JlSC~~ American Society A tIiii of Civil Engineers
DESIGN OF FOUNDATIONS
FOR DYNAMIC LOADS
2008
Design of Foundations for Dynamic Loads
Learning Outcomes • • • • •
Comprehend the basics of soil dynamics as related to soilstructure interaction modeling Know the major field tests that are used for evaluating the dynamic soil properties Design shallow and pile foundations for rotary machine and seismic loads Know the major steps for determining the seismic stability of cantilever and gravity retaining walls Apply the capacity spectrum method for evaluating the seismic performance of large caissons
Assessment of Learning Outcomes Students' achievement of the learning outcomes will be assessed through solved examples and problem-solving following each session.
UNIVERSITY OF WESTERN ONTARIO Faculty of Engineering
Design of Machine Foundations
Professor M.H. EL NAGGAR, Ph.D., P. Eng. Department of Civil and Environmental Engineering Geotechnical Research Centre
LECTURE NOTES M.H. EL NAGGAR
M.NOVAK
DEPARTMENT OF CIVIL ENGINEERING THE UNIVERSITY OF WESTERN ONTARIO LONDON, ONTARIO, CANADA, N6A 5B9
Course Content
1. Basic Notions: Mathematical models , degrees of freedom , types of dynamic loads, types of foundations, excitation forces of machines. 2. Shallow Foundations: Definition of stiffness , damping and inertia , circular and non circular foundation , soil inhomogeneity, embedded footings , impedance function of a layer on half-space. 3. Pile Foundations: PHe applications, mathematical models, stiffness and damping of piles, pile groups, impedance functions of pile groups, nonlinear pile response, pile batter.
4. Dynamic Response of Machine Foundations: Response of rigid foundations in 1 OaF, effects of vibration, coupled response of rigid foundations, 6 OaF response of rigid foundations, response of structures on flexible foundations. 5. Dynamic Response of Hammer Foundations: Types of hammers and hammer foundations, design criteria , stiffness and damping of different foundations, mathematical models, impact forces , response of one mass foundation, response of two mass foundation, impact eccentricity, structural design .
6. Vibration Damage and Remedial Measures Damage and disturbance, problem assessment and evaluation , remedial principles, examples from different industries, sources of error. 7. Computer Workshop - DYNA5 Types of foundations, types of soil models, types of load , types of analysis and types of output, practical considerations, computer work on DYNA5.
2
1
BASIC NOTIONS
3
BASIC NOTIONS Statics deals with forces and displacements that are invariant in time. Dynamics considers forces and displacements that vary with time at a rate that is high enough to generate inertia forces of significance. Then, the external forces, called dynamic loads or excitation forces , produce time dependent displacements of the system called dynamic response. This response is usually oscillatory but its nature depends on the character of the dynamic forces as well as on the character of the system. Thus , one system may respond in different ways depending on the type of excitation. Conversely, one type of excitation can cause various types of response depending on the kind of structure . Mathematical models. The systems considered in dynamics are the same as those met in statics, i.e. buildinqs, bridges, towers, dams, foundations, soil deposits etc. For the analysis of a system a suitable mathematical model must be chosen. There are two types of models, which differ in the way in which the mass of the structure is accounted for. In distributed mass models, the mass is considered as it actually occurs, that is, distributed along the elements of the structure . In lumped mass models the mass is concentrated (lumped or discretized) into a number of points. These lumped masses are viewed as particles whose mass but not size or shape is of importance in the analysis.
There is no
rotational inertia associated with the motion of the lumped masses and translational displacements suffice to describe their position. Between the lumped masses the structural elements are considered as massless . Examples of distributed and lumped mass models are shown in Fig. 1.1. As the number of concentrated masses increases,
4
the lumped mass model converges to the distributed model. In rigid bodies (Fig. 1.1c), mass moment of inertia is considered as well as mass .
Figure 1.1
--
-~
(a) distributed models
-A......
•
.::IL
/
t (b) lumped mass models
.,
(c) - I.:
~
rigid bodies
Degrees of freedom The type of the model and the directions of its possible displacements determine the number of degrees of freedom that a system possesses. The number of degrees of ----~- --
.:
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)~
l, ;'
t
\
r:
freedom is the number of independent coordinates (components of displacements) that
---------
-
---
- -- -- -
must be specified in order to define the position of the system at any time. One lumped mass has three degrees of freedom in space corresponding to three possible
5
translations, and two degrees of freedom in a plane. If a lumped mass can move only either vertically or horizontally it has one degree of freedom. Thus, if the vertical motion of a bridge is investigated using a model with three lumped masses and axial deformations are neglected, there are three degrees of freedom (displacements). However, a rigid body such as a footing has significant mass moments of inertia and hence rotations have to be considered as well . Three possible translations and three possible rotations represent six degrees of freedom for a rigid body in space . A distributed mass model can be viewed as a lumped mass model with infinitesimal distances between adjoining masses . Such a system has an infinite number of degrees of freedom . This does not necessarily complicate the analysis however.
Types of dynamic loads. The type of response of a system depends on the nature of the loads applied. The loads and the responses resulting from them can be periodic, transient or random. :+-
--------
Periodic Loads can be produced by centrifugal forces due to unbalance in rotating and reciprocating machines, shedding of vortices from cylindrical bodies exposed to air flow
and other mechanisms. The simplest form of a periodic force is a harmonic force. Such a force may represent the components of a rotating vector of a centrifugal force in the vertical or horizontal directions.
6
Figure 1.2: Harmonic time history
_t
T
If the vector P rotates with circular frequency w, the orientation of the vector at time t is given by the angle wt (Fig. 1-2) The components of the vector P in the vertical and horizontal directions, respectively, are:
P; (t)
I
= Psin(wt)
p,,(t) = PCOS({tJt)
These forces are harmonic with amplitude P and frequency to. The period measured in seconds is T
=Znk»,
which follows from the condition that in one period one complete
oscillation is completed and thus wT = 2n:.
The frequency measured in cycles per
second and expressed in units called Hertz (Hz) is
f=~=~
T
21f
Consider now the joint effect of two harmonic forces having different amplitudes P1 and P2 , different frequencies
(01
and
W2
and a phase shift ¢. The resultant is
r .-
.
r j
f
I
I (
I
,I
I
,I 1
,'"'1
. f
" "
-
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)
)
r
t,
~
7
I
J I
;
-,
.
I
;r,. (
The time history of the resultant can be generated by projecting the resulting, rotating vector R horizontally.
The character of the time history depends on the ratio of the
amplitudes, the ratio of the frequencies and the phase shift. When the two frequencies are equal, the resultant force is harmonic (Fig. 1.3a). When the frequency ratio
W2 /W1
an irrational number, the resultant force is not periodic (Fig. 1.1 b). When the ratio
is
0)2 /OJ1
is a rational number, the resultant force is periodic but not harmonic (Fig. 1.1 c). However, the envelope of the resultant force is always periodic. When the two frequencies and the two amplitudes do not differ very much, a phenomenon called beating occurs; the force periodically increases and diminishes, similarly to Fig. 1.3b, with frequency of the peaks being
W2 - W1
In general, a periodic function can be represented by a series of harmonic components whose amplitudes and frequencies can be established using Fourier analysis. Therefore, knowledge of the harmonic case facilitates the treatment of more complicated types of excitation.
'.1
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r,
-(
--
r
/
('
8
Fig. 1.3: Basic types of processes composed of two harmonic components
./'
/
--
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'
P2{tr=P2 Sin (wt"'f) p, ft)=-Pr Sin wi
/ I
\ \ -,
-
.-
. ;]R(t)=p'{t).,.~m \. J1Rftl"RslO (cat» 'fll)
\.../.
a)
,/
/
---t ,, P, (t)-:.A sin wit ~ (f)"B sin(w2t fJr)
,/
/
.......
/
P, =P2 6)2
e
W, . 1, 188. . "
b}
",
-, -,
/
\,
'\
/
'\
\
I
\
\
I I
\ \
I I \
\
P, (t) ~ P, sm ~ (I )~f3
W, t
sin(w2t.,.. Jf)
,,
I
/ /
.....
/
I
/
'\J
I
--,\ I
",'\
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\,
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w{
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P'="S ~ wr!,5w,
c)
9
Transient Loading is characterized by a nonperiodic time history of a limited duration and may have features such as those indicated in Fig. 1.4. A smooth type of loading such as the one shown in Fig. 1.4a is produced by hammer blows, collisions, blasts, sonic booms etc. and is called an impulse . Earthquakes or crushers generate more irregular time histories, similar to that shown in Fig. 1.4b. It is presumed that such a process is determined accurately either by an analytical expression or by a set of digital data. A process so defined is called a deterministic process . Often, the duration of an impulse,
~t,
is much shorter than the dominant period of
the foundation response, T (Fig. 1.5). Such loading is characteristic of impacts associated with the operation of hammers and presses . The limited duration of the impact makes it possible to base the analysis of the response on the consideration of the collision between two free bodies.
Random Loading is an irregular process that cannot be predicted mathematically with accuracy, even when its past history is known, because it never repeats itself exactly. Fluctuating forces produced by mills, pumps, crushers, waves and by wind or traffic flow are typical of this category (Fig. 1.6a). A random force and its effect is most meaningfully treated in statistical terms and its energy distribution with regard to frequency is described by a power spectral density (power spectrum), Fig. 1.6b. Earthquake forces can also be treated in this way.
The advantage of the random
approach over the deterministic approach is that the analysis covers all events having the same statistical features rather than one specific time history .
10
Figur e 1.4: Transient loading
p( t)
P(f)
t
o
t
Figure 1.5: Impact loading
. _.
'-r
k
I
(
fJt
o
(.,
I
r(1 'J :' V.
1J.f« T - 1/0
t Figure 1.6: Random loading
P(f)
Sp(f)
o a) Time History
Frequency, f
b) Power Spectrum
11
Types of foundations.
Machine foundations are designed as block foundations, wall foundations, mat foundations or frame foundations. Block foundations, the most common type, and wall foundations behave as rigid bodies . Mat foundations of small depth may behave as elastic slabs.
Sometimes the foundation features a joint slab supporting a few rigid
blocks for individual machines. The foundations can rest directly on soil (shallow foundations) or on piles (deep foundations).
The type of foundation may result in considerable differences in
response .
Notations and sign conventions
The vibration of rigid foundations is characterized by three translations, u, v, w and three rotations,
S, \V, 11 - These
are expressed with regard to the three perpendicular
(Cartesian) axes X, Y, Z. The origin of this system is most conveniently placed in the joint centre of gravity (CG) of the foundation and the machine (Fig. 1.7). The orientation of the axis and the signs of all displacements and forces are governed by the right-hand rule. The translations u, v, wand the forces P, , P, positive if they follow the positive directions of the axis.
The rotations
I
P, are
S, 'V' 11 and
moments Mx , My ,Mz are positive if they are seen to act in the clockwise direction when looking in te positive directions of the corresponding axis, i.e. away from the origin .
12
Figure 1.7: Notations and Sign Convention
Z,w,F;
Examples of typical machine foundations
Basic types of foundations for typical machines are shown in Figures 1.8 to 1.17.
13
Figure 1.8: Block Foundation for Two-cylinder Compressor
Figure 1.9: Block Foundation with Cavities for Horizontal Compressor
14
Figure 1.10: Two Compressors on Joint Pile Supported Mat
~/1:::\.-~·,
o
w
/'1·" ........ :r::J:l' r:.... , , / _+1 ~ IY '-.! -"'",H ;,. ~ ~ . .., . :" .~': . ' ,,,
<,
(-{-0)(_-'
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V I I:z·'-;w~:'''''''' @.f~; i"'l'" ~4, n' : ,. . ',.//:~/, H".%,~';. .~./;?"
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Figure 1.11: Foundation Block on Springs
15
Figure 1.12: Cascade Millon Piles
Figure 1.13: Hammer Foundation
PAD
FOUNDATION
BLOCK
16
Figure 1.14: R.C. Frame Foundation for Turbine Generator
:.. l,]t'
~ ~ t - - -l,,-
f
- - - - - - ·-----r ... . ~
'."
'
-~"1
-11-
Figure 1.15: Pile Supported R.C. Frame Foundation for Turbine Generator
- _. "/4"-
17
Figure 1.16: Steel Frame Foundation for Turbine Generator
" -~
-" r I
Ii
r-. t - - - - - - , I
a) elevation view •
•
.
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:
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b) plan view
18
Figure 1.17: Very Light Steel Frame Foundation for Turbine Generator '- ' -
=
...... -.--
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-
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- --
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I I I I I
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= -l
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r---
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19
e xWe .....• (I
(V
L
.l....t. "
c'
Excitation Forces of Machines
= In rotating machines the excitation forces stem from centrifugal forces associate
~o
"-'1..
':;"'/-with
residual unbalances. Their magnitude can be estimated on the basis of balancing experiments or experience. The centrifugal force is represented by vertical and horizontal forces, the amplitude P for rigid rotors is usually defined as:
)
'" \ \.,\ •.1
:
• rl ('~ '
ftf, e.
Q. . \
~ ,
I." J
7
(,...\ " \ J t."
'I .
,F
t ' l ( ".. )
and co = circular frequency of rotation. The magnitude of e is typically a fraction of a " .tr . C; , .( millimeter such a s,I Q1-0~rnrn.] I !
{ it
,
In reciprocating machines the excitation forces stem from inertial forces and centrifugal forces associated with the motion of the pistons, the fly wheel and the crank mechanism . Many of these forces can be balanced by counterweights but often, higher harmonic components and couples remain unbalanced.
In design situations the
excitation forces should be provided by the manufacturer of the machine.
DESIGN OBJECTIVES The design of foundations for vibrating equipment is always governed by displacement considerations. The displacement of foundations subjected to dynamic loads depends on i) the type and geometry of the foundation ; ii) the flexibility of the supporting ground; and iii) the type of the dynamic loading. The main objective of the design is to limit the response amplitudes of the foundation in all vibration modes to the specified tolerance. Usually the tolerance is set by the machine manufacturer to ensure a satisfactory performance of the machine and minimum disturbance for people working in its
r
I
ru
"",J <
immediate vicinity.
Another objective which could be extremely importan t in some
cases is to limit vibration propagating from the footing into the surroundings.
DESIGN CRITERIA
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"
Factors that may be included in the design requirements .
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.
1,_ static requirements for bearing capacity and settlement. .
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2. Dynamic behaviour , '
,1 -. ft
• limiting vibration amplitude • limiting velocity
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• limiting acceleration • maximum dynamic magnification factor • maximum transmissibility factor • resonance conditions
t
0
3. Possible modes of vibration
vertical; horizontal; torsional; rocking; pitching and possibility of coupled modes.
4. Possible fatigue failures in the machine, in the structure, or in connections.
5. Environmental considerations
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,
• physical and physiological effects on people • effects on nearby sensitive equipment
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,
• possible resonance of structural components
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• consideration of foundation isolation 6. Economy
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• initial cost L
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• maintenance costs • down time costs
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• replacement costs . ~1 , r
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)
Ijt't-· ,~ DESIGN PROCEDURE
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It is a trial-and-error procedure which includes :
l~
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e L a 1'1..-- /pi J""
I
c
• ,
r j '
-T
.
l. ~ 1- Estimating the dynamic loads . -
f.rJ
)
2- Establishing the soil profile and determining the soil properties required for the analysis (Shear modulus, mass density, Poisson's ratio and material damping ratio). 3- Select the type and trial dimensions of the foundation and with clients input , establish the performance criteria.
r
4- Compute the dynamic response of the trial foundation (step 3) supported by the
1,.'1
~ ~
given soil profile (step 2) due to the estimated load (step 1) and compare the response with the performance criteria. If the response is not satisfactory, modify the dimensions of the foundation (step 3) and repeat the analysis until satisfactory design is achieved .
DESIGN INFORMATION
To carry out the design of a foundation system to support a vibration producing equipment, certain loading and site parameters must be known or evaluated.
The
information required for the design can be generally categorized into three main groups: machine properties, soil and foundation parameters, and environmental requirements.
22
(~
.
.."
;.. ,
,
_
Machine Properties The machine properties required for the determination of the loading function include: 1- Outline drawing of machine assembly 2- Weight of machine and its rotor components (or head for hammers) 3- Location of center of gravity both vertically and horizontally 4- Speed ranges of machine and components or frequency of unbalanced primary and secondary forces 5- Magnitude and direction of unbalanced forces both vertically and horizontally and their points of application 6- Limits (tolerance) of deflection (total or differential) and vibration amplitudes to satisfy the machine functions. To calculate the magnitude of the unbalanced forces, the eccentricity of the rotating parts is required . Arya et aI., 1979 give some guidelines to establish the design ~ ',
t
f:.,eccentricities for different types of machines.
