 m
Check of the Foundation's Design
The response of the foundation was computed for the two conditions of investigations as defined earlier. Postdesign. This analysis was conducted as though the foundation were being designed for the first time to ensure vibration amplitudes smaller than their permissible values. The size of the foundation is shown in Fig. 13.1, and dynamic soil properties were obtained from Fig. 4.47, plot C, as follows:
Maximum permissible amplitude= 0.0125 mm Average width of the foundation (Fig. 13.1a)
=
2924; 1285 =
. mm 2104 5
"':
. .. . 0.0125 10-6 . = 5.94 X Shear stram amphtude, 'Yo= 2104 5 6
From Fig. 4.47, plot C, the value of G at 'Yo= 5.94 X 10- and (j0 = 1.0 kg/ cm 2 is 1050 kg/ cm 2• The mean effective confining pressure at a depth equal to one-half of the width of the foundation is 0.593 kg/cm 2• The value of G for associated shear strain levels and confining pressure is given by (4.15)
560
CASE HISTORIES
CASE HISTORY OF A COMPRESSOR FOUNDATION
The value of G thus computed is 808.5 kg/ cm 2 The value of C" is then computed as C = 1.13 X 2G 1 (4.29) " (1- v) v'A
4. Mass ratio B,
The value of C" = 10.25 kg/em' for area of the compressor (v = 0.33).
5. Damping ratio
Postmonitoring. The soil properties in this investigation were computed according to the shear strain actually induced in the soil. This shear strain was calculated from observed amplitudes. The value of the shear strain 'Yo 4 . was 1.49 x 10- . The values of G and C" were computed and were found to 2 2 he G = 400 kg/cm and C" = 5.07 kg/cm for area of the compressor. The procedure for designing foundations for reciprocating machines was discussed in Chapter 6 (Section 6.8). Based on this discussion, the dynamic response of the foundation is estimated below hy using the elastic half-space method and the linear spring method.
1- v m 4 pro
B=----, z
561
(6.17)
B, = 1.190 ~'
0.425
~.
= VB;
~'
= 0.3899
(6.22)
6. Amplitude of vertical vibration A, (6.23)
A =0.0031mm
Elastic Half-Space Method
..::.;.•:.:t;
z
Case 1. Postdesign: the response is calculated as though the foundation were being designed for a limiting amplitude of 0.0125 mm, the induced shear strain level of 5.94 X 10- 6, and G = 808.5 kg/cm 2 Appropriate equations derived in Chapter 6 are used to interpret the behavior of the machine's foundation. ·
Sliding Vibrations
7. Spring stiffness 32(1- v)Gr0 , 7 -8v k, = 59890.1 tim
k
Vertical Vibration
Y
=
(6.26)
8. Limiting natural frequency in sliding
1. Equivalent radius r0 (6.58a)
Wny =
r 0 = 1.5036 m
wny
2. Spring stiffness 4Gr k = - -0 z l-v
{§,
(6.30)
= 108.62/sec
fuy
= 17.2 Hz
=
7-8v m -,32(1-v) pr 0,
9. Mass ratio B, (6.18)
k, = 72711.4 t/m
B
'
(6.25)
BY= 1.4513 3. Vertical natural frequency 10. Damping ratio in sliding, y (6.20)
wn, = 119.69/sec fn• = 19.05 Hz
g,= ~y
0.285
VB;
= 0.239
(6.28)
562
CASE HISTORIES
563
CASE HISTORY OF A COMPRESSOR FOUNDATION
Rocking Vibrations
w~ 1 = 43678.4/sec'
11. Equivalent radius
"'nt
fnt = 33.26 Hz
=(41)114 71"
roq,
l=3.93m
= 208.9/sec
17. Amplitudes in combined rocking and sliding,
4
The damped amplitudes in sliding and rocking occasioned by the exciting moment MY are, respectively,
roq, = 1.4956 m
12. Spring stiffness, k"'
MY AY =
8Gr~"'
= 3(""1---v7) 7
k"'
(6.34a) '
A
kq, = 107851.2 t m/rad.
13. Limiting natural frequency in rocking
"'""' =
A;
V(w~y)
2
+ (2/;x Wny) 2
(6.55a)
Mm!J.(w) 2
_ MYV (w~Y- w ) + (21;,wnyw) •Mm!J.(w) 2 2 2
2
(6.55b)
in which
fiF-
(6.37)
mO
"'no = 72.35 sec - l fn> = 11.51 Hz
14. Inertia ratio B •
(6.56) B
=
3(1- v) _1_ 8 5 pr O
(6.32)
Ay = 0.0071 mm A 1 = 3.47 x 10- 5 rad
s. = 3.293
Ah=A;=A,+hA•
0.15
15.
t;
1
)Vl'f;,
(6.36)
= 0.0192
A;= 0.1091 mm 18. Maximum vertical amplitude
Combined Rocking and Sliding
2
n
(
Wny
(6.130a)
'"''A; = 0.06156 mm
2
+ Wnq.,) w 2 + '}'
A;
A v =A*= A z + (a/2)A z
16. Undamped natural frequencies w4 -
(6.130b)
2
2
wnyWnq,
n
w~ 2 = 4740.5/sec 2
'Y
=
0
(6.54a)
Torsional Vibrations
19. Equivalent radius r 0 •
"'"' = 68.85 /sec
roo~= (21,/71")114
fnz = 10.96 Hz
r,, = 1.606 m
(6.58c)
564
CASE HISTORIES
20. Spring stiffness k•
CASE HISTORY OF A COMPRESSOR FOUNDATION
565
Table 13.1. Computed Natural Frequencies and Amplitudes by Elastic Half-Space Method ( 6.40a)
Quantity Frequency or Amplitude
Postdesign Yo ~ 5.94 X 10- 6
Postmonitoring y0 ~ 1.49 X 10-'
Free Vibrations Y, ~ 1 X 10- 6
1
2
3
4
5
1 2 3 4 5 6
fnz Hz !,,Hz !,, Hz f, 1 Hz f, 2 Hz f,. Hz
21. Natural frequency of torsional vibrations w
, {7<;"
Wn.p
=
\jM
""
S. No. (6.41a)
m'
w,, ~ 152.68 sec_, !,.= 24.3 Hz 22. Inertia ratio B•
B
= •
Mmz 5 P'o.p
( 6.38)
s. = 3.4388 23. Damping ratio ~.
7 8 9 10 11 12
Azmm
AYmm A, rad
A;mm A;mm A• rad
19.05 17.28 11.51 33.26 10.96 24.30 0.0031 0.0071 3.47 x w-' 0.1091 O.Q§U6 1.5 x '10- 6
13.43 12.18 8.12 23.65 7.05 17.12
58.07 13.01
0.00686 0.523 2.42 X 10- 4 0.785 0.3031 2.396 X 10-'
(6.42) =
0.0635
24. Amplitude A, (daniped)
A,,= 1.5 X 10- 6 rad The values of the frequencies and amplitudes for different modes of vibration are listed in Table 13.1, column 3.
Case 2. Postmonitoring: The foundation response was also calculated using the values of dynamic shear modulus corresponding to the shear strain levels in !?e soil induced by the operation of·the compressor, i.e., Yo= 1.49 x 10 . The calculatwns for thts case are performed in the same manner 2 as for Case 1 by using G = 400 kg/ cm • The values of the natural frequencies and amplitudes for different modes of vibration are listed in Table 13.1, column 4. ~ computation w~s also performed to find the natural frequency of the honzontal free vtbra!ton. This was done by using the value of G at Yo = 10- 6.
The shear strain level of Yo~ 10- 6 corresponds to the free vibration condition. These values of computed natural frequency are shown in Table 13.1, column 5. The amplitudes of the horizontal and vertical vibrations computed by using the elastic half-space method for the post design-type investigation (Table 13.1, column 3) are 0.1091 and 0.06156mm, respectively. These amplitudes are several times greater than the permissible peak to peak amplitude of 0.025 mm. For the post monitoring-type investigation, the values of the computed horizontal and vertical amplitudes (Table 13.1, column 4) are 0.785 and 0.303 mm, which are also more than the permissible amplitudes. A computation based upon the realistic values of the soil springs at the design stage would have indicated an unsatisfactory performance for the machine. This would have resulted in an effort to redesign the foundation before the machine was installed to avoid later interruptions. It must , however ~be noted that the shear strain with computed values .of amplitudes are different than the values with which the natural frequenctes and amplitudes have been computed. In actual design of machine foundations another trail may be performed. Linear Spring Theory
The foundation response was also calculated by using the linear spring theory and the same cases as for the computation with Elastic Half-Space
566
CASE HISTORIES
theory were considered. The appropriate equations derived in Chapter 6 were used for the computations. Case 1. Post design: The foundation response is calculated as though it were designed for a permissible amplitude of 0.00125 mm, "Yo = 5.94 x 10- 6, 3 and C" = 10.25 kg/cm for area of the compressor.
567
CASE HISTORY OF A COMPRESSOR FOUNDATION
5. Natural frequency in combined rocking and sliding Frequency equation 2
2 )
4 - (wny
+ (J)nrb 1'
wn
w
Vertical Vibrations
w2
+
2
2
wny(J)n
0
'Y
n
(6.112)
!2 = 2775/ sec 2
w" 2 = 52.68/sec
1. Natural frequency of vertical vibrations w"'
w"'
=
~C:nA
fu 2 = 8.38Hz
w:, (6.63a)
w"' = 119.76 sec
6. Amplitude of vibration occasioned by coupled rocking and sliding _ C,AL M A,- Ll.(w2) x
2. Amplitude of vertical vibrations A, (undamped) P, =
2
2
m(wn,- w )
28709.8/sec
fu 1 = 26.90 Hz
fu, = 19.06 Hz
Az
=
(6.64b)
2 2 Ll.(w 2) = m;I~(w:,- w )(w! 2 - w ) Ll.(w 2) = 9.734 X 10 8
(6.127) (6.122)
A y =0.1333mm
A,= 0.00322mm
2
_ C,A-mw M A1 Ll.(w2) x
3. Limiting natural frequency in sliding, wnx
Wny
=
~C:nA
A.p = 6.16 x 10- 5 rad
(6.66a)
A•= A;= A, +hA.;
wuy = 84.68/sec
A~= A,+ ai2A
fuy = 13.74 Hz
A,= 0.0787 mm
4. Limiting natural frequency in rocking wu
mO
C
fu
= 9.95 Hz
(6.130b)
A*= 0.1998mm y
C,= 112C"
Wu
(6.128)
(6.130a)
Torsional Vibrations (6.74c)
7. Natural frequeflt:y of torsional vibrations w""'
_ ~c.I,
Wn.p-
M
m•
c.=0.75C"
wu.p = 102.309/sec fn;• = 16.28 Hz
(6.81a)
5&8
CASE HISTORIES
8. Amplitude of torsional vibrations A
•
type investigation, the values of the horizontal and vertical amplitudes are
z
(6.83)
m' Wn1>- W )
A•= 2.78 X 10- 6 rad
. The values of the computed response are tabulated in Table 13.2, column
Case 2. Postmonit?ring: The foundation response has been calculated for the ~peratmg cond1l!o~s of the compressor. The shear strain 'Yo for this cond1t1~n IS 1.49 x 10- , :nd the value of C" for the area of compressor's foundatiOn IS 5.13 kg/em. The response for this case was calculated as in Case 1. The computed values are shown in Table 13.2, column 4. .. In add1l!on to computing the vibration characteristics of the foundation for the postdes1gn an? postmonitoring-type investigations, the frequency of honzontal free Vlbral!oll~ was also calculated by using the value of c" that corresponds to 'Yo= 10 . The results are shown in Table 13.2, column 5. . The precedmg computations show that for the postdesign-type invesl!gal!on, the amplitudes of horizontal and vertical vibrations (Table 13.2, column 3) are 0.1998 and 0.0787 mm, respectively. For the postmonitoring-
Ta~le 13.2 Computed Natural frequencies and Amplitudes by Linear Sprong Theory Quantity Frequency
Free
or Amplitude
Postdesign 'Yo = 5.94 X 10- 6
Postmoni to ring 'Yo - 1.49 X 10-'
1
2
3
4
1 2 3 4 5 6
fnz Hz fny Hz fn• Hz fnl Hz fnzHZ fn• Hz
19.06 13.74 9.95 26.90 8.38 16.28
13.48 9.53 7.05 19.07 5.93 11.53
7 8 9 10 11 12
A .. mm AYmm
0.0032 0.133 6.16 X 10- 5 0.1998 0.0787 6 2.78 x
0.0075 0.340 1.049 x w-• 0.4552 0.135 6 7x
S. No.
A<~>
rad A;mm
A;mm A• rad
w-
5&9
0.4552 mm and 0.135 mm, respectively. These values are more than the
A M, 1>-M(z
3
CASE HISTORY OF A HAMMER FOUNDATION
w-
Vibrations 'Yo
1 X 10- 6
allowable values, thereby necessitating a redesign of the foundation. This could have been taken care of in the beginning if realistic values of the soil parameters had been used to evaluate the design. Use of the linear spring theory yields natural frequency value for the horizontal vibration of 14.75 Hz, which is in reasonable agreement with the observed natural frequency of 17.5 Hz. It can be seen that whether the elastic half-space theory or linear spring theory is used, the final conclusions, in this case, are more or less identical. Nevertheless, there is an urgent need to monitor the performance data of machine foundations, but these data are meaningful only if the dynamic properties of the supporting soil are also obtained.
13.2
CASE HISTORY OF A HAMMER FOUNDATION
General
"5.,-:.;t-
The performance of a 1. 55 t forging hammer is reviewed here. Its foundation was designed by Prakash and Gupta (1970), who based their calculations on the machine data supplied by the manufacturer and the soil data obtained by in situ tests (Prakash et al., 1966). Earlier the supplier had suggested a foundation. size that, according to the soil conditions at the site, was found to be inadequate. A new foundation was therefore designed, taking into account the soil conditions and permissible amplitudes of vibrations. The amplitude response of this foundation was monitored under working conditions. The natural frequency of the foundation was also computed from recorded observations. These details are briefly discussed below along with the calculations for the design and the computed and observed responses. The hammer had the following specifications:
5
36.77 105
= 1150kg Tup weight without die Maximum weight of top die = 400 kg Maximum falling weight W = 1550 kg = 900 mm Maximum tup str~fe h Supply steam pressure p = 6-8 atm Cylinder diameter (internal)= 410 mm Anvil block weight without die holder = 22.5 tons Anvil weight including hammer frame = 34:5 tons 2 Base area of anvil over pad = 2.00 x 1.20 = 2.4 m
570
CASE HISTORIES
The limiting amplitudes of vibration of the anvil and foundation block had not been specified. The allowable amplitudes of vibration of the foundation and anvil were selected from Table 7 .1. A foundation = 1.2 mm A'"'" = 1.0 mm for 1-t hammer and 2.0 mm for 2-t hammer
CASE HISTORY OF A HAMMER FOUNDATION
I
Silty clay, CL N value at 1.5 m
=
571
20
2.7 m Sandy silt, SP- SM N value at 3 m = 14
Soil Data The following tests (Prakash et al., 1966) were conducted to determine the properties of the soil at site: (1) boring and sampling, (2) standard penetration tests, (3) cyclic plate load tests, and (4) dynamic tests. A site plan showing the locations where the various tests were performed is given in Fig. 13.2. The soil was angered to a depth of 6.75 m below the surface. Standard penetration tests were conducted every time there was a change in ihe strata. A log of the auger test is shown in Fig. 13.3. Based upon an evaluation of the data, the following values for the soil parameters were adopted for the design: Allowable bearing capacity of 31.2 tim 2 and the corrected value coefficient of elastic uniform compression C" for a 10m 2 area was 6.10 x 103 tim 3.
3.15 m Silty sand, SM N value at 3.6 m = 14
4.5 m Medium to coarse sand, SW
N value at 4.8 m
= 14
4.95 m Sandy,$ilt, SP
N value at 5.4 m
=
15
6.0 m Medium to coarse sand, SW N value at 6.6 m = 15 6.75 m
1.5m Cyclic plate
load test
Design of the Foundation
9'" t N
The size of the foundation adopted for analysis is shown in Figs. 13.4. The computations for the estimated response are given below: 1. Data Assumed for Design:
3.4 m
T
0.6~0 Auger
l~'~;:_
To
5 25
o.375 m r---3 m
.lmL__0_._37_5_m _test _ __ J Dynamic
r---4.425 Figure 13.2.
Figure 13.3. Boring log at site, Courtesy, Ind. Geot. Soc.
~r load
T
1.5 m
1.5 m
m---1
Location of field tests, Courtesy, Ind. Geot. Soc.
Material of pad below anvil-hard wood 4 2 Modulus of elasticity of pad E 2 = 5 X 10 tim Thickness of pad below anvil t = 0.4 m Dimensions of the foundation block= 6.50 X 5.70 X 1.30 m Dimensions of reinforced cement concrete walls= 0.50 m x 1.13 m ...\ all around an~il Unit weight of reinforced cement concrete= 2.4 tim' Unit weight of backfill= 1.76 tim 3 Coefficient of elastic uniform compression for impact loading C" = 6.1 x 103 tim 3 Coefficient of restitution e = 0.5 Coefficient, which takes into account counterpressure and frictional forces, or efficiency of drop '7 = 0.65
573
CASE HISTORY OF A HAMMER FOUNDATION
2. Soil Contact Area and Weight of Foundation. The foundation area in 2 contact with the soil A 1 = 6.50 x 5.70 = 37.05 m The combined weight of the foundation and backfill was,
Expansion joint filled with asphalt Anvil Floor level Timber pad 400 mm
Foundation block
r- l" ;~ ·+'r_·~-" -~=" -:-= ~;" g " _" _" _ " _z _z ':i_·~ ~:lz•=1zz4oo-1=·,!, 22
t
fT il T +2430
~-~-~--- 2100--___..;1-1--1 100 500 300 300 500
14oo,-'"f,
r-------------6500·---------~--+1
3
Block= 6.50 x 5.70 x 1.30 48.165 m 3 Walls= 2 X 3.70 X 0.50 X 1.13 = 4.181 m 3 Walls= 2 X 1.90 X 0.50 X 1.13 = 2.147 m 54.493 m 3 x 2.40 = 130.78 tons Backfill= 2 X 6.50 Backfill= 2 X 2.90
X X
1.40 X 1.13 20.5660 1.40 X 1.13 = 9.1756 3 29.7416 m x 1.76 = 52.34 tons
The total weight of the foundation and backfill W1 was equal2 to 183.12 tons. Therefore, the total mass m 1 was 183.12/9.81 = 18.65 t sec /m. 3. Naturql Frequencies of the Foundation for the Hammer System. Appropriate eqhations derived in Cli'.1l"pter 7 were used for the computations. 4 2 The modulus of elasticity of the pad E 2 = 5 X 10 t/m Thickness of the pad t = 0.4 m 2 Area of the anvil over the pad= A 2 = 2.4 m Coefficient of rigidity of the pad k 2
Longitudinal section
1,1
level
EA 2
(7.1)
k,= -tk,=30X 10 4 t/m 2
The mass of the anvil and frame m 2 = 3.5 t sec /m The limiting natural frequency of anvil on pad wn 1,
k2
2
Wnt2= m2
1400~
w~ 1 , = 8.57
100
r------------5700--------~~ .. Transverse section (b)
Figure 13.4. section.
572
Details of the hammer foundation: (a) Longitudinal section, (b) Transverse
fn 1,
=
30 X 10 = 3.5 X
4
4
(2.99b) 2
10 /sec
46.6 Hz
The limiting natural frequency of the entire system resting on the soil wnll'
2
The area of the foundation in contact with the soil was 37.05 m , which is 2 3 3 larger than 10m 2 Therefore, Cn = 6.1 X 10 t/m for a 10m area was selected for the design. Limiting natural frequency of the anvil and foundation system on soil wn 11
574
CASE HISTORIES
CASE HISTORY OF A HAMMER FOUNDATION
(2.99a) 5m
k 1 = C"A
(4.27)
k 1 = 22.6 x 10 4 tim
W~ 11 = 1.0203 X 10 /sec 4
fnll
EJ 1
S75
\---5.7
I
OPunching press
l
4.57 m 2
!(A Observation
16.07 Hz
=
m--J
\U locations
The ratio of mass of anvil to that of the foundation
!L =
m,!m
= 1
0.1875
Figure 13.5.
Layout of vibration measuring points, Courtesy, Indian Geot. Soc.
The combined n_atural frequencies of the anvil and foundation system Frequency equatwn: · (7.5b) (2.98)
Monitored Response of the Foungation
fn2
=
15.90 Hz
Wn 1
=
322.18/ sec
fn1
=
51.27 Hz
Amplitudes of anvil and foundation Velocity of dropping parts Vr,
I
VTi=11y2g
(W+pA) W
n h
(7.7)
Vr, = 7.63 m/sec
Initial velocity of anvil motion Vn V,
=
V, =
1+e -1-V:T. +s ' 0.497 m/sec
Amplitude of vibration of foundation block A
z =A= 1
'
2
(7.1la)
,· The behavior of the foundation was carefully observed after it was installed. There were no undesirable vibrations transmitted from the foundation to the adjoining area as a result of the hammer's operation. The vibratory response of the foundation was monitored under normal operating conditions. Acceleration transducers were used to sense the vibrations, and their output was amplified through universal amplifiers and recorded by oscillographs. Records of vibrations were obtained for the anvil and the foundation as well as for the surrounding area at locations as shown in Fig. 13.5. A typical record obtained during the observations is shown in Fig. 13.6. Displacement of the anvil and the foundation was obtained by integrating the recorded acceleration time history records twice. The values of the computed and observed vibration amplitudes (Prakash and Gupta, 1970) are compared in Table 13.3.
I
g
~:..
'
2 2 2 )( 2 ( wntz-wnz wntz-wn '('
A,=0.956mm
Wnf2 Wnl- Wnz)wn2
1
)V a
(7.5a)
A,=0.844mm Amplitude of vibration of the anvil A n
Paper speed 125 mm/sec
Figure 13.6. A typical vibration record.
577
CASE HISTORIES
576
Table 13.3. Comparison of Computed and Observed Amplitudes Amplitudes, mm Location Anvil
Foundation block
Computed
Observed
0.956 0.844
1.08 0.423
The computed values of the amplitude of vibration of the anvil and the observed value under normal operating conditions are of the same order of magnitude. The amplitude of vibration of the foundation block was computed to be 0.844 mm, whereas its observed value was 0.423 mm. The computed amplitude of the foundation block is thus greater than the observed value (Table 13.3). This was to be expected, because the damping had not been accounted for in computing the amplitude of the foundation. The design of the foundation in this case was adequate, and the foundation's performance has since been satisfactory, thereby implying the adequacy of the design for this case.
13.3
FINAL COMMENTS
The two cases of monitored performance of prototype machine foundations that have been described in this chapter stress the importance of an adequate design using realistic values for the soil parameters. It is necessary to procure data on performance so that a rational evaluation of the design procedures can be made. Such an attempt is meaningful only if data on the dynamic soil properties is also obtained. The construction aspects have a very important effect on the performance of the machine's foundation. Guha (1984) published data for several block foundations, and Wang (1984) described a case study of decreasing vibration amplitudes for machine foundations and structures. These data need to be interpreted more realistically for one to be able to arrive at a meaningful conclusion.