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~
~
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\ . '{J
Soil and Foundation Parameters Knowledge of the soil formation (soil profile) and its properties is required for the dynamic analysis. The information is to be obtained from field borings (or soundings)
~"
and laboratory tests. The following parameters are required for the dynamic analysis : density of soil, "/' or mass density, p. 1- Poisson's ratio, v.
2- shear modulus of soil, G, at several levels of strain.
3- material damping ratio, 0, at several levels of strain .
23
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,)
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.
F
'"'l -~
V'
~ "'l
0....
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6-
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(-
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21; ~ ", . ' \ ' Foundation requirements may include:
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1- minimum depth of foundation.
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2- base dimensions for the machine and other components attached to it.
3- type of foundation system to be used (recommended by the geotechnical consultant) .
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4- configuration and layout of the foundation (width, length and depth) .
For piled
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,
f
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/
foundations, the number of piles, pile geometry (diameter or width and cross-sectional
c
area), pile length and spacing between piles are required on top of the configuration of the foundation block. 5- the material properties of the foundation (unit weight of the concrete or steel, the
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n /
Poisson's ratio and elastic modulus).
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-
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Environmental Requirements The machinery produces vibrations that may travel to the neighboring vicinity.
If the
vibration amplitudes are significant, some measures have to be taken to minimize the environmental impact of the machine (this is a major concern for shock producing equipment). On the other hand, there can be some situations where the machine is installed in the vicinity of vibration sources such as quany blasting, vehicular traffic or in a seismic active area. In this case , the information requested should include the character of the vibration and the attenuation at the installation site . The effects of seismic forces have to be addressed using special techniques that deal with the wave propagation and ground response analyses.
24
2
STIFFNESS AND DAMPING OF
SHALLOW FOUNDATIONS
2. STIFFNESS AND DAMPING OF SHALLOW FOUNDATIONS
The dynamic response of foundations, just as the response of other systems, depends on stiffness and damping -characteristics. This chapter presents a general introduction to this subject and a summary of approaches and formulae that can be used to evaluate the stiffness and damping of shallow foundations.
Examples of
structures with shallow foundations are shown in Fig. 2.0. Figure 2.0: Typical Structures with Shallow Foundations a) offshore rigs
Ice
b) nuclear power plants
c) buitdings
; ... -
...-~ _. .-~
-~ . ~
24
d) machine foundations
2.1 Basic Notions of Stiffness and Damping
Stiffness The basic mathematical model used in the dynamic analysis of various systems is a lumped mass with a spring and dashpot (Fig. 2.1). If the mass, rn, is free to move in only one direction, e.g. vertical, it is said to have one degree of freedom. The behavior of the mass depends on the nature of both the spring and the dashpot. Figure 2.1: Basic Model of Single Degree of Freedom System
~
+v
T
k
I
m
k .'---;---~-' ----1--~.-
t
~V
==
I
c=J c
The spring, presumed to be massless, represents the elasticity of the system and is characterized by the stiffness constant k. The stiffness constant is defined as the
25
force that would produce a unit compression (or extension) v of the spring in the positive direction of the displacement of-the mass . For displacements other than unity, the force in the spring (the restoring force) is kv. In dynamics, displacements vary with time, t, and thus v = v(t). However, because the spring is massless, the stiffness constant is equal to static stiffness k
= k st. and k, as well as the restoring force,
kv, is independent
of the velocity or frequency with which the displacement varies . Figure 2.2: Effect of Mass on the Dynamic Stiffness p (f)
P{ t)
. -. . -····,j ·--r;.'.."::.... .: . .:.;.:. .... . .. . " 05]5 ~..~ .. .. ::.';. .:~ ~~
. .
m s -,
k
?0'Y..J" ;///.
.
r- --
"
.~ ~. ~ .~ ".:".:.: ~. . . . . .,..
k = canst.
~/. ;~~~~ ':.:~~ ~L ,/-
o
k
W
FREO UElvCY W
a)
b)
c)
Now apply the same concept of stiffness definition to a harmonically vibrating column, which possesses mass and has its mass distributed along its length (Fig. 2.2a) . For an approximate analysis at low frequency, the distributed mass can be replaced by a concentrated (lumped) mass m., This mass is attached to the top of the column and the column itself can be considered massless (Fig. 2.2b). Consequently, the stiffness of this massless column can be described by a static stiffness constant, kst, which is independent of the frequency . The elastic force in the column just below the mass is kstv for any displacement, v. However, the total restoring force generated by the column at the top of the lumped mass is the sum of the elastic force in the column and the
26
inertial force of the mass. If the displacement varies in harmonic fashion, such that;
vet) = V cas( CO t)
(2.1 )
in which v is the displacement amplitude and
CD
is the frequency, the acceleration is:
2
~ ~ .f.
(j--Ilf ~~
I
.. d 2v =-vw 2 COS ( cat ) V =-dt
J I V
"
- r[
and the inertial force is:
msv.. = -msvw 2 COS ( cat ) In the absence of damping, the relation between the external harmonic force and the displacement is:
or for the amplitudes;
Stiffness, being the constant of proportionality between the applied force displacement, becomes : 'H JO
"t- ~ I,~, ~ . ,1 "y'A f_l.J .
Iii p
-
..
-~
'./.
i
. l ~....... '
c: t r
.
~t
" CA..
f'- ~.
r
""
v
1. ',C:
( ~:......)
"_'
and
,
i,.-""-
o
,~
.;1.-
•
tV"
(2.2) -;...
(I
Thus, with vibration of an element having distributed mass, the dynamic stiffness constant generally varies with frequency. At low frequency this variation is sometimes close to parabolic as the example considered here and presented in Fig. 2.2c suggests. The column used in this example may be a column of soil and it thus appears obvious that a soil deposit may feature stiffness constants that are frequency dependent. The magnitude and character of the effect of frequency depends on the size of the body,
27
vibration mode, soil layering and other factors. Another way of accounting for the parabolic variation of dynamic stiffness with frequency is to add the lumped mass , rn., representing the mass of the supporting medium, to the vibrating mass, m and consider the stiffness as constant and equal to the static stiffness. Equation 2.2 suggests this approach . However. a quadratic parabola only in some cases can represent the variation of dynamic stiffness with frequency and the added mass is not the same for all vibration modes . For these reasons, it is usually preferable to consider dynamic stiffness as frequency dependent and to forego the added mass.
Finally , if the frequency range of interest is not very wide, it is often
sufficient to replace the variable dynamic stiffness by a constant representative of the true stiffness in the vicinity of the dominant frequency (Fig. 2.2c)
Damping
The dashpot in the model shown in Fig. 2.1 represents damping caused by energy dissipation. Like the restoring forces, the damping forces oppose the motion but the energy dissipated through damping cannot be recovered. A characteristic feature of damping forces is that they lag the displacement and are out of phase with the motion. In soils, the energy of vibration is dissipated through two mechanisms: propagation of elastic waves away from the source and inelastic deformation of soil. The former mechanism results from the practical infinity of the soil medium and is referred to as geometric or radiation damping. It is close to viscous in character. Inelastic deformation of soil manifests itself in the form of a hysteretic loop and is considered as material or hysteretic damping .
28
Viscous damping is proportional to vibration velocity and its magnitude is
. dv cv=c dt
(2.3)
in which c is the constant of viscous damping. The damping constant c is defined as the force associated with a unit velocity. Viscous damping describes quite well the resistance p of viscous fluids to motion of subrnerqed bodies, hence the term.
To
commemorate the scholars who were the first to extensively employ this type of damping model, viscous damping is also known as Kelvin or Voigt damping . If the mass indicated in Fig. 2.1 vibrates in a harmonic fashion described by Equation 2.1, the viscous damping force is:
cv = -CVoOJ sineOJt)
(2.4)
and its amplitude (peak value) is cVO(J). Hence, for a given constant c and displacement amplitude vo. the amplitude of the viscous damping force is proportional to frequency (Fig. 2.3).
Figure 2.3: Comparison of Viscous and Hysteretic Damping
C)
z
iL----- ~
HYSTERUIC
o
FREOUUJCY W
Hysteretic or material damping results from the dissipation of energy due to the
29
imperfect elasticity of real materials, which under cyclic loading, exhibit a hysteretic loop (Fig. 2.4) . The amount of energy dissipated is given by the area of the hysteretic loop. For most materials, including soil, the amount of the dissipated energy depends on strain (displacement) but is essentially independent of frequency just as the hysteretic damping shown in Fig. 2.3. Figure 2.4: Hysteretic Loop T
w
The hysteretic loop implies a phase shift between the stress and strain because there is a stress at zero strain and vice versa as can be seen from Fig. 2.4. Thus, the basic stress-strain relations of elasticity such as r
= Gy
have to be extended to
accommodate the phase shift . This is conveniently achieved by the introduction of the complex shear modulus in which the real shear modulus G is complemented by an imaginary (out-of-phase) component Gr. Then, the complex shear modulus can be defined as:
G* =G+iG'=G(l+i
G') G
(2.5)
The dimensionless ratio G'/G may be expressed in terms of the "loss angle" 0 such that
tan(5)
G' G
=
(2.6)
30
or in terms of the material damping ratio;
/l = -
G'
2G
=
1 -tan(tS) 2
With these dimensionless measures of material damping, the complex shear modulus IS
G* = G(l + itan(6))
(2.7)
= G(l + i2/l) The magnitude of the material damping can be established experimentally using the hysteretic loop and the relation
1
sw
/l=--. 4JZ" W
(2.8)
in which I1W is the area enclosed by the hysteretic loop and W is the strain energy (Fig. 2.4) , a typical value of is 0.05 (5%). The material damping of soils is constant for small strains (y ~10-2%) but increases with strain due to the nonlinear behaviour of soils . Conversely, the shear modulus decreases with strain. The terms material or hysteretic damping are usually meant to imply frequency independent damping. If material damping proportional to frequency is to be described, the definition of the complex modulus can be modified by replacing
f3 by W(0 where f3' is another
constant. Such damping would actually be viscous as in Eq. 2.4 . In experiments, the difference between the two types of material damping can be clearly recognized in the area: of the hysteretic loop measured at a constant level of strain at different frequencies (Fig. 2.5). Such experiments indicate that frequency independent hysteretic damping is much more typical of soils than viscous damping because the area of the
31
hysteretic loop does not grow in proportion to frequency. Figure 2.5: Hysteretic Loops at 2 Different Frequencies: T
b) Viscous
a) Hysteretic
Methods for Describing Stiffness and Damping Until recently, stiffness and damping constants of foundations were most often described
using
empirical
formulae
derived
experimentally. The experimentally
established total stiffness was divided by the base area to define the so-called subgrade modulus, which was considered to depend only on the type of soil.
Sometimes
corrections were introduced to allow for some variation of subgrade modulus with base area and the direction of vibration. Alternatively to the experimental method, stiffness constants were obtained by means of static analysis of the continuum and damping was estimated . Presently, the prevalent trend is to obtain the stiffness and damping of foundations using dynamic analysis of a three dimensional or two dimensional continuum representing the soil medium . The continuum is modeled as an elastic or viscoelastic halfspace whose surface limits the extent of the soil medium. The halfspace can be homogeneous or nonhomogeneous (layered) and isotropic or anisotropic. The
32
governing equations are solved analytically or by means of numerical methods such as the finite element method. The analytical solutions were initiated by Reissner (1936) and have undergone rapid development since then. Many researchers contributed to this development: Bycroft, 1965; Luco and Westman, 1971; Veletsos and Verbic, 1973; Kobori et al, 19 71 to name only a few. The finite element method was applied to the dynamics of a continuum by Lysmer and Kuhlerneyer, 1969, Kausel et al., 1975 and others. The refinement of both the analytical
and numerical techniques and the extension of their versatility contributed greatly to the increase in the popularity as well as the credibility of the continuum approaches. Although further improvements and corrections of these approaches are needed, their principal advantages are that they account for energy dissipation through elastic waves (geometric damping), provide for systematic analysis , and describe soil properties by constants, which can be established by independent experiments. In the continuum approaches, the stiffness and damping constants of foundations are obtained by the theoretical determination of the relationship between a harmonic force acting on a massless disc resting on the surface of the halfspace and the resulting displacement of the disc.
For mathematical convenience, the harmonic force is
considered complex, i.e.
Pet) =Pe im! =P(COS(OJt) + isin(OJt)) in which v(t)
(L)
is circular frequency. The resulting displacement of the disc is also complex,
=v eiu)t.
Canceling the time function eicot from P(t)
= P eico t and v(t) =ve imt for
brevity,
the relation between the applied force and displacement is obtained as:
33
(2.9a) in which the complex stiffness (impedance function) is (2.9b) The complex stiffness has a real part K1 = ReK and an imaginary part
K2 =
ImK. The
real part represents the true stiffness and defines directly the stiffness constant of the base
k=K 1 =ReK
(2.10)
The imaginary part of the complex stiffness, K2 describes the out-of-phase component and represents the damping due to energy dissipation in the halfspace. Because this damping generally grows with frequency, resembling viscous damping as in Fig. 2.3, it can also be defined in terms of the constant of equivalent
VISCOUS
damping
lInK
K
2 c=--= -- (j)
(2.11)
(j)
Then the complex stiffness can be rewritten as
K
=
k + iaic
(2.12)
and the force-displacement relation either as
P
=
(k + ioicyv
(2.13)
or
P = kv + CV in which both k and c are real and
(2.14)
v:::
dv/dt is velocity.
Eqs. 2.13 and 2.14 are
equivalent, as can be checked by evaluating the out-of-phase (damping) part of these
34
equations for harmonic motion For v(t)= voeiGJt the damping force in Eq. 2.13 becomes
iaicv -- ito cv 0 e
i (()t
c
and the damping force in Eq. 2.14 becomes
, V I I
These are the same .
/
l'
r:
I.
\1 t- . Co
C'.
t
-
--.
t
J -
.
/
The stiffness K as well as the constants k and c generally depends on frequency and other factors. The elastic halfspacetheory indicates that the effect of frequency can be lumped with a few other factors into
bdimensionless parameter, the so-called \
dimensionless frequency,
.J
. ~
~
r .
(2.15)
in which R = radius of the base , Vs = shear wave velocity of the soil and p = mass density of the soil.
Material damping can be incorporated into the stiffness and damping of the footing in a few ways. The most direct way is to introduce the complex shear modulus (or, more generally, complex Lame's constants) into the governing equations of the soil medium at the beginning of the analysis and to carry out the whole solution with material damping included. Another way is to carry out the purely elastic solution first and to introduce material damping into the results of the elastic analysis by means of the correspondence
principle of viscoelasticity. With steady-state oscillations considered in the derivation of footing stiffnesses, the application of the correspondence principle is quite simple and
35
consists of the replacement of the real modulus G by the complex shear modulus G* (Eq. 2.7). This replacement must be done consistently wherever G occurs in the results of the elastic solution. This implies even in the shear wave velocity V s and the dimensionless frequency given by Eq. 2.15 which, consequently, also become complex. Therefore, all functions which depend on ao are complex as well. The substitution of G* can easily be done if analytical expressions for the stiffness K or constants k and care available from the elastic solution. With the material damping included, the constants K1 , K2 and k, c have the same meaning as before but depend also on tano (or (3). The above procedures for the inclusion of material damping into an elastic solution are accurate but not always convenient. When the elastic solution is obtained using a numerical method, the impedance functions are obtained in a digital or graphical form and analytical expressions are not available . Then, an approximate approach is often used whereby the complex modulus replaces only the real modulus occurring in front of the dimensionless expressions for stiffness and damping but not in the dimensionless frequency ao. Thus for the complex stiffness described using the true stiffness and damping constants of Eq. 2.12 hysteretic damping may be accounted for by multiplying the complex stiffness evaluated without regard to material damping by
(1 + i2(3) to give:
K,
=
(k + iOJc) (1 + i2fJ)
(2.16)
Performing the multiplication,
KJz =k-2j3mc+im(c+ 2j3k) OJ
(2.17)
Defining again the true stiffness as kh = ReKh and the constant of equivalent viscous
36
damping as ch = ImK/w, the stiffness and damping constants incorporating material damping are:
k h -:=k-2j3cOJ
(2.18a)
2f3k
Ch =C+-
OJ
(2.18b)
in which k and c are calculated assuming perfect elasticity with c accounting only for geometric damping. Comparison with the accurate approach indicates that the approximate Eqs. 2.18 give sufficient accuracy at low dimensionless frequencies but that the accuracy deteriorates with increasing frequency. Eqs. 2.18 are very illustrative. They indicate that material damping reduces stiffness but increases damping. The degree of these effects depends on the magnitude of material damping and on whether this damping is defined as hysteretic (by the constant p) or as viscous (by Ww) These effects are shown in Fig. 2.6. The effect of frequency independent hysteretic damping is shown as well as the effect of material damping proportional to frequency (viscous damping). The assumption of viscous damping
may overestimate the effect of material damping particularly at high
frequencies . The constant hysteretic damping results in the equivalent viscous damping constant 2Pk/w which varies with frequency and approaches infinity for co
---t
O.
37
Figure 2.6: Effect of Material Damping on Stiffness, Equivalent Constant of
Viscous Damping and Imaginary Part of Complex Stiffness
\ \ /
HYSTERETIC
--~----
.....
------ --- ----- VISCOUS DAMPING
..... "
- --------
NO MAT. DAMPING
"
FREQUENCY
2.2 Stiffness and Damping of Shallow Foundations Using the theory of the elastic halfspace and assuming linearity, stiffness and damping
------
constants can be evaluated for various shapes of the foundation base and different types of soil medium. However, the basic case is one of a circular disc. The results of the theory can be used even for non-circular shapes if an equivalent circular base of suitable radius replaces the real noncircular base. The radius of the equivalent circular base , the equivalent radius for brevity, is usually determined by equating the areas of the actual and equivalent bases for vertical and horizontal translation, the moments of inertia (second moment of area) for rotation in the vertical plane (rocking) and the polar
38
moments of inertia for torsion about the vertical axis.
From these conditions, the
following equivalent radii are obtained for rectangular bases having dimensions a and b (Fig. 2.7): Figure 2.7: Notation for Calculation of Equivalent Radii of Rectangular Bases
tZ,W
-t
»:
. I
"
\
R
~Ca
X,U
/ 7 7 / / / / / / / 1 / // / / /
---i
~ Y,v
./
<,
~
~if.t
777 - --
I
/
"
lb --p,u
I•
..1
.1
a
Table 2.1 - Equivalent Radii /for Rectangular Footings Translation
Rocking
R=f!
2.19a
R=~ 31Z"
2.19b
Iff
Torsion
R
2.19c
2+b 2
-4
'7 - ,
ab(a 61r
)
For rocking, two different equivalent radii are needed for the horizontal directions. The equivalent radius works very well for square areas and quite well for rectangular
39
areas with ratios alb of up to 2 (Kobori et aI., 1971) With increasing ratio alb, the accuracy of this approach decreases. For very long foundations the assumption of an infinite strip foundation may be better suited.
Surface Foundations
For circular bases or equivalent circular bases the complex stiffness Kj associated with direction i can be expressed in terms of the true stiffness constant, kj ~ r l.: ' 1.(,", '- r r. : 'c _ I. and damping constant, c as: /'/ --..~
f
.
s, =k[kj'(ao)+iaoc\(ao)J
(2,20)
-
in which k, is static stiffness, ao = dimensionless frequency and k'j and e'i are stiffness and the damping constants normalized as follows :
k' .= I
k. I
IeI
v
c'.=~c" I
k .R
I
(2.21)
I
In the case of an isotropic homogeneous halfspace, the static stiffness constants for the vertical translation, v, horizontal translation sliding) u, rocking, and torsion, n, are
k = 4GR v
1 -v
(2.22a)
8GR
2-v
(2.22b)
8GR 3
k= - - (2.22c) VI 3(1- v)
40
(2.22d) in which v is Poisson's ratio. These expressions are approximate and torsion in particular can be in error because, in reality, the torsional stiffness can be affected by slippage of the foundation . There is also a small coupling stiffness between 'V and u but this can usually be neglected . The normalized, dimensionless stiffness and damping constants kr and
Ci
are shown in Fig. 2.8. Figure 2.8 indicates how the stiffness and
damping constants vary with dimensionless frequency, ao, and Poisson's ratio, v, and suggests the frequency ranges in which the constants can approximately be taken as frequency independent. The horizontal motion is most favourable in this respect. The most frequently used frequency range for machine foundations is from ao
=0.5 to about
2.0 but higher dimensionless frequencies are also met, e.g. with large turbine generators, compressors or buildings . For the results plotted in Fig. 2.8, material damping was neglected . Yet it is important in some cases, particularly for rocking and torsion.
Also, analytical
expressions or numerical data are needed for more detailed calculations . Such expressions as well as tabulated data can be found in the papers by Luco and Westmam (1971), Veletsos and Wei (1971), Veletsos and Verbie (1973), Veletsos and Nair (1974) and Wong and Lueo (1978).
For impedance functions of foundations
resting on the surface of a viscoelastic halfspace, analytical expressions incorporating material damping in a more accurate way than Eqs. 2.18 are given in the appendix.
41
Figure 2.8: Dimensionless Stiffness and Damping for Circular disk on Surface of
Homogeneous Halfspace (Velestos and Verbic, 1974; Vetestos and Nair,1974:
Poisson's Ratio v = 0, 1/3 J "'12 )
'-,
.
~
c
~--- _. -
..
-:.c,_-- ."
,
\.
--t..:-rr:
~
~
It'
u .
k'
v
,
t I
M~ -=- -j1
\
\
I 2 1_ .
..... e
'0
cr c' v
r
:.::::..:.:..::...._ .. --~
U~'rl?
,~."
:
- _ ., ..
0)
~.'
"--~_,_-____;::----:-~......,.
t
t}L--y-----.~-.__-.-~
ao
a) vertically excited disk
a
o
b) horizontally excited disk
x'11
C
J
n
0'
0.'
.
•• ' .:!.
"
c) disk in rocking motion
d) disk in torsion
I I
<1:5
i
L, r:
.
j
(
~r
. .
I
42
(I
t. ~\
. I
Inhomogeneity - In real situations, the soil shear modulus often increases with depth due to increasing confining pressure, i.e., the soil is not homogeneous.
A
theoretical study by Werkle and Waas (1986) has shown that in such cases the geometric damping can be greatly reduced compared to that of an homgeneous halfspace. This may explain the fifty percent reduction in damping observed in field experiments (Novak, 1970).
Embedded Foundations Most footings do not rest on the surface of the soil but are partly embedded. Embedment is known to increase both stiffness and damping but the increase in damping is more significant. These effects of embedment were observed in early experiments by Novak (1964) and by others. It is very difficult to extend the elastic halfspace solution to include embedment although progress has been made. The finite element approach is well suited to the determination of stiffness and damping of embedded foundations and has been used by a number of investigators, e.g. Urlich and Kuhlemeyer (1973) and Kausel and Ushijima, 1979) An approximate but versatile approach can be formulated using the assumption that the soil reactions acting on the base of an embedded foundation can be taken as equal to those of a surface foundation (halfspace) and the reactions acting on the footing sides as equal to those of an independent layer overlying the halfspace (Fig . 2.9). The evaluation of the reactions of the layer can be simplified even further if they are calculated using the assumption of plane strain .
43
Figure 2.9: Schematic of Embedded Foundation
I.
2R
C1 ,h
.1f
ut
C ' '
j
.1
This means that these reactions are taken as equal to those of a rigid, infinitely long, massless cylinder undergoing a uniform motion in an infinite, homogeneous medium. This assumption can also be interpreted as meaning that the layer is composed of independent, infinitesimally thin layers. This concept was first employed by Baranov (1967) and was further developed by Novak and his associates (Novak and Beredugo, 1972; Beredugo and Novak, 1972; Novak and Sachs , 1973). It was found that this approximate approach works quite well and that its suitability and accuracy in creases with increasing frequency . The plane strain approach to the side reactions has many advantages: it accounts for energy radiation through wave propagation , leads to closed form solutions and allows for the variation of soil properties with depth. It can also allow for a slippage zone around the footing (Novak and Sheta, 1980; Lakshmanan and Minai, 1981). Finally, the approach is very simple and makes it possible to utilize the well established
44
solutions of surface footings since the effect of the independent side layer actually represents an approximate correction of the halfspace solutions for the embedment effect. The side reactions are described by complex, frequency dependent stiff nesses for a unit length of the embedded cylinder in a way analogous to the surface disk For a unit length of the embedded cylinder; vertical stiffness (2.23a) horizontal stiffness
K,
=
Gs [SuI (a o' v,D) + ia OS u 2 (a o,v,D)]
(2.23b)
rocking stiffness (2.23c) torsional stiffness (2.23d) In these expressions, the shear modulus Gs and material damping 0 = tano are those of the side layer which may represent the backfill. The dimensionless parameters SJ and 8 2 relate to the real stiffness and the damping (out of phase component of the stiffness), respectively. They all depend on the dimensionless frequency,
45
in which psis the mass density of the side layer. Only the horizontal stiffness depends on Poisson's ratio, v. The mathematically accurate analytical expressions for the parameters 8 1 and 52 are given by the author et al. (1978) . The variation of parameters 8 1 and 8 2 with frequency is shown in Fig. 2.10. For the embedment I, the total stiffness is Kd. Figure 2.10: Stiffness and Damping Parameters of Side Reactions (5/2
= Siz i a o;
Novak et al. 1978)
~
~6
z;
s
"' . t~
"'- 0.2
-I
ot-I-------c,,--~o
0.5
0.'
""'- r, - - - - - - ,I ~
1.0
DIl1£loSl 0Nl.E:SS rp<:w
Vertic:al Stiffnoss and Damping Parameters Sw
Horizontal Stiffness and Damping PatamatarsSu
o:J r0 """,
~
~
1 1 ,
i
~ . J
~~ 1-1 ~ i
0 '
4
::1
~ :I~--. . : : "--~-:------- :" :" --" " " -"
c
=.
~5
Torsional Stiffness and Damping ParamBters Sl;,
O!5 ,
) .(1
r ' I''''
OIr",(1iJ;I!X. ( 'S$ r ~«,(t·C'l' <'10 • ~;
'/>
Rocking Stiffness and Damping Parameters SIj,t
It may be seen in Fig. 2.10 that as ao ---+ 0 the stiffness parameters 8 v 1 and 8 u 1 vanish. This is a consequence of the plane strain assumption. For practical application,
46
this may be empirically corrected by extending a suitable. nonzero value of Sj such as S1(ao=0.3), towards the origin as shown later herein. Such a correction is needed only for bodies featuring a very small diameter. The stiffness constants of the side layer Kv and K, depend on the radius R only through the dimensionless frequency Go = RwNs This seemingly, surprising result is, however, correct for plane strain and has been confirmed by finite element analysis. The versatility of the plane strain approach can be enhanced further if a cylindrical weakened zone is considered around the footing to account, in an approximate way, for the lack of bond between the footing and soil, non-linearity due to high strain or different properties of the backfill. With the weakened zone , the side reactions are still described by Eqs. 2.23 except that parameters S1,2 are modified. These more general parameters are available in Novak and Sheta (1980). The most prominent effect of the weakened zone is a reduction of radiation damping, which increases with frequency. The inclusion of the weakened zone may improve the agreement considerably between theory and experiment. The complex stiffness of embedded foundations can be evaluated approximately by adding the stiffness generated by footing sides and defined by Eqs. 2.23 to that generated in the base and given by Eqs. 2.20. For the vertical direction and torsion, the total stiffness and damping of the embedded foundation thus implies a simple addition of the two reactions. For the horizontal direction and rocking, coupling between the two motions has to be considered .
47
Table 2.2: Stiffness and Damping Parameters (0=0)
I Soil
Motion Sliding
Side Layer
Halfspace
Cohesive
SuI = 4.1
S"2 = 10.6
CUI = 5.1
Cu 2 = 3.2
! Granular
SuI = 4.0
SI/2 = 9.1
C,I! = 4.7
C"2 = 2.8
S.... I = 2.5
SIJI 2 =1.8
CIJI] =4.3
C'1' 2 = 0.7
CIJII =3 .3
C¥l2 = 0.5
S"2 = 5.4
C'I[ =4.3
C,12
Sv2
Cv1
; ;
i ~
Rocking
Cohesive
; ; ; ;
Granular
I Cohesive
Torsion
S"I = 10.2
I
= 0.7
I
t
Granular
l
Vertical
Cohesive
Svl
= 2.7
Granular
:=
6.7
= 7.5
C"2 = 6.8
C,.! = 5.2
CV2 = 5.0
An embedded footing implies a body of a certain depth and therefore, a horizontal translation is resisted not only by horizontal soil reactions but also by moments.
This gives rise to coupling between translation and rotation and the
corresponding "off-diagonal" or cross stiffness and damping constants such as k UIfI and
CU \jT
::::
klflu
= cljiu, For coupled horizontal and rocking motion the qeneration of the stiffness
and damning constants is indicated in Fig. 2.11.
48
Figure 2 .1 1: Translation and Rocking Components of Coupled
otion and Related
Stiffness and Damping Coefficients for Embedded Footings
s
......-
(-) CdJu
u' t
..--...
M
r'
c ...j. :::--
- - -,-,
lj!~-------
1 I
(a)
~
(e)
Y.,u
t,:
To identify the coupling, double subscripts are needed.
Stiffness
~j
or damping
Cij
indicate the force acting at the reference point in the direction i and associated with a sole unit displacement or unit velocity in the direction j. Applying the sign convention indicated in Fig. 2.11c, the stiffness and damning constants of embedded footings are: For vertical vibration
V,
the stiffness constant
(2.24a) and the damning constant
(2.24b)
For torsional vibration 11 > the stiffness constant
(2.25a) and the damping constant
49
For coupled horizontal translation u and rocking \!f the stiffness constants are:
(2.26a)
(2.26b)
G t5 2 2 +_ s t5(-+ ~-t5~)S ] G 3 R2 R ul
(2.26c) and the damping constants are
CUll
G, S 1/2 ) ="\j~R2(P c u 2 + U.<' ~ PsP G LT
(2.27a)
(2.27c)
50
In the equations, the embedment ratio 0= I/R, where I is the embedment deptl and Yc = the vertical distance of the reference point CG from the base. Parameters C relate to the reactions acting in the base and can be extracted from Eq. 2.20. Their variation with frequency can be seen from Fig. 2.8 Parameters S relate to the side reactions and are given by Eqs. 2.23. They are shown in Fig. 2.10. Parameters C and S are frequency dependent.
However, given all the
approximations involved in modeling dynamic soil behaviour it is often sufficient to select suitable constant values to represent the parameters, at least over a limited frequency range of interest.
Such constant values are suggested in Table 2.2. The
values are given for cohesive soils as well as granular soils with Poisson's ratio presumed as 0.4 and 0.25 respectively . The values shown in Table 2.2 correspond to dimensionless frequencies between 0.5 and 1.5, which are typical of average machine foundations . For other dimensionless frequencies numerically more accurate values can be computed.
If a large frequency-range is of importance, parameters C and S
should be considered as frequency dependent and calculated from the forming expressions . Such expressions are given in polynomial form in Beredugo and Novak (1972), and in the Appendix to this chapter. Material damping is not included in Table 2.2 but it can be accounted for either accurately, i.e. evaluating parameters C and S with regard to material damping, or approximately by means of Eq. 2.18. Equations 2.24 to 2.27 hold for cylindrical footings that feature only one radius , R. For rectangular footings} the equivalent radius for translation, R, differs from that for rocking, R, as Eqs. 2.19 suggest and this should be incorporated into constants k'l'o/ and
51
CIjI'lf
whose generation implies both sliding and rocking . Hence, for rectangular footings,
these constants can be rewritten as
(2.28a)
-o~)S R u2 ] IfI
(2.28b)
For the torsional costants k'l'l and ~'1 the radius given by Eq. 2.19c should be used. These measures are, of course, approximate but the solution for rectangular embedded bodies is particularly difficult and the theory is approximate anyway. Experiments indicate that the theoretical values of stiffness and damping coefficients should be adjusted.
Adjustment of Theoretical Values First, experience has shown that the theory tends to considerably overestimate the damping in the vertical direction (Novak, 1970) . This is caused by the usual presence of interfaces between soil layers that reflect the waves back to the vibrating
52
body, reducing geometric damping. An empirical reduction of C v2 (D=O) to about one half of the values valid for homogeneous halfspace appears advisable for practical applications. On the other hand, the first resonant amplitudes of coupled response of surface footings to horizontal forces are often overestimated by several hundred percent if material damping is neglected. This discrepancy can be eliminated by the inclusion of material damping, desirable because of the low level of radiation damping in rocking. Torsional response is often predicted very poorly because of slippage. Slippage reduces stiffness and increases damping of surface foundations, (Weissmann, 1971) but reduces damping of embedded foundations (Novak and Sachs, 1973). The inclusion of the weakened zone around the footing may improve the agreement between the theory and experiments (Novak and Sheta , 1980.). On the whole, embedment effects are often overestimated because soil stiffness (shear modulus) diminishes toward the soil surface due to diminishing confining pressure. This is particularly so for backfill with which no stiffer surface crust is present and whose effects are always much less pronounced than those of undisturbed soil. The lack of confining pressure at the surface often leads to separation of the soil from the foundation and to the creation of a gap, as indicated in Fig. 2.9, which signi ficantly reduces the effectiveness of embedment. Considering an effective embedment depth smaller than the true embedment can be used as an approximate correction for this effect.