REFERENCES Guha, S. K. (1984). Vibration studies of block type foundations. Proc. Int. Conf. Case Hist. Geotech. Eng., St. Louis, MO. Vol. 3, 1!47-1!54.
Kumar, K., Prakash, S., Dalal, M. K., and Bhandari, R. K. M. (1985). Dynamic analysis and performance of compressor foundations. Proc. Symp. Vib. Probl. Geotech. Eng., Am. Soc. Civ. Eng., Annu. Conv., Detroit, 286-300. Madshus, C. F., Nadim, A., Engen, and Lerstol, A. M. (1985). Low tuned compressor foundations on soft clay. Proc. Symp. Vib. Probl. Geotech. Eng., Am. Soc. Civ. Eng., Annu. Conv., Detroit, 117-136.
REFERENCES
upta D. c. (1970). Design and performance of a 1.55 t forging hammer P ra k as h , S ., an d G ' 9(2) 128-142 foundation. J. Indian Nat. Soc. Soil Mech. Found. Eng. • · 4) "Final report on Dynamic Analysis and Settlement (198 K Prakash S an d K umar • · · · " 1 28 Inv~sti~ations of Foundations for Air Compressor (~ffsite) at BRPL, Bongatgaon, - · Central Building Res. lost., Roorkee, Roorkee, Indta. . . · y K (1981) Observed and predicted response of a machme foundatiOn. Prakash, S. , an d Pun, · · · 1 3 269 272 Proc. Int. Conf. Soil Mech. Found. Eng., 9th, Stockholm, Vo · , . · .. · y K (1984) Behavior of compressor foundattons: Predtctlons and P d Prakash, S., an un, · · · · MO VII 4 1705observations. Proc. Int. Conf. Case Hist. Geotech. Eng., St. Lows, ' 0 • • ~. c . d , S G D c and Agarwal, S~ L. (1966). "Report on Beanng apactty an A 1 d t · Yamunanagar" Prakash, ., upta, · ., Dynamic Soil Constants for Forging J:Iammer of Jamna uto n us nes, · . University of Roorkee, Roorkee, Indta. . d w d R D (1982). Foundations for auto shredders. Int. Conf. Soli Rtchart, F. E., Jr., an oo s, . · Dyn. Earthquake Eng., 1st, Southampton, Vol. 2, 811-824. . . Wan X. K. ( 1984). A case study on decreasing vibrations of ~achme foundations and ;;ructures. Proc. Int. Conf. Case Hist. (}eotech. Eng., St. Louts, MO, Vol. 2, 787-792.
14
CONSTRUCTION ASPECTS OF BLOCK FOUNDATIONS
14.1
579
CONSTRUCTION ASPECTS OF BLOCK FOUNDATIONS
Concreting
The Construction of Machine Foundations
The proper construction of a machine foundation is very important, because if care is not exercised during the construction, all of the effort put forth in rigorously designing the foundation and the dollars spent in its development will be wasted, Besides that, the commissioning of the machine may be delayed or early unscheduled interruptions may cause a loss of production, the value of which may far exceed the cost of the machine and its foundation. Thus, sound construction coupled with a realistic design can contribute significantly to long-term performance and trouble-free operation. Some construction-related defects that can lead to the malfunctioning of a machine are improper curing of the concrete, unrestricted height of the pour, undesired and lengthy interruptions in pouring the concrete, negligence in building the construction joints, and imperfect alignment of the machine frame with the foundation. Because of careless construction, the alignment of openings and pockets may shift during concreting and a lot of chipping may be required later. Cracks and seprations at cold joints may cause difficulties. Thus, it can be seen that the construction of machine foundations is a matter needing great care, and precautions in addition to those normally observed in the construction of reinforced concrete to achieve a well compacted, dense matrix that has a good bond with its steel reinforcement. The standard requirements for reinforced concrete construction are given in the code ACI 318-83 (AmericanConcrete Institute, 1983), but for the specific problems of machine foundations, additional considera-
tions are discussed below. Machine foundations are either of the block type or frame type. Both are discussed below. Sometimes it may be necessary to alter the size of an existing foundation, and for this purpose the method of bonding new
concrete to old concrete is included in the discussion. The erection and interfacing of a machine to the foundation is also described. 578
1. The selection of the aggregates and proportioning of concrete mix should be made according to specifications laid down in ACI 301 (American Concrete Institute, 1975) or similar prevalent codes dealing with use of concrete for general building construction. 2. The ultimate compressive strength of concrete should be in accord with the criteria set forth in ACI 318-83 (American Concrete Institute, 1983). If no information is available, the ultimate strength of concrete should not be less than 150 kg/cm 2 or 2.2 ksi. 3. The concreting should be done in horizontal lifts. The first pour should be done in a 300-mm (12-in) layer and subsequent pours in 400-mm (16-in) layers. 4. The height of the pour should be as low as possible, and one must make sure.that the concrete does not segregate. 5. The foundation should t;'<;',';,oncreted in a single pour to avoid cold joints. If it is necessary to have a time gap between two successive pours, the time should be short and should not exceed 30 min. 6. Because of practical difficulties, sometimes a single pour may not be possible, and a cold joint becomes unavoidable. In this case, it should be considered as a construction joint and its location chosen with care. The monolithicity of the structure at this construction joint should be ensured by providing a suitable number of dowels and shear keys through the joint, and quality control and supervision during the operation. The dowels should be long enough to assure a full capacity bond. Their length beyond the joint should be 4.0 diameters or 12 in, whichever is more. The dowels may be made using #5 or #6 bars. To attain an adequate joint in mass concrete construction, such as in casting a block, one must provide shear connectors (U-bars) at the level of the joint. A strong bond between old and new concrete can be made by roughening or honeycombing the upper surface of the old concrete. The upper surface should be cleaned with a hard wire brush and then covered with a thin layer of Cfment grout before the new concrete is poured. For specific guidelines fo?rorming a proper joint, one should follow those given in the building codes or ACI: 318-83 (American Concrete Institute, 1983).
7. Care should be taken to avoid bulging of the concrete at offsets by using suitably designed form work. 8. The areas around openings and pockets should be concreted with care.
9. The foundation should be properly cured. Improper curing may lead to shrinkage cracks, which may widen after the machine is in operation.
580
THE CONSTRUCTION OF MACHINE FOUNDATIONS
Reinforcement
Reinforcement for a Foundotion Block. Massive block foundations do not have the same structural requirements as beams or columns. Therefore, they are provided with only minimal reinforcement to take care of temperature and shrinkage effects. According to ACI 318 (American Concrete Institute, 1983), the minimum steel reinforcementt should be approximately 0.0018 times the gross concrete area in each direction. The steel bars should be spaced no farther than 18 in center to center. The minimum cover of concrete for protection of reinforcement should be 75 mm (3 in) at the bottom and 50 mm (2 in) on the sides and tops. Details for a typical block foundation, which was provided to support a compressor unit of medium capacity, are shown in Figure 14.1a, and b. The reinforcement details for the foundation block of an impact machine are similar to those discussed above except for the top portion of the block below the anvil where additional reinforcing bars are required to take care of stresses occasioned by impact. The spacing of the bars in this part of the block is usually kept on 100-mm ( 4-in) centers. Typical reinforcement details for a hammer foundation are shown in Figure 14.2a and b. Reinforcement around Openings and Cavities. A steel reinforcement equal to 0.5 to 0. 75 percent of a cross-sectional area of an opening or cavity should be installed around all such features. This must be provided in the form of a cage. In the case of circular openings, the reinforcement should overlap for a length equal to 40 times the bar diameter or should extend 300 mm (12 in) beyond the point of intersection. Typical reinforcement details around a circular opening are shown in Figure 14.3.
~ ...•
0 0
~~
~
c
"'
e
.,;
i:' Q
N
t:
••
a'6
~~ J.
':!..
v
t. ""
t
_j
---
~-==]
0
00
"'
"\ •---,
··.·
"- ''\.___~
':1+g
~
••
D
t~< "" ~
,;.;
"'
..
~\
.,;
g
0
lo= F==o~
0 0
Q
~
"'
l-- L--_j·
.,;
·~c
0 0
~
v
;
.
~~~
:--.-..._
"'
c
~
!c
0
t;
••
v
0
•
v
•• :9 '0 ~
>
~
2
~
~
t; "0
N
~
E
'!i ~ • -~ E .:! -c:
• =-= >-..
""'c
t ·i. ~
!
.i i ' ~
:':I ~~
~
c .5 •~ E •v •< • .ec .•
0 N
continuous pour in the same manner as for a block foundation. t For grade 60 steel bars.
....~
"'"'
Concreting
The base slab is usually concreted in a single,
N D
----
.,;
14.2 CONSTRUCTION ASPECTS OF FRAME FOUNDATIONS
Concreting the Base Slab.
0
"' ~
~~ The construction of a frame foundation involves the concreting of the base slab, columns, and deck slab. The concrete mix should be blended to ensure the strength required by the design. This can be done by following the recommendations of ACI 301 (American Concrete Institute, 1975) or other relevant building codes. In contrast to the construction. of block foundations, flexural strength is a very important factor in the design of frame foundations and should receive the utmost consideration.
,-f-.s0
1: 581
CONSTRUCTION ASPECTS OF FRAME FOUNDATIONS 12-mm diam. 200 mm c/c
10 bars 12m~ 6 Dowels 12-mm-diam.
0-mm-diam. bars\ 100 mm ~/c
200m200
Opening
~
'.I /
300
~
~~
1-
~ v
/
'xfX
- t-300
583
cement
lb
~
1-300
1- 1-250
3
f
/
::----::
"'
IX'!>
Figure 14.3. Typical reinforcement details around a circular opening.
I/ Vi'
150
'
[\ 12-mm~iam. bars 200 mm c/c
~ 22~Im-diam. bars-
lJ
}
20-mmlam. 200 mm c/c '}-mm diam. 120 mm c/c . , { 4 bars 12 mm 250 mm c/c
6500 (a)
12-mm diam. 200 mm c/c
\
200
1-H 200
mm~r.==rn=~F't;:=r=r=;=or=;=ililr==;r
10 bars 12 10-mm-diam. bars 100 mm
1-~~~
n~
Concreting the Columns and Deck Slab. Although the concreting of the base slab, columns, and deck slab in a single continuous pour is desirable, it is usually not possible from a practical standpoint. As a result, a construction joint is formed between the columns and the base slab. Details of a typical construction joint formed between the columns and the base slab are shown in Fig. 14.4. The concreting of the columns and the deck slab is then completed in one pour with the necessary precautions being taken to ensure the monolithjcity of the structur~.,J:Vhen the column heights are more than 6 m (20ft) concreting of the superstructure in one pour may not be feasible, and it may be necessary to provide a second construction joint, but generally such a joint is not recommended. The provision of a construction joint in the top part of a column near the deck will aid in the construction and reduce the height of the pour, thereby reducing the chances of the concrete becoming segregated. A construction joint is a weak plane from the standpoint of shear strength, and in
1-
c/c 4 bars 12 mm diam. 2'mm c/c
300
t- 250 ~
d.
Main column reinforcement
v ?Dowels
I
20-mm diam. 200 mm c/c Dowels 12-mm diam. (spacing)
~-
--22-mm diam. bars
20 mni diam. 120 mm c/c
p.---
1
iY
12-mm diam. bars 200 mm c/c
i------------5700--------4
-column vshear key
L'
"
(b)
Figure 14.2. (a) Reinforcement details for a hammer foundation: (a) longitudinal section; (b) cross section. (After Prakash and Gupta, 1970.)
Base slab
lJ Figure 14.4. Typical details of a base for clarity). 582
~c-
u._ slab~column
joint (other reinforcement details not shown
584
THE CONSTRUCTION OF MACHINE FOUNDATIONS
t
To base slab (a)
A
I I I
~
~
0
0
u
•
:
"' 30
~
"
r B
u:
I I I I I I I I
:
v
v
I I I
v J_
v v /
I I I I I I I
I
A
' (b)
585
structural considerations alone. The reinforcement in the base slab is generally set at a minimum with regard to anticipated temperature changes and shrinkage. This minimum for block foundations is 0.0018 of the concrete area in all three directions as set forth by the ACI-318 (American Concrete Institute, 1983) specifications and is provided in the form of 15 to 25 mm (0.6 to. 1 in) diameter bars spaced on 18 in centers in all three directions. For the slabs, the amount of reinforcing steel should be designed and provided according to structural requirements. A minimum concrete cover of 100 mm ( 4 in) should be provided on all sides, Typical details of reinforcement in a base slab of a frame foundation are shown in Fig. 14.6.
Frame column
30
CONSTRUCTION ASPECTS OF FRAME FOUNDATIONS
-16 mm 10
mm Stirrups
4-16 mm
:
Reinforcement in Columns and Deck Slab. The amount of reinforcement for columns and a deck slab is determined by the structural design. The ties and stirrups should be designed to satisfy the requirements of the building codes or ACI 318 (American Concrete Institute, 1983). The minimum diameter of the longitudinal steel for beams should be selected so that the spacing of bars does not exceed 150 mm ( 6 in). The minimum amount of shear reinfor~~>ment in the beams and deck slab should be as per ACI 318 (American Concrete Institute, 1983) requirements. The steel reinforcement in the columns should be determined from structural requirements. The minimum number of reinforcement bars in columns shall be four when rectangular or circular ties are used and six when spirals are used. The minimum cover of concrete over the reinforcement bars should be 50 mm (2 in) on all sides for columns and pedestals and 40 mm (1.5 in) for beams. The vertical reinforcing bars in the columns should have adequate embedment in the base slab to ensure their full strength in bond. A schematic of the reinforcement for the components of a turbogenerator frame is shown in Fig. 14. 7.
Figure 14.5. Details of a typical construction joint in frame columns: (a) frame column; (b) section of column with reinforcement; (c) plan of joint. Dowel bars
/(/\
reinforced concrete construction, such joints are placed at points of zero or minimum shear. The shear force on the frame columns is uniform,. and if the provision of a second construction joint in the columns of a frame foundation is accepted in principle, its location on the columns is immaterial. A typical joint in a column may, if necessary, be designed as shown in Fig. 14.5. Reinforcement Reinforcement in the Base Slab. The base slab of a frame foundation generally has a depth which is much greater than that required from
Shear stirrups
v I--
Shear key
v
/-
Trans verse ba
Base slab
!'- Longitudinal bars
Figure 14.6. Typical reinforcement details for base slab of a frame foundation.
586
THE CONSTRUCTION OF MACHINE FOUNDATIONS
ERECTION AND INTERFACING OF A MACHINE TO THE FOUNDATION
587
r--b--+-b---1 I
1
II
a
IaI
T
lei
(b)
,--+-0
a
X--- -
I
0
'
0
-
---x
0
I (a)
~I
Foundation
Anchor bolts
(b) (d)
Figure 14;7.
Figure
(e)
Typical reinforcement in members of a frame foundation: (a) and (b) for cross
beam; (c) longitudinal beam with cantilever projection; (d) column; (e) becim column joint.
14.3 ERECTION AND INTERFACING OF A MACHINE TO THE FOUNDATION
The machine, base plate, and other equipment that have precise tolerance for alignment cannot be placed directly on the finished concrete surface. The irregularities on the concrete surface and the machine base cause alignment problems. After the concrete is set, steps are initiated to install a machine on the foundation. The machine is fixed to the foundation with the help of a base plate and anchor bolts. The concreting of the foundation is terminated at the level of the base plate, and the gap is filled later with mortar after the base plate is leveled. The thickness of the grout below the base plate varies from 20 to 50 mm (0. 75 to 2 in) depending upon the size of the base plate. The base plate is fixed to the foundation by anchor bolts. The anchor bolts are kept in position before concreting with the help of a template (Fig. 14.8), which is removed after the concrete is set. Alternatively, by using suitable form work, pockets may be left in the concrete at the predetermined positions of the anchor bolts. In this case, the bolt holes are filled with mortar after placing the base plate and aligning the bolts. Holes for the bolts should not be too large. A 150 mm x 150 mm (6 in x 6 in) hole is generally
14·:a.
Positioning of anchor bolts.
adequate. The minimum clearance between the bolt holes and the edge of the foundation should not be less than 80 mm (3 in), as shown in Fig. 14.9. The length of the bolts is usually decided with regard to the bond and should be 40 times the diameter of the bolts. In case it is not possible to provide for the full length of the anchor bolts, an arrangement of the anchor bolts similar to that shown in Fig. 14.10 should be used. The position of the anchor bolts should invariably be fixed with reference to the axis of the machine.
After the base plate has been leveled, the space under it should be grouted. The grout used should be nonshrinking type and should provide complete contact with the top surface of the concrete foundation and
I I I 1->< I I I I I I
I I
rJ L I
L __
80
•~ !
1 J2000mm
I I
U H+3oo+H mm
200mm Figure 14.9.
the edge.
Details of a bolt hole close to
588
THE CONSTRUCTION OF MACHINE FOUNDATIONS
INSTALLATION OF SPRING ABSORBERS
589
opening should be insulated to a length of at least 150 mm (6 in) from the intersection point to avoid any stray currents occasioned by induction.
'I
I
X
Split
X
14.4
GAP AROUND THE FOUNDATION
half
w=~
Section at YY
Piece of rod welded to split ends
L[jbJ
To minimize the transmission of vibrations to adjoining structures, a gap should be provided around the foundation as shown in Fig. 14.1. For frame foundations, a clear gap should be provided around the base as well as around the deck slab. The gap around the foundation should be kept free from debns .. ~f contact of the machine foundation with an adjoining structural umt IS unavOidable, two layers of a resilient material such as felt may be used at the interface.
II 11
II II
Plan at XX
Figure 14.10. Typical details of an anchor bolt.
maintain a uniform support. Cement grouts cons1stmg of expansive and hydraulic cements and additives to compensate for shrinkage effects or epoxy grouts with specially blended aggregates and fillers that will reduce or eliminate shrinkage can be used. Epoxy grouts have the advantage of chemical resistance, high early strength, and impact resistance, but are sensitive to increase in temperatures. Grout should be placed below the base plate without trapping air or water. Preblended grout mixes are commercially available and a proper choice should be made after a careful consideration of properties of the grout mix and the specific requirements of a given job. To prevent the concrete from spalling at the edges of the foundation, a border of steel angle irons may be provided, as shown in Fig. 14.11. Holes are generally left in the body of the concrete for the lugs of the angle iron. These are subsequently grouted along with the floor finish. For generator foundations, the reinforcement on either side of the bus Angle
14.5
BONDING OF FRESH TO OLD CONCRETE
It may be necessary to bond freSh to old concrete to repair a defective concrete surface brought about through an unforeseen interruption in the concreting or as a result of a defective casting or improper curing. Also, if for some reason the surface after concreting is loose, it would be necessary to provide a hard surface for proper machine performance. In such cases, the affected area should be chipped off up to 100 mm ( 4 in) and cleaned. Shear keys should then be cut into the surface. The number and size of shear keys depend upon the extent of the surface being repaired. A minimum of four shear keys should be provided. The size of the shear keys should be 75 X 75 X 600 mm (3 X 3 X 24 in). They should be thoroughly cleaned. The surface so exposed, including the grooves for the shear keys, may be treated with epoxy. This would consist of Araldite (100 parts), hardener (40 parts), and filler silica (4 parts) and be applied in three thin coats. When the last coat is sticky, rich concrete mix should be poured and vibrated. Additional steel bars 10 mm (0.375 in) in diameter may be placed in the grooves for the shear keys. This process of bonding fresh concrete to old concrete is quite expensive and is feasible for small areas only.
...:,
corner
14.6
Figure 14.11. Typical detail at insert.
INSTALLATION OF SPRING ABSORBERS
The methods of installing spring absorbers for a machine foundation depend on the type of the absorber system. There are two types of spring absorber systems: the supported and the suspended type. These have been described in Chapter 10 (Section 10.1).
590
THE CONSTRUCTION OF MACHINE FOUNDATIONS
INSTALLATION OF SPRING ABSORBERS
591
Machine
Supported Type
Heavy Mass above the Springs Not Required. For spring-supported foundation systems for which a heavy mass above the springs is not needed, the machine can be directly mounted on a rigid metal frame resting on springs (Fig. 14.12). This arrangement is used for well-balanced machines that are essentially unaffected by the unbalanced exciting forces associated with the high harmonics of operating frequency. The principal stages of constructing such a foundation are as follows: 1. The base slab construction is similar to that of a block foundation. Its thickness depends upon the machine foundation design and generally varies from 0.3 to 1.2 m (1 to 4ft). 2. Before the base slab is cast, anchor bolts are placed at predetermined locations for attaching the lower plates of the spring absorber. When the concrete is set, the lower· plates of the absorbers are fixed at the proper locations. 3. A rigid prefabricated metal frame (consisting of rolled s(eel sections) is installed above the lower plates of the absorbers (Fig. 14.12.). 4. The springs are placed on the lower plates and covered at the top with the upper plates, which are bolted to the upper metallic frame. 5. The upper part of the foundation (the metal frame) is leveled by adjusting the regulating bolt.
Heavy Mass above the Springs Is Required. A heavy mass above the springs is required in cases where the operation of the machine induces large unbalanced forces at higher harmonics of operating frequency. The heavy mass is provided hy placing another concrete block above the springs, as shown in Fig. 14.13. The principal stages of constructing such a foundation Machine
Upper plate lower plate
Frame beam {embedded in foundation block) Upper plates
_Springs
Figure 14.13. Supported-type spring absorber with upper foundation block.
are as follows: 1. The base slab construction is similar to the previous case. Anchors for holding the lower plates of the absorbers are fixed in position before the slab. is cast. 2. When the concrete is set, the top surlace of the base slab is covered with tar paper to prevent direct contact with the upper foundation block, which is subsequently cast over it. (Fig. 14.13). 3. The lower plates of the spring absorbers are fixed at predetermined locations. 4. A rigid metal frame consisting of rolled steel sections is installed over the lower absorber plates. 5. The form work for the upper foundation is then constructed, and the foundation is cast. The beams of the metal frame are also set into the lower part of the foundatiol'K block. Depending upon the arrangement of the absorbers and type of frame used, cavities may be needed in the lower part of the upper foundation block to provide access for the springs. 6. The springs are placed in position after the concrete in the upper block is set. The tops of the springs are covered by plates, which are bolted to the beams.
Figure 14.12. Supported-type spring absorber system with machine attached to metal frame.
7. The upper block is leveled by adjusting the regulating bolts of the springs.
THE CONSTRUCTION OF MACHINE FOUNDATIONS
592
The spring assembly shown in Figure 10.4b can be set directly between the lower slab and the upper foundation block. In such a case, the upper block is cast over the spring casing, which is kept in position. The spring is kept compressed during construction, but is loosened afterward so that the weight of the upper foundation block and machine is transferred to the springs.
Appendixes
Suspended Type
A foundation on a suspended type absorber system (Fig. 10.4) can be constructed along similar lines to those of supported-type systems. The principal stages of construction are as follows: 1. The foundation below the springs (Fig. 10.4) is concreted first.