Shallow Layers. - Another correction of the halfspace theory may be required if the deposit is a shallow layer. In such a case, the stiffness increases and geometric
53
damping decreases or even vanishes. These effects can be seen from Fig. 2.12 in which the stiffness and damping parameters are plotted in dashed lines for layers of different depth, h, with material damping neglected . These parameters were calculated from the results due to Warburton (1957). The parameters for the halfspace and side layers are also shown for comparison. (The subscript v is deleted.) Similar behaviour is observed in other vibration modes as well and is confirmed by more general solutions of layered med [a presented by Bycroft (1956), Luco (1974) and others .
54
Figure 2.12: Stiffness and Damping Parameters for Vertical Vibr ation of Footings on Halfspace and Strata of Limited Depth (ro =R)
- -L
~
12- _
, , 'F'.
Ii" 10
I
o oL--L--'------'-
--->O.4
0.2
0 .6
OIM t: NS JONL £.SS
c~
lJ1
5 I
6
(f:
4
w :E -a. cr:.
Itl l ,
0.8
L_
I
1.0
FREOUENCY
1.2
' - ----'----'
1.4
0"
=CD,,, = 0,5)
~.:;lo~·-=CD::..(:..:":-~-.-:O=.2:::.5:::J:-
- - - - - -
( b)
0 .2
0.4
0 .6
DlI,,'.E NSIONLE5S
0.8
1.0
1.2
1-4
FREOU ENCY
55
It can be seen from Fig. 2.12 that the geometric damping of strata is quite small or even absent at low frequencies. Then, material damping may be the principal cause of energy dissipation. It can be evaluated using Eq. 2.18b. Studies of the behaviour of strata suggest that geometric damping may completely vanish if the frequency of interest, e.g. the excitation frequency, is lower than the first natural frequency of the soil layer (Kobori et aI., 1971; Nogami and Novak, 1976). For a homogeneous layer with soil shear wave velocity V s the first natural frequencies are
OJ,
= Jr
~ ~2(1
2h
u)
(2.29 a)
1-2v
for the vertical direction and
(2.29b) for the horizontal direction. At frequencies lower than
Wv
and w u , only material damping
remains because no progressive wave occurs to generate geometric damping in the absence of material damping and only a very weak progressive wave occurs in the presence of material damping. The damping parameters
8 2 generated
damping alone can be established from Eqs. 2.23 by neglecting
82
by material
and substituting the
complex shear modulus defined by Eq. 2.7 for Gs . This yields constant damping parameters
Su2 = 2/3 SuI ao
(2.30a)
56
S v2 =2/3 Svl
(2.30b)
Qo
to be used for frequencies lower than
Wh
and
Wv ,
respectively (Fig. 2.13). This
correction is most important for both the vertical and horizontal directions in which the geometric damping of the halfspace is highest. Figure 2.13: Correction of Parameters Su and Sv for Low Frequencies
Stiffness and damping from finite element solutions. Other theoretical approaches and formulae for embedded foundations have also been reported, most of which were obtained using the finite element method. The results were evaluated by Roesset (1980) who also compared the stiffness constants obtained by Elsabee (1977) and Kausel and Ushijima (1978) with those of the writer described by Eqs. 2.24 to 2.27. For a halfspace (H ---7 0:), v = 0.4 and Yc
=0, Roesset's comparison is given in Table 2.3.
57
Table 2.3: Comparison of Stiffness Constants for Embedded Foundations (after Roesset, 1980; 8 = IIR) Stiffness
Elsabee - Kausel
Novak
kw
4GR (I + 0.470) I-v
4GR (1 + 0.415)
kTl T1
1- v
3
3
16GR (l + 2.675) 3
kuu
16GR 3
8GR (l + 0.670)
8GR
2-v
2-v 3
klj/\{'
8GR
ku'l1
(1 + 2.400)
(.1 + 0.805) 3
(1 + 2.05)
8GR
(1+ 0.65 + 0.35
3(1- v)
3(1- v)
1.6GI (l + 0.470 ) I-v
2GI (I + 0.415)
3
)
1- v
Given all the approximations involved , the agreement between the two solutions is very good except for the rocking constant
k'V~ ,
for which substantially larger values are
obtained from the Elsabee formula than from the author's over most of the embedment ratio range. With respect to this agreement, Eqs. 2.24 to 2.27 appear quite adequate for practical application and have the advantage of allowing for the various corrections discussed. Recently, a number of efficient solutions for embedded foundations were formulated using the boundary element or boundary integral methods (e.g., Kobayashi and Nishimura, 1983; Wolf and Darbre, 1984, Karabalis and Beskos, 1985). A few investigators using the finite element method have studied embedment in a layer of Limited thickness. The static stiffness for the horizontal and rocking modes was
58
derived by Elsabee (1977) and the vertical and torsional modes were derived by Kausel and Ushijima (1979) . The empirical expressions derived by these authors for the static stiffness of circular foundations embedded in a homogeneous soil layer of total depth H are:
k ==8GR(1+~~)(1+26)(1+~~) 2-v
uu
2H
3
4H
(2.31 a)
kYIJI == (0.4£5 - 0.03) R k uu
(2.31b)
8GR 3 1R I
klJllf/ == 3 (1- v) (1 + 6 H)(1 + 28)(1 + 0.7 H)
(2.31 c)
I
k;
=
4GR R HI] (1 + 1.28 H)(l + 0.478)[1 + (0.85 - 0.288) I-v 1- H (2.31 d)
16 3 k ll Tj == -GR (1 + 2.678) 3
(2.31 e)
These stiffnesses are referred to the centre of the base and are valid for 0 = fIR.::: 1.5,
I/H .::: 0.75 and R/H .::: 0.5 . It may be noticed that the first factor in all the expressions except ku vr is the static stiffness of a circular disk on the surface of the halfspace given by Eqs. 2.22. Complex dynamic stiffnesses are then defined as
K ==k(k'+ ia oc')(1 + i2f3)
(2.32)
59
k are the static stiffnesses determined by Eqs. 2.31, k' and c' are
in which
dimensionless stiffness functions depending on dimensionless frequency ao ;:: RroN s and
p is material damping
ratio of soil. Kausel and Ushijima recommend taking k' and c'
as equal to the halfspace functions except for the function c' in the low frequency range. The stiffness functions are shown in Fig. 2.14. In the low frequency range, the stiffness functions of layers differ substantially from those of the halfspace because the geometric damping vanishes below the first layer resonance (Fig. 2.14).
Figure 2.14: Stiffness Functions for Embedded Foundations (after Elsabee) a) embedded foundation - finite layer b) surface foundation - finite layer c) surface foundation - halfspace
/ } 7;;; ", r r ' / ;;,...·T77"T
b
j
. o
o
a
_
!_ .... _
O.!
'"
2 , l.--...L1_ --'-::---7'-::-0.4 0.2 0 .3
-J1
-=-'":::-
0 .5 f
o
"i
'[ __ (~a 5~ ,
e,
lfI
05
b 71";
,2
0. 2
't;
,
I
0. 3
SWAYING
0.4
00
0.5 f o
o
~
1
r ~--oo ~
'0
-----,..-.-.-...
--- ,~
0.1
0 .2
0 .3
0.4
fo
ROC K iN G
In this range, Kausel and Ushijima suggest estimatin g the radiation damping coefficient
60
as
c=
aj3t; t;<1
1- (1- 2j3)t;2'
with the parameter a. depending on the mode of vibration and the frequency ratio
s as
shown in Table 2.4 Table 2.4 a.
~
Swaying
0.65
fff u
Rocking
0.50
f/fv
Vertical
0.67
flf v
Torsion
0.15
f/f u
The natural frequencies of the layer fu and fv are evaluated using Eqs. 2.29 with the total depth H substituted and f = w/2n.
However, it would be safe to ignore
geometric damping constant c completely below the first layer resonance.
Then,
material damping can be established as a fraction of stiffness, giving the complex part of stiffness (2.33a) and the damping coefficient
(2.33b) Similar data on embedded foundations can be found in Kausel et al. (1978) and
61
Gazetas (1983) presented a detailed review of stiffness and damping constants available for foundations . With the stiffness and damping constants established using the approaches outlined above, the response of footings and structures to dynamic loads can be predicted. The methods suitable to this end will be presented later.
Footing on a Layer Overlying a Halfspace (Composite Medium)
In this case, the footing base rests on the surface of a shallow layer underlain by a halfspace (Fig, 2,15a and b), The layer may be uniform (Fig. 2.15c) or non-uniform with linearly varying shear wave velocity (Fig. 2.15d). The halfspace is homogeneous. The footing can also be embedded in overlying layers as shown in Fig. 2.15b, The properties of the embedded layers may vary independently. The layer under the footing base and the halfspace should satisfy the conditions mentioned later. The base soil reaction is calculated using equivalent shapes, the impedances are evaluated approximately using equivalent dimensions obtained by equating the geometric properties of the base area of the actual footing with those of a square base. The effect of embedment is evaluated using the plane strain theory as described for embedded foundations. The impedance functions may undulate as depicted in Fig. 2,16
Note on Limitation The impedance functions are exact for the ratio of layer thickness to halfwidth of the square footing (H/a) equal to 0.5, 1,2,3 and 4 for uniform layers (Fig. 2.15c) and equal to 2,3,4,5 and 10 for nonuniform layers (Fig. 2.15d), If the ratio (H/a) doesn't coincide with one of the above values, choose the closest (H/a) ratio avallable (interpolation is not implemented because of the strong non-monotonic variations at high frequencies).
62
Accurate values of stiffness and damping are used at frequencies equal to 0.10, 0.25,0.50 ..., 4.75 and 5.0 times (Vs 'fa) where Vs' is the shear wave velocity at footing base level and a is halfwidth of the square base(or the equivalent square base). For a frequency less than 0.10 Vs'la , use the minimum value (0.10 Vs'/a) and for frequencies in the range (0.10-5.0) Vs'/a, a linear interpolation is implemented . If the frequency is greater than 5 (Vs'/a) use the maximum value of 5 (Vs'/a).
Poisson's ratio of the
halfspace is assumed to be 0.33 and two values for Poisson's ratio of the layer are available 0.33 and 0.45 (select the closer one for your actual value).
The material
damping of soil is assumed 0.03 and 0.05 for the layer and the halfspace, respectively. Vs' is the shear wave velocity at footing base. Three values for the shear wave velocity ratio are available 0.8, 0.6 and 0.3. If a different value is encountered set it the closest one. The ratio of unit weight of the halfspace to that of the layer is assumed 1.13. Figure 2.15: Rigid Footing Resting on Composite Medium
a) Surface Footing
b) Embedded Footing
c) Uniform Layer Profile
d) Non-Uniform Profile
layer / / . / / . / , ./
'./././
halfspace halfspace
63
Figure 2.16 Stiffness and Damping for Composite Medium
H / c
10
:c
.z.
0 .3
>
2.
v :: 0.33
9
J,
J
8
\....
--E
Q)
Q)
7
o\.... o
6
5
0....
UJ (f) (l)
C
'+ '+ ~
_ _ Uniform Loyer
(f)
o (l) n::::
-----.. Nonuniform Lay er
o
v
- 1
o
2
1
O;m e n s ion fes s Freq uenc y
5
4
.3
/
0
a
= -JJ. G j V
s
.
2S N
H/
J
v
\....
Q)
0
= 2.
- 2 c-~
:.-
= 0.33
i
.i H
20
~
Q)
E
0 "
0
c,
1 ft.
IS
! I
en
(f) ())
c
'+-
! : il
i ,
,
I
LJ
r 10
~
U1 >..
r
'
0
c
S ....
CJl
0
E 0
t 0
I
2
.3
5
Dim ens io nless Frecu e ncy
64
o
i
o
•
•
I
-o I
I
o
N ,
I
~ '
a
en
I i .
a
~
Imaginary Stiff ness Parameter f
N
I
(j)
Ul'
........ ~
<
0
8
II
0 0
'< (.,.l
o
:J
ru
C
ctl ..Q
-,
-q
(j) (j)
(D
0
:::J
(jj
:1
ro
3
o~
o
N
I
/
-\
\
•
. (,..l
I
N
p..J
_
. ~
n
0
<,
::t:
I
~
,
o-
Do
II-
P
P
•
o
//~
e l(>
•
V /1\
l>.,
]
,/\
\-"" "
-~ f
0
/\
\
HI
\jf
i \i \
P
V1 /l/
..
0
/
:f / ! /i /. i V/ /0
o.
0
/. !\1 I- i r
-
•
i
1>.. (
/I
I
CD
•
I
•
/-
VI
(,I
9
-
"II
/-
/-
a
<
~
::.;
C
II
~
"'
0'< ~
.co
-, g ",,< 3
III
0
~
KVV,
/j. l i // I /
I
0>
: : I I I to/co\ /
ooool'l
A
_
1'_
I
0
•
/ I-
,
Real Stiffness Parameter
t-.:
7
:c I
~ L
6
Q) -.J
Q)
E
oL o
5
-;~;;;:;:;::::::~l:>',:o:o'"<;>,,, ---~'"""'"'" .x~~ . -, "<, o-.....;~ f <,•
e"'-,
4
e
e_e_e
0.... (IJ
H/o = 0 .5
(J)
Q)
C .......
Uniform Loyer
....... :;:;
v s'
(j')
o
/v« = 0.8
].1 '=
Q)
a::::
0 .33
1~O
• -
2.0
0-0
3.0
. t . ..
4 .0
e
b,-"
o o
2
4
• Dirne rision l1e s S Frequency 0 0 - w.o/vS '
. : 2S
Y::
~
~
(l)
0 _
0.5
1.0 2.0
20
o
e-e 0-0
3.0
f
Q) E
4.0
r i
~
o
HI
15
] tt .....
\0
~J
c
Uniform Loyer v 5'
/v« = 0 .8
;y
o
v'= 0 .33
E o
o
66
"4 H / a = 2.
J
21
v = 0.33 ___ Uniform Loyer
18 L
Q)
""""' Q)
____ Nonuniform Loyer
15
E 0
I
12
0
IL
en
0.3
9
(f) Q)
C "
6
"
""""'
(/)
.)
0 Q)
n:.:
0
-.3 0
1
:2
J
o
H/ a II
I
=:
5
4
Dimens ionless Fre q ue ncy a= w.a / vs
I
2.
= 0.33
W -'-'
w
E
o o
L
0
en en
20
Q.J
C " "
--'
(/)
C o
10
_
_
Uniform Loyer
C
en
Nonuniform Loyer
o
E
o o
1 2 3
5
Dimensionless Freq uency
67
Interaction between footings Interaction between adjoining footings is sometimes important.
It can be
evaluated using the paper by Triantafyllidis and Prange (1989).
TRIAL SIZING OF SHALLOW FOUNDATIONS The design of a shallow foundation for a centrifugal or reciprocating machine starts with trial dimensions of the foundation block (Step No .3 in the design procedure). The trial sizing is based on guidelines derived from past experience.
The following guidelines may be used for the trial dimensions of the foundation block: 1. Generally, the base of the foundation should be above the GWT.
It should be
resting on competent native soil (no backfill or vibration-sensitive soil).
r " I .
\ "
/\~ . ) \
\
2. The mass of the block should be 2-3 times the mass of the supported centrifugal
\
machine, and 3-5 times the supported reciprocating machine. 3. The top of the block should be 0.3 m above the elevation of the finished floor.
./'/
4. The thickness of the block should be the greatest of 0.6 rn, the anchorage length of the anchor bolts and 1/5 the least dimension of the footing.
/ "
\ ./ 5. The width should be 1-1.5 times the vertical distance from the base to the machine centreline to increase damping in rocking mode.
./
6. The length is estimated from the mass requirement and estimated thickness and width of the foundation .
The length should then be increased by 0.3 m for
maintenance purposes .
68
7. The length and width of the foun dation are adjusted so that the centre of gravity of the machine plus equipment lies within 5% of the foundation dimension in each
j
direction , from the foundation centre of gravity.
8. It is desirable to increase the embedded depth of the foundation to increase the damping and provide lateral restraint as well. 9. If resonance is predicted from the dynamic analysis, increase or decrease the mass of the foundation to change its natural frequency (try to undertune for rotating machines and overtune for reciprocating machines).
Important note: trial dimensions are only preliminary and a complete dynamic analysis must be carried out to check that the performance is within the acceptable limits . If the predicted response from the dynamic analysis exceeds the tolerance set by the manufacturer, the foundation dimensions have to be adjusted and the dynamic analysis be repeated until satisfactory performance is predicted .
\
I
L
o
.
"
t , L
I.
t, 'J /
~
~
69
REFERENCES (Shallow Foundations) Baranov, V.A (1967) - lion the Calculation of Excited Vibrations of an embedded Foundation," (In Russian) Voprosy Dynamiki i Prochnocti, No. 14, Polytechnicallnstitute of Riga. Beredugo , Y.O . and Novak, M. (1972) - "Coupled Horizontal and Rocking Vibration of Embedded Footings," Canadian Geotechnical Journal, Vol. 9, No.4, pp. 477-97 . Bycrott, G.N. (1956) - "Forced Vibrations of a Rigid Circular Plate on a Semi- Infinite Elastic Half Space and on an Elastic Stratum," Philosophical Transactions of the Royal Soc., London, Series A, Vol. 248, No. 948, pp, 327-368. Elsabee, F. and YDrray, J .P. (1977) - "Dynamic Behavior of Embedded Foundations," Research Report R77-33, Civil Engineering Department, Massachusetts Jnstitute of tech nology, September. Gazetas, G. (1983) - "Analysis of Machine Foundation Vibrations: State of the Art," J. Soil Dynamics and quake Engineering, Vol. 2, No.1 , pp. 2-42 . Kausel , E., Roesset, LM. and Waas, G. (1975) - "Dynamic Analysis of Footings on Layered Media," J. Eng. mechanics. Div., ASCE, Vol. 101,06, pp. 679-693. Kausel, E. and Ushijima, R. (1979) - "Vertical and Torsional Stiffness of Cylindrical Footing ," Civil Eng. Dept. Report R79-6 , MIT, Cambridge, Massachusetts. Kausel , E., Whitman, R.V., Morray, J.P. and Elsabee, F. (1978) - "The Spring Method for Embedded Foundations ," Nuclear Engineering and Design 48, pp. 377-392 . Karabalis, D.L. and Beskos, D.E. (1985) - "Dynamic Response of 3-D Embedded Foundations by the Boundary Element Method ," 2nd Joint ASCE/ASME Conference, Albuquerque, June 1985, p. 34. Kobayashi, S. and Nishimura, N. (1983) - "Analysis of Dynamic Soil-Structure Interactions by Boundary Integral Equation Method," Proc. of Third int. Symposium on Numerical Methods in Engineering , March 1983, Paris, pp. 353-362. Kobori, T., Minai, R. and Suzuki, T. (1971) - '''The Dynamical Ground Compliance of a Rectangular Foundation on a Viscoelastic Stratum," Bulletin Disaster Prevention Research Institute, Kyoto University, Vol. 20, pp. 289-329. Lakshrnanan, N. and Minai, R. (1981) - "Dynamic Soil Reactions in Radially Non homogeneous Soil Media, Bull. of the Disaster Prevention Res .lnst., Kyoto University, Vol. 31, Part 2, No. 279 , pp. 79-114. Luco, J.E. (1974) - "Impedance Functions for a Rigid Foundation on a Layered
70
Medium, " Nuclear Engineering and Design 31, pp. 204-217. Luco, J.E. and Westmann, R.A. (1971) - "Dynamic Response of Circular Footings," J. Eng. Mechanics Div., ASCE, EMS, pp. 1381-1395. Lysmer, J. and Kuhlerneyer , R.L. (1969) - "Finite Dynamic Model for Infinite Media ," J. Eng. Mech. Div., ASCE, Vol. 95, No. EM4, pp. 859-877. Nogami, T. and Novak, M . (1976) - "Soil-Pile Interaction in Vertical Vibration," International Journal of Earthquake Engineering and Structural Dynamics, Vol. 4, No.3, January-March, pp. 277-293 . Novak, M. (1960) - "The Vibrations of Massive Foundations on Soil," Publications of the International Association for Bridge and Structural Engineering, No. 20, Zurich, pp. 263 281. Novak, M. (1970) - "Prediction of Footing Vibrations," J. of the Soil Mechanics and Foundations Division, Proceedings of the ASCE, Vol. 96, No. SM3, May, pp. 837-861 . Novak, M. (1974) - "Effect of Soil on Structural Response to Wind and Earthquake," Inter. J. Earthquake Engineering and Struct. Dyn., Vol. 3, No.1, pp, 79-96. Novak, M. and Beredugo, Y.O. (1972) - "Vertical Vibration of Embedded Footings," J. Soil Mechanics and Foundations Division, ASCE , SM12, December, pp. 1291-1310. Novak, M., Nogami, T. and Aboul-Ella, F. (1978) - "Dynamic Soil Reactions for Plane Strain Case, J. Engrg. Mech . Div., ASCE, Vol. 104, No. EM4, pp, 953-959. Novak, M. and Sachs, K. (1973) - "Torsional and Coupled Vibrations of Embedded Footings," Inter. J. Earthquake Engrg. and Struct. Dyn., Vol. 2, No. 11, p. 33 . Novak, M. and Sheta , M. (1980) - "Approxiniate Approach to Contact Problems of Piles, "Proc. Geotech. Engrg. Div. ASCE National Convention , Dynamic Response of Pile Foundations: Analytical Aspects,"October, pp. 53-79. Reissner, E. (1936) - "Stationare, Axial-Symmetrische Durch Eine SchQttelnde Masse Erregte Schwingungen Eines Homgenen Elastischen Halbraumes," Ingenieur-Archiv, Vol. 7, Part 6, December, pp. 381-396. Thomson, W.T. and Kobori, T. (1963) Dynamical Compliance of Rectangular Foundations on an Elastic Half-Space," Journal of Applied Mechanics, ASIC , Vol. 30. Triantafyllidis, T. and Prange, B. (1989) - "Dynamic Subsoil-Coupling Between Rigid, Circular Foundations on the Halfspace," Soil Dynamics and Earthquake Engineering, Vol. 8, No.1, pp. 9-21.
71
Ulrich, C.M. and Kuhlermyer, R.L. (1973) - "Coupled Rocking and Lateral Vibrations of Embedded Footings," Canadian Geotechnical J., 10, pp. 145-160. Veletsos, A.S. and Nair, V.V.D. (1974) - '" Torsional Vibration of Viscoelastic Foundation," J. Geotech. Div., ASCE, Vol. 100, No. GT3, March, pp. 225-246. Veletsos, A.S. and Verbic, B. (1973) - "Vibration of Viscoelastic Foundations," J. Earthquake Engrg. and Struct. Dyn., Vol. 2, pp. 87 ~1 02. Veletsos, A.S. and Wei, Y.T. (1971) - "Lateral and Rocking Vibration of Footings," J. Soil Mech. and Found. Div., ASCE, SM9, September, pp. 1227-1248. Warburton, G.B. (1957) - "Forced Vibration of a Body on an Elastic Stratum," J. Applied Mechanics, March, pp. 55-58. Weissman G. (1971) - " Torsional Vibration of Circular Foundations," J. Soil Mech. Founds. Div., ASCE, SM9, Sept., pp. 1293-1316. Werkle, H. and Waas, G. (1986) - "Dynamic Stiffness of Foundations on In homogeneous Soils," Proc. 8th European Conference on Earthquake Engineering, Lisbon, September 1986. Wolf, J.P. and Darbre, G.R. (1984) - "Dynamic-Stiffness Matrix of Soil by the Boundary Element Method: Embedded Foundations," Earthq. Eng. and Struct. Dyn., Vol. 12, pp. 401-416. Wong , H.L. and Luco, J.E. (1978) - "Tables of Impedance Functions and Input Motions for Regular Foundations," Univ . of Southern California, Dept. of Civil Engrg., Report No. CE78-15, p. 92. Wong , H.l. and Luco, ...I.E. (1985) - "Tables of Impedance Functions for Square Foundations on Layered Media," Soil Dynamics and Earthquake Engineering, Vol. 4, No.2, pp. 64-81.
72
Appendix - Analytical expressions for impedance functions for surface disc.
The complex stiffness constants of the surface footing described by Eq. 2.20 can be
rewritten in the following form :
Vertical stiffness (2.34)
Horizontal stiffness (2.35)
Rocking stiffness (2.36)
Torsional stiffness (2.37) The cross stiffness is less significant and can be neglected. In Eqs. 2.34 to 2.37, R = disc radius , G
= the
shear modulus of the soil and ao
= RwN s where
Vs
= shear wave
velocity of the soil below the disc . The viscoelastic solution by Veletsos and Verbic (1973) yields the following closed form expressions that incorporate the effect of frequency and material damping:
Horizontal parameters
Rocking parameters
73
(2.39)
where
(2.40)
and
(2.41 )
where
R' =:. .Jl + D 2
1
D = material damping = tanf = G'/G, and v = Poisson's ratio
Vertical parameters
+ I. a o (~R'+I r 4 + 'II" + -D )] 2
ao
(2.42)
Xv and \J.fv in Eq. 2.42 are calculated from Eqs. 2.40 and 2.41 by replacing the coefficients ~i with the coefficients Yi in the expressions of X~J and 'P1jI. cr, ~i and Yi are numerical coefficients which depend on Poisson's ratio, v, as shown in Table 2 .5.
74
Table 2.5:
Values of
o.i ,
~i
and Yi
v=O
v = 1/3
v = 0.45
v =0.50
0.1
0.775
0.650
0.600
0.600
Uz
0.525
0.500
0.450
0.400
~1
0.800
0.800
0.800
0.800
pz
0.000
0.000
0.023
0.027
Y1
0.250
0.350
-
0.000
Y2
1.000
0.800
-
0.000
Y3
0.000
0.000
-
0.170
14
0.850
0.750
-
0.850
Torsional parameters
(2.43)
where
b, [R' +
A
~ R /_(b, ao)](b,ao)'
= 1- ------::::'::::-------- R'+
2~~'-!(b 2
a ) + (b2 a0 )2 2 0
(2.44)
and
75
(2.45)
The constants bl and b2 are taken as (Veletsos & Nair 1974): b- = 0.425 and b2 = 0.687.
NUMERICAL EXAMPLE: Calculation of Stiffness and Damping Constants of a Machine Foundation The stiffness and damping constants of the shallow foundation shown in Fig. 2.17 are to be evaluated for the data given below:
1.The Machine:
Weight
2000 Ib (88.96 x1 03N)
Height of horizontal excitation
12 ft (3.657 m)
2.The Footing:
R.C. density
150 Ib/ft3 (23.57x 103 N/m3 )
Dimensions:
a= 10 ft (3.048 m)
b= 16 ft (4.87 m)
c= 8 ft (2.44 m)
height of centroid of system (Yc)
4.75 ft (1.448 rn)
76
Figure 2.17: Machine Found ation Used in Example
r------ I
I
I
I
I I
Ye
~
I-
u,
• • (\J
Yc
/ / ////
= I.45m =4 .7 5 ft
/ / / // / /.
~
E \Cl co
rri
. / / / // / /
-~
I
-il:CG ~ 1
I
~
X ~
l"~
E
'
I
" U
/// / / /77
,I !
Y
b = 4 .87 m ( 16 f r)
!
~l
( , I
I Px(r)
t
I
~
- - 1z:-t) - - - - ~ 17
o
>
o E
- I
~
~ o
4 .877m ( 16 f t )
77
3. The Soil : Unit weight (y)
100 Ib/ft (15.714x10N/m)
Unit mass (p)
3.105 slug/ft (1602 kg/m)
Shear wave velocity (Vs )
492.1 fUsec (150 m/sec)
Material damping (tanc)
0.1
Poisson's ratio (v)
0.25
4. Masses: from 1 and 2
6583 slug (9.60x10 4 kg)
Total mass of the system, m
Solution
The stiffness and damping constants will be calculated for the following vibration modes:
1. Vertical mode 2. Coupled horizontal and rocking vibration in X-V plane
3. Torsional mode.
Equivalent radii: from formulae (2.19)
Translation :
Ru , R" =
Rocking:
R'I'
lab
V-; == 2.174 m == 7.132 ft
A~
= v3"; =1.96 m =6.43 ft
78
Torsion:
RTI:::
4
ab (a 2 + b 2 ) 6Jr
:::
2 .26 m ::: 7.414 ft
Shear modulus of soil :
G = p V} :::
1602 X (150)2::: 3.6045x107 N/m ::: 7.519x10s Ibltf
CASE (1) - SHALLOW FOUNDATION Shallow foundation overlying a deep homogeneous soil layer (halfspace) with no embedment.
a. Soil Material Damping Neglected: -
-
-
Use formulae (2 .24 - 2 .27). Setting all constants SV1, 5 "2' S'11, 5"/2 S",1 , 5 'f/ 2' SU1, and I
Su 2
equal to zero (Le. no embedment cont ribution) and reading the values of the other
constants CV1, C"2'" from Table 2.2 for granular soil the stiffness and damping constan ts are:
Vertical Motion : = 3.604x10 7 x 2.174 x 5.2
=4.074x108 N/m
= 2.788x10 7 Iblft = (2 .174)2 x (1602 x 3.60 x 10 7)1/2 x 5 6
::: 5.68x10 Nrrn/sec
>
s 3 .887x10 Iblftlsec
79
Coupled Motion: 8
=3.604x10 7 x 2.174 x 4.7 = 3.683x10 N/m 7
= 2.520x10 Ib/tt ;::: (2.174)
X
2.403x10 5 x 2.8
= 3.18x10 6 N/m/sec;::: 2.176x10 5Ib/fUsec
=1.668x10 9 N.m/rad ;::: 2.857x10 8 Ib.fUrad
4
4
1.45"1 2.17
= 2.403x10 5 [(1.96) x 0.5 +(2.174} x (- ) - x 2.8] ;::: 1.15x10 7 Nrn/rad/sec
>
1.97x106Ib .ftJrad/sec
7
= -3.604x10 x 2.174 x 1.45 x 4.7
= -5.34x10 8 N/rad
r:-;::;
CUl{/
2
=- '\j P G Ru
YcCu 2
=-1.199x10 8Ib/rad
= -2.403x1 05 x (2.174)2 x 2.8 x 1.45
= - 4.611x10 6 N/rad/sec = -1.035x10 6Ib/rad/sec
80
Torsion: k'7'1 :::;;
GR~ C'11
=3.604x10 7 X (2.26)3 X 4.3 =1.789x10 9 N.m/rad =3.065x10 8Ib.fUrad
=(2.26t X 2.403x10 5 x 0.7 = 4.39x10 6 Nrn/rad/sec = 7.52Ix10 5Ib.ftIrad/sec
b.Soii Material Damping included (tano
=2~ =0.1)
Use Eqs, 2.18. For hysteretic material damping frequency w is needed. If the footing response is to be evaluated for a given operating frequency this operating frequency is substituted for w. If whole response curves are to be calculated, the frequency is better taken as equal to the natural frequencies of the footing, l.e.
(0
=
WI.
The natural frequencies are calculated in Chapter 4, in which the effect of material damping on the stiffness and damping constants of the footing is accounted for.
81
3
STIFFNESS AND DAMPING
OF PILE FOUNDATIONS
) . .,
Ilrrl'U:."~
'"'I......... _. _
Examples of pile supported structures are shown in Fig. 3.0. Figure 3.0: Examples of Pile Supported Structures a) offshore towers
b) nuclear reactors
; i
.IU '-1.\
-
_.
~~
J r<--'"'
i
t:
" e-
o.
I
~J
r.:
'""--'
I
J
!
I
I
.:.;
-
"...
"-'-
-._. . "--'-
-.
o ~
.
=
d) machine foundations
c) buildings
I
I
-
I
i
1
I
I
I I I
i
I
;
I
,
!
Stiffness and damping of piles are affected by interaction of the piles with the surrounding soil. In the past, consideration of this interaction was limited to the
Q '1
determination of the length of the so-called equivalent cantilever which was a free standing bare pile shorter than the embedded pile. Pile damping was estimated . More recent approaches consider soil-pile interaction in terms of continuum mechanics and account for propagation of elastic waves.
The solution is conducted
using a few approaches as shown in Fig. 3.1; the continuum approach (Tajimi, 1969; Kobori et aI., 1977; Novak and Nogami, 1977), the lumped mass model (Penzien, 1964; Matlock et al., 1978) and the finite element method (Kuhlemeyer,1976; Blaney et aI., 1976; Wolf and von Arx, 1978). More recently, the boundary element, or boundary integral, method has also been used, e.g. by Davies et al. (1985). Such studies indicate that dynamic soil-pile interaction modifies pile stiffness making it, in general, frequency dependent and generates geometric damping as with shallow foundations. In groups of closely spaced piles, the character of dynamic stiffness and damping is further complicated by interaction between individual piles known as pile-soil-pile interaction or the group effect. It is useful therefore to discuss single piles first. Small amplitudes and linear behaviour are assumed in most of the studies referred to.
Figure 3.1: Mathematical Models used for Dynamic Analysis of Piles
CONTi ~JU Ur'i
kI., .
..
..
FEr"
MASS
------,-~-i-IT:...........~\.J\.<---
.. .
-
LUt·:PED
I----eE HE-a-r\Nl'1
H T I
U .