2. When the concrete has hardened, two or three layers of tar paper are placed on the foundation slab (Fig. 10.4) and on that part where the slabs of the absorber are to be placed. 3. A prefabricated frame of rolled steel beams is installed above the foundation slab. The projecting sills of this frame serve as a support for the anchor plates of regulating bolts of the absorber. 4. When the form work for the upper part of the foundation is laid, cavities are left for each absorber. The concrete is then cast. 5. When the concrete is set, the absorbers are mounted. The springs are placed on the lower slab of the absorbers and are covered by upper supporting slabs, which are bolted to the girders. 6. The foundation is leveled by adjusting the regulating bolts of the springs. In the preceding sections of this chapter the authors have tried to project the salient features which need special attention during construction of a machine foundation. However this discussion is by no means comprehensive so far as total construction is concerned. The reader must, however follow accepted construction/installation procedures as per relevant codes of practice in a particular country.
REFERENCES
The computer programs in Appendixes 1 and 2 are included for the convenience pf the reader. The B[OJ>;rams were written for personal computers using FORTRAN 77 Compile(: The computer program in Appendix 1 is for calculating the steady-state response of a rigid block foundation in different modes of vibration. Similarly, the program in Appendix 2 can be used for calculating the natural frequencies and vibration amplitudes of a hammer foundation considering an undamped two-degrees-of-freedom system. Some problems in the text were solved using these programs. Detailed comments are included in the program. Additional aid may be obtained from the solved text examples that have been listed in each of the two appendixes to help in determining the order of input quantities. Disclaimer for Computer Programs
The programs in Appendixes 1 and 2 are intended for instructional purpose only. These programs are not sophisticated and are not meant to compete with commercial programs. The authors and the publisher are not responsible for any damages arising out of the use of these programs. The use of these programs to solve problems other than those displayed in these appendixes and correct or incorrect interpretation of the results obtained is the sole responsibility ~f the user.
American Concrete Institute (1975). "Specifications for Structural Concrete for Buildings," ACI Comm. 301. ACI, Detroit, MichigaJl. American Concrete Institute (1983). "Building Code Requirements for Reinforced Concrete," ACI Comm. 318-83. ACI, Detroit, Michigan. Major, A. (1980). "Dynamics in Civil Engineering: Analysis and Design," Vol. 2.
Akademiai Kiad6, Budapest. Prakash, S., and Gupta, D. C. (1970). Design and performance of a 1.55 t forging hammer foundation. J. Indian Natl. Soc. Soil Mech. Found. Eng. 9(2), 129-142.
593
APPENDIX
1
Computer Program for the Design of a Block Foundation This program calculates the undamped natural frequencies and damped or undamped amplitudes of vibratio~ p.f a rigid-block-type foundation for different mo'des of vibration usfii'if the elastic half-space analogs and the procedure of Section 6.8. The options available are:
....................................................................... Case l=vertical vibrations along Z axis occasioned by a force Pz Case 2=torsional vibrations about Z axis occasioned by a moment Mz Case3=translation along X axis and rotation about Y axis, occasioned by a horizontal force Px and a moment My Case 4=translation along Y axis and rotation about X axis, occasioned by
a force PY and a moment Mx •
0
0
0 0.
0
0
0
0
0
0 •••••••••••••••••••••••••••••••••••••••••••••••••••••••••••
Option !=undamped vertical amplitude (Case 1) Option 2=damped vertical amplitude (Case I) Option 3=undamped amplitude of torsional vibrations (Case 2) Option4=damped amplitude of torsional vibrations (Case 2) OptionS=undamped amplitudes for translation along X axis and rotation about Y axis ~Px+Mr; Case 3) Option6=damped amplitudes for translation along X axis and rotation about Y axis (Px+My; Case 3) Option 7=undamped amplitudes for translation along Y axis and rotation about X axis (Py+Mx; Case 4) Option 8=damped amplitudes for translation along Y axis and rotation about X axis (Py+Mx; Case 4)
595
596
APPENDIX 1
The input quantities are: W=weight of the foundation block including weight of the machine in tons A=area of the foundation in. contact with the soil in m2 L=height of the combined center of gravity of the machine and the foundation above the base in m Lx=maximum distance of the p~int where horizontal amplitude is to be calculated from the axis of rocking (this distance is to be given as input when excitation is due to a horizontal force Px and moment My. Lx will be measured parallel to X axis in m, Ly=maximum distance of the point where horizontal amplitude is to be calculated from the axis of rocking. (This distance is to be given as input when the excitation is due to a horizontal force P7 and moment Mx. Ly will be measured parallel to axis of Y in m. R=maximum horizontal distance of the point from Z axis where horizontal amplitude occasioned by torsional vibrations is to be calculated G=dynamic shear modulus in t/m 2 v=Poisson' s ratio lx=moment of inertia of the area of the foundation about an axis passing through its centroid and parallel to x axis in m4 ly=moment of inertia of the area of the foundation about an axis passing through its centroid and parallel to y axi~ in m4 lz=polar moment of inertia of the area of the foundation about a vertical axis passing through its centroid (Z axis) in m4 M~=mass moment of inertia of the foundation and machine about an axis
passing through the combined centre of gravity and parallel to X axis in t m/ sec 2 M~=mass moment of inertia of the foundation and machine about an axis
passing through the combined center of gravity and parallel to Y axis in t m/ sec 2 ~=polar mass moment of inertia of the foundation about a vertical axis passing through its center of gravity (Z~axis) in tm/sec 2 Px=horizontal unbalanced force parallel to X axis in tons Py=horizontal unbalanced force parallel to Y axis in tons Pz=vertical unbalanced force parallel to Z axis in tons Mx=vertical moment causing rotation about X axis in tm My=vertical moment causing rotation about Y axis in tm Mz=horizontal moment causing rotation about z-·axis (torsional vibrations) in tm
r:unit weight of soil in t/m3 QPA=allowable soil pressure in t/m2 N=operating speed of machine in rpm
h=HH=height of top of the foundation above the center of gravity of the system and Aa=permissible amplitude in mm
COMPUTER PROGRAM FOR THE DESIGN OF A BLOCK FOUNDATION
597
IMPLICIT REAL(A-Z) INTEGER I CHARACTER'! Q,Y,TITLE(120) CHARACTER'14 FILEOP DATA CASE1,CASE2,CASE3,CASE4/0.,0.,0.,0.I DATA & NO PI ,NOP2 ,NOP3 ,NOP4 ,NOPS ,NOP6,NOP7 ,NOPB/0, ,0, ,0. ,0. ,0. ,0. ,0. ,0.1 DATA YI'Y'i WRITE(',')'IIP 0/P-DATAFILE NAME' READ(',915)FILEOP 915 FORMAT(A14) OPEN( 2,FILE = FILEOP, STATUS= 'NEW') WRITE(',')'IIP PROBLEM TITLE(I)' READ(',6669) (TITLE(I),I=1,120) WRITE(2,6668) (TITLE(I),I=1,120) 6668 FORMAT( lOX, 'TITLE=' ,120AI,I,I,I,72('•'),11) 6669 FORMAT(120Al) WRITE(',') 'DO YOU WANT CASE !(TRANSLATION -Z)?' READ(',914)Q 914 FORMAT(Al) IF(Q.EQ.Y) CASEI=l.O WRITE(',') 'DO YOU WANT CASE 2(TORSIONAL -Z)?' READ(',914)Q IF(Q.EQ.Y) CASE2=1. WRITE(',') 'DO YOU WANT CASE 3(TRANSLATION -X,ROTATION -Y)?' READ(',914)Q IF(Q.EQ.Y) CASE3=1 WRITE(',')' DO YOU WANT CASE 4(TRANSLATION -Y,ROTATION -X)?' READ(',914)Q IF(Q.EQ.Y) CASE4=1.
c IF(CASEl.EQ.l)THEN WRITE(',*)'***********CASEl'***********'
WRITE(',')' DO YOU WANT THE UNDAMPED CASE?' READ(',914)Q IF(Q.EQ.Y)NOPl=l.O WRITE(',')' DO YOU WANT THE DAMPED CASE?' READ(',914)Q IF(Q.EQ.Y)NOP2=1.0 END IF _, IF(CASE2.EQ.l,THEN WRITE(* , *)'****'******CASE2************' , WRITE(',')' DO YOU WANT THE UNDAMPED CASE? READ(',914)Q IF(Q.EQ.Y)NQP3=1.0 WRITE(',')' DO YOU WANT THE DAMPED CASE?' READ(',914)Q IF(Q.EQ.Y)NOP4=1.0 END IF IF(CASE3.EQ.l)THEN WRITE(*,*)'***********CASE3*'**********'
598
APPENDIX 1
WRITE(',')' DO YOU WANT THE UNDAMPED CASE?' READ(',914)Q IF(Q.EQ.Y)NOPS=l.O WRITE(',')' DO YOU WANT THE DAMPED CASE?' READ(',914)Q IF(Q.EQ.Y)NOP6=1.0 END IF IF(CASE4.EQ.!)THEN WRITE(*,*)'***********CASE4************'
c
WRITE(',')' DO YOU WANT THE UNDAMPED CASE?' READ(',914)Q IF(Q.EQ.Y)NOP7=1.0 WRITE(',')' DO YOU WANT THE DAMPED CASE?' READ(',914)Q IF(Q.EQ.Y)NOP8=!.0 END IF
!IX, 'POISSONS RATIO, END IF
c
C ******TYPE* ******
WRITE(2,1!44) 1144 FORMAT(ISX,IS('•'), 'INPUT VARIABLES:',IS('''),/72('''),///) WRITE(',')'I/P A' READ(',')A WRITE(',')'l/P W' READ(',')W WRITE(2,11SS) A,W !ISS FORMAT( !IX, 'AREA OF THE FOUNDATION, A=',F10.4.,1X,'m2.',1/, 21X, 'WRIGHT of THE FOUNDATION',!, 31X, '(-including weight of the machine), W=' ,F10.4,1X, 't.' ,I)
NU=' ,F!0.3/)
C ******TYPE2******
IF(CASE4.EQ.l.O)THEN WRITE(',')'I/P IX' READ(',')IX WRITE(',')'I/P MX' READ(',' )MX WRITE(',')'I/P MMX' READ(',' )MMX WRITE(',')'I/P PY' READ(',' )PY WRITE(',')'I/P LY' READ(',') LY WRITE(2,1!66) IX,MX,MMX,PY,LY 1166 FORMAT( 11X, 'MOMENT OF INERTIA, 21X, 'UNBALANCED MOMENT,
C VARIABLE INPUT SECTION
WRITE(',')'I/P GAMMA' READ(',' )GAMMA WRITE(',')'I/P G' READ(',')G WRITE(',')'!/P N' READ(',')N WRITE(',')'I/P QPA' READ(',') QPA WRITE(2,1112) GAMMA,G,N,QPA 1112 FORMAT( !IX, 'UNIT WRIGHT OF THE SOIL, 2!X, 'DYNAMIC SHEAR MODULUS, 3!X, 'OPERATING SPEED OF MACHINE, 41X, 'ALLOWABLE SOIL PRESSURE,
COMPUTER PROGRAM FOR THE DESIGN OF A BLOCK FOUNDATION
IX=' ,F10.4,1X, 'm4. ',II, MX=' ,FlO. 4, lX, 't-m. ',' ',
3!X, 'POLAR MASS MOMENT OF INERTIA, MMX=' ,FI0.4,1X, 't-m.sec2' ,I 4!X,'UNBALANCED FORCE, '>r'ir PY=',FI0.4,1X,'t.',//, SIX, 'MAXIMUM DISTANCE OF THE'.POINT' ,I, 6'FROM THE AXIS OF ROCKING(HORIZONTAL),LY=' ,F!0.4,!X, 'm,' ,I) END IF C ******TYPE6******
IF(CASE4.EQ.!.O.OR.CASE3.EQ.!.O) THEN WRITE(',')'I/P L' READ(',')L WRITE(',')'I/P HH' READ(',') HH WRITE(2,1!77) L,HH 117 7 FORMAT( L=' ,F10.4, lX, 'm.' ,1 I, !IX, 'HEIGHT OF THE CENTER OF GRAVITY, h=' ,F10.4,tX, 'm.' ,I) 21X, 'HEIGHT OF THE TOP OF THE FOUNDATION, END IF C ******TYPE3******
GAMMA~,' ,F10.4, lX, 't/m3.', 11,
G=',F10.2,1X,'t/m2.',1/, N= I ,FlO. 2, lX; 'RPM'' I I'
QPA=' ,F!0.2,1X, 't!m2. ',/)
C ******TYPEl******
IF(CASEI.EQ.!.O.OR.CASE3.EQ.l.O.OR.CASE4.EQ.!.O)THEN WRITE(',')'I/P NU' READ(',')NU WRITE(2,19SS) NU !9SS FORMAT(
IF(CASE3.EQ.!.O)THEN WRITE(',')'I/P IY' READ(',')IY WRITE(',')'I/P MY' READ(',')MY WRITE(',')'Il'P MMY' READ(',')MMY WRITE(',')'I/P PX' READ(',') PX WRITE(',')'I/P LX' READ(',') LX WRITE(2,1!88) IY,MY,MMY,PX,LX 1100 FORMAT( IY=' ,FI0.4,1X, 'm4.' ,II, !IX, 'MOMENT OF INERTIA, MY=',F10.4,1X,'t-m.',l/, 21X, 'UNBALANCED MOMENT, 31X, 'POLAR MASS MOMENT OF INERTIA, MMY=' ,F10.4, lX, 't-m.sec2', I
599
APPENDIX 1
4!X, 'UNBALANCED FORCE, PX=' FlO 4 !X 't ' 11 ' ' ' ) ' ' SIX, I MAXIMUM DISTANCE OF THE POINT' I 6 ' FROM THE AXIS OF ROCKING(HORIZONTAL),LX=' ' ' F!0.4 !X 'm ' /) END IF
'
'
'
' '
C ******TYPE4****** IF (CASE2.EQ.l.O)THEN WRITE(',')'I/P IZ' READ(',')IZ WRITE(',')'I/P MZ' READ(',')MZ WRITE(',')'I/P MMZ' READ(',' )MMZ WRITE(',')'I/P R' READ(',') R WRITE(2, 1100) IZ ,MZ,MMZ,R 1100 FORMAT( l!X, ,'MOMENT OF INERTIA, IZ =, ' FlO •4,IX, ' m4. ' ,11, 2!X, , UNBALANCED MOMENT, MZ=' , FlO • 4, !X , 't- m. ' , 11 , 3 !X,,POLAR MASS MOMENT OF INERTIA MMZ=' ,F!0.4,1X, 't-m.sec2',/ 41X, MAX. HORIZONTAL DISTANCE(TORSION) R=' F!0.4 !X 'm ' f)
END IF C******TYPES******
'
'
'
'
. '
IF (CASE!.EQ.!.O)THEN WRITE(',')'I/P PZ' READ(',')PZ WRITE(2,!11!) PZ 1111 FORMAT( !!X, 'UNBALANCED FORCE, PZ=' ,F10.4,1X, 't. ',I) END IF
-cc..................................................................... .. . · ........ , .. , .. CASE ONE, .. , .. , .. , .. C .. · ....... TRANSLATION ALONG Z·AXIS . . ........ , . " .......... ,.,,., .. '. C .•........ NOPT=! UNDAMPED AMPLITUDE C ...•...... NOPT=2 DAMPED AMPLITUDE GA=9.8! ZZ=3.14!592654 IF(CASE!.EQ.O) GO TO 124 ROZ=SQRT(A/ZZ) KZ=4.'G'ROZ/(1.-NU) OMGNZ=SQRT(KZ'GA/W) FNZ=OMGNZ/(2'ZZ) OMEGA=2'ZZ'N/60. BZ=(l.-NU)'W/(4.'(ROZ''3)'GAMMA) ZETAZ=0.425/SQRT(BZ) IF(NOP!.EQ.O) GO TO 123 AA=PZ/(KZ'(l.-(OMEGA/OMGNZ)''2)) AZ=AA'!OOO. WRITE(2,!3) ROZ,KZ,OMGNZ,FNZ,AZ 13 FORMAT(I!J,20X, '***********UNDAMPED VERTICAL CASE**********'/ ; 1 15X, 'EQUIVALENT RADIUS, ROZ=' FlO 3 !X • • 11 ' ' '
. '
' m. '
'
COMPUTER PROGRAM FOR THE DESIGN OF A BLOCK FOUNDATION
25X, 'EQUIVALENT SPRING, 35X, 'NATURAL FREQUENCY, 45X,
I
601
KZ=' ,FlS.S,lX, 'tim.'.//, OMGNZ=' ,Fl0.3,1X, 'RAD/S' ,1,1 FNZ= I ,Fl0.3, lX, 'HZ. I ' I' I
55X, 'VERTICAL AMPLITUDE, AZ=' , F!S. 9, !X, 'mm. ' , 1//1/) 123 IF(NOP2.EQ.O) GO TO 124 PIN=OMEGA/OMGNZ AAD=PZ/(KZ'SQRT((!.-(PIN)''2)''2+(2'ZETAZ'PIN)''2)) AZD=AAD'!OOO. WRITE(2,14) ROZ,BZ,KZ,ZETAZ,OMGNZ,FNZ,AZD 14 FORMAT(///,20X,'**********DAMPED VERTICAL CASE**.********',I,!,/ 15X, 'EQUIVALENT RADIUS, ROZ=' ,F!0.3,1X, 'm,' ,II,
25X 'MASS RATIO, 35X,' 'EQUIVALENT SPRING, 45X 'DAMPING FACTOR, 55X,' 'NATURAL FREQUENCY(UNDAMPED), 65X
BZ=',F!0,3,1,1
KZ=' ,F15.5,1X, ' tim ' ,II, ZETAZ=',F10.3,1,/ OMGNZ=' ,F!0.3,1X, ' RAD/S ' ,1,1 FNZ=' ,F10.3,1X, 'HZ. I ,I ,I
I
75X:'VERTICAL AMPLITUDE,
AZD=' ,F15.9,1X, 'mm.' ,IIIII)
c...................................................................... . C ............. , .. CASE TWO .............................. ,,,, ........... , C ....... ,,, .. TORSIONAL VIBRA~J,Q!i ABOUT Z·AXIS C ..•........ NOPT=3 UNDAMPED:AMPLITUDE C ........... NOPT=4 DAMPED AMPLITUDE 124 IF(CASE2.EQ.O) GO TO 126 ROSI=(2'IZ/ZZ)''0.25 KSI=(!6.'G'ROSI''3.)/3, OMEGA=2'ZZ'N/60. OMGNSI=SQRT(KSI/MMZ) FNSI=OMGNSI/(2'ZZ) IF(NOP3.EQ.O) GO TO 125 ASI=MZ/(KSI'(l.-(OMEGA/OMGNSI)''2.)) WRITE(2,15) ROSI,KSI,OMGNSI,FNSI,ASI 15 FORMAT(///,20X, '***********UNDAMPED TORSIONAL CASE***********',!,
ll,i5X,'EQUIVALENT RADIUS, 2 5X,'EQUIVALENT SPRING, 3 SX, 'NATURAL FREQUENCY, 4/ /5X,' 5 5X, 'TORSIONAL AMPLITUDE, 125 IF(NOP4.EQ.O) GO TO 126 BSI=MMZ'GA/(GAMMA'ROSI''S) ZETASI=O.S/(1.+2.'BSI)
ROSI=',F!0.3,1X,'m.',i/ KSI=',F!5.5,1X,'tim',// OMGNSI=' ,F!0.3,1X, 'RAD/S', FNSI=' ,F!0.3,1X, 'HZ.' ,II ASI=' ,F!5.9, 'rad.' ,Ill)
ASID=MZ/(KSI'S~T((l,-(OMEGA/OMGNSI)''2)''2
+ (2.'ZETASI'OMEGA/OMGNSI)''2.)) WRITE(2,!6) ROSI,BSI,KSI,ZETASI,OMGNSI,FNSI,ASID 16 FORMAT(///,20X, '***********DAMPED TORSIONAL CASE**********' ,1,1,
115X, 'EQUIVALENT RADIUS, ROSI=' ,F10.3,1X, 'm.' ,II 2 SX 'INERTIA RATIO, BSI=' ,F!0,3,1,1 3 SX,' 'EQUIVALENT SPRING, KSI=' ,F!S.S,!X, ' tim ' ,II 4 5X 'DAMPING FACTOR, ZETASI=',F!0.3,i,/ 5 SX,' 'NATURAL FREQUENCY (UNDAMPED),OMGNSI=' ,F10.3,1X, ' RAD/S ' ,II 6
sx
I
7 SX,' 'TORSIONAL AMPLITUDE,
FNSI=',F10.3,1X, 'HZ.',!,!
ASID=' ,F15.9,1X, ' rad. ' ,II )
602
APPENDIX 1.