I
I
,
I I
j J
,I
84
3.1 Single Piles Dynamic behaviour of embedded piles depends on frequency and the properties of both the pile and soil. The pile is described by its length, bending and axial stiffness, tip and head conditions, mass and batter . Soil behaviour depends on soil properties and their variation with depth and layering.
Figure 3.2: Generation of Pile Stiffness in Individual Directions
Hc :::'::on ,-" I
Rot.a :.i on
Tc r5 . ~.::>;}
As with shallow foundations, the prediction of the response of pile supported footings and structures requires knowledge of dynamic stiffness and damping of piles. These properties can either be described in terms of complex stiffness (impedance functions) as in Eq. 2.9a, that is as K
= K1 + iK2 or by means of true stiffness, ki, and the
constant of equivalent viscous damping , c. , as in Eq. 2.12. The single subscript indicates the properties of a single pile. The constants ki
= K1 and Cj = K2 /0), These
constants can be determined experimentally or theoretically. The latter approach is preferred because experiments, though very useful, are difficult to generalize. In the theoretical approaches, dynamic stiffness is generated by calculating the forces needed
85
to produce vibration of the pile head having a sole, unit amplitude in the prescribed direction (Fig. 3.2 Such theoretical studies have shown (Novak, 1974) that the stiffness constants, ki • and the constants of equivalent viscous damping, c, of single piles can be described for individual motions of the pile head as follows: Vertical translation:
(3.1a)
Horizontal translation:
(3.1b)
Rotation of the pile head in the vertical plane:
(3 .1 c)
Coupling between horizontal translation and rotation :
(3.1d)
Torsion:
G J
kTJ Rp
I.
7]1 ,
GpJ
e7]
= V j,
7]2
(3.1e)
s
In these expressions, Ep is the Young's modulus of the pile, A and I its crosssectional area and moment of inertia (second moment of area) respectively and R pile radius or equivalent radius; GpJ is torsional stiffness of the pile. Finally, the symbol f 1 ,2
86
represents dimensionless stiffness and damping functions whose subscript 1 indicates stiffness and 2 indicates damping.
These functions
depend
on the following
dimensionless parameters: (1) dimensionless frequency ao = wRNs , (2) the relative stiffness of the soil and pile, which can be described either by the modulus ratio G/E p or velocity ratio v =V s /VC in which V s
and v,
:::;
shear wave velocity of soil
= longitudinal (P-wave) velocity in the pile equal 10 ~ E P
where PP
Pp
=pile
mass density, (3) the mass ratio (4) the slenderness ratio l/R in which I :::; pile length (5) material damping of both the soil and pile. Finally, the functions f also depend on the tip condition , fixity of the head and the variation of soil and pile properties with depth.
For a mathematically accurate
consideration of all these factors , the use of a computer is necessary.
A method
suitable for such calculations and accounting for an arbitrary soil profile was presented by Novak and Aboul-Ella (1978) and extended to include pile separation due to lack of bond between the pile and soil by the writer and Sheta (1980). These solutions are based on the plane strain soil reactions defined by Eqs. 2.23 and an efficient computer program, DYNA5, facilitates their use. However, all factors affecting the function f are not of equal importance in all situations.
Often some of them can be neglected, making it possible to present
numerical values of functions f for some basic cases in the form of tables or charts.
87
The effect of dimensionless frequency can be seen from Fig. 3.3. The real pile stiffness (Fig. 3.3a) diminishes with frequency quickly if the soi) is very weak relative to the pile (curve 1). This happens when the soil shear modulus is very low or when the pile is very sturdy.
Figure 3.3: Example of Variation of Pile Vertical Stiffness with Frequency and Soil Stiffness (Nogami and Novak, 1976)
1.2-
o ;;. o.or
1.0
o ,. 0.04
tC
0.6
V)
0.4
.~
or
THE Pfl.El
0
k· • kif STATICAL U
I.
o
RE AL PART
VI
...
b,. w/lFIRST NAT URAL FRE O.
0 .8
@
@
0.8 .....
0
Q) ; . 0 .08
'"
....
...z
O.Z
'
'-
0.2
I
0 .4
.... '0.2 - 0 ,4 - 0 .6
, 0 .a L..
0 ,4 j • 0 .6 D • OOZ Y
0 .6 0 .6 FREOUE NCY b l
1.0
0 .6
VI
....
0 .4
V>
0 .2
•
~
0 .4
0 .6 0 .6 FRECUENCY b,
1.0
l.z
For slender piles in average soils, dynamic stiffness can be considered to be practically independent of frequency as indicated by curves 2 and 3. The troughs visible on curves 2 and 3 are caused by soil layer resonances but they completely disappear for higher values of soil material damping , 0 = tano. The imaginary part of stiffness (pile damping) grows almost linearly with frequency and therefore can be represented by constants of equivalent viscous damping
Cj
which are also almost frequency
independent. Only below the fundamental natural frequencies of the soil layer given by Eqs. 2.29 does geometric damping vanish and material damping remains the principal source of energy dissipation; then the soil damping can be evaluated using Eq. 2.30b.
88
The disappearance of geometric damping may be expected with low frequencies, shallow layers and/or stiff soil.
Apart from these situations , frequency independent
viscous damping constants and functions f 2 which define them, are sufficient for practical applications. The mass ratio PP is another factor whose effect is limited to extreme cases. Pile stiffness and damping changes significantly with the mass ratio only for very heavy piles (Novak and AbouJ-Ella, 1978b). The Poisson's ratio effect is very weak for vertical vibration, absent for torsion and not very strong for the other modes of vibration unless the Poisson 's ratio approaches 0.5 and frequencies are high. The effect of Poisson's ratio on parameters f 1,2 can be further reduced if the ratio E/Ep rather than G/Ep is used to define the stiffness ratio.
Figure 3.4: Comparison of Vertical Stiffness and Damping Parameters of Floating Piles with End Bearing Piles (Novak, 1977, ao
=0.3)
D Ob ,..
I
\
- - fl OA""O PIl[
\
-
- - [NO
\
a~AFl'I'j(; PIL£.
\ I ,? l OA....PIN al
,
\ \
" ....
- '\.----
__~'"'r-- -
I
40 Pll£.
&0
Sl C N O <: R N ~S S {f
~c
lDO
R
89
The slenderness ratio, I/R, and the tip conditio n are very impo rtant for short piles particularly in the vertical direction in which the piles are stiff . Floating piles have lower stiffness but higher damping than end bearing piles (Fig. 3.4). In the horizontal direction, the piles are very flexible and consequently parameters f 1,2 become practically independent of pile slendemess (length) and the tip condition for I/R ratios greater than about 25 if the soil medium is homogeneous (Fig. 3.5 ). If soil stiffness diminishes upward , as in Gibson's medium, parameters f 1,2 level off at higher I/R ratios.
Figure 3.5: Variation in Stiffness and Damping Parameters with Slenderness for Pinned Tip and Fixed Tip Piles (Novak, 1974, ao
= 0.3)
- - P!NNED TlP
1
,
~
~
~~ ~
-- - - ", XE D r IP
I
<:> Z
1
o.Olr rI
\
0 .0)
...
.
,/ \
.
)(
oL. J
o
f
\ ~"'7_-----::t u2 );?\ """""-- -----..,-----..,- .. . /
10
''-..,..
U1 f 20
,
!
,
:30
40
~o
PIL =: SLE ND=:;:lNE SS
2./R
The above observations suggest that the most important factors controlling the stiffness and damping functions f 1.2 are: the stiffness ratio relating soil stiffness to pile stiffness, the soil profile and, for the vertical direction , the tip condition.
90
With the qualifications outlined above, stiffness and damping parameters f 1.2 appearing in Eqs. 3.1a to 3.1e are given for a few basic cases in Table 3.1 and Figs. 3.6 and 3.7. All data are given for homogeneous media as well as parabolic variation of soil shear modulus with depth and vertical piles of circular cross-section. For other cross sections, the same data can be used after an equivalent pile radius has been established. This radius relates to soil reactions and is therefore best evaluated from Eq. 2.19a in which a and b are external dimensions such as width and depth; A, I and J are used as they really are. Table 3.1 gives parameters f 1.2 for horizontal translation and rocking for piles whose slenderness ratio ZlR
~
25. In the vertical response , parameters
f 1,2 depend strongly on slenderness and tip condition. Fig. 3.6 gives the vertical parameters for end bearing piles while Fig. 3.7 corresponds to floating (friction) piles.
All data were calculated using the program
PILAY2 and are numerically accurate for ao ;:::0.3, Pp ;:::1, tans (soil)
= 0.05, tanf (pile) =
0.01 . The functions fu~. 2 give the values for pinned head piles.
91
Figure 3.6: Stiffness and Damping Parameters of Vertical Response for End Bearing Piles
..J
20 tI1
~ !£? 006 r---~~~==:jZ::'...i:~-===t=====l ~ 0 ~ w ~ z .:<
w
~ ~ O.O~ I
~ .
I
o.0 2 C-
I
0 .1 0
L~ O'O~
60
80
10 0
I
II
ur 0 .08 ;"' ; _ -\._!-_ _ ...J
40
,~
!-!
I -------'
- -=c--i
20
"
2~OO ~ (
o _ _-:.- :
o
]"T
/T/
f V1
-
f V2 ....l._
S"TI .fFNESS DA:I.P ING
,
, I
I
u,
o
I
..,:.... Q:
0.. (j)
0::
...J
W
0
I-
\f)
~
u
w
0::
0.0 6
r77~~
\-----1~--.,.L+--
500
...J
I
] /
'777777
H
<: CD 0 0<:
GSOll
0::
<:r:
c,
00
92
Torsional parameters are needed less often because they are significant only for caissons and small groups of very massive piles. They can be found in Novak and Howell (1977 1978). The data given in Table 3.1 and Figs. 3.6 and 3.7 are from the I
paper by Novak and EI Sharnouby (1983) in which the effect of limited stratum depth is also described. For more general soil properties, i.e. arbitrary layering , the functions f 1.2 have to be calculated using the approaches mentioned above. (Program DYNA5 is available)
Table 3.1 Stiffness and damping parameters of horizontal response (UR>25 for homogeneous soil and LlR>30 for parabolic soil ~rofil~)j
r
I ( . t
l
U. ~ , r,"
r
Soil Pro file
v
IT
Ep/G
10000 0.25 2500 1000 500 250 1 0.4 10000 so i1 2500 homo 1000 500 250 10000 0.25 2500 1000 500 250 bSQlil 0.4 10000 2500 parab 1000 500 250
b
IT
filii
fd
fU1
fu l P
f",2
F c2
fU 2
ful
0.2135 0.2998 0.3741 0.4411 0.5186 0.2207 0.3097 0.3860 0.4547 0.5336 0.1800 0.2452 0.3000 0.3489 0.4049 0.1857 0.2529 0.3094 0.3596 0.4]70
-0.0217 -0.0429 -0.0668 -0.0929 -0.1281 -0.0232 -0.0459 -0.0714 -0.0991 -0.1365 -0.0144 -0.0267 -0.0400 -0.0543 -0.0734 -0.0153 -0.0284 -0.0426 -0.0577 -0.0780
0.0042 0.0119 0.0236 0.0395 0.0659 0.0047 0.0132 0.0261 0.0436 0.0726 0.0019 0.0047 0.0086 0.0136 0.0215 0.0020 0.0051 0.0094 0.0149 0.0236
0.0021 0.0061 0.0123 0.0210 0.0358 0.0024 0.0068 0.0136 0.0231 0.0394 0.0008 0.0020 0.0037 0.0059 0.0094 0.0009 0.0022 0.0041 0.0065 0.0103
0.1577 0.2152 0.2598 0.2953 0.3299 0.1634 0.2224 0.267 7 0.3034 0.33 77 0.1450 0.2025 0.2499 0.2910 0.3361 0.1508 0.2101 0.2589 0.3009 0.3468
-0.0333 -0.0646 -0.0985 -0.1337 -0.1786 -0.0358 -0.0692 -0.1052 -0.1425 -0 .1896 -0.0252 -0.0484 -0.0737 -0.1008 -0.1370 -0.0271 -0.0519 -0.0790 -0.1079 -0.1461
0.0107 0.0297 0.0579 0.0953 0.1556 0.0119 0.0329 0.0641 0.1054 0.1717 0.0060 0.0159 0.0303 0.0491 0.0793 0.0067 0.0177 0.0336 0.0544 0.0880
0.0054 0.0154 0.0306 0.0514 0.0864 0.0060
0.0171
0.0339 0.0570 0.0957 0.0028 0.0076
0.0147
0.0241
0.0398
0.0031
0.0084
0.0163 0.0269 0.0443
93
Figure 3.7: Stiffness and Damping Parameters of Vertical Response for Floating Piles
0 .:0
~-'-----.,--
0 .08
PILE
94
3.2 PILE GROU PS
Piles are usually used in groups. The behaviour of the group depends on the distance between the piles . When the distance between the piles is large, say twenty diameters or more, the piles do not affect each other and the group stiffness and damping are simple sums of contributions from the individual piles. If, however, the piles are closely spaced , they interact with each other and this pile-soil-pile interaction or group effect exerts considerable influence on the stiffness and damping of the group. These two basic situations may be treated separately.
Pile Interaction Neglected
When the spacing between the piles is large, their interaction can be neglected and the stiffness and damping of the group are determined by the summation of stiffness and damping constants of the individual piles. In the vertical and horizontal directions this is straight-forward; for coupled sliding and rocking as well as torsion, the position of the reference point such as the centre of gravity , CG, and the arrangement of the piles in plan comes into play. For example , the group stiffness and damping in rotation derives from the horizontal, vertical and moment resistances of individual piles because the unit displacement \V = 1 occurs at the reference point (Fig. 3.8).
95
Figure 3.8: P Ie Displacements for Determination of group Stiffness and Damping Related to Rctatlon e
=1
- .-.:..------ ~
Sf
J
~..ti.
.
).....' --:'-
Consequently, the pile head undergoes a horizontal translation translation
Vh ;:: Xr
and the rotation
'IIh ;::
1.t
fA c.-.
a
Uh :::
:?
I {' .c
.
_\..
Ye, a vertical
1. For torsional stiffness and damping of the
group, the unit twist '11 ::: 1 applied at CG twists the pile by the same angle and translates
the head horizontally by a distance equal to
(Fig.3.9) . With these
considerations, and the notation of Figs. 3.8 and 3.9 the stiffness and damping
constants of the pile group for the individual directions are as follows:
Vertical translation:
(3.2a) r
(3.2b) r
96
t" -
_
III
I
(
u-
r
J
'-~'
L
•
17 ) •
I
. ~.
c..
I
-
I
I
,
1
.
I
Horizontal translation:
(3.3a) I~ ({"
, "
,
A '"
... (
, (/
Rotation of the cap in the vertical plane:
klf/If/
= L (klf/ .+kv x;)- ku Y;- Zk; Yc) r
u
(3.3b) ,
\~. /
+ c; x,2 + Cu Y c2_2Cc Yc )
- ,
" If ,
(3.4a) 'i
_,,( CIfIf - L...J CIf
t
_
<....
(3.4b)
r
Coupling between horizontal translation and rotation:
r
r
Torsion about vertical axis: /' . r'
kryry
;\ . . ....-
= L [kry ~ ku ~y + zj )] r
(3.6a)
(3.6b) r
The summation extends over all the piles. The distances x r, z. and Yc refer to the reference point as indicated in Fig. 3.8. The torsional constants k l1 and ct') can usually be
97
•
neglected. If the pile heads are pinned, k'll =
kc
=0 and e.v = cc = 0 in the above formulae
and k, has to be evaluated for pinned head piles. Only the vertical constants labeled v are the same for fixed as well as pinned heads.
Figure 3.9: Pile Displacements for Determination of Group Stiffness and Damping in Torsion 11
'J
'- ":
\ \
.J -
1
I
I
I
Pile Interaction Considered When piles are closely spaced, they interact with each other because the displacement of one pile contributes to the displacements of others. The study of these effects calls for the consideration of the soil as continuum. For static loads , pioneering research in this field has been conducted by Poulos who published his results in a number of papers (e.g. 1968, 1971. 1974, 1-979) and in an extensive monograph (Poulos and Davis, 1980).
Other data on static interaction effects were reported by
Banerjee (1978) as well as Butterfield and Banerjee (1971). These studies indicate that the main results of static pile interaction are an increase in settlement of the group , the
98
' ,.