C , , , ............. CASE THREE .... ,,, ... ,,, .......... , ... ,,., .. , ...... , .. . C ... , , ..... TRANSLATION ALONG X-AXIS and ROTATION ABOUT Y-AXIS C .......••. NOPT=S UNDAMPED AMPLITUDE C .......... NOPT=6 DAMPED AMPLITUDE 126 IF(CASE3.EQ.O) GO TO 131 ROX=SQRT(AIZZ) ROPHIY=(IY'41ZZ)''0.25 MMOY=MMY+((WIGA)'L''2) BPHIY=(3.'(1.-NU)'MMOY'GA)I(8'GAMMA'ROPHIY''5) RIY=MMYIMMOY KX=32'(1-NU)'G'ROXI(7-8'NU) KPHIY=(8'G'ROPHIY''3)1(3'(1-NU)) OMGNX=SQRT(KX'GAIW) ONPHIY=SQRT(KPHIYIMMOY) W3=SQRT( ( (OMGNX' '2+0NPHIY' '2) IRIY)' '2) I RIY)' '2, -4, ' ( OMGNX' '2)' (ONPHIY''2,)/RIY) W6=(0MGNX''2+0NPHIY''2)1RIY X7=(W6+W3)12. X8=(W6-W3)12. ONI=SQRT(X7) ON2=SQRT(X8) FNI=ONII(2'ZZ) FN2=0N21(2'ZZ) OMEGA=2'ZZ'NI60. IF(NOPS.EQ.O) GO TO 127 Y5=(0MEGA''4-(0MEGA''2)'((0NPHIY''2+0MGNX''2)1RIY)+OMGNX''2'0NPHIY 1"21RIY) DELTA=YS M=WIGA PINA=(PX'L'OMGNX''21DELTA'MMY) MURA=(PXIM'MMY)'(-MMY'OMEGA''2+KPHIY+L''2'KX)IDELTA AO=(MYIMMY)'(OMGNX''21DELTA)+MURA AX=AO'!OOO, APHII=(MYIMMY)'((OMGNX''2-0MEGA''2)1DELTA)+PINA WRITE(2,17) ROX,ROPHIY,BPHIY,KX,KPHIY,OMGNX,ONPHIY,ON!,FN!,ON2,FN2 I ,AX,APHI! 17 FORMAT(II,ISX, '''''''''''UNDAMPED SLIDING AND ROCKING CASE'''''' 1 * • • • ', 11 ,35X, ' •... . x-z PLANE ..... ', 11,
25X, 'EQUIVALENT RADIUS, 35X,'
ROX=',F12.3,1X, 'm. ',II, ROPHIY=',F12.3,1X,'m.',;;,
45X, 'MASS RATIO, 55X,'EQUIVALENT SPRING,
BPHIY=',FI0.3,11, KX=',FI5.5,!X,'tlm',ll,
65X,'
KPHIY=',FlS.),lX,'t/m',l/,
75X, 'NATURAL FREQUENCY, 85X,' 95X, 'COUPLED NATURAL FREQUENCY,
OMGNX=' ,FI0.3,1X, 'RADIS' ,II, ONPHIY=' ,FIS.S,IX, 'RADIS' ,II, ONI=',FI0.31X, 'RADIS' ,II,
@SX,'
FNl=',Fl0.3,1X,'HZ.',I/,
:JFX,
ON2=',F10.3,1X, 'RADIS',I!,
I
$5X,' %5X, 'SLIDING AMPLITUDE, &SX, 'ROCKING AMPLITUDE,
FN2=',Fl0.3,1X,'HZ.',II, AX=',F15.9,1X, 'mm.,ll, APHI=' ,F15.9,1X, 'rad',!!l/11)
COMPUTER PROGRAM FOR THE DESIGN OF A BLOCK FOUNDATION
603
127 IF(NOP6.EQ.O) GO TO 128 BX=((7-8'NU)'W)1(32'GAMMA'(ROX''3)'(1-NU)) ZETAX=.27BSI(SQRT(BX)) ZIPHIY=O.l51((1.+BPHIY)'SQRT(BPHIY)) Ul=(ONPHIY''2+0MGNX''2 -4.'ZIPHIY'ZETAX'OMGNX'ONPHIY)IRIY U2=(0MEGA''4-0MEGA''2'Ul+OMGNX''2'0NPHIY''21RIY)''2 U3= (ZETAX'OMGNX'OMEGA'(ONPHIY''2-0MEGA''2)1RIY) U4= ZIPHIY'ONPHIY'OMEGA'(OMGNX''2-0MEGA''2)1RIY U5=4'(U3+U4)"2 DELTE=SQRT(U2+U5) M=WIGA PIN=(PXIM'MMY)'((-MMY'OMEGA''2+KPHIY+L''2'KX)''2+4'0MEGA''2'(ZIPHI !Y'(SQRT(KPHIY'MMYO))+L''2'ZETAX'SQRT(KX'M))''2)''0.2SIDELTE MUR=(PX'LIMMY)'SQRT(OMGNX''2+4'ZETAX'OMEGA''2)'0MGNXIDELTE AXX=(MYIMMYO'(SQRT(OMGNX''4+4'ZETAX''2'0MGNX''2)1DELTE)+PIN AXD=AXX'IOOO. APHIDI=(MYIMMY)'(SQRT((OMGNX''2-0MEGA''2)''2+(2'ZETAX'OMGNX'OMEGA) 1''2)1DELTE)+MUR WRITE(2,18) ROX,ROPHIY,BX,BPHIY,KX,KPHIY,ZETAX,ZIPHIY,OMGNX, 1 ONPHIY ,ON! ,FNI·t-ON2,FN2,AXD,APHIDI 18 FORMAT(/ II ,ZOX, '*"'*'******-~'DAMPED SLIDING AND ;ROCKING CASE****** 1**** 1 II 37X,' ..... X-Z PLANE ..... ',!! ' ' ' ' II 2 SX,'EQUIVALENTRADIUS, ROX= ' ,FlO. 3 ,I X,,m.,,
3 SX ' ROPHIY=', FlO. 3, IX, m. , II 4 SX,' 'MASS RATIO, BX=,' ,F10.3, I I 5 SX 'INERTIA RATIO BPHIY= ,FI0.3,11 ' ' II KX= I ,Fl5,5,1X, I t I m, 6 SX' 'EQUIVALENT SPRING ' 7 SX ' ' RPHIY= I ,F15.5,1X, I t I m' , II ' 8 SX,' 'DAMPING FACTOR, ZETAX=,,F10.3,11 9 SX 'DAMPING FACTOR, ZETAPHIY= ,F10.3,11 * si. 'NATURAL FREQUENCYOMGNX=',FI0.3,1X, 'RADIS',II $ SX ', UNDAMPED ONPHIY=' ,F10.3,1X, 'RADIS' ,I I @ SX 'COUPLED NATURAL FREQUENCY- ON!=' ,FI0.3,1X, 'RADIS' ,II 1 SX ', UNDAMPED FNI=',F10.3,1X, 'RADIS',II
~ sx',
,&
sx'•
% sx' 'SLIDING AMPLITUDE
ON2=',F10.3,1X,'RAD/S',fl
FN2=',Fl0.3,1X,'HZ.',I/
AXD=',F15.9,1X,mm.' ,II APHID=' ,FIS, 9, IX,' rad.', I I I I I)
sx: 'ROCKING AMPLITUDE: 128 CONTINUE C CASE FOUR ....... ,,,,, ..... ,,, ........... , ........... · .. • c : : : : : : : : : :TRANSUTi"6N ALONG Y-AXIS AND ROTATION ABOUT X-AXIS C ,,,,,,,,,,NOPT=7 UNDAMPED AMPLITUDE C ,,,,,,,,,,NOPT=8 DAMPED AMPLITUDE 131 IF(CASE4.EQ.O) GO TO 431 ROY=SQRT(AIZZ) ROPHIX=(IX'41ZZ)''0.25 MMOX=MMX+((WIGA)'L''2) BPHIX=(3,'(1.-NU)'MMOX'GA)I(8'GAMMA'ROPHIX''5) RIX=MMXIMMOX KY=32'(1-NU)'G'ROYI(7-8'NU) KPHIX=(8'G'ROPHIX''3)1(3'(1-NU))
bU4
APPENDIX 1
OMGNY=SQRT(KY'GAIW) ONPHIX=SQRT(KPHIXIMMOX) X3=((0NPHIX''2+0MGNY''2)/RIX)''2 X4=((4.'(0MGNY''2)'(0NPHIX''))IRIX) X5=SQRT(X3-X4) X6=(0MGNY''2+0NPHIX''2)/RIX X7=(0.5)'(X6+XS) XS=(0.5)'(X6-X5) ONI=SQRT(X7) ON2=SQRT(XS) FNI=ONI/(2'ZZ) FN2=0N21(2'ZZ) OMEGA=2'ZZ'NI60, IF(NOP7.EQ.O) GO TO 427 Y4=(0MEGA''4-0MEGA''2' (ONPHIX''2+0MGNY''2)/RIXtOMGNY''2'0NPHIX''2/RIX) DELTA=Y4 M=WIGA ATA=(PYIM'MMX)'(-MMX'OMEGA''2+KPHIX+L''2'KY)IDELTA ATAV=PY'L'OMGNY''21(MMX'DELTA) AQ=(MXIMMX)'(OMGNY''21DELTA)+ATA AY=AQ'IOOO APHI2=(MXIMMX)'((OMGNY''2-0MEGA''2)/DELTA)+ATAV
1****',1!,37X,' ..... Y-Z PLANE •...• ',!!
2 SX,'EQUIVALENT RADIUS,
'
'
' ..... Y-Z PLANE
' 6 5X, 'EQUIVALENT SPRING, 7
sx '
I
II
· , ,
t/m ·,I I
8 5X>NATURAL FREQUENCY KPHIX:>F15.3,1X, 'tim' ,II 9 5X ' ' OMGNY- ,FI0.3,1X, 'RADIS' II ONPHIX=' FlO 3 IX 'RA '' ' @ 5X,'COUPLED NATURAL FREQUENCY ON!=',FI0'3'Jx''RADDIISS',I/
* sx,
$ SX,'
I
% SX ' & sx:'SLIDING AMPLITUDE,
'
I '
•
'
'
'
FN1=,,F10.3,1X,'HZ.',II
II
ON2= ,F!0.3,JX, 'RADIS' ,II
FN 2 ~: ,F10.3, lX, :Hz.:, I 1
' 5X 'ROCKING AMPLITUDE AY- ,F15.9,JX, mm, ,II ' • APHI=' F!5 9 IX ' d • ) 427 IF(NOPS.EQ.O) GO TO 428 ' ' ' ' ra · ,Ill! BY=((7-8'NU)'W)I(32'GAMMA'(ROY''3)'(1-NU)) ZETAY=.2785/(SQRT(ABS(BY))) · ZIPHIX=O.l51((1.tBPHIX)'SQRT(BPHIX))
~~=i~~~~::~~~~~:;: 2•;U)IIR+OIMXG-NY4.:~ 2IPHIX'ZETAY'OMGNY'ONPHIXIRIX _ * *ONPHIX**2/RIX)**2
U3- (ZETAY OMGNY'OMEGA'(ONPHIX''2-0MEGA''Z)IRIX) U4= ZIPHIX'ONPHIX'OMEGA'(OMGNY'' 2-0MEGA'' 2)IRIX U5=4'(U3tU4)"2 DELTE=SQRT(U2tU5)
4 SX, 'MASS RATIO, 5 SX, 'INERTIA RATIO, 6 5X, 'EQUIVALENT SPRING,
BY=' ,FI0.3,11 BPHIX=' ,FI0.3,11 KY=' ,F15.5,1X, 'tim' ,II
7 SX,'
KPHIX:::',FlS.S,lX,'t!m',/1
8 5X, 'DAMPING FACTOR, ZETAY=' ,FJ0;3, II 9 5X, 'DAMPING FACTOR, •;.JIETAPHIX=' ,FI0.3,11 5X, 'NATURAL FREQUENCY· ,· OMGNY=' ,FI0.3, 'RADIS.' ,II $ 5X,' UNDAMPED ONPHIX=' ,FI0.3, 'RADIS.' ,II @ 5X, 'COUPLED NATURAL FREQUENCY· ON!=' ,FI0.3, 'RADIS.' ,II
*
··•·· ROY~' FlO 4 IX ' ' ' • ' , m. , I I ROPHIX=',FI0.4,JX,'m.' II BPHIX-' Fl ' KY:',F!50.53,1/XI ' '
'
ROY=',FI0.3,1X,'m.',ll ROPHIX=', FlO. 3, IX, 'm. ',II
3 SX,'
77 FORMAT(// lSX '**********UND 1* * **' II 3sx : AMPED SLIDING AND ROCKING CASE**"'"'**
sx, 'EQUIVALENT RADIUS, 3 SX ' s sx'•
605
M=WIGA SUK=(PY'L/MMX)'OMGNY'(OMGNY''2+4'ZETAY'OMEGA''2)''0.5 SU=(PYIM'MMX)'((-MMX'OMEGA''2+KPHIX+L''2'KY)''2+4'0MEGA''2'(ZIPHIX l'SQRT(KPHIX'MMXO)+L''2'ZETAY'SQRT(KY'M))''2)''0.5/DELTE AQD=(MX/MMX)'(SQRT(OMGNY''4+(4'ZETAY''2'0MGNY''2))1DELTE)+SU AYD=AQD'IOOO. APHIDE=(MX/MMX)'(SQRT((OMGNY''2-0MEGA''2)''2+(2'ZETAY'OMGNY'OMEGA) 1''2) IDELTE)+SUK WRITE( 2, 58) ROY,ROPHIX,BY ,BPHIX,KY ,KPHIX,ZETAY ,ZIPHIX,OMGNY, I ONPHIX,ON!,FNI,ON2,FN2,AYD,APHIDE 58 FORMAT('l' ,20X, '***********DAMPED SLIDING AND ROCKING CASE******
~~;~i~:~~~I~OY,ROPHIX,BPHIX,KY,KPHIX,OMGNY,ONPHIX,ONI,FNI,ON 2 , 2
COMPUTER PROGRAM FOR THE DESIGN OF A BLOCK FOUNDATION
! SX,'
UNDAMPED
+ 5X,' & 5X,' % 5X,'SLIDING AMPLITUDE, + SX,'ROCKING AMPLITUDE, 428 CONTINUE 431 CONTINUE
FNl=',Fl0.3,'HZ.',//
ON2=' ,FI0.3, 'RADIS' ,II FN2=' ,F10.3'HZ. ',II AYD=',F15.9,1X,'mm.',ll APHID=',F15.9,1X,'rad.',lllll)
c .................................................................... .. WRITE(2, 7901) 7901 FORMAT( Ill, 72('' '),I, 72('' '),Ill) IF(NOP2.EQ.l.O.OR.NOP4.EQ.J.O.OR.NOP6.EQ.l.O.OR.NOP8.EQ.l.O)THEN C ...................... TOTAL DAMPED AMPLITUDE ......................... .. C MAX HORIZONTAL AMPLITUDE DUE TO TORSIONAL VIBRATION AHSID=ASID'R'IOOO. C MAX.'VERTICAL AMPLITUDE DUE TO PZ and (PXtMY) AVDI=AZD+LX'APHIDI'IOOO. C MAX VERTICAL AMPLITUDE DUE TO PZ and (PY+MX) AVD2=AZD+LY' APHID!l":1000. C MAX HORIZONTAL AMPLITUDE DUE TO COMBINED ACTION OF PX AND MY AMDI=AXD +APHID I 'HH'IOOO. C MAX HORIZONTAL AMPLITUDE DUE TO COMBINED ACTION OF PY AND MX AHD2=AYD+APHIDE'HH'IOOO. WRITE(2,2234) ASID,AHSID,AVDI,AVD2,AHDI,AHD2 2234 FORMAT(20X, '**********TOTAL DAMPED AMPLITUDE**********' ,II,
@SX, 'MAX TORSIONAL AMPLITUDE, ASID=' ,FIS.9,1X, 'rad',ll ISX, 'MAX HORIZONTAL AMPLITUDE, (TORSIONAL)AHSID=' ,FIS.9,!X, 'mm. ',II 25X,'MAX VERTICAL AMPLITUDE, (PZ,PX+MY) AVDI=',F15.9,1X,'mm,',ll
606 APPENDIX 1
35X, 'MAX VERTICAL AMPLITUDE, (PZ,PY+MX) AVD2=' ,F15. 9, IX, 'mm. ',I I 45X, 'MAX HORIZONTAL AMPLITUDE, (PX and MY) AHDI=' ,F15.9,1X, 'mm. ',II 55X,'MAX HORIZONTAL AMPLITUDE, (PY and MX) AHD2=',F15.9,1X,'mm,',l! 611 II) END
IF
EXAMPLE 6.9.1 (CHAPTER 6)
Input Variables Area of the foundation
IF (NOPI.EQ.I.O.OR.NOP3.EQ.I.O.OR.NOP5.EQ.I.O.OR.NOP7.EQ.I.O) THEN
c ...................................................................... . C ..... , .... , , ........ TOTAL UNDAMPED AMPLITUDE, . , .. , ..... , ....... , .. , ... . C MAX HORIZONTAL DUE TO TORSIONAL VIBRATION AHSI=ASI'R C MAX VERTICAL DUE TO PZ and PX+MY AVI=AZ+APHII'LX'IOOO, C MAX VERTICAL AMPLITUDE DUE TO PZ and PY+MX AV2=AZ+APHI2'LY'IOOO. C MAX HORIZONTAL AMPLITUDE DUE TO COMBINED ACTION OF PX and MY AHI=AX+APHII'HH'IOOO. C MAX HORIZONTAL AMPLITUDE DUE TO COMBINED ACTION OF PY AND MX AH2=AY+APHI2'HH'IOOO. WRITE(2,2567) ASI,AHSI,AVI,AV2,AHI,AH2 2567 FORMAT(20X, '*'*'*"'"'**"'TOTAL UNDAMPED AMPLITUDE****"**"**',!/,
#5X, 'MAX TORSIONAL AMPLITUDE, 15X, 'MAX HORIZONTAL AMPLITUDE, (TORSIONAL) 25X,'MAX VERTICAL AMPLITUDE, (PZ,PX+MY) 35X, 'MAX VERTICAL AMPLITUDE, (PZ,PY+MX) 45X, 'MAX HORIZONTAL AMPLITUDE, (PX,MY) 55 X, 'MAX HORIZONTAL AMPLITUDE, (PY, MX) 61!111) END IF
COMPUTER PROGRAM FOR THE DESIGN OF A BLOCK FOUNDATION
ASI=' ,F15. 9, IX,' rad', I I AHSI=' ,F15,9,1X, 'nun.', I I AVI=',FI5.9,1X,'mm.',i! AV2=' ,F15. 9, IX, 'mm. ',I I AHI=',FI5.9,1X, 'mm. ',!! AH2=' ,F15, 9, IX, 'mm. ',I I
cc "'"'"'. * ****"'*****."'. ** * * * ** **********"'** * *. * * * ** *** ***********"' * * * * * *. * * * c
QP=W/A IF(QPA.LE.QP) GO TO 4455 WRITE(2,4445) QP,QPA 4445 FORMAT(68('•'),ii, I 5X,' SOIL PRESSURE, QP='FI0,4,1X, 'tim2. 'IX, 'and SMALLER THAN 2', I ,5X, 'THE ALLOWABLE SOIL PRESSURE QPA=' ,FI0.4, IX, 'tim2. ',I I ,68( '3• ')) GO TO 1477 4455 WRITE(2,4466) QP,QPA 4466 FORMAT(68('•'),/i, 5X, 'SOIL PRESSURE,QP=' ,FI0.4,1X, 't/m2. ',IX, 'and MORE THAN', I 25X, 'THE ALLOWABLE SOIL PRESSURE QPA=',FI0.4,1X, 'tim2. ',11,68('•')) 1477 STOP END
12.0000m'
A=
Weight of the foundation . (including weight of :he machlne) Unit weight of the so1l
37.1000t 1.6500 tim' G= 5000.00 tim' N= 600.00 rpm QPA= 20.00t/m2 v= 0.330 0. 2000 t P= ' W=
y=
Dynamic shear modulus . Operating speed of mach1ne
Allowable soil pressure Poisson's ratio, Unbalanced force
Damped vertical case
r,,=
1. 954m 0.504 k'=58340, 60000 tim ,:= 0.598
Equivalent radius Mass ratio
B=
Equivalent spring Damping factor
Natural frequency(undamped)
w11 z=
124, 203 rad/ sec
f,= Vertical amplitude Max.
Max. Max. Max. Max
19.768Hz O.Q03573769mm
A=
' Total Damped Amplitude . . de =O.OOOOOOOOOrad torslonal amplltu 00 horizontal amplitude (torsional) =0.0000000 mm . . d (P p +M) =0.003573769mm vertlcal amphtu e '' x ' ) -0 003573769mm vertical amplit~de (P,,P,+Mxd M) ooooooooomm
horizontal amplltude (Px an
'"y
=a·
-
•
Max.. horizontal amplitude (P, and Mx)
=O.OOOOOOOOOmm Sol'1 pressure, QP-3 - • 0917 tim' and smaller than 2 the allowable soil pressure QPA=20.000 0 t 1m
EXAMPLE 6.9.2 (CHAPTER 6)
Input Variables Area of the foundation Weight- of the foundation . (including weight of :he machlne) Unit weight of the s~ll Dynamic shear modulifil . Operating speed of mach1ne Allowable soil pressure Poisson's ratio
.
Height of the center of gravlty . Height of the top of the foundatlon Moment of inertia
Unbalanced moment Polar mass moment of inertia Unbalanced force,
A=
12.0000m'
38.5000t 1. 6500 tlm3 G=6000.00t/m2 N= 450.00 rpm QPA= zo .. ootim' v= 0.330 L= 0.9344m h= 1.2156m I= !6.0000m' y M,= 0.5000tm M.,= 4, 7051 t m/ sec' Px= O.OOOOt W= y•
607
APPENDIX I
bUH
Damped Torsional Case
Maximum distance of the point Lx=
from the axis of rocking(horizontal)
2.0000m
Damped Sliding and Rocking Case .... .x-z plane ••••. Equivalent radius rox= 1.954m roq,y= 2.125m Mass ratio B,= 0.636 Inertia ratio B,y= 0. 281 Equivalent spring kx= 57664.06000 tlmn k"=228990. 70000 tim Damping factor tx= 0.349 Damping factor t,y= 0.221 Natural frequency-undamped wnx= 121.215rad/sec wn¢y=
Coupled natural frequencyv undamped
wn 1 :::
f,,= w, 2= f,= Axi' A.,=
Sliding amplitude Rocking amplitude
Max. Max. Max. Max. Max. Max.
COMPUTER PROGRAM FOR THE DESIGN OF A BLOCK FOUNDATION
167.811 rad/sec 250.295 rad/s 39.836Hz 106.839 radl sec 17 .004Hz 0.002466732 mm 0.000002198 rad
Total damped amplitude =0. 000000000 rad horizontal amplitude (torsional) =0. 000000000 mm vertical amplitude (P.,P,+M) =0.004396916mm vertical amplitude (P~,P7 +~) =0. 000000000 mm horizontal amplitude (P, and M) =0.005139177m • ~ y =0. 000000000 mm hor1zontal amplitude (PY and Mx)
torsional amplitude
Soil pressure,. QP=3. 2083 t/m 2 and smaller than the allowable soil pressure QPA=20.0000 t/m 2
EXAMPLE 6.9.3 (CHAPTER 6) Input Variables
Area of the foundation
A=
Weight of the foundation (including weight of the machine) Unit weight of the soil
W=
12 .oooom'
36.0000 t !. 6500 t 1m3 Dynamic shear modulus G= 6000.00 tl m' Operating speed of machine 600.00 rpm N= Allowable soil pressure 20.00 tim' QPA= Moment of inertia 25.0000m4 J~= Unbalanced moment 0.5000tm M~= Polar mass moment of inertia, 5.4434 tmlsec 2 Mm~= Max. horizontal distance(torsion) 2.5000 m R= y=
Equivalent radius Inertia ratio Equivalent spring Damping factor Natural frequency(undamped)
Torsional amplitude Max. Max. Max. Max. Max. Max.
r,.=
1. 997 m 1.018 k,=254985. 30000 tim t,= 0.165 w,,= 216.432radlsec f,,= 34. 446Hz A,= 0.000002130 rad B,=
Total damped amplitude =0.000002130 rad torsional amplitude =0.005324474m horizontal amplitude (torsional) =0. 000000000 mm vertical amplitude (P.,P,+M) =0. 000000000 mm vertical amplitude (Pz,Py+M,) =0 •000000000 mm horizontal amplitude (P, and MY) =0. 000000000 mm horizontal amplitude (Py and M,) Soil pressure, QP=3 .0000 t/m and smaller th~n the allowable so~J,,Pressure QPA=20.0000 tim 2
609
APPENDIX
2
COMPUTER PROGRAM FOR THE DESIGN OF A HAMMER FOUNDATION
Computer Program for the Design of a Hammer Foundation
This program calculates the undamped natural frequencies and amplitudes of a hammer foundation soil system using the simplified model shown in
Fig. 7.4. Problem title
C •• A1=the area of the foundation block C •• A2=the area of the anvil in contact C .. AP=AP=area of piston in m2 C •• g=AG=acceleration due to gravity in C •• b=B=total thickness of the absorber
in contact with the soil in m2 with the absorber pad in m2 m/sec 2 pad below the anvil in m
C .• C=Option to use either Cu or G Values
C ••3 Cu=CUS=coefficient of elastic uniform compression of the soil in t/m for !Om' area. Option C=l. C .. E=YOUNG modulus of the pad material in t/m' C .. e=ER=coefficient of elastic restitution (suggested value=O.S) C ··~=efficiency of the drop (suggested value=0.65) C .• G=dynamic shear modulus of the soil in t/m
2
,
Option C=2.
C •. H=height of drop of the tup in m C .. v=Poisson's ratio of the soil C .. P=steam or air pressure in t/m 2
C .. q,=QPA=allowable soil pressure in t/m' C .. W,=weight of the foundation block (including weight of the frame if supported on the block) in t C .• W 2 =weight of the anvil in t
C •. Wo=weight of the tup (total falling weight including upper half of the die) in t C .. Zl =Z.. =permissible amplitude of anvil vibrations in mm C •• Z2 =Z£=permissible amplitude of foundation vibrations in mm
610
611
CHARACTER'14 FILEIP,FILEOP CHARACTER'! TITLE(30),YES,ASK REAL Kl,K2,NU,MU DATA YES/ 'Y' I 17 CONTINUE WRITE(',')' I/P INPUT DATA FILE NAME' READ(',987)FILEIP 987 FORMAT(A14) WRITE(',')' I/P OUTPUT DATA FILE NAME' READ(',987)FILEOP OPEN(5,FILE=FILEIP) OPEN(6,FILE=FILEOP,STATUS='NEW') READ(5,l)(TITLE(I),I=1,30) 1 FORMAT(30Al) READ(5,')Wl,W2,WO,Al,A2,B,H,AP,P,QA,QPA,C,PM,E,NU,ER,ETA,ZA,ZF AG=9.81 PI=3,14 WRITE( 6, 361) (TITLE( I), I= 1, ~-0), WRITE(6,36)Al,A2,AP,AG,B -- 361 FORMAT(/,/ ,lOX, 'THIS PROGRAM CALCULATES THE UNDAMPED NATURAL' ,I ,5~ 1 'FREQUENCIES AND AMPLITUDES OF VIBRATION OF A HAMMER FOUNDATION. ' ,SO('-'), II, lOX, 'TITLE=' ,30Al, I, 10X,38( ' =' ) , II ) 2,/ 36 FORMAT( 20X 'THE INPUT QUANTITIES ARE .... :' ,11,2X, 'THE AREA 3 OF THE FOUNDA~ION BLOCK IN CONTACT WITH THE SOIL Al=' ,F10.3,1X, 'm 42. ',11,2X, 'THE AREA OF THE ANVIL IN CONTACT WITH,THE,ABSORBER P~D 5 A2=',F9.4,1X,'m2',1/,2X,'AREA OF PISTON,',43X, AP= ,F10.4,1X, m 62.', II, 2X, 'ACCELERATION ON DUE TO GRAVITY,', 25X, 'g=AG=', F9. 4, lX,' 7m.!sec2',1,1,2X, 'TOTAL THICKNESS OF THE ABSORBER PAD BELOW THE ANVIL, I'
SX, 1 b=-B= 1 ,FlO. 4, lX, 1 m.
1 )
WRITE(6,77) E,ER,ETA,H 77 FORMAT(I,/,2X, 'YOUNG MODULUS OF THE PAD MATERIAL,' ,25X, 'E=' ,F10.2, l,lX, 't/m2' ,//2X, 'COEFFICIENT OF ELASTIC RESTITUTION(suggested value 5) ' ' ' 'e=E-R=' ' F5 3 ' I ' I ' 2X ' EFFICIENCY OF THE DROP (Suggested value. .,65), '13X, 'ETA=' ,F5.3,1,1,2X, 'HEIGHT OF DROP OF THE TUP,' ,33X,
.
1
4 1 H= 1 ,F10.4,1X, m. 1 )
WRITE(6,9876) NU~P,QA,QPA,Wl,W2,WO 9876 FORMAT(I,/,2X, 'POISSONS RATIO OF THE SOIL,' ,31X, 'NU=' ,Fl0.4,/,1,2X !,'STEAM OR AIR PRESSURE,',37X,'P=',F10.4,1X,'t/m2,',1,/,2X, 'ALLOWABLE SOIL PRESSURE ',31X, 'qa=QA=' ,F10.4,1X, 't/m2' ,1!,2X, 'ALLOWABLE STRESS IN THE' PAD BELOW THE ANVIL,' ,9X, 'qpa=QPA=' ,Fl0.4,1X, 't/m2.' 4 I I 2X 'WEIGHT OF THE FOUNDATION BLOCK,' ,27X, 'Wl=' ,F10.4, lX, 't. ', ' ' ' ' I I I 5/,1,2X,'WEIGHT OF THE ANVIL,',38X,'W2=',Fl0.4,1X, t. ,//,2X, WEIGHT
612
APPENDIX 2
OF THE TUP(Total falling weight including upper half of the die 7. ) '
I '
I' 60X, I WO=' 'Fl 0. 4' lX, It. I)
WRITE(6,2110) ZA,ZF 2110 FORMAT(I,2X, 'PERMISSIBLE AMPLITUDE OF ANVIL VIBRATIONS, ',IBX, 'ZA=', IFI0.4,1X, 'rom.' ,11,2X, 'PERMISSIBLE AMPLITUDE OF FOUNDATION VIBRATION 2S,', 13X, 'ZF=' ,F10.4, lX, 'mm. ',I)
IF(C.EQ.2) GO TO 611 WRITE(6,54) PM 54 FORMAT(2X, 'COEFFICIENT OF ELASTIC UNIFORM COMPR. OF THE SOIL CU=' ,F 110.4,1X,'tlm3',i;l,80('-')) GO TO 939 611 WRITE(6,610) PM 610 FORMAT(2X, 'DYNAMIC SHEAR MODULUS OF THE SOIL, ',24X, 'g=',F9.4,1X, I 'tim2', II ,80('-' )) 939 WRITE(6,901) 901 FORMAT(10X,' .•••••• THE RESULTS ARE ......• ',1,80('-'),1,) c *"'"'"'"" * *" ************* * * •••••••••• ***************** * * * •••• ************ C •...... CHECK FOR SOIL PRESSURE Q=(W1tW2)/A1
c
* * * * ••••• *. * * * * *. * ••••••••• * * *. * •• * * ••••••• * * ••• * * * * •••••• *
*. ******** *
C •... CALCULATIONS OF EQUIVALENT SPRINGS KI,K2 C •... K1=EQUIVALENT SOIL SPRING C .... K2=EQUIVALENT SPRINGS OF PAD BELOW THE ANVIL IF(C.EQ.2) GO TO 31 CUS=PM IF(A1.GE.10) CU=CUS GO TO 445 IF(AI,LT.IO) GO TO 93 93 CU=SQRT(AI I 10.) 'GUS 445 KI=CU'AI WRITE(6,86) CUS, CU 86 FORMAT(3X, 'COEFFICIENT OF ELASTIC UNIFORM COMPRESSION FOR 10 SQUAR E METERS AREA ',i,48X, 'GUS=' ,E10.4,11,3X, 'COEFFICIENT OF ELASTIC UNIFORM COMPRESSION FOR THE SIZE OF THE FOUNDATION' ,1,49X,' 3CU=' ,E10.4,11) GO TO 32 31 G=PM RO=SQRT(AII PI) K1=(4'G'RO)I(I-NU) WRITE(6,87) RO 87 FORMAT(3X, 'EQUIVALENT RADIUS OF THE FOUNDATION, RO=' ,E10. 13, 'm.' ,I)
32 K2=(E/B)'A2 WRITE(6,3) K1,K2
COMPUTER PROGRAM FOR THE DESIGN OF A HAMMER FOUNDATION
613
3 FORMAT(3X, 'EQUIVALENT SOIL SPRING, Kl=' ,E10. 14,1X, 'tim.' ,11,3X, 'EQUIVALENT SPRING FOR THE PAD BELOW THE ANVIL 2K2=' ,EI0.4,1X, 'tim.' ,I)
c *********** ••• *. * ••••••••••••• *. ******** ••••••••••••• *'*** ••• ****** ••• C ••• CALCULATION OF LIMITING NATURAL FREQUENCIES (OMG)NLI AND (OMG)NL2 C , .. LIMITING NATURAL FREQUENCY OF THE ENTIRE SYSTEM OF SOIL=(OMG)NLI C ••. LIMITING NATURAL FREQUENCY OF THE ANVIL ON THE PAD=(OMG)NL2 OMGNLI=SQRT((KI'AG)I(WitW2)) OMGNL2=SQRT(K2'AGIW2) WRITE(6,435) OMGNLI,OMGNL2 OMGNLI=' ,EIO, 435 FORMAT(3X, 'LIMITING NATURAL FREQUENCY 0 14,1X, 'radlsec.' ,11,3X, 'LIMITING NATURAL FREQUENCY 2MGNL2= ',EI0.4, IX, 'rad/sec', /) C ... NATURAL FREQUENCIES OF THE RAMMER FOUNDATION SOIL SYSTEM (OMG)NI and C ... (OMG)N2 MU=W2/WI OI=OMGNLI 02=0MGNL2 OPP=(ItMU)'(01''2+02''2) OPF=SQRT(((ItMU)'(01''2+02''2))''2-4'(1tMU)'(OI''2)'(02''2)) PIN=O.S'(OPPtOPF) OMGNI=SQRT(PIN) FNI=OMGNI/(2.'PI) PINA=O.S'(OPP-OPF) OMGN2=SQRT(PINA) FN2=0MGN21(2.'PI) WRITE(6,41) MU 41 FORMAT(3X, 'THE RATIO OF THE WEIGHT OF ANVIL AND FOUNDATION BLOCK IMU='FI0.4,/)
c
***** * •••••• * •• ****** ••• **********************"**** * * * ***** •••••••
C •••. AMPLITUDES OF VIBRATION OF THE ANVIL AND FOUNDATION C .... AMPLITUDE OF FOUNDATION BLOCK=ZI C •.•. AMPLITUDE OF THE ANVIL=Z2 C ...• VELOCITY OF TUP AT THE TIME OF IMPACT=VT C , ... VELOCITY OF THE ANVIL JUST AFTER IMPACT=VA VT=SQRT( (2, ;AG'H' (WOtP' AP) IWO)) 'ETA SU=W2/WO . VA=((I.+ER)/(I.+SU))'VT WRITE(6,5) VT,VA 5 FORMAT(3X, 'VELOCITY OF THE TUP AT THE TIME OF IMPACT,' ,ISX, 'VT=',EI0.4,1X, 'misec' ,11,3X, 'VELOCITY OF THE ANVIL IMMEDIATELY AFTER IMPACT, VA=',EI0.4,1X, 'm/sec',i/80('''),/) SUR=(OI)"2
"·
614
APPENDIX 2
SURA=(02)"2 Zl=ABS( ( SURA-PINA)' (SURA-PIN) 'VA/ ( (PIN-PINA) • SURA 'OMGN2)) Z2=ABS( (SURA-PIN) 'VA/ ( (PIN-PINA) '0MGN2)) c * *' ********* * * * * * ********** * ** * *.'' ********"' * *' * *' ******* ** * *' * ******* C.••..... COMPARISION OF AMPLITUDES OF MOTION WITH PERMISSIBLE VALUE C WRITE(6, 74) C 74 FORMAT(80('•')) C IF(Zl.LE.ZF) GO TO 193 C WRITE(6,476) C476 FORMAT(5X, '! .... ,THE AMPLITUDE OF THE FOUNDATION VIBRATION (Zl) IS C IMORE',i,5X,'THAN THE PERMISSIBLE VALUE .... UNSAFE.',/,1) C GO TO 194 C!93 WRITE(6,466) C466 FORMAT(5X, '! ..••• THE AMPLITUDE OF THE FOUNDATION VIBRATION (Zl) IS C !LESS' ,1,5X, 'THAN THE PERMISSIBLE VALUE ••• ' ,1,1) C194 CONTINUE C IF(Z2.LE.ZA) GO TO 198 C WRITE(6,874) C874 FORMAT(5X, '2 .... THE AMPLITUDE OF ANVIL VIBRATION (Z2) IS BIGGER C !THAN' ,1,5X, 'THE PERMISSIBLE VALUE ... UNSAFE. ',1,1) C GO TO 121 C198 WRITE(6,873) C873 FORMAT(5X, '2 ••.•.. THE AMPLITUDE OF ANVIL VIBRATION (Z2) IS SMALLER C !THAN' ,1,5X, 'THE PERMISSIBLE VALUE ... ' ,1,1) Cl21 WRITE(6,75) C 75 FORMAT(82('•'))
c * ******'. * * * ******** * * * * * ** ''********** *. * * * *. * ******* * * * * * * * * * ******* C••••.... STRESS IN THE PAD BELOW THE ANVIL qp=QP QP=(K2'(Zl-(-Z2)))/A2 C WRITE(6,761) QP C761 FORMAT( lOX, 'THE COMPRESSIVE STRESS IN THE PAD BELOW THE ANVIL IS Q C l=',FI0.4,/,80('-')) C IF(QP.LE.QPA) GO TO 555 C WRITE(6,554) QP C554 FORMAT( lOX, 'THE COMPRESSIVE STRESS IN THE PAD BELOW THE ANVIL IS Q C l=',EI0.4, 'AND IS BIGGER THAN THE PERMISSIBLE VALUE' ,I,) C STOP C555 WRITE(6,553) QP C553 FORMAT( lOX, 'THE COMPRESSIVE STRESS IN THE PAD BELOW THE ANVIL' 1 C 15X, 'IS QP=',E!0.4, 'AND IS SMALLER THAN THE PERMISSIBLE VALUE',;,;) c •••• * * * * •••••••• ** * ** ••••••••••••••• * * * * •••••••••••• * * * * * * * * * ••••••••• WRITE(6,2) Q 2FORMAT(IOX, 'THE SOIL PRESSURE, Q=',F!0.4,1X, 'tim2 !. ',I)
WRITE(6,605) OMGNl,FNl,OMGN2,FN2
COMPUTER PROGRAM FOR THE DESIGN OF A HAMMER FOUNDATION
615
605 FORMAT( lOX, 'NATURAL FREQUENCY,' ,12X, '(OMG)Nl=' ,FI0.4,1X,'rad/sec' !,II ,lOX, 'NATURAL FREQUENCY, FNI=' ,FlO. 4,1X, 'HZ.' 2,fi,!OX, 'NATURAL FREQUENCY, (OMG)N2=' ,FI0.4,1X, 'rad/sec 3,' ,II, !OX, 'NATURAL FREQUENCY, FN2=' ,FlO .4,1X, 'HZ.' 4,/)
ZMI=IOOO. 'Zl WRITE(6,88) ZMI 88 FORMAT( lOX,' AMPLITUDE OF THE FOUNDATION BLOCK Zl=',EI0.4,1X,'mm.' I, I)
ZM2=1000. 'Z2 WRITE(6,111) ZM2 Ill FORMAT( lOX, 'AMPLITUDE OF THE ANVIL
Z2=' ,EI0.4,1X, 'mm.'
I, I)
c ..................................................................... . WRITE(6, 761)QP 761 FORMAT( lOX, 'THE COMPRESSIVE STRESS IN THE PAD BELOW THE ANVIL IS Q IP=',F9.4,1X,'tim2',/i,80('•'),//) IF(Q.LE.QA) GO TO 107 WRITE(6,!2) Q,QA 12 FORMAT(3X, 'THE SOIL PRESSURE IS Q=' ,EI0.4,1X, 'tim2' ,2X, 'and IS !MORE THAN' ,1,3X, 'THE ALLOWABLE SOIL PRESSURE QA= • ,ElO. 4, lX, 't/m2 .... 2.,,, .. UNSAFE!!!' ,II)
GO TO 533 107 WRITE(6,207) Q,QA 207 FORMAT(3X, 'THE SOIL PRESSURE IS Q=' ,EI0.4,1X, 't/m2.' ,2x, 'and ISLE ISS THAN' ,1,3X, 'THE ALLOWABLE SOIL PRESSURE QA=' ,EI0.4,1X, 't/m2, ', 2'., .. , .SAFE!' ,II) 533 IF(ZMI.LE.ZF) GO TO 193 WRITE(6,476) ZMI,ZF 476 FORMAT(3X, 'AMPLITUDE OF THE FOUNDATION VIBRATION IS Zl=' ,EI0.4, 'mm I.' ,1,3X, 'and IS MORE THAN THE PERMISSIBLE VALUE ZF=' ,EI0.4,1X, 'mm. 2..... UNSAFE!!!' ,I,/) GO TO 194 193 WRITE(6,466) ZMI,ZF 466 FORMAT(3X, 'AMPLITUDE OF THE FOUNDATION VIBRATION IS Zl=',EI0.4, 'mm I. ',1,3X, 'and IS LESS THAN THE PERMISSIBLE VALUE ZF=',EI0.4,1X, 'mm. 2.,,,, .SAFE!' ,II)"'· 194 IF(ZMZ.LE.ZA) GO TO 198 WRITE(6,874) ZM2,ZA 874 FORMAT(3X, 'AMPLITUDE OF ANVIL VIBRATION IS Z2=' ,EI0.4,1X, 'mm. and liS MORE THAN' ,1,3X, 'THE PERMISSIBLE VALUE ZA=' ,EI0.4,1X, 'mm .. , .UN ZSAFE! ! ! ! ' ,II) GO TO 121 198 WRITE(6,873) ZM2,ZA
616
APPENDIX 2
873 FORMAT(3X, 'AMPLITUDE OF ANVIL VIBRATION IS Z2=',E10.4,1X, 'mm and liS LESS TRAN' ,1,3X, 'THE PERMISSIBLE VALUE ZA=' ,E10.4,1X, 'mm ... SAFE 2! ',I I) 121 IF(QP.LE.QPA) GO TO 555 WRITE(6,994) QP,QPA 994 FORMAT(3X, 'THE COMPRESSIVE STRESS IN THE PAD BELOW THE ANVIL IS QP l=',E10.4,1X, 'tim2. ',! ,3X, 'and IS MORE TRAN THE PERMISSIBLE VALUE QP 2A=' ,E!0.4, !X, 'tim2 •.. UNSAFE!!!!', I ,80( '•')) STOP 555 WRITE(6,553) QP,QPA 553 FORMAT(3X, 'THE COMPRESSIVE STRESS IN THE PAD BELOW THE ANVIL IS QP 1=' ,E10,4,1X, 'tim2. ',I ,3X, 'and IS LESS TRAN THE PERMISSIBLE VALUE QP 2A=' ,E!O. 4, !X,' t/m2. . ... SAFE!', I I ,80(' • ')) WRITE(',') 'CONTINUE? ENTER EITHER A"Y" OR A "N".' READ(',S12) ASK IF(ASK.EQ.YES) GO TO 17 512 FORMAT(Al) STOP END
EXAMPLE 7.4 (CHAPTER 7)
The input quantities are:
Area of the foundation block in contact with the soil Area of the anvil in contact with the absorber pad Area of piston Acceleration due to gravity, Total thickness of the absorber pad below the anvil
A,= 48.000m2
4.0000m' 0.1400m2 g=AG= 9.8100m/sec' A,= A,=
b=B= 0. 6000 m Young modulus of the pad material E=SOOOO.OO tim' Coefficient of elastic restitution (suggested value .5), e=ER= 0.500 Efficiency of the drop (suggested value .65), ~· 0.650 Height of drop of the tup H= 0.7500m Poisson's ratio of the soil v= 0.3300 Steam or air pressure p= 70.0000 tim' Allowable soil pressure q,=QA= 25.0000 tim' Allowable stress in the pad below the anvil, q,,=QPA= 400.0000 t/m 2 Weight of the foundation block W,= 266.8800 t Weight of the anvil W,= 45.5000 t
COMPUTER PROGRAM FOR THE DESIGN OF A HAMMER FOUNDATION
617
Weight of the tup (total falling weight including W,= upper half of the die.), Permisible amplitude of anvil vibrations Z2= Za=
2.0000t
Permissible amplitude of foundation vibrations Z1= Zt=
1. 2000mm.
Dynamic shear modulus of the soil
l.SOOOm
G= 4600.0000t/m2
The results are: Equivalent radius of the foundation Equivalent soil spring Equivalent spring for the pad below the anvil Limiting natural frequency Limiting natural frequency Ratio of the weight of anvil and foundation block Velocity of the tup at the time of impact Velocity oFthe anvil immedfateJ.yr after impact Soil pressure Natural frequency Natural frequency Natural frequency Natural frequency Amplitude of the foundation block Amplitude of the anvil Compressive stress in the pad below the anvil
r 0 =3.91m k,=0.1074x 10 6 t/m k,=O. 3333 x !0 6 tim
w," =0. 5807 x 102 rad/ sec wn 12 =0.2681
x 10 3 rad/sec
~=0.1705
V,=6.056m/sec VA=0.382Sm/sec q= 6.5079t/m2
w,,=29!.0662 rad/ sec f,= 46.3481Hz
w,,=
57.8632 rad/sec
f,= 9.2139Hz Z,= 0.9954mm
Z2 = 1.044mm q,=169.9560t/m2
The soil pressure is q=6.508t/m2 and is less than the allowable soil pressure q,=25. 00 t/m2 .... ,,. Safe! Amplitude of the foundation vibration is Z,=0.9954mm and is less than the permissible value Z,=1.200mm ....... Safe! Amplitude of anvil vibration is Z2=1.044mm and is less than the permissible value Z,=l.SOOmm ... Safe! The compressive stre~ in the pad below the anvil is q,=0.1700xl0 3 t/m2 • and is less than the permissible value q,,=O. 4000 x 10 3 tim'. .. .. Safe! CASE HISTORY OF THE HAMMER FOUNDATION IN CHAPER 13 (SECTION 13.2)
The input qUantities are: Area of the foundation block in contact with the soil
A,= 37.050m2
618
Area of the anvil in contact with the absorber pad Area of piston Acceleration due to gravity
Total thickness of the absorber pad below the anvil Young modulus of the pad material
APPENDIX 2
A,= A= p g=AG=
2.4000m2 0.1320m2 9.8100misec 2
b=B= 0.4000m E=50000.00 tim'
Coefficient of elastic restitution
(suggested value .5), Efficiency of the drop (suggested value .65), Height of drop of the tup Poisson•s ratio of the soil Steam or air pressure Allowable soil pressure
Allowable stress in the pad below the anvil Weight of the foundation block Weight of the anvil Weight of the tup (total falling weight including upper half of the die) Permissible amplitude of anvil vibrations
e=ER=
COMPUTER PROGRAM FOR THE DESIGN OF A HAMMER FOUNDATION 2
q= 5.8737tim w01 =321. 7216 radisec
The soil pressure Natural frequency Natural frequency Natural frequency Natural frequency
0. 650 0.9000m v= 0.3300 P= 79.8800 tim 2 q,=QA= 31.2000 tim' ~=
H=
q,,= 400.0000 t/m 2
W1= 183.1200t 34.5000 t
W,=
I. 5500 t
Z,=
2. 0000 mm
Z,=
I. 2000 mm
Coefficient of elastic uniform compression of the soil
C.,= 6100.0000 tim'
Z2= 0. 9586 mm
Com·pressive stress in the pad below the anvil is
q,= 225.6349 tim'
Equivalent spring for the pad below the anvil Limiting natural frequency Limiting natural frequency
Ratio of the weight of anvil and foundation block
'
kl=
Amplitude of the foundation vibration is Z1=0.8465mm and is less than the permissible value Zt=1.200mm • •..... Safe! Amplitude of anvil vibration is Z2 =0.9586mm and is less than
the permissible value Z,=2.000mm .•• Safe! The compressive stress in the pad below the anvil is qP=O. 2256 X 10 3 t/m2 and is less ,than the permissibl~;/Yllue Qpa=0.4000xl0 3 t/m2 • • • • • Safe!