IJVr;'v J
'"
(u
'j
{
• I
-flil ._ I '
I)
\
I
-
~1 ':
C,- of
I
0
I
\
redistribution of pile stresses and, with rigid caps, redistribution of pile loads. The studies of dynamic pile-soil-pile interaction are only recent and few in number. Various approaches have been used, all limited to linear elasticity: the finite element method (Wolf and von Arx, 1978, 1981), a semi analytical solution (Waas and Hartmann , 1981), the boundary integral procedure (Aubry and Chapel, 1981) and approximate analytical solutions (Nogami, 1980 and Sheta and Novak, 1982). These studies suggest a number of observations; dynamic group effects are profound and differ considerably from static group effects. Dynamic stiffness and damping of piles groups vary with frequency and these variations are more dramatic than with single piles. Group stiffness and damping can be either reduced or increased by pile-soil-pile interaction. These effects can be demonstrated if the group stiffness and damping are described in terms of the group efficiency ratio GE defined as:
.~
,\ .~ , :~
Ll
group stiffness group eJJlClency =- - - - - - - - - - - - - - sum of stiffness of individual piles ,+I' .
i.e. GE
i·1
IJ
=~roup II. k,
where k is stiffness of individual piles considered in isolation. When the pile-sail-pile interaction effects are absent GE = 1. The group efficiency damping can be defined in the same way. For the basic group of two piles, the group efficiency of vertical stiffness of two end bearing piles is presented in Fig. 3.10. The efficiency ratio is shown for different dimensionless frequencies and varying dimensionless separation sId in which s is the distance between the piles and d their diameter.
I
The static efficiency calculated by
99
means of Poulos' (1974) results is also plotted for comparison. such as ao
=0.01, the dynamic group efficiency increases
At very low freq uencies.
monotonically just as static
group efficiency. However, as the frequency increases, the group efficiency starts to fluctuate about unity. This fluctuation is even stronger for damping for which the group efficiency can be either much greater or much smaller than unity (Fig. 3.11). The weakened zone around the pile, characterized by the ratio Gm IG and tm JR, in which Gm and tm are the shear modulus and thickness-of the weakened zone , has a strong influence on damping.
r. \
"-~
'.
, . r-.
(
100
Figure 3.10: Group efficiency of vertical stiffness of two end bearing piles for varying pile separation (sId) and different frequencies (ao) (Sheta & Novak, 1982)
4
.
U'>
,
, ";
<:»
\.81
-l ~.:. 1 ,-- s ~ -J
E N D 8£ :'RI N G
....;
0-
n..
1'""7'>
VERTi CAL L/ D"2S · DO
!:1:- ~_--·-·~
~ - o_o,
uJ
~ c: =""
~
\
' - STAT IC ( POU LO S J974)
,
, --~----,----""",,-----,-----'------r-'
0-
_.-.--_ .--., -
30 ·
20
1 O.
' --" -' -' -' -' -. -. -' j
+0·
s id
Figure 3.11 : Group Efficiency of Vertical Stiffness and Damping of Two Floating
Piles for Different Separations and Weakened Zones Around Piles (rO
~o
t / :l
c.'a
~ t
-1 0
r l. O;' ''
i
1. ,
~r: .
30.
=R)
1 ·~ C.
.0,_
s Id
101
Another remarkable feature of dynamic behaviour of pile groups is the oscillatory variation of stiffness and damping with frequency (Fig. 3.12) . Curves numbered 4 and 5 were calculated including pile-soli-pile interaction while for the other curves this interaction was neglected.
Different soil profiles were considered as well as a
composite soil medium that incorporates the weakened zone (curves 5).
This zone
reduces the sharp peaks observed in the homogeneous medium (curve 4) but does not eliminate them. Obviously, dynamic group-effects are quite complex and there is no simple way of alleviating these complexities. The use of suitable computer programs appears necessary to describe the dynamic group stiffness and damping over a broad frequency range.
However, a thorough experimental verification of the phenomena
indicated by theoretical analysis is still lacking. The only simplifications available are the approximate approach due to Dobry and Gazetas (1988) in which the interaction problem is reduced to the consideration of cylindrical wave propagation . The replacement of the group by an equivalent pier, considering the dynamic interaction as equal to static interaction or using dynamic interaction factors. The equivalent pier cannot yield the peaks shown in Fig. 3.12, may be applicable only for very closely spaced piles and may overestimate the damping (Novak and Sheta, 1982).
The static interaction may be sufficiently accurate for
dynamic analysis if the frequencies of interest are low and especially if these frequencies are lower than the natural frequencies of the soil deposit given by Eqs . 2.29 .
102
Figure 3.12: Variation of (a) Stiffness and (b) Damping of Group of Four Piles with Frequency and Soil Profile (Sheta and Novak, 1982)
...
ir '--
~
"2 .
~
\0 ;
z
I:
f';
ec
'"
,.
~
<"00
-~
,0
.0 o
~
~
~
r"'ro", ...cy
~
~
~J=t&,jlA.'iS '" 5,t(.'
~
~
!
~
~ .
A
rf.a',o.J"~C ¥
Or.>
'_='~A ~
~-r. ~.;1I4"'S I
__.;.,
IT:.
"'e" k
s.r: }
Evaluation of Group Stiffness Using Static Interaction Coefficients An accurate analysis of static behaviour of pile groups also has to be done using a suitable computer program, i.e. Poulos and Randolph (1982), EI Sharnouby and Novak (1985). However, a simplified approximate analysis suitable for hand calculation can be formulated on the basis of interaction factors , o, introduced by Poulos (1971). The interaction factors derive from the deformations of two equally loaded piles and give the fractional increase in deformation of a pile due to deformation of an equally loaded neighbouring pile. The flexibility and stiffness are then established by superposition of the interaction between individual pairs of piles in the group. The approximation comes from neglecting the stiffening effect of the other piles when evaluating the factors o , The accuracy of the approach appears adequate, at least for small and moderately large groups (Poulos and Randolph, 1982, EI Sharnouby and Novak, 1985).
103
The interaction factors for both axial and lateral loading can be found in the form of charts in Poulos and Davis (1980). Some of them are shown in Figs. 3.13 and 3.14. Fig. 3.13 gives the interaction factors for the vertical direction,
Ctv,
for three values of the
length to diameter ratio, lid. Soil stiffness variation with depth is accounted for by the ratio p ::: the average shear modulus, Gave /shear modulus at the pile base, and G1• The relative stiffness of the pile and soil is defined by the stiffness ratio A. = Ep I Gl. There is an approximately linear relationship between offshore structures,
a;
=
0.5 In(l/ s) ( I In --
J
Ct can
Ctv
and Ig (sId).
For A. - 500, typical of
be estimated from the formula (Randolph and Poulos, 1982)
(3.7)
for s ::; I
dp
For lateral loading, the pile behaviour depends on 4L- h e length of the upper part of the pile which deforms appreciably under lateral loading.
This critical length may be
estimated as (see Randolph , 1981)
(3.8)
In which R
=pile radius
and Ge
= the average value
of shear modulus of the soil over
the critical length, Ie. A few iterations may be needed to find corresponding values of Ge.
and Ie Define further pc as the ratio. Pc
=
G, /4
G
in which G1e 14 is the shear modulus at
c
depth z = [cf4. Then, the interaction factors for horizontal translation, u, and rotation , 'V,
104
may be estimated as (Randolph and Poulos, 1982) 1
a uf:::::
E J'7'-(1 R + cos" fJ) 0.6 r. _P (
Gc
s
(3.9)
2
3
ar;;H = a uH , a/f/.!vf :; ; ; a uH in which a allowed); a'llH
auf
auH
= horizontal
interaction factor for fixed-headed piles (no head rotation
= horizontal interaction factor due to horizontal force (rotation allowed);
=rotation due to horizontal force and u'lIM
= rotation
due to moment. Finally, the
angle between the direction of the loading and the line connecting the pile centres (Fig. 3.14) When the interaction factor, a, calculated exceeds 1/3, its value should be replaced by:
a' = 1-
2 -J27a
This correction is made to avoid
(3.10)
U ---7 00
when s ---7 O.
Eqs. 3.8 and 3.9 and the charts make it possible to correct the group stiffness given by Eqs. 3.2 to 3.6 for the interaction effects using the interaction factors as flexibility coefficients normalized by the flexibility of the isolated pile f
=1/k.
105
Figure 3.13: Interaction Factors for Vertically Loaded Piles (Poulos, 1979)
o
)., = Ep/G O!
c
J
x lD~
et"
0t01[-- __
0 .6
P =0.)
il L
' f~
o0·]
G,
'
os
f) :
- -' ---h-
- -'" I
C
-
p; 0.5
- ---.
I
;
l
)
1
..£nl. /dl
{J =
L.,vc )
l
0·75
--- -''---...'--. - -.- .~ !
•
I
. _-~- - "'''''''-
)
J
C&
J
L
L" 11 /0 1
a.V
0 .&
O.L
--'
0.1
p= I
---.
--'--~
' - r - •.::--:::
. ..
o
Q
10 1
lie = to
-"'?"
Ib J 1/d =2s
k "'" (. ~
Q.\
t L'
• I"
l c)
~
I
C'
l/d -::
50
<,
'r
("
106
Figure 3.14: Interaction factors for Horizontally Loaded Fixed-Headed Piles (Randolph and Poulos, 1982)
o.e ~l
0 .(,
~
~.
0.7 ' \
~,\
~B x
-
4-
P
A
'Ff
~----
I
\ \
0.5
+\
+1\ +
\ £
... \
Of.
\
...
X X
++,
\
x
x
X
C.l
++
Xx
+
--- ---- -------:...t.-
o
2
L
6
e
10
12
11..
16
x ~
__
18
~
20
pde- spacIng
~ ~(~) r _/' G c
For the vertical stiffness of a symmetrical pile group, a very simple formula results from this consideration if it is assumed that all piles carry the same load. Then, the group stiffness may be estimated as
(3.11) r
in which k, is the vertical stiffness of the isolated pile, available from Figs. 3.6 and 3.7 and
Uv
are the interaction f actors between a reference pile, it and pile r with r = 1, 2,
107
... .n where n is the number of piles. The reference pile should not be in the centre or at the periphery and has aii = 1. If a-rigid cap is assumed which implies the same displacements for all piles but different individual stiffnesses, a somewhat different formula is obtained (Novak, 1979), i.e.
(3.12) r
in which Eir are the elements of the inverted matrix, [airr 1
::::
[Eir] of all interaction factors,
air . This matrix [airJ lists the interaction factors a between any two piles and all diagonal terms aii
= 1. The
difference between the results obtained by Eqs. 3.11 and 3.12 is
usually not great. For the horizontal stiffness, the approximate correction may be done in a similar fashion, using factors
a.uf
or C1. u H.
For rotation of a thin rigid cap , the rocking stiffness comes primarily from the vertical stiffness of the piles. This part of the group stiffness becomes (Novak, 1979)
klf'If' = k;
LL
Cir Xi X,.
(3.13)
r
in which x is the horizontal distance of the pile from the axis of rotation. For thick caps, these corrections can be introduced into Eq. 3.4a . For torsion of the cap ignoring the contribution from individual pile twisting, the group stiffness can be written analogously as:
(3.14)
in which x and z are the pile coordinates indicated in Fig. 3.9; if C1.ir(x} and air(z), horizontal
108
interaction factors between piles i and r in direction X and Z, respectively, elements in [eir J = [airr
1
Sjr
are the
.
Another approach to the application of the interaction factors to stiffness evaluation was formulated by Randolph and Poulos (1982) who presented formulae particularly well suited to offshore pile groups . Examples of the group effect on the efficiency ratio evaluated by means of Eqs. 3.12 to 3.14 are shown in Figs. 3.15.
Figure 3.15; Static Group Efficiency Ratio for Groups of 4 and 16 piles: Vertical, Horizontal, Torsional, and Rocking Modes (Novak, 1979)
-'CI=
= 25
- •
sic
:J _
16 _
_
_
M O'
__
_
_
. ,_
P il ES
__
_
.~ ••
,
--= _
"-4 _
. ~~~-------==-
:{==--~--- H
.. . )1:.--
~
. -=-:=:::~
.... ••
~---
- r-- - - -
a_I_
,~ ~ _,••• •_ _
-I5~
. . . _ , -_,----,-_ I
~
- ~ - --"-- ' '' -''-.~---r l ,....,{- '1""7"1
si d
109
The static procedure does not offer any guidance as to the effect of interaction on group damping. Indications are that group interaction usually increases the damping ratio (not necessarily the damping constant c). To account for this approximately, the group damping constants may be taken as:
A better estimate may be obtained using dynamic interaction factors discussed below. Additional discussion of the above formulae for pile group stiffness will be given later.
Evaluation of Group Effects Using Dynamic Interaction Coefficients To extend the interaction factors approach to dynamic situations, Kaynia and Kausel (1982) presented charts for dynamic interaction. In the solution, the soil reactions acting on the piles were evaluated numerically. The dynamic interaction factor is a dimensionless, frequency dependent complex number, aU
=Uij(1) + ia.ij(2) defined as:
. 17 Dynamic Displacement of Pile 2 Lnteraction r actor = -.,;~--_----:=---------=----Static Displacement of Pile 1 in which the displacement of pile 2 is caused by a unit harmonic load of pile 1 and the static displacement of pile 1 is established for an isolated pile. The displacement means either a translation or a rotation . These dynamic interaction factors are used in association with stiffness and damping of single piles given above in the same way as static interaction factors are. Examples of the dynamic interaction factors are given for a limited range of parameters in Figs. 3.16 and 3.17. The oscillatory character of the interaction curves is again evident. The interaction factors shown in Figs. 3 .16 and 3.17
110
can be used in lieu of static interaction factors in Eqs. 3.12 and 3.13. This substitution yields complex group stiffnesses, k = k1 + ik2 whose imaginary part defines the group damping constant c = k2 I
(I).
An increase in damping and strong variation with
frequency is often obtained. A derivation of all the complex stiffness constants including the coupling terms is described for flexible caps, rigid caps and piles with separation (gapping) in full detail in Novak and Mitwally (1987).
In this paper, the closed form
formulae for group complex stiffness analogous to Eqs. 3.12 to 3.14 are derived. For example, analogous to Eq.3-12, the vertical dynamic group stiffness is:
(3.15) r
in which k is static stiffness of a simple pile and
Sjr
are the elements of the inverted
matrix [ar 1 = [s] in which the matrix [a] lists all the complex dynamic interaction factors air between any two piles in the group. For purely horizontal vibration, the Eq 3.15 also holds . Equations 3.11 and 3.12 hold for both vertical stiffness and horizontal stiffness with pertinent values SUbstituted. Eq. 3.11, which is easy to use, is accurate if all the piles carry the same load; otherwise, it can be used as approximate assuming that the pile loads are equal. In the latter case, the results somewhat depend on the choice of the reference pile. Equations 3.12 to 3.14 are accurate for rigid caps but require matrix inversions. To further illustrate how the simpler solution by Eq. 3.11 is formulated for dynamic analysis consider a group of n piles whose displacements and loads are identical. As in the case of a doubly symmetrical group of four, with vertical (or
111
horizontal) harmonic load on each pile, Pi exp (kot). interaction factors u =
U1
+
iU2
Define the complex dynamic
as on p. 3.25 and in Figs. 3.16 and 3.17, i.e.
.t,
a lJ..
f
where f ij , is the complex dynamic deflection of pile j due to harmonic loading of pile j and
I
is the static deflection of a single pile due to its own load . Assume further, that the
deflections fij and
I
correspond to a unit load and are, therefore, flexibility coefficients.
The total response of each pile is the sum of the displacement due to its own loading, v11, and the displacements caused by the loading of the other piles, Vij , Omitting the common time factor, exp (kot), the total displacement of one pile can be written as:
=f / I
where f
is the ratio of dynamic flexibility to static flexibility with f
=I /
K
= the
inverse of single pile complex impedance, K (p. 3.2). For the definition of group stiffness, all pile displacements v == 1. Then , the force on one pile Pi, is equal to the stiffness of one pile, i.e.
P, I
= k. = ~ f I
1 11
t' + La l ) )':=2
The group stiffness is a sum of individual pile stiffnesses. Introducing the single pile
112
static stiffness
k =1.., f
the complex stiffness of a group of n piles becomes
nk (3.16)
In which n
11
a = !r.~) +
L
a 1j (I ) ,
)=2
b = 1(;) +
L
a lj (2)
)=2
where the subscripts (1) and (2) indicate the real and imaginary parts of f and a1j respectively.
From K G the true group stiffness follows as kG :: Re KG and the
coefficient of equivalent viscous damping cG :: ImK GI co Formulae analogous to Eq.3 .15 can be readily formulated for other vibration modes as long as the pile loads can be assumed to be either equal or proportional. More general formulae not limited by these assumptions are given by EI Naggar and Novak (1995).
If the interaction factors were defined as dynamic deflections,
normalized by dynamic single pile flexibility , f, rather than static flexibility,
j, k would
be
replaced by dynamic stiffness and f :: 1 in Eq. 3.16 ; f:: 1 also for static loading . The interaction factor approach would be mathematically accurate, if the interaction factors, and the single pile properties were calculated with all piles present in the system, which is not normally done . Nevertheless, the results are quite adequate for most applications .
More significant errors, overestimating the interaction effects,
may occur in the vertical response of endbearing piles (EI Sharnouby and Novak 1985).