0.6100x 10 4 tim3 0.6100X10 4 tim 2 0. 2260 x 10 6 tim
0.3000X 10 6 tim 0.1009 x 10 3 radi sec - {t)nll = 0.2921x 10 3 radisec Wn12= k,=
~=
0.1884
Velocity of the tup at the time of
impact Velocity of the anvil immediately after impact
Z1= 0.8465mm
The soil pressure is q=5.874t/m2 and is less than the allowable soil pressure Q8 =31. 20 t/m 2 • • • , , •• Safe!
compression for the size of the C=
= 99.8921 rad/sec
Amplitude of the foundation block Amplitude of the anvil
The results are:
foundation Equivalent soil spring,
= 51.2295Hz
f 02 = 15.9064 Hz
W,=
Gus=
[ 01
W112
0.500
Permissible amplitude of foundation vibrations
Coefficient of elastic uniform compression for 10=m2 area Coefficient of elastic uniform
619
VT=
7.630misec
VA=
0.492lmisec
""'
APPENDIX
3
Brief Description of Some Available Computer Programs
BRIEF DESCRIPTION OF SOME AVAILABLE COMPUTER PROGRAMS
621
The group action of piles is generally (1988) accounted for in the program by using static interaction factors. The soil reactions on the sides of the pile cap are also taken into consideration in evaluating the performance of the pile group. For further details, reference may be made to the users' manual (Novak et al., 1981).
REFERENCES Novak, M., and Aboul~Ella, F. (1978a). Impedance functions for piles embedded in layered media. J. Eng. Mech. Div., Am. Soc. Civ. Eng. 104 (EM-3), 643-661. Novak, M., and Aboul-Ella, F. (1978b). Stiffness and damping of piles in layered media. Proc. Earthquake Eng. Soil Dyn., Am. Soc. Civ. Eng., Spec. Conf., Pasadena, CA, Vol. 11, 704-719.
Novak, M., and Howell, J. F. (1978). Dynamic response of pile foundations in torsion,
PILAY 2
The computer program PILAY 2 can be used to calculate the dynamic stiffness, damping, internal forces, and displacements for a vertical pile undergoing vibrations in any mode. The theoretical basis for this program is provided by the published works of Novak and Aboul-Ella (1978a,b), Novak and Howell (1978), Novak and Sheta (1980), and Novak et al. (1978). The program can be used for a pile embedded in a uniform soil layer or in a layered soil medium. The pile may have a constant cross section or a stepwise variabl~ cross section and may have a fixed or pinned head. The pile can project above the ground level. The pile is considered to be perfectly bonded to the soil, and void elements are used to simulate any separation between the pile and the soil. The degree of fixity of the pile tip as well as the damping of the pile material are also considered. The program can take into account up to 30 different horizontal soil layers, each of which is assumed to be linearly elastic. Soil properties required to characterize a layer are the shear wave velocity v;, unit weight y, Poisson's ratio v, and the material damping (;. The complex modulus approach has been used to define the amount of material damping in the soil. The variation of soil properties in a deep homogeneous deposit is accounted for by subdividing the deposit into several layers of . small thickness, each with different but constant characteristics. The change in the pile section is considered by introducing auxiliary interfaces. The soil below the pile tip is defined by its shear modulus G •. The effects of soil remolding, nonlinearity or pile separation are accounted for by considering a weakened (or stiffened) zone around the pile. The program can also be used for embedded rigid-block-type foundations. 620
Geotech. Eng. Div., Am. Soc. Civ. Eng. 104 (GT-5), 535-552. Novak, M., and Sheta, M. (1980) Approximate approach to contact effects of piles. Proc. Dyn. Response P;ile Found.: Anal. Aspea,..g-.J'i._.,A.m. Soc. Civ. Eng., Hollywood, FL, 53-79. Novak, M., Nogami, T., and Aboul~Ella;
STRUDL-11
STRUDL II is a multipurpose program capable of solving a wide variety of problems involving linear elastic, static frame analysis, finite element analysis, nonlinear analysis, frame optimization, dynamic analysis, and proportioning and design of reinforced concrete structures. The capabilities of STRUDL for conducting static and dynamic analysis of linear elastic structures undergoing small displaGements make it a very important tool in the design of frame foundations for supporting turbogenerator units. The frame can be analyzed as a plane or a space frame. The static stiffness
+ [C]{i} + [K]{x} = F(t)
(1)
in which [M] is the mass matrix of the system, [ CJ is the damping matrix, [K] is the stiffness matrix, and F(t) is the time-dependent load function.
&22
APPENDIX 3
{x}, {x}, and {i} are the time-dependent displacement, velocity, and acceleration vectors respectively. The equation of motion (1) can be solved in two ways, by direct integration or by modal analysis by transforming the system to a new coordinate system to yield uncoupled linear equations. The results of modal analysis yield eigen values and mode shapes. The response may then be determined by modal superposition. The results of response analysis are the dynamic displacements and forces. The dynamic loads can be specified in the form of initial conditions,. time histories, or response spectra. An approximate frequency analysis can be carried out by Rayleigh's method. The soil effects can be modeled with equivalent springs. The results of static and dynamic analysis may then be combined to obtain the design conditions.
REFERENCES ICES STRUDL II (1979a). "Engineering Users' Manual," Vol. 1. Frame Analysis. School of Civic Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts. ICES STRUDL II (1979b). "Engineering Users' Manual," Vol. 2. Additional Design and Analysis Facilities. School of Civil Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts.
ADINA
ADINA is a finite element program for computing the static and dynamic displacements and stresses induced in solids and structural systems. Both the linear and nonlinear problems can be solved. ADINA provides the facility of using a variety of element configurations such as two-dimensional plane strain and plane stress elements, three-dimensional truss elements, threedime.nsional plane stress elements, three-dimensional solid elements and isoparaffietric beam elements, etc. The nonlinearity in behavior may be induced by large displacements, large strains, and material property. Material model used may be linear elastic, nonlinear elastic, elastoplastic, curve description model, etc. The linear dynamic analysis can be used to determine the frequencies of the system and the response is evaluated by using mode superposition method or by direct time integration using the Newmark or Wilson method, or central difference method. The nonlinear analysis is performed by using an incremental approach. The soil behavior can be represented using Drucker Prager-Cap model. Tension cutoff may be provided if desired. The output can be asked at any point. Further details of this program may be obtained by referring to the users' manuals.
BRIEF DESCRIPTION OF SOME AVAILABLE COMPUTER PROGRAMS
&23
REFERENCES ADINA (1984a). "Theory and Modelling Guide," Rep. No. AE 84-4. ADINA Engineering, Inc., Watertown, Massachusetts. ADINA (1984b). "Users Manual," Rep. AE 84-1. ADINA Engineering Inc., Watertown, Massachusetts. Bathe, K. J. (1982). "Finite Element Procedures in Engineering Analysis." Prentice~Hall, Englewood Cliffs, New Jersey.
APPENDIX
625
COMPUTATION OF MOMENT OF INERTIA
4
Mass moment of inertia
Shape of the Element
Formula for
'
Computation of Moment of Inertia
Mm,
Figure
Rectangular block
rrFx I
Th
1
I
I
y
~a~:_j
r ....._' '--
--1-
/ ---, y
'-----"
m(b'+h')
j?'
CG.
Circular block
Im,
Mmy
X ~
12
m (a ' 12
+ h' )
m (a ' 12
b
T
m ed' ) 12 4 +h'
h
l
f-d~
m ed'
12 4+h
')
'"·~:'·
Illustration of Calculation of Mass Moment of Inertia Moment of inertia of area
Shape of the Area
Rectangle
Given Data Formula for Figure
I,
I,
ab'
ba'
12
12
1. Concrete block as shown in Fig. (A-4-1) I,
ab(a'
+ b')
12
2. 3. 4. 5.
Machine mounted symmetrically on the block Weight of machine Wm Height of machine CG above the top of the block h 1 Unit weight of concrete y.
Required
Solution
Circle
Divide the concrete ~lock into three parts as shown in Fig. (A-4-1) Mass m 1 ~ (ax 1 )(a, 1 )(a>~) I g Hollow circle
.!:_ d') 64 (d'0 ;
Mass m 2
~
m3
~
(ax 2 )(a, 2 )(a, 2 ) -'Y g
Mass of the machine m 4 ~ Wm
g
624
+ b' )
md' 8
626
APPENDIX 4
627
COMPUTATION OF MOMENT OF INERTIA
M
rnyl
= mass moment of inertia of mass m 1 about an axis parallel to y-axis and passing through its own centroid + transfer to parallel axis about the common centroid of the system
(a) Elevation
Similarly
Mrnyz
=
Mrny3
+ Mm 4
=
mz 2 2 J2 (ax,+ a,,)
m,[ (a, + az, - L )' + (axz- X) 1
= m 4 (0) +"'iii,(a, +a,,+ h 1 -
L)
2
]
2
M
= mass moment of inertia of the foundation and machine about an axis parallel to y-axis and passing through center of gravity of the base contact area M = M + mL 2 in which m = m 1 + m 2 + m 3 + m 4 my ' . M myO= Polar mass moment of iner!la of the mach'me f ound at10n sys t em m' about an axis parallel to z-axis and passing through the combined center of gravity myo
X (c) Plan
Figure A~4.1.
Layout of block and machine.
= Mmz1 = Mmzl
Coordinates of Combined Center of Gravity
Because of symmetry of the system needs to be calculated
-
X= ax 1 12; Y = aY 1 /2,
and only
i
!m.z. !m,
Z=L=--'-'
m 1 a, 1 /2 + 2m 2 (a, + a, 2 /2) + m 4 (a, 1 + a, 2 + h 1 ) m 1 +2m 2 + m 4 Mass Moment of Inertia
Mrny =mass moment of inertia of foundation and machiue about an axis parallel to y-axis and passing through the common centroid of the system Mmy = Mmyl + Mmy2 + Mmy3 + Mmy4 = Mrnyl
+ 2Mrny2 + Mrny4
M mz1
+ Mmz2 + Mmz3 + Mmz4 + 2Mmz2 + Mz4 ,
.
=polar mass moment of inertia of mass m 1 about an axis parallel to z-axis and passing through its own centrm'd + any trans f er moment of inertia to the z-axis through the combined centroid
628
APPENDIX 4
Similarly 2 M mz2 =Mmz3 -_m 12
(
2
ax2
5
APPENDIX
') + ay2
+m,[ (az, -X)'+ (a~, - y) '] _ mz z -12 (ax,+ ay2 2 )
+ m,[ ( a, _ az 1 ) ' + ( a~ 2
2
_ mz
_
a~ 1 ) ' ]
z
-12 (ax,+ ay 2 ) 2
+ m2 (ax22
ax1)' 2
Conversion Factors
To
To Convert from
and
Multiply by
Length
M m''-
m, (O' + O') + m, (a2xi 12
-)' -X
'"":';millimeters (mm) meters (m) meters (m)
inches (in) inches (in) feet (ft)
=0
25.4 0.0254 0.3048
Area square square square square
2
square square square square square
inches (in ) feet (ft') yards (yd 2 ) miles (mile')
centimeters (cm 2 ) meters (m2 ) meters (m 2 ) 2 meters (m ) kilometers (km')
6.4516 0.0929 0.8361 4047 2.59
Volume
cubic inches (in. cubic feet (ft')
3
cubic centimeters (em 3 ) cubic meters (m 3 )
)
Mass
pounds (lb) tons (ton)
kilograms (kg) kilograms (kg)
0.4536 907
Force newtons (N) newtons (N)
one pound force (J4lf) one kilogram force (kgf)
4.4482 9.8066
Pressure or Stress
pounds per square foot (psf) pounds per square inch (psi)
kilogram force per square centimeter (kgflcm')
kilonewtons per square meter (kN I m 2 ) or kilopascals (kPa) kilonewtons per square meter (kN lm') or kilopascals (kPa) kilonewtons per square meter (kN I m 2 ) or kilopascals (kPa)
0.0479 6.895
98.066
629
630
APPENDIX 5
Convers!on Factors
To Convert from
To
Multiply by
Notation
Liquid Measure cubic meters (m 3 ) cubic meters (m 3 )
gallon (gal) acre-feet (acre-ft)
0.0038 1,233
Quantity of Flow gallons per second (gal/sec) cubic feet per second (ft' I sec)
cubic meters per second (m 3 /sec) cubic meters per second (m 3 /sec)
0.0038 0.0283
Mass Density pounds per cubic feet (pcf)
megagrams per cubic meter (Mgim')
0.0157
Definition
Symbol
A
foundation bas~Atfea in contact with the soil cross-sectional area of the joint between the condenser and the turbine exhaust
AD
Aim
A Mm
A, A, A, A< Ah A;(r) AP Ar A" Ax Ax! A"' A, Azl' Az2
Au A• A•,
Unit
L' L'
in Eq. (4.39)
L'
area of the foundation area of the pad in contact with the anvil cross section area of the beam cross-section area of the column maximum horizontal amplitude amplitude of ith mass in mode shape "r" net area of the piston vertical area of the trench perpendicular to direction of wave propagation maximum amplitude in vertical vibrations maximum amplitude of vibrations in horizontal direction horizontal displacement occasioned by rocking maximum horizontal amplitude of vibrations of an emb~llded foundation maximum amplitude in vertical vibrations amplitudes of vertical vibration of two-degrees-offreedom system maximum amplitude of vertical vibrations of an embedded footing maximum amplitude in rocking maximum amplitude of the embedded foundation in rocking
L' L' L' L' L L' L' L L
L L L
L L
631
632
NOTATION
Symbol
Definition maximum amplitude of vibrations in yawing (torsional vibrations) maximum amplitude of the embedded foundation in torsional vibrations A,1,Mm)(meemre) =dimensionless amplitude of torsional vibration with quadratic excitation length of foundation one-half of the depth of the beam for a frame without haunches horizontal distance from Y axis of rotation
A,,
A,l,e a. a
Unit
C,
L L
r0 (w!V,) ~ r,(w!Vb) ~ r,wyPTG ~dimensionless
a, at2• at3
aJ a, a, a,
B B, B, B• B• b
c [C]
cr, cz Cr, Cz
c" c"' c", C,ot, C, 2 cxl'
cx2
cx1'
cx2
frequency factor coefficients in the flexibility matrix distance of jth mass from the mass center dimension along x axis dimension along y axis dimension along z axis Vertical acceleration width of trench width of foundation modified mass ratio in sliding modified mass ratio in vertical vibrations inertia ratio in rocking vibrations inertia ratio in torsional vibrations width of foundation mass ratio thickness of pad one-half of the column width for a frame without haunches numerical base shear coefficient damping matrix integration constants frequency-dependent parameters of vertical vibrations frequency-independent parameters for vertical vibrations coefficient of elastic uniform compression equivalent value of coefficient of elastic uniform compression for an embedded footing coefficient of elastic uniform compression for the side layer dimensionless parameters of half space frequency-dependent parameters for horizontal translation frequency-independent parameters for horizontal translation
c,, c.,
L
c. c<~>l' c.P2
c~l,
c,, L L L LT-' L L
c. c>frl' co/12
c., c cc c, I
cw
c~ c~
c, c~
c; FL _,
c;' ex.
FL - 3 FL _,
coefficient of elastic resistance of pile pile stiffness at resonance coefficient of elastic uniform shear equivalent value of coefficient of elastic uniform shear for an embedded footing coefficient of elastic uniform shear on the sides of the footing coefficient of elastic nonuniform compression frequency-dependent functions of the elastic half space for rocking vibrations frequency-independent values of c>1 and c>2 equivalent coefficient of elastic nonuniform compression for the embedded foundation coefficient of elastic nonuniform compression for side layers coefficient of elastic nonuniform shear frequency dependent elastic half space stiffness
and damping
c"ll' coJ>2
L
L
c>2
c.,
c,,
L
Definition
Symbol
c, c," L
633
NOTATION
I
c,. c, c'X c. c' t c.
pat~eters
Unit FL-' FL-' FL - 3 FL - 3 FL _, FL- 3
FL - 3 FL- 3 FL- 3
for torsional vibration
frequency indepen"dent values of c>/11 and co/12 equivalent coefficient of elastic nonuniform shear of embedded foundation coefficient of elastic nonuniform shear on the sides of the foundation coefficient of internal damping damping constant critical damping damping parameter constant of equivalent viscous damping of one pile in vertical vibrations constant of equivalent viscous damping of pile cap in vertical vibrations damping coefficient of pile group damping coefficient in H-sliding damping constant of single pile in H-translation constant of equivalent viscous damping of pile cap in translation damping constant of pile group in H-translations cross coupled damping factor for coupled rocking and s'Tfding (see Eq. 11.40a) cross damping constant of a single pile damping coefficient in vertical vibrations equivalent geometric damping ratio for pile group in vertical vibrations damping in rocking mode damping constant of single pile in rotation damping coefficient of pile cap in rocking
FL- 3 FL- 3 FL-'T FL-'T
FL-'T FL-'T FL-'T FL-'T FL-'T
FL-'T
Fr 1
FL-'T FL- 1 T FL-'T FLT FLT
634
Symbol
c•" ct c, D D~ d dN!dt
E E, E" Ep E. e
e
em
F
Fo Ft, Fz
F' F" FA FAX FB FT Fr(A) F, Fol> Fa2
F" Fd Fp F, F,
Definition
critical damping in rocking damping constant of piles or footing in torsion constant of equivalent viscous damping of one pile in torsional vibrations distance between .centerlines of pistons geometric damping ratio for a single pile diameter of the wire of a helical spring rate of change of speed Young's modulus bulk modulus constrained modulus Young's modulus of pile total strain energy, area under the hysteresis loop eccentricity eccentricity of unbalanced mass horizontal distance of point of rotation of mass me from the center of gravity of footing base of natural logarithms coefficient of elastic restitution voids ratio eccentricity of rotating mass exciting force, maximum value displacement function lateral force due to earthquake compliance function force transmitted, maximum value modified displacement function primary force secondary force unbalanced force due to rotation of mass lumped at the crankpin centrifugal force in the direction of x axis acting at the crankpin force due to translation of mass lumped at the piston head thermal load complex torsional stiffness parameter dynamic compliance function (a= z, x, y, 1/>, 1/J, xrf>, yi/>) real and imaginary components of Fa centrifugal force due to rotation of equivalent mass Ma 1 of crank rod force of damping inertia force due to translatory motion of the piston force transmitted to the support or foundation vertical soil reaction on the sides
635
NOTATION
NOTATION
Unit
Symbol
FLT FLT
frequency functions centrifugal fOrce in the direction of X force in the direction of X primary component of Fx F'X secondary component of Fx F"X unbalanced force in the x direction due to Fxi operation of ith cylinder centrifugal force in the direction of z axis F, unbalanced force in the direction of z axis due to Fzi operation of ith cylinder frequency (in Hz) f displacement functions for vertical vibrations f, !2 natural frequency f" two natural frequencies of coupled rocking and U"n)" (!",,), sliding of an embedded footing natural frequency in horizontal sliding f"x natural frequency of sliding vibrations of fnxe embedded found,at!on natural frequency'' in vertical vibrations f"x natural frequency of vertical vibrations of fnze embedded foundation natural frequency in pure rocking f". natural frequency of rocking vibrations of fne embedded foundation natural frequency in yawing f •• natural frequency of torsional vibrations of fmf'e embedded foundation torsional stiffness and damping parameters, fn,rz respectively vertical stiffness and damping of pile fwl> fwz horizontal (sliding) stiffness and damping f~l' ~~2 parameters of a pinned head pile cross stiffness and cross damping parameters fx>l' fx>2 parametrs of a pinned head pile rocking stiffness and damping parameters of a pile f.,t.2 shear modulus of soil G shear modulus at limiting strain 'Yr G, shear modulus beyond the limiting strain 1', G2 G*=G 1 +iG2 complex shear modulus of soil real and imaginary parts of complex shear Gt,z modulus of soil shear modulus of soil beneath the pile tip G, maximum value of shear modulus Gmax shear modulus of pile Gp shear modulus of the soil on the sides of the G, embedded footing
FLT L L
y-' FL- 2 FL - 2 FL- 2 FL - 2 L L L
L F F F
F F F F F F
F F
F F
F
F, Fx
Definition
Unit
F F F F F F
y-' yyy-' y-' y-' y-' y-' y-' y-'
FL - 2 FL-' FL- 2
FL- 2 FL-2
FL- 2
636
NOTATION
Symbol g H
Hz
h
Definition
Unit
acceleration due to gravity height of the foundation block, thickness of the elastic layer depth of trench Hertz effective height of column distance of top of foundation above its center. of
LT-'
gravity
h, h, I
I, I, Ic Ia Ip IR I, I, I,
drop of the tup depth of embedment height of the column from the top of the base slab to the center of the frame beam clear height of the column moment of inertia of the base contact area about the axis of rotation moment of inertia of pile cross section impulse in Eq. (2.65) importance factor 3.46 II Mmo in Eq. ( 4.39) moment of inertia of the beam moment of inertia of the column coeffi,cient of shear modulus increase with time polar moment of inertia of the area impedance ratio moment of inertia of the area about the x axis moment of inertia of the area about the y axis polar moment of inertia of the area
Vi lo, lt
J, K
[K] K, K,(w) Kal> Ka2
K'
Bessel functions of first kind of order 0 and 1, respectively polar moment of inertia of the base contact area relative stiffness factor Impedance function l'!G(EPA) ~dimensionless constant factor in Eq. (12.15) stiffness matrix coefficient of earth pressures at rest frequency dependant impedance function, a= x, y, z,x>,y>, >1/1 real and imaginary parts of Ka EpA. E G/10 q. (12.30) b
KW
k k, k,
L L T-' L L L L L L L' L'
FT L' L' FL-' L' L' L' L'
Symbol
kequ
k, kh< kij
k" kw
k~
k: k'w kx k' k{' k; kx• k~
k, k, kt k1 k, k'
•
L'
kt k. L
r,
kilowatts spring constant spring constant of soil for hammer foundation total vertical stiffness of both columns of a
FL _, FL - I
637
NOTATION
l,
I,
Definition
transverse frame equivalent spring of the pad below the anvil equivalent spring lateral stiffness of all frames (Eq. 8.19) combined stiffness of all the frames in bending stiffness coefficient total vertical spring stiffness stiffness of pile in vertical direction stiffness constant of one pile in vertical direction stiffness constant of pile cap in vertical direction stiffness constant of pile group in vertical direction stiffness constant for translation along x axis equivalent spring constant of the soil in horizontal X direction spring constant of single pile in translation spring constant of pile cap in translation stiffness constant of pile group in translation cross coupled stiffness factor for coupled rocking and sliding cross spring stiffness of single pile stiff constant for translation along y axis spring constant in vertical vibrations equivalent spring constant of the soil in vertical direction spring constant in rocking vibrations spring constant of single pile in rocking spring constant of pile cap in rocking spring constant of pile group (piles only) in rocking complex stiffness of soil-pile system torsional stiffness of footing stiffness constant of one pile in torsion distance of center of gravity of the system from the base length of the spring length of trench length of pile, thl'&ness of soil layer effective span distance from centerline length of connecting rod any distance center-to-center distance between adjacent columns clear distance between columns
Unit
FL FL-' FL _, FL- 1 FL- 1 FL- 1 FL- 1
FL-' FL- 1
FL-' FL-'
F
FL_, FL _,
FL FL
FL L L
L L L L L L L
638
Symbol
I, M M= M 0 cos wt
Definition
distance of crankpin from center of gravity of connecting rod distance of piston head from center of gravity of connecting rod moment
excitation moment mass matrix M, mass of eccentric [or 50% of crankshaft mass (see Table 9.1)] M 0 = meemrcw 2 amplitude of moment M for quadratic excitation Ml moment due to displacement of center of gravity MA total mass lumped at the crankpin M'A total mass lumped at the crankpin for cylinder i M,, moment of soil reaction at the sides due to C M-. moment of soil reaction at the sides due to c""s ML magnification factor !/Is MR moment due to soil reaction M•• moment of the soil reaction at the base due to C MW megawatts "' M,l equivalent mass of crank rod lumped at the crankpin M., mass of the connecting rod lumped at A M,, mass of crank rod lumped at piston head MB total mass lumped at piston head mass of moving (crushing) jaw (see Table 9.1) MBi total mass lumped at the piston head for cylinder i M, mass of the connecting rod lumped at B M. mass of connecting rod (Table 9.1) mass of the crank acting at center of gravity mass of the connecting rod M"' Md total mass of counterweights (see Table 9.1) M, moment due to inertia force, torque Mm mass moment of inertia about an axis through the center of gravity of the system MmO mass moment of inertia of the machine and foundation about the horizontal axis passing through the contact area of the base Mmx polar mass moment of inertia of the system Mp mass of piston, piston rod and crosshead M,, short-circuit moment MStf! moment of soil reaction at the sides due to C Mw moment due to displaced position of center gravity of machine-foundation system Mx unbalanced moment about X 'axis M'X primary moment about X axis
NOTATION
NOTATION
Unit
Symbol M~
M, M,(t)
L M' M"'y M, M;
[M]
o't
M~
m
FL FL -IT2 FL - 1 T 2 FL FL FL FL
m, ml
m, m.
FL - 1 T 2 FL -ty2 FL-IT' FL -tT2
m, m, N
FL -IT2 FL -tT2 FL - 1 T 2 FL - 1 T 2 FL - 1 T 2 FL FLT'
NG Nx(f) N(z) N,(t) N•(t) N,(t)
n FLT' FLT' FL - 1 T 2 OCR
FL FL FL FL
p
P, PA
639
Definition secondary moment about X axis unbalanced moment about Y axis time-dependent applied moment inducing rocking about Y axis primary moment about Y axis secondary moment about Y axis unbalanced moment about Z axis primary moment about Z axis secondary moment about Z axis mass of footing mass of pile and static load on pile rotating mass mass of the main shaft including mass of crushing cone [Eq. (9.2a)] mass of foundation block mass of pile per unit length mass of camshaft, gears, and counterweights [Eq. (9.2b)] mass of anvil mass of the cap plus machinery or portion of the structure vibrating in phase with the cap unbalanced rotating mass eccentric mass jth mass dynamic factor magnification factor operating speed in rpm speed standard penetration value normalized shear modulus increaSe with time dynamic horizontal soil reaction at the base axial force in the pile dynamic vertical reaction along the sides resisting moment due to soil reactio!l at the sides moment of the resistiv~ force on the sides of the foundation about z axis degrees of freedom of a multidegree system IlJ,l;mber of cycles number of cylinders number of vanes in the impeller overconsolidation radio concentrated load load on pile sum of loads due to machine, condenser, pipes, and normal torque vertical load power transferred by coupling A
Unit FL FL FL FL FL FL FL FL FL -IT- 2
FL -tT2
FL -IT 2
FL -IT- 2
T_,
F FL FL
F F F F
KW
640
Definition
Symbol
PB Pc PI P, px
power transferred by coupling B power transferred by coupling C plasticity index horizontal dynamic force
horizontal (shear) force on a plane
P,(t) p
PP P2 p" Pc
Q Q,
q
R
horizontal exciting force load due to vacuum in condenser maximum unbalanced force in vertical direction
vertical component of resultant inertia force time~dependent vertical force vertical stress vertical soil reaction along a pile air pressure atmospheric pressure vacuum pressure in condenser vertical exciting force vertical maximum force uniformly distributed load per unit area radius of plate x~ + y~ = distance from pile to centroid of
V
footing
Ro R, Ri R, R,(t) R,(t) R,(t) R,(t) R<(t)
soil reaction acting at pile tip distance from source of vibration to centerline of trench vertical soil reaction at the base lateral reaction of jth frame horizontal resisting force time-dependent horizontal soil reaction along xaxis time dependent horizontal soil reaction in the direction of y-axis time-depen~ent vertical ·soil reaction at the base time~dependent resisting moment due to soil reaction at the base time~dependent moment of the resistive force at the base of the footing about Z axis
r
crank radius horizontal distance eccentricity radius of gyration VM) m radius of rotor in Eq. (8.5) frequency ratio wlwn, flfn
r,
NOTATION
Unit
Symbol
KW KW
641
distance between center of gravity of the main shaft and crusher cone from the axis of crusher distance from axis of rotation to center of gravity of counterweights or crank distance between crusher axis and center of
r,
r, F F F F F F F FL _, FL-' FL -z FL _, FL_,
gravity of the camshaft frequency ratios
r1,2 r' r,
lever arm for rotating mass crarik radius of ith cylinder equivalent radius in slidin_g
fox•'oy
s
.
L L L
L L L L L L
parameter depending on soil type in Fig. 4.8
frequency-dependent parameters of the side layer for vertical .:vfQration frequency-i~de-_pendent parameters of side layer for vertical vibration clear distance between adjacent piles frequency dependent dimensionless parameters of vertical resistance of soil along a vertical pile
sl, sz sl> s2
L
sw1,w2
s" sx1'
sl> s<~>z
for horizontal sliding frequency-independent parameters of side layer for horizontal sliding .
sl' Sq, 2 sl/>1' s,,,z
F
sl/>1' s,,,z
L
frequency-dependent parameters of the side layer
sx2
sx1' sx2
L F F F
.
equivalent radius for vertical vibrations equivalent radius in rocking vibrations equivalent radius in torsional vibrations distance between geophones
r "' r,_ r,.
F F FL _,
F F
Unit
Definition
F
horizontal component of resultant inertia force (Table 9.1)
P, P,
NOTATION
frequency-dependent parameters of the side layer for rocking vibrations frequency-independent values of sl' and s2 frequency~dependent side layer parameters for torsional vibrations frequency-independent values of SIJ1 1 and S,,_ 2 for torsional vibrations
F
s
constant in Eq. (2.24) ratio of W2 /W, in Eq. (7.11b)
FL
s, s(z, t)
elastic settlement time~dependent soil reaction per unit length on
L
.q,'!>rtical side of the footing
F T FL FL FL FL FL T
FL L L L L
T
tiine period torque
TA TB Tc
torque due to high pressure (H.P.) turbine torque due to intermediate pressure (J.P.) turbine torque due to low pressure (L.P.) turbine
T" Tc T,(t)
transmissibility
r,
effective radius of one pile equivalent radius
L
radius of circular contact area
L
torque due to generator natural period time dependant soil reaction for rocking vibrations
642
Symbol
T•(t)
u u
u v v, v1
vl, v2
VR
Vn
v. v,
v: vc vp V,
v, v
w w, w. WL WT
w. we
wm W, w
w, w1.2
ww w(z) w(x, t)
X X,
Definition time dependant soil reaction for torsional vibrations time thickness of vibration absorbing pad displacement amplitude of pile displacement function displacement in x direction velocity in x direction velocity initial velocity velocity of rebound of the tup volume [in Eq. (10.6)) Rayleigh's wave velocity initial velocity of the tup velocity of the anvil VG,Ip, =shear wave velocity of soil beneath pile tip longitudinal or compression wave velocity in infinite medium VEP/pP =longitudinal wave velocity in pile shear wave velocity of pile longitudinal wave propagation velocity in rod shear wave velocity \[GTjJ =shear wave velocity of soil adjacent to pile displacement in y direction weight, total load, weight of machine minus the moving crosshead weight of the tup weight of the transverse beam load transferred by the longitudinal beams to the columns total weight of deck slab and machine weight of anvil weight of the two columns constituting a transverse frame weight of machine and bearing rated capacity of turbogenerator unit in megawatts vertical displacement weight per unit length amplitude of vertical vibration of footing real and imaginary parts of displacement displacement in Z direction complex amplitude of pile vibration at depth z complex pile displacement function at depth z axis of X distance of mass center from the end horizontal soil reaction
NOTATION
NOTATION
Unit
Symbol
XC xcj
T L L
X,
X
L LT- 1 LT- 1 LT- 1 LT- 1 L' LT- 1 LT- 1 LT- 1 LT- 1 LT- 1 LT- 1 LT- 1 Lr' LT- 1 LT- 1
L
F F F
X
x, xP .i;p
ip x, y
Yo, Yt Y,
y
Yo Y,
z
{Z} z, z1 z, z, zc z,
F F F F
zmax
Z, Z,
(Z,), z
F L FL - t
i {i) ii
{ii) IX
L L
a,
F
at, O'z
643
Definition distance of stiffness center from end distance of centerline of frame j from the end projection of all external forces acting on the foundation on the x-axis force in the direction x-axis axis of x horizontal displacement or distance displacement at the base displacement of the piston in the direction of x axis velocity of the piston in the direction of x axis acceleration of the piston in the direction x axis coordinate of pile axis of Y Bessel functions of the second kind of order 0 and 1, respectively force in the direction of y axis displacement ~~•..:-:."maximum value of y horizontal coordinates of pile axis of Z, maximum amplitude of simple displacement vector initial displacement maximum amplitude of hammer foundation maximum amplitude of the anvil or absorber, amplitude vertical soil reaction height of center of gravity of pile cap above its base projection of external forces on the z axis maximum amplitude amplitude of frequency w, (see Fig. 4.41) static deflection static deflection of an elastic layer displacement in vertical direction displacement of simple harmonic motion ve~~ity
veloCity vector acceleration acceleration vector coefficient of attenuation coefficient in Fig. 8.6 crank angle 1 + e'!r' [see Eq. (8.40)] phase angle correction factor terms defined by Eqs. (!1.65) and (11.66)
Unit L L
F F L L L LT- 1
F L L
L L L L L
F L
F L
L L L LT- 1 LT-' L-1
644
Symbol ex. ex. exL ex,, ex, {3 {3, {32 'Y
'Y, y, 'Yxy 'Yn
'Y,, y, 0
B BL 5stat
a,, a"
dE.
a, a, a,
dz1 dz 2
.6.z3
az, A
AR IL
A Al,A2
Ao A, €
•
Definition axial displacement interaction factor for a typical reference pile in a group phase angle, a= x, y, z, t/J, 1/J lateral displacement 'interaction factor for a typical reference pile in a group horizontal seismic zone coefficient crank angle for the ith cylinder soil foundation factor terms defined by Eqs. (11.67) and (11.68) weight density or unit weight Mm/ Mrno in Eq. ( 4.39) unit weight of concrete unit weight of soil shear strain in the xy plane shear strain in the xz plane shear strain in yz plane. shear strain angle of twist any angle displacement logarithmic decrement loss angle [see Eq. (4.48b)] static deflection specific damping capacity for decaying vibrations specific damping capacity for steady-state vibrations area enclosed under the hysteresis loop change in thickness change in volume vertical deflection vertical deflection due to concentrated load vertical deflection due to distributed load vertical deflection due to shear vertical deflection due to axial compression in column Lammes' constant wavelength eigen value wavelength of Rayleigh waves coefficient of friction m 2 1m 1 = W2 1W1
complex frequency parameter of a pile real and imaginary parts of A, respectively real frequency parameter of pile dimensionless parameter longitudinal strain Ex+ Ey
+ Ez
NOTATION
NOTATION
Unit
Symbol
645
Definition
€1,2
Ex
e, e,
,., v
g FL-'
FL-'
gx g, g. g, p
.
p, Pp P,
0'
L
u,
0'1 0'2
u, L
O'p
L'
O'x
L L L L
u, 0', T
Txy, Tyz, Tzx
L FL- 2 L T-2
"'
"'>,, · w w. lt)n1 •
Wnd
w.,
lt)n2
terms defined by Eqs. (11.69) and (11.70), respectively longitudinal strain in x direction longitudinal strain in y direction longitudinal strain in z direction efficiency of the drop degree of absorption Poisson's ratio normal coordinates (dimensionless) damping factor damping factor in horizontal sliding damping factor in vertical vibrations damping factor in rocking damping factor in torsional vibrations mass density of the soil adjacent to pile mass density of the soil ratio of amplitudes Ax!A mass density of'Soit beneath pile tip mass density of pile material mass density of the soil on the sides of the embedded footing principal stress normal pressure effective all-around stress major principal stress intermediate principal stress minor principal stress stress in the pad normal stress in x direction normal stress in y direction normal stress in z direction vertical pressure below the base time for which a square pulse acts shear stresses angle of internal friction phase angle rotation (rocking) of a footing apparent angle of internal friction aver~ rotation of a flexible contact area angular velocity circular frequency operating frequency operating speed in rad I sec circular natural frequency first and second natural circular frequencies natural frequency of damped vibrations natural frequency of the embedded foundation
Unit
FL _,T 2 FL _,T2 FL -4T2 FL _,T 2 FL- 4 T 2 FL- 2 FL- 2 FL - 2 FL - 2 FL- 2 FL-2 FL-2 FL- 2 FL - 2 FL- 2 FL- 2 T FL- 2
T-1 T-1 T-1 T-1 T-1
646 Symbol wnh wnll' wn/2
(wnll)e, (wnl2)e
NOTATION
Definition
natural frequency of horizontal vibrations limiting natural circular frequencies two natural circular frequencies of the embedded
Unit
Author Index
footing in coupled rocking and sliding
average natural circular frequency in vertical vibrations natural circular frequency in horizontal sliding natural circular frequency of sliding vibration of embedded foundation
natural circular frequency in vertical vibrations natural circular frequency of vertical vibrations of embedded foundation
"'1' o/2 %
"''
"'''' I.P,,,I
natural circular frequency in pure rocking natural circular frequency of embedded footing in rocking vibrations natural circular frequency in torsional vibration natural circular frequency of embedded foundation in torsional vibrations real and imaginary parts of .p( z) torsional amplitude of footing resonant amplitude of pile rotation complex amplitude of pile rotation at elevation z real torsional amplitude of pile at elevation z
Abdel-razzak, K.G., 113, 187 Abel, J.F., 437 Aboui-Ella, F.,'542, 553, 620, 621 Adiar, A., 400, 436 Afifi, S.E.A., 112, 183 Agarwal, S.L., 577 American Concrete Institute, 9, 578, 579, 580, 585, 592 American Society of Civil Engineers, 337, 363 American Society For Testing Materials, 144 Anandakrishnan, M., 438, 490 Anderson, D.G., 106, 110, 111, 113, 114, 115, 116, 117, 126, 183 Annaki, M., 133, 134, 183, 188 Arango, 1., 187 Arnold, R.N., 224, 225,236, 239, 303 Arya, A.S., 553 Arya, S.D., 363, 374 Aubry, D·., 549, 552 Babcock, F.M., 437 Balakrishna, R.H.A., 258, 302, 303 Ballard, R.F., Jr., 139, 183 Baranov, V.A., 438, 439, 44~, 488, 490, 503, 542, 552 ... Barkan, D. D., 3, 10, 89, 93, 121, 122, 144, 150, 154, 177, 183, 240, 258, 260, 302, 303, 312, 314, 328, 329, 339, 349, 350, 374,382,388,398,400,414,428,436, 439,459,471,490,494,499,547,552 Barnett, N.E., 437 Bathe, K.J ., 623 Baxa, E., 400, 409, 436 Baxter, R.L., 8, 9, 10
Beredugo, Y.O., 438, 439, 440, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 454, 455, 471, 488, 489, 490,-491, 492, 515, 552, 553 . Bernhard, D.L., 8, 9, 10 Beskos, D.E., 420,436, 542 Bhandari, R.K.M., 576 Bhargava, S., 553 Biot, M.A., 117, 183 Bishop, A.W., 99, 183 Bjerrum, L., 128, 183 Black, W.L., 104, 105, 112, 117, 184 Blake, R.L., 6, 7, 8, 10 Bolt, B.A., 419, 437 Brooker, E.W., 159, 178, 183 Bycroft, G.N., 219, 224, 225, 234, 236, 237, 238, 303, 441, 446, 457, 491, 506, 552 Carrier, W.D., 439, 491 Castellani, A., 217, 304 Castro, G., 133, 183 Chae, Y.S., 236, 303, 438, 491 Cho, Y., 131, 132, 183 Christian, J.T., 188, 439,491 Christiano, P., 491 Chung, R.M., 187 Crandell, F.J., 6, 10 Crede, C.E., 409, 437 Crockett, J.H.A., 400, 437 Cunny, R.W., 177, 183 Curran, J.W., 185 Dalal, M.K., 576 Dasgupta, B., 236, 303, 436, 439, 491 Davis, E.H., 544, 553
647
••• Day, S.M., 439, 491 Demello, V.B.F., 145, 183 Dietrich, R.J., 124, 187, 188 Dobry, R., 236, 303, 488, 489, 491, 550, 552 Doebelin, E.O., 138, 184 Dolling, H.J., 414, 436 Dominguez, J., 439, 491 Drnevich, V.P., 118, 123, 124, 125, 126, 127, 131, 132, 149, 155, 156, 179, 180, 184, 205, 211 Drweyer, R., 363, 374 Dutko, P., 180, 185 Dyvik, R., 181, 184 Ebisch, R., 400, 409, 436 Ehlerchritof, O.M., 302, 303 El Hifnawy, L., 328, 329 Elorduy, J., 234, 303 El-Sharnouby, B., 511, 512, 518, 519, 541, 543, 544, 545, 546, 547, 548, 549, 552, 553 Engen, A., 576 Epstein, R.I., 491 Espana, C., 183, 184 Evans, J.C., 185 Ewing, W.M., 93 Firestone, Inc., 410, 437 Fletcher, G., 145, 184 Fry, Z.B., 139, 177, 183, 184, 438, 491, 552 Fugro, Inc., 113, 184 Funston, N.E., 302, 304 Gazetas, G., 236, 303, 482, 485, 488, 489, 491, 550, 552 Gle, D.R., 541; 543, 544, 552 Goodier, G.N., 78, 94 Griffith, A.H., 185 Grigg, R.F., 514, 541, 547, 553 Guha, S.K., 576 Gupta, B.N., 438, 491 Gupta, D.C., 491, 556, 569, 575, 577, 582, 592 Hadjian, A.A., 485, 491 Hall, J.R., Jr., 91, 94, 118, 124, '148, 150, 153, 184, 186, 205, 211, 215, 224, 225, 226,227,229,235,238,264,302,304, 305, 329, 497, 553 Hall, W.J., 304 Hansen, W.E., 186 Harder, L.F., 187
AUTHOR INDEX
Hardin, B.O., 81, 84, 93, 104, 105, 106, 112, 117, 131, 148, 149, 150, 155, 156, 179, 180, 184 Harris, C.M., 409, 437 Hartmann, H. G., 544, 546, 553 Harwood, M., 329 Hatheway, A.W., 113, 188 Haupt, W.A., 419, 437 Hayashi, K., 240, 304 Henkel, D.J., 99, 183 Heteyni, M., 240, 304 Hoar, R.J .. , 135, 187 Horst, W.D., 304 Hausner, G.W., 217, 304 Howell, J.F., 521, 522, 523, 525, 526, 527, 528, 529, 549, 553, 620, 621 Hsieh, T.K., 220, 221, 258, 302, 304 Hudson, W.R., 186 Humphries, W.K., 183 Hurty, W.C., 363, 374 Hvorsley, M.J., 128, 184 ldriss, I.M., 109, 187 !ida, K., 124, 184, 185 Imai, T., 136, 146, 185 Indian Standard Institution, 9, 11 Ireland, H.O., 145, 159, 178, 183, 185 Ishibashi, I., 131, 132, 185 Ishihara, K., 107, 108, 131, 132, 185 Ishimoto, M., 124, 185 Iwasaki, T., 107, 131, 132, 178, 185 Jakub, M., 485, 491 Jardetzky, W.S., 93 Johnson, G.R., 439, 489, 491 Joshi, V.H., 186 Judd, S., 378, 398 Kaldijan, M.J., 489, 491 Kaufman, R.I., 128, 184 Kausel, E., 439, 491 Kausel, F., 544, 546, 552 Kaynia, A.M., 544, 546, 552 Kjellman, W., 128, 185 Kleinlogel, 345, 375 Klien, A.M., 400, 437 Knox, D.P., 187 Ko, H.Y., 134, 185 Kobori, T., 234, 304, 305 Koerner, R.M., 185 Kolsky, H., 78, 93 Korfund, Inc., 404, 405, 406, 437 Kovacs, W.D., 146, 185
649
AUTHOR INDEX
Kranthammer, T., 542 Krishnaswamy, N.R., 438, 490 Krizek, R.J., 489, 491 Kuhlemeyer, R.L., 238, 304, 489, 491, 492 Kumar, K., 555, 576, 577 Ladd, R.S., 132, 180, 185 Lamb, H., 85, 90, 93, 215, 234, 304 Lambe, T.W., 178, 185 Landra, A., 128, 183 Lawrence, F.V., 107, 188 Lee, K.L., 133, 134, 183, 188 Lee, P.C.Y., 437 Lee, S.H.H., 187 Leonards, G.A., 98, 99, 185 Leontovich, V., 345, 375 Lerstol, M., 576 Li, S., 131, 185 Liang, H., 547, 552 Liao, S., 422, 423, 437 Lodde, P.J., 113, 116, 117, 185 Lord, A.F.~·;Jr., 126, 185 ,~;,~;.:?' Luco, J.E., 491 Luco, Y.E., 441, 491 Lorenz, H., 154, 155, 185 Lysmer, J., 205, 211, 221, 222, 223, 304, 328, 329,419,437,439,489,491,492 McNeill, R.L., 301, 304, 414, 437 Madhav, M.R., 499, 552 Madshus, C.F., 181, 184, 555, 576 Major, A., 328, 329, 344, 349, 375, 386, 390, 391, 394, 398, 407, 437, 592 Marcuson, W.F., III, 112, 179, 185, 186 Marganson, B.E., 437 Martin, G .R., 187 Maxwell, A.A., 499, 500, 501, 547, 552 May, T.W., 419, 437 Meister, F.J., 6, 11 Melore, V.R., 183 Miller, G.F., 92, 93 Mohr, H.A., 145, 186 Moore, P.J., 303, 304 Moretto, 0., 185 Nadim, F., 576 Nagendra, M.V., 437 Nagraj, C.N., 302, 303 Nandakumaran, P., 186, 553 Nazarian, S., 139, 140, 186 Newcomb, W.K., 201, 203, 211 Nieto, J.A., 303 Nogami, T., 547, 549, 552, 621
Novak, M., 302, 303, 304, 310, 311, 318, 328, 329, 438, 439, 440, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 454, 455, 456, 458, 471, 488, 489, 490, 491, 492,499,502,503,504,507,508,509, 510, 511, 512, 514, 515, 516, 517, 518, 519, 521, 522, 523, 525, 526, 527, 528, 529,531,541,542,543,544,545,546, 547, 548, 549, 552, 553, 620, 621 Oh-Oka, H., 131, 132, 188 O'Neill, M., 374 Palacios, A., 146, 186 Park, T.D., 133 Parmelee, R.A., 491 Parthasarathy, T., 437 Pauw, A., 258, 302, 304 Peacock, W.H., 128, 129, 130, 186, 187 Peck, R.B., 145, 186 Pincus, G., 374 Poplin, J .K., 552 Postel, M., 549, 552 Poulos, H. G., 498, 514, 521, 544, 553 Poulos, S.J., 133, 183 Prakash, S., 1, 4, 11, 81, 83, 93, 96, 107, 108, 109, 110, 128, 140, 142, 145, 146, 147, 165, 172, 176, 178, 180, 186, 232, 258, 302, 303, 304, 328, 329, 439, 459, 471, 492, 494, 517, 548, 553, 555, 556, 559, 569, 575, 576, 577, 582, 592 Press, F., 93 Puri, V.K., 1, 11, 81, 83, 93, 107, 108, 109, 110, 133, 146, 147, 165, 172, 176, 178, U6,n2,258,3ro,W4,C9,~9,471,
492, 494, 553, 556, 559, 577 Pursey, H., 92, 93 Pyke, R.M., 187 Quinlin, P.M., 216,236, 304 Ranjan, G., 186 Rao, N.S.V.K., 236, 303, 439, 491, 499, 552 Rausch, E., 6, 8, 11, 341, 375, 393, 398 Rayleigh, L., 85, 93 Reiher, H., 6, 11 Reissner, E., 215, 216, 228,236,239, 304, 441, 457, 492 Richart, F.E., Jr., 5, 6, 8, 11, 80, 81, 84, 90, 91, 93, 94, 106, 110, 112, 113, 118, 123, 124, 136, 150, 151, 152, 153, 178, 181, 183, 184, 186, 187, 188, 205, 211, 215, 217, 218, 221, 222, 228, 229,230, 235,
650
Richart, F. E., Jr. (Continued) 236, 238, 260, 264, 302, 304, 305, 312, 328, 329, 414, 437, 438, 492, 496, 497, 498, 553, 555, 577 Rizzo, P.C., 183
Roesset, J.M., 439,485,491 Roscoe, K.H., 128, 186 Rubinstein, M.F., 363, 374 Sachs, K., 439, 456, 458, 471, 488, 492, 522, 553 Sagessor, R., 437 Sagoci, H.F., 228, 304, 441, 457, 492 Sangrey, D.A., 422, 423, 437 Saran, S., 186 Schmertmann, J.H., 145, 146, 186 Scott, R.F., 134, 185 Seed, H.B., 109, 128, 129, 130, 131, 133, 134, 146, 178, 179, !86, 187, 188 Segol, G., 419, 437 Shannon, W.L., 124, 187 Shannon and Wilson-Agbabian Associates, 106, 187 Shen, G.T., 363, 375 Sherif, M.A., 131, 132, !85 Sheta, M., 499, 517, 547, 553, 620, 621 Shippy, D.J., 184 Silver, M.L., 132, 133, 134, 185, !87 Singh, B., 186 Skempton, A.W., 146, 179, 187 Snowdon, J.C., 407, 437 Sridharan, A., 419, 437 Srivastava, L.S., 186 Stephenson, R.W., 127, !87 Stevens, H.W., !50, 187 Stokoe, K.H., II, 110, 111, 113, 114, 115, 116,117,135,140,178,181,183,186, 187, 188, 303, 438, 492 Stone, N.E., 363, 375 Sung, T.Y., 216,217,234,305,441,492 Szekley, E.M., 303
Takagi, Y., 185 Tassoulas, J.L., 439, 492 Tatsuoka, F., 107, 178, 185 Taylor, D.W., 100, 168, 188 Terzaghi, K., 240, 305
AUTHOR INDEX
Thiers, G.R., 128, 129, 130, 131, 188 Thomson, W.T., 27, 31, 43, 61, 234, 305 Thornburn, T.H., 186 .Timoshenko, S., 78, 94 Ting, J.M., 541, 553 Tokimatsu, K., 187 Trudeau, P.J., 113, 188 Tsai, N.C., 491 Tucker, R.L., 517, 548, 553
Subject Index
Urlich, C.M., 489, 492 Ushijima, R., 439,491 Vardoulakis, I. G., 436, 542 Vargas, M., 185 Vijayvergiya, R.C, 438, 492 Wahls,H.E., 112, 179, 185, 186 Wang, X.K., 576, 577 Warburton, G.B., 236, 237, 303, 305 Wass, G., 419, 437, 439, 489, 492,544,546, 553 Weissman, G.F., 150, 188 Westman, R.A., 441, 491 White, S.R., 150, 188 Whitman, R.V., 107, 150, 178, 185, 188, 217, 218, 228, 230, 236, 302, 305, 312, 329, 498, 553 Wilson, S.D., 124, 188 Wolfe, W.E., 134, 188 Wong, R. T., 187 Woods, R.D., 91, 92, 94, 112, 113, 115, 124, 125, 126, 127, 131, 132; 134, 135, 137, 138, 140, 145, 150, 153, 183, 186, 187, 188, 215, 229, 235, 238, 264, 305, 329, 413, 414, 415, 416, 417, 418, 420, 421, 422, 425, 437, 497, 541, 542, 543, 544, 549,552,553,555,577 Yamane, G., 187 Yang, Z., 113, 188 Yasuda, S., 131, 132, 185 Yoshimi, Y., 131, 132, 188 Yoshimura, M., 136, 185 Yu, P., 181, 188 Zoeppritz, K., 93, 94
Absorber, 44 installation o~, 589,590,591,592 pneumatic, 399, 404, 408 principle of, 48, 404 types: 404 supported, 405, 406, 589, 590, 591 suspended, 405, 406, 589, 592 Acceleration, 15 pickups, 44 Accelerometer, 43, 44 Active Isolation, 399, 413, 415, 425 ADINA, 363, 374, 622 Amplitude, 5, 13, 26, 29, 30, 39, 47, 48, 53 of foundation: 213 rocking, 233, 246, 257, 450, 454, 467, 468, 469, 474, 476 sliding (horizontal), 233, 243, 246, 248, 257, 267, 448, 454, 464, 468, 474, 476, 477 torsional, 229,248, 267, 457, 471, 473, 475 vertical, 224, 242, 246, 256, 266, 267, 313, 314, 322, 402, 403, 411, 412, 442, 462, 469, 472, 473 of frame: ~, horizontal, 348, 356, 371 rotation, 356, 370 vertical, 346, 351, 352 reduction factor, 414, 415, 416, 418, 419, 420, 421 rotating maSs excitation, 313, 332 Analog: Hall's, 224 Hsieh's, 220 Lysmer's, 221, 223
Analysis: dynamic, 366 methods, see Methods of analysis preliminary, 363 Anchor bolts, 586, 587, 588 Anvil, 306, 307, 308, 309, 310, 311 amplitude, 313, 322 Aperiodic motion, 12 Attenuation, see Damping Auxiliary equipment, 330 Bandwidth method, 40 Barriers: pile, 399,420,425 trench, 399, 414, 434 Base, slab, 330, 331 Beam: longitudinal, 330 transverse, 330 Bilinear models, stress-strain curves, 128, 129, !30 Block foundation, mode of vibration, 213 Bulk modulus, 102, 103 Case history: compressor foundation, 556 hammer foundation, 569 Chemical soil stabilization, 427 Coefficients: of earth pressure at rest, 159 of elastic non-uniform compression, 121 of elastic non-uniform shear, 121 of elastic resistance of piles, 494 of elastic restitution, 316 651
652 Coefficients (Continued) of elastic uniform compression, 121, 122, 142 of elastic uniform shear, 121, 143 of shear modulus increase, 111, 112 Comments, final, 93, 436, 547 Compliance, functions, 483, 485, 490 Compliance Impedance functions, 482 Computer programs: ADINA, 363, 374, 622 for block foundations, 595 for hammer foundations, 610 PILAY II, 542, 543, 620 STRUDL II, 363, 374, 375, 621 Condenser, 330, 331, 333 loads, 332, 333 Confining pressure, mean, 84, 105 Constrained modulus, 79, 95, 102, 103 Construction of foundation: block, 579 frame, 580 Construction joint, 579, 583, 584 Contact pressure, 216 parabolic, 216, 217 rigid base, 216, 217 uniform, 216, 217 Continuum approach for piles, 217 Conversion factors, 629 Coulomb damping, see Damping, Coulomb Counterbalancing, 194, 195, 426 Counterweights, 194, 202, 381, 382 Criteria: for design, see Design, criteria for vibrations, 7 Critical damping, 21, 22 Crushers: gyratory, 380 hammer, rotary, 380, 383, 384 jaw, 380, 381 primary and secondary, 383 Cycle, 12 Cyclic·plate-load test, 144 Damper, vibration, 429, 430 Damping, 14, 20, 28, 38 coefficient of attenuation, 151 constant, geometrical, rigid circular footing: on elastic half space: rocking, 227 sliding, 225 torsional, 228 vertical, 221, 223
SUBJECT INDEX embedded in elastic. half space: cross coupled, 453, 455, 473 rocking, 450, 453, 455, 473 sliding, 446, 448, 453, 455, 473 torsional, 457, 458 vertical, 442 Coulomb, 14 critical, 21, 22 equivalent viscous, 153 factor, 22, 264. See also Damping, ratio factors affecting, 155, 156 geometrical, 151, 152, 219 material, 147, 179, 150 specific damping capacity, 150 viscous, 20, 24, 25, 40, 147 parameters, 519 ratio, 95, 153, 180 rigid geometrical circular footing: on elastic half space: rocking, 153, 228, 232 sliding, 153, 225, 232 torsional vibration, 153, 154, 229 vertical vibration, 153, 223 embedded in elastic half space: rocking, 450 sliding, 448 torsional, 458, 459, 472 vertical, 442, 443, 472 hysteretic, 485 viscous, 14 Deck slab, 330, 331 Degree of freedom systems, 13 multi degree, 13, 50 single degree, 13 two degree, 44, 45 Degree of saturation, effect on soil modulus, 117 Design: criteria, 4, 318, 339, 377, 379, 385 parameters for vertical vibrations of pile, 511, 512, 513 procedure: embedded foundation, 471 foundations on absorbers, 410 hammer foundation, 319 pile foundation, 529 reciprocating machines, foundations for, 260 turbo generator foundation, 363 wave barriers: pile barrier, passive isolation, 423 trench barrier, active isolation, 423 trench barrier, passive isolation, 423
SUBJECT INDEX Dip angle, 92 Displacement: functions, rigid circular footing on elastic half space, 215, 216, 217, 219-222, 235, 441 pick up, 44 Drainage during shear, 97 Dynamic: compliance, 482, 485 impedance, 482, 484, 485 real and imaginary, 484 Dynamic loads, see Forces, Unbalanced; Unbalanced moments Dynamic prestrain, effect on soil modulus, 118 Eccentricity, 201, 426, 427 Effective: height, 344, 366 mass, 302 mass moment of inertia, 302 span, 344;,'"367 >\;,,--·~1'· Elastic constants, 100 Elastic half space approach, 260, 264, 472, 488. See also Elastic half space method Elastic half space method: embedded: coupled sliding and rocking, 451, 473 rocking, 448, 449 sliding, 443, 445, 447, 448 torsional, 456, 457, 458, 472 vertical, 440, 444, 472 rigid circular footing, vibrations of, 438, 439 surface: 214, 301 coupled sliding and rocking, 230 rocking, 225 sliding, 224 torsional, 228 vertical, 215 Elastic layer, rigid circular footing: torsional oscillation, 239 vertical oscillation, 237 Embedded foundations, vibrations of: coupled sliding and roiJRing, 451, 468, 473 rocking, 448, 449, 464, 465, 467 sliding, 443, 445, 447, 448, 462, 463 torsional, 456, 457, 458, 469, 470, 472 vertical, 440, 444, 459, 460, 461, 472 Equivalent radius, 234, 264, 321 Equivalent spring, 19, 312, 320, 342, 355, 370, 402 pad below anvil, 312
653 soil: cross coupled, 453, 455, 473 rocking, 227, 246, 264, 450, 453, 455, 473 sliding, 121, 225, 242, 264, 448 soil pile system, see Pile stiffness torsional, 228, 248, 264, 457, 458, 473 vertical, 121, 219, 241, 264, 312, 401, 411, 442, 443, 472 Fatigue factor, 394 Field methods, 135 crossMbore hole test, 135 cyclic plate test, 144 down-hole test, 136 free footing vibration test, 144 horizontal footing resonance test, 143 standard penetration test, 145, 178, 179 surface wave test, 137 up-hole test, 136 vertical footing resonance test, 140 Finite rcids, end conditions: fixed-fixed, 73, 76 fixed-free, 72, 76 free-free, 70, 71, 76 Flexibility influence coefficient, 50, 52 Flexibility matrix, 52 Footing shape, effect on vibratory response, 234 Force: due to impact,.205 primary and secondary, 194, 202 Forced vibrations, 13, 39 with viscous damping, 24, 25 Forces, unbalanced, 2, 29 jaw crushers, 382 multicylinder machines, 195 in reciprocating machines, 189 in rotary machines, 201 single cylinder machines, 190 Form of vibrations associated with coupled rocking and sliding, 256 ..• Foundations, embedded block, See 'Embedded foundations Foundations: for crushing mills, 380 for fans and blowers;· 379 for impact machines, 306 for mills, grinding, 391 for mills, rolling, 386 for motor generators, 376 for presses, stamping, forging and impact, 392
b54
Foundations (Continued) for pumps, centrifugal, 378 for reciprocating, 212 for rotary machines, high speed, 330 for rotary machines, low speed, 376 for tools, machine, 391 Frame foundation, 330, 331, 332, 363 combined horizontal and rotation, 352, 353 horizontal vibration, 347, 352, 358 vertical vibration, 342, 343, 349, 357 Free vibration Of spring mass, 16 with viscous damping, 20 Frequency, 12 circular, 15, 17 damped, 24, 259 Frequency dependent excitation, 29 Frequency determinant, 51, 53, 57 Frequency equation, 57, 232, 253, 312, 351, 355' 402, 468 Frequency ratio, 5, 14, 26, 27, 28, 31 dimensionless, 216,441, 456 Functions, compliance and impedance, 301, 482
Geometrical damping, 91, 92
Hammer: counter blow, 306 . double acting, 306 drop, 3, 306 forging, 3 Harmonic motion, 14 Heat exchanger, 330
ICES STRUDL, 363, 374, 375, 621 Impedance, 483 function, 482, 485 Inertia ratio, 226, 228, 264 Instruments, vibratory measuring, 42, 43 Interaction factors, piles, 514 Interfacing machine with foundation, 586 International standard, 15 Isolation active, 399, 413, 415, 423 passive 399, 414, 415, 423 Laboratory methods, 122 cyclic simple shear test, 128 cyclic torsional shear test, 131 cyclic triaxial compression test, 133 resonant column test; 123 end conditions, 125 ultrasonic pulse test, 127
SUBJECT INDEX Lame's constants, 78 Limiting amplitudes, 6 Linear elastic weightless spring approach, 260, 266, 474, 488. See also Linear elastic weightless spring method Linear elastic weightless spring method, 214, 240, 301 embedded foundations, vibrations of: coupled rocking and sliding, 468, 476 rocking, 464, 465, 467 sliding, 462, 463 torsional, 469, 470, 475 vertical, 459, 460, 461, 475 surface footing, vibrations of: coupled rocking and sliding, 251 rocking, 243 sliding, 242 torsional, 247 vertical, 240 Logarithmic decrement, 38, 148. See also Damping Longitudinal vibrations of rods, 63, 69 Machines on floors, 394 Magnification factor, 27, 32, 222, 226, 227, 238 Mass effective, 302 Mass ratio, 216, 217, 224, 226, 443 modified, 221, 224, 225, 264 Measures, remedial, 401, 406 Method, Rayleigh's, 35 Methods of analysis, 214, 307, 340 elastic half space, 214, 301, 439, 471 linear elastic weightless spring, 214, 240, 301, 439, 459, 471 Mode of vibration, 54 normal, 14, 70 principal, 14, 51 Modulus, see also Field methods; Laboratory methods secant, 101 shear, 74, 84, 95, 101, 102 complex, 15~, 523 tangent, 101 Young's, 63, 64, 95, 102 Moment of inertia: of area, 247, 262, 263, 624 of mass, 262, 263, 625, 626, 627 of pile cross section, 518, 524 polar, 74, 263, 369 Multidegrees of freedom, 50 Natural frequency, 4, 12, 18, 19, 35, 49, 346, 348
655
SUBJECT INDEX of foundations: coupled rocking and sliding, 232, 251, 267,476 limiting, 313, 320, 367, 411 rocking, 228, 246, 450, 467, 476 torsional, 229, 248, 265, 370, 457, 471, 475 vertical, 242, 264, 266, 442, 461, 462, 475 Natural period, 17 Nonlinearity, effect on foundation effect, 302 Normalized shear modulus, 109 Normalized shear modulus increase, 111, 112 Normal mode: method, 55 of vibration, 14, 70 Overview, 177, 301, 328, 488 Pads, absorber: cork, 404, 406 neoprene, -404, 408 rubber, 404, 407 stresses in, 317, 322 timber, 404, 408 Particle motion: longitudinal waves, 66 surface waves, 90 Particle velocity: definition, 66 equation, 66 Passive isolation, 399, 414, 415, 423 Period, 12 natural, 17 Phase angle, 27, 29, 31,44 Pickup: acceleration, 44 displacement, 44 PILAY 2, 542, 543, 620 Pile barriers, 420 isolation effectiveness, 420, 421 Pile cap: damping, 515 stiffness, 515 Pile damping: ·•·~ cross,-518 rotation, 518 torsion, 524 translation, 518 vertical, 508 .._ Pile stiffness: cross, 518 parameters, 519 rotation, 518
''j,":.;f"
torsion, 524 translation, 518 vertical, 508 Piles: coefficient of elastic resistance, 494 damping, 508 end bearing, 495 footing stiffness, 515, 524 friction, 497 interactiOn factors, 514, 521 lumped mass model, 499 pile groups, 514, 520 predicted response, 541 resonant frequency, 498 rocking, 517 stiffness parameters, 541, 512 torsion, 521 translation, 517 vertical stiffness, 508 vertical vibrations, 499, 502 Poisson's ratio, 78, 81, 95, 101 Principal modes of vibration, 14, 51 Radius of footing, equivalent, see Equivalent radius Ratio, impedance, 422 Rayleigh wave (R~wave), 85 wave length, 424 Rayleigh's method, 35, 36 Reciprocating machine, 2 Reinforcement, steel, 580, 581, 582, 583, 584, 585, 586 Resonance, 14 ResonantMcolumn test, 123, 149 hollow specimen, 126 Retrograde ellipse, 90 Rotary machine, 4 Scaled area, of trench, 419 Secant modulus, 101 Service factor, 6, 8 Shear modulus, 74, 84, 95, 101, 102, 104 cohesionless soils, 105, 106, 107 effect of prestrain, 118 factors affecting, 104 strain level, 107, 108, 109 time effects, 110, 111, 112, 113, 114, 115, 116, 117 cohesionless soils, equations for 106, 107,
424 complex, 152, 523 Shear tests: consolidated~undrained,
drained test, 100
98
656
SUBJECT INDEX
Shear tests (Continued) unconsolidated-undrained, 98 Shear wave velocity, 75 Simple harmonic motion, 14 Soil stabilization: with cement, 428 chemical, 427 with silicates, 428 Spectral analysis of surface waves, 139
Springs: pneumatic, 409 rubber, 407 steel or metal, 404 Square pulse, 33 Static deflection, 18, 19, 26 Stiffness, cross coupled, 453 Stiffness coefficient, 50 Stiffness and damping parameters, embedded foundations:
half space: rocking, 449
sliding, 447 torsional, 448 vertical, 444 side layer: rocking, 449 sliding, 447 torsional, 458 vertical, 444 Stiffness matrix, 50, 362 Stiffness method, 51 Surface footing, vibrations of co~pled rocking and sliding, 230, 251 rocking, 225, 227, 243, 244 rocking, sliding, and vertical, 248 sliding, 224, 226, 242 torsional, 228, 229, 247 vertical, 213, 215, 218, 240, 241 Tangent modulus, 101 Test data, evaluation, 146 Torsional vibrations of rods, 74, 76 Transducer, acceleration, 140, 141 Transient loads, 31 Transmissibility, definition, 41 Triaxial compression test, 96 Tuning of foundation, high; low, 5 Tup,306, 307,309,310,311 Turbines, 330, 331, 332 Turbogenerator: foundation, 331, 332 layout, 331 loads on foundation: due to emergency condition, 332, 337
due to normal operation, 332 Two degree of freedom systems, 44 coupled rocking and sliding, 230, 251 coupled translation and rotation, 352 forced vibration, 44, 45 Unbalanced forces, see Forces, unbalanced Unbalanced moments, 197, 202 primary, 199, 200, 202 secondary, 200, 202 Velocity: compression wave, 79, 31, 82 longitudinal wave propagation, 65 Rayleigh wave, 81, 90 shear wave, 75, 81, 82, 84, 424 wave propagation, 62, 66, 88 Vibration: absorber, 44 absorption, principle of, 401 isolation: active, 399 passive, 399 measuring instruments, 42 modes of rigid foundation block, 213, 214 of elastic rods of finite length: fixed-fixed, 71, 73, 76 fixed-free, 71, 72, 76 free-free, 70, 71, 76 rigid circular footing on elastic layer: torsional, 239 vertical, 237 severity data, 9 Viscous damping, 14 determination by band width method, 39 Waves due to surface footing, 91 Wave equation: for longitudinal waves in a rod, 65, 66 solution for finite rods, 70, 76 Wave front, cylindrical, 91 Wave length, measurement, 63, 86, 424 Wave propagation, 62 elastic half space, 84, 85 elast.ic infinite medium, 76 elastic rods, 63 fixed end, 69 free end, 68 longitudinal vibrations, 63 torsional vibrations, 74 velocity of, 62, 65 Young's modulus, 63, 64, 95, 102