113
Figure 3.16: Interaction Curves for Horizontal and Vertical Displaceme t of Pile 2 due to Horizontal and Vertical Force on Pile 1 (Kaynia & Kausel, 1982) (ao = droNs )
~ d = 15 •' IU;t.Fx
u O.er I x.Fx
( s" 0 .0)
0 .51-
0 .5
L
0.4
o.4L
0 .3
0 .3
0 .2
0.2
0.1
0 .1
0 ,0
0 .0
,/
-oll~T"'----""'~~ r~,
.
~ ~::l-
,/
10
-0.4
- 0.5 __ . . 0 ,0
"-=:! -----;-;f" ... ·
f""1:. - =~~t 1'[-,--- ~ -0. I
_ O. 2
»>
[,,-
7
»:
..,
'0
t
~!
0.5
1.0
0 .0
0 .5
00
1.0
°0
0.7
0,4
0.3
."i'eoI po r t
-- -
i ma go part
- 0.1
- 0 .2 - 0 .3 - 0 .4 - 0.5 <-_ _ 0 .0
I'
0.5
La
_ hori zont al displ acement of pi le 2 due to horizontal fo rce on pile 1 ~ ver t i c al displace ~ent of pile 2 due to ve rtical 'f or ce on pile 1.
00
114
Figure 3.17: Interaction Curves for Rotation of Pile 2 due to horizontal Force and Moment on Pile 1 (Kaynia and Kausel, 1982)
(ao ~
t:.. s
i5 ,
:
d
Ep
1 ¢-'l. ::' }. :; j
= 10 -
L:J; M
~
.
=droNs)
.. s
;: p
= 0 il)
X
03
';
d
(6=7Tf2l
( 8 : C.Ol
.- -
0 .2
Co
0.1
/
/' ./
/'
I
1.0
0 .5 0 0
Rea l . 15
Po r t
L rn n q . PorI
I c;.xMx
o.J
( B = O.Ol ) . 10
~=2 d
( $ = 11/2)
-0.05
-
- 0 . 10
[
0 .5
°0
1.0
- 0 . 1 5 L0 .0
..!I
0 .5
-'
1.0
00
115
3.3 LARGE DISPLAC EMENTS For large displacements, piles behave in a nonlinear fashion, which manifests itself by the lack of proportionality between the applied force and displacement. It is very difficult to incorporate nonlinearity into rigorous dynamic solutions based on continuum consideration; the inclusion of the weakened zone around piles or the adjustment of soil shear modulus and damping according to strain level are about the only practical corrections available. The finite element method could handle nonlinearity but the solution is very costly and inaccurate. The most practical model for nonlinear analysis is the lumped mass model in which the soil stiffness and damping are discretized and represented by isolated springs and dashpots featuring various nonlinear characteristics. Such models are popular in offshore technology where large displacements are expected.
An example of the
nonlinear lumped mass model is shown in Fig. 3.18. The lumped mass models can reproduce the complex nonlinear behaviour observed in experiments.
However, the
selection of nonlinear elements beforehand is difficult and group effects have not been incorporated in these models as yet. A useful FEM study of pile behaviour under large displacements is reported by Trochanis et al. (1988) who investigated static monotonic and cyclic loading on single piles and a pair of piles.
1)6
Figure 3.18: (a) Lumped Mass Model of Pile, (b) Observed Cyclic ReactionDeflection Characteristics (Matlock et al., 1978)
//
I I
, I I I
~
rt
rt.!
I
I FI crd
\
\ D:lmpil'l9
I
,\~ ri
\~L.! \\ \ \
a)
R ~;]:!Ion
9s
\]±....>-..-ol.
b)
117
Pile Batter Pile batter (Fig. 3.18a) can be accounted for approximately by calculating first the pile stiffnesses for a vertical pile, assembling them as the stiffness matrix [K J in element coordinates and transforming this matrix into global coordinates being horizontal and vertical (Novak 1979). This gives the pile stiffness matrix
[K] = [TJT[K] [T]
(3.16)
in which the transformation matrix, [T] depends only on direction cosines. I
When the horizontal coordinate axis lies in the plane of the batter (Fig. 3.18a), the transformation matrix is:
~]
cosa sma
[T];::= - sin a cosa
[
o
0
(3.17)
and the pile stiffness matrix in global coordinates becomes
K""
[
x;
«:
K
lJW
K ww K\I'It!
K
2
-
.?
cos aK Uli + Sln- aK ww
U lfI
cosa sina(K11/I -KwJ cos a K1fIU] ? sin - a K + cos- a K WH' sina K'I/II ?
K ww = KIf/
][
cos a sin a(K"" - K w cosa K 11/II
)
-
lJU
«:
sina K'I/'I (3.18)
The element impedance 'functions, are calculated assuming that the pile is vertical ;
Il8
Figure 3.18a: Stiffness Constants in Element and Global coordinates
l
I ~ 7
TRIAL SIZING OF PILED FOUNDATIONS
The design of a deep foundation for a centrifugal or reciprocating machine starts with trial dimensions of the pile cap, and size and configuration of the pile group (Step No.3 in the design procedure).
The trial sizing is based on guidelines derived from past
experience . The following guidelines may be used for trial sizing the pile cap: 1. The pile cap (block) mass should be 1.5-2.5 times the mass of the centrifugal machine and 2.5-4 times the mass of the reciprocating machine. 2. The top of the cap should be 0.3 m above the elevation of the finished floor. 3. The thickness of the block should be the greatest of 0.6 m, the anchorage length of the anchor bolts and 1/5 the least dimension of the block. 4. The width should be 1-1.5 times the vertical distance from the base to the machine centreline to increase damping in rocking mode . 5. The length is estimated from the mass requirement and estimated thickness and width of the block. The length should then be increased by 0.3 m for maintenance purposes.
119
6. The length and width of the block are adjusted so that the centre of gravity of the machine plus equipment lies within 5% of the block dimension in each direction, from the block centre of gravity. 7. It is desirable to increase the embedded depth of the foundation to increase the damping and provide lateral restraint as well. The following guidelines may be used for the trial configuration of the pile group: 1. The number and size of piles are selected such that the average static load per pile ~
% the pile design load.
2. The piles are arranged so that the centroid of the pile group coincides with the centre of gravity of the combined structure and machine. 3. If battered piles are used to provide lateral resistance (they are better than vertical piles in this aspect), the batter should be away from the pile cap and should be symmetrical.
j
;/
I
\ ;:
J /
o/'V<
V 'I/'
•
~'
4. If piers are used, enlarged bases are recommended. 5. Piles and piers must be properly anchored to the pile cap for adequate rigidity (common assumption in the analysis).
Important note: trial dimensions are only preliminary and a complete dynamic analysis must be carried out to check that the performance is within the acceptable limits. If the predicted response from the dynamic analysis exceeds the tolerance set by the manufacturer, the foundation dimensions have to be adjusted and the dynamic analysis be repeated until satisfactory performance is predicted .
120
REFERENCES (Piles) Aubry, D. and Chapel, F. (1981) - "3-D Dynamic Analysis of Groups of Piles and Comparisons With Experiment," SMIRT , Paris, pp. 9. Banerjee, P.K. (1978) - "Analysis of Axially and Laterally Loaded Pile Groups ," Chapter 9 in "Developments in Soil Mechanics ," Ed. C.R. Scott, Applied Science Publishers , London, pp.317-346. Blaney, G.W., Kausel, E. and Poesset, J.M .' (1976) - "Dynamic Stiffness of Piles," 2nd Int. Conf. Numerical Methods in Geomech., ASCE, New York, P. 1001. Butterfield , R. and Banerjee , P.K. (1971) - "The Elastic Analysis of Compressible Piles and Pile Groups ," Geotechnique, Vol. 21, pp. 43-60. Davies, T.G., Sen, R and Banerjee , P.K. (1985) - II Dynamic Behavior of Pile Groups in Inhomogeneous Soil," J. of Geotechnical Engineering, Vol. 111, No. 12, December, pp. 1365-1379. Kaynia, A.M. and Kausel, E. (1982) - "Dynamic Behavior of Pile Groups," 2nd Int. Conf. on Num. Methods in Offshore Piling, Austin, Texas . Kobori, T., Minai, Rand Baba, K. (1977) - "Dynamic Behaviour of a Laterally Loaded Pile," 9th Int. Conf. Soil Mech., Tokyo, Session 10, 6. Kuhlemeyer, R.L. (1979) - "Static and Dynamic Laterally Loaded Piles," J. Geotech. Eng. Div., ASCE, Vol. 105, No. GT2, pp. 289-304. Matlock, H., Foo, H.C. and Bryant, L.M. (1978) - "Simulation of Lateral Pile Behaviour Under Earthquake Motion," Proc. Am. Soc . Civ. Engrgs. Specialty Conf. on Earthq. Engrg. and Soil Dyn., Pasadena, Calif., II, pp. 600-619 . Mitwally, H . and Novak, M. (1987) - "Response of Offshore Towers With Pile Interaction," J. of Engrg. Mech., Vol. 113, No.7, July, pp. 1065-1084. Nogami, T. (1980) - " Dynamic Stiffness and Damping of Pile Groups in Inhomgeneous Soil," Proc. of Session on Dynamic Response of Pile Foundations: Analytical Aspects, ASCE Nat. Conv., Oct., pp. 31-52. Nogami, 1. and Novak, M. (1976) - "Soil-Pile Interaction in vertical vibration," J. Earthq. Engrg. & Struct. Dyn., Vol. 4, pp . ,277-293. Novak, M. (1974) - "Dynamic Stiffness and Damping of Piles," Canadian Geotechnical Journal, Vol. II, pp. 574-598.
Novak, M. (1977) - "Vertical Vibration of Floating Piles," Journal of the Engineering
121
Mechanics Division, ASCE, Vol. 103, No. EM1, February, pp. 153-168. Novak, M. (1979) - "Soil Pile Interaction Under Dynamic Loads," Proceedings of International Symposium on Numerical Methods in Offshore Piling, London , England, May, pp. 41-50. Novak, M. and About-Ella, F. (1978a) - "Impedance Functions of Piles in Layered Media," Journal of the Engineering Mechanics Division, ASCE, Vol. 104, No. EM3, Proc. Paper 13847,June, pp.643-661 . Novak, M and Aboul- Ella, F. (1978b) - "Stiffness and Damping of Piles in Layered Media", Proc. Earthg. Engrg. and Soil Dyn., ASCE Specialty, Conf., Pasadena, California, June 19-21, pp. 704-719. 1
Novak, M. and Grigg, R.F. (1976) - "Dynamic Experiments With Small Pile Foundations", Canadian Geotechnical Journal, Vol. 13, No.4, November, pp. 372-385. Novak, M. and Howell, J.F. (1977) - "Torsional Vibration of Pile Foundations", Journal of the Geotechnical Engineering Division, ASCE, Vol. 103, No. GT4, April, pp. 271-285. Novak, M. and Howell, J.F. (1978) - "Dynamic Response of Pile Foundations in Torsion" , Journal of the Geotechnical Engineering Division, ASCE, Vol. 104, No. GT5, pp. 535-552. Novak, M. and Nogami , T. (1977) - "Soil Pile Interaction in Horizontal Vibration", Int. J. Earthquake Engrg. Struct. Dynamics, 5, July-Sept., No.3, pp. 263-282. Novak, M. and Sheta, M. (1980) - "Approximate Approach to Contact Problems of Piles", Proc. Geotechnical Engineering Division ASCE National Convention "Dynamic Response of Pile Foundations: Analytical Aspects", Oct. 30, pp. 53-79 Novak, M. and Sheta, M. (1982) - "Dynamic Response of Piles and Pile Groups", 2nd Int. Conf. on Numerical Methods in Offshore Piling, Austin, Texas, April. Novak, M. and EI Sharnouby, B. (1983) - "Stiffness and Damping Constants of Single Piles", J. Geotechnical Engineering Division , ASCE , July . EI Sharnouby, B. and Novak, M. (1985) - "Static and Low Frequency Response of Pile Groups", Canadian Geotechnical Journal, Vol. 22, NO.1 . Penzien, J., Scheffey, C.F. and Parmelee, R.A. (1964) - "Seismic Analysis of Bridges on Long Piles", J. Eng. Mech. Div., ASCE, EM3, pp. 223-254. Poulos, H.G. (1968) - "Analysis of Settlement of Pile Groups", Geotechnique, Vol. 18,
pp. 449-471.
Poulos , H.G . (1971) - "Behaviour of Laterally Loaded Piles II - Pile Groups", J. Soil
122
Mech . Foundations Div.,ASCE, 97 (SM5), pp. 733-751. Poulos, H.G. (1974) - Technical Note, J. Geotech. Engrg. Div., ASCE, Vol. 100, No. GT2, Feb., pp. 185-190. Poulos, H.G. (1979) - "Group Factors for Pile-Deflection Estimation", J. Geotech. Engrg. Div., ASCE, GT12, pp. 1489-1509. Poulos, H.G. and Davis, E.H. (1980) Pile Foundations Analysis and Design", John Wiley and Sons, p. 397. Novak, M and EI Shamouby, B. (1984) - "Evaluation of Dynamic Experiments on Pile Groups ," J. Geotech. Engrg, Vol. 110, No.6, June, pp. 738-756. Novak, M- and Mitwally, H. (1987) - "Random Response of offshore Towers With Pile-Soil-Pile Interaction ," Proc. of 6th Inter-national Symposium on Offshore Mechanics and Arctic Engineering (CMAE), Houston , @s, March, Vol. 1, pp. 329-336. Poulos, H.G. and Randolph, M.F. (1982) - "A Study of Two Methods for Pile Group Analysis," J. Geot. Engrg. Div., ASCE. Randolph, M.F. (1981) - "The Response of Flexible Piles to Lateral Loading" Geotechnique, 31(2), pp. 247-259. Randolph, M.F. and Poulos, H.G. (1982) - "Estimating the Flexibility of Offshore Pile Groups ," Proc. of the Conf. on Numerical Methods in Offshore Piling", Univ. Of Texas , Austin, May, p. 16. Roesset, J.M. (1980) - "Stiffness and Damping Coefficients of Foundations." Proc. of Session on Dynamic Response of Pile Foundations: Analytical Aspects' ASCE National Convention , Florida, Oct., pp. 1-30. Sheta, M. and Novak, M. (1982) - "Vertical- Vibration of Pile Groups," J. Geotech. Engrg. Div., ASCE , Vol. 108, No. GT4, April, pp. 570-590. Tajimi, H. (1969) - "Dynamic Analysis of a Structure Embedded in an Elastic Stratum" , Proc. 4th World Conf. Earthquake Engineering , Chile. Waas , G and Hartmann , H.G. (1981) - "Pile Foundations Subjected to Dynamic Horizontal Loads," European Simulation Meeting "Modelling and Simulation of large Scale Structural Systems, Capri, Italy, Sept pp. 17 (also SMIRT , Paris). Wolf, J.P. and von Arx, G.A. (1978) - "Impedance Functions of a Group of Vertical Piles," Proc . ASCE Specialty Conf. on Earthquake Engrg. and Soil Dynamics, Pasadena, Calif., II, pp. 1024-1041.
123
Wolf, J.P., von Arx, G.A., de Barros, F.e.p. and Kakubo, M. (1981) - "Seismic Analysis of the Pile Foundation of the Reactor Building on the NPP Angra 2", Nuclear Eng. and Design, Vol. 65, No.3, pp. 329-341. Dobry, R. and Gazetas, G. (1988) - "Sirnple Method for Dynamic Stiffness and Damping of Floating Pile Groups, " Geotechnique 38, No.4, pp. 557-574 . Trochanis, A.M., Bielak, J and Christiano , P. (1988) - "A Three - Dimensional Nonlinear Study of Piles Leading to the Development of a Simplified Model," Research report R 88-176, Dept. of Civil Eng., Carnegie Mellon University, Pittsburgh, PA. EI Naggar, M.H. and Novak, M., 1996. Nonlinear analysis for dynamic lateral pile response. Journal of Soil Dynamics and Earthquake Engineering, Vol. 15, No.4, pp. 233-244. EI Naggar, M.H. and Novak, M., 1995. Non-linear lateral interaction in pile dynamics. Journal of Soil Dynamics and Earthquake Engineering , Vol. 14, No.2, pp. 141-157.
124
Example: Evaluate the stiffness and damping constants of the pile foundation shown in Fig. 3.19. The footing is the same as the one used in the example in Chapter 2, and thus has the same mass, mass moments of inertia and the position of the centroid . The soil is also the same for comparison.
The Soil: Homogeneous Unit weight (y)
100 Iblft 3 (15.714x10 3 N/m)
Unit mass (p)
3.105 sluq/ft" (1602 kg/m3 )
Shear wave velocity 0Is)
492.1 ftlsec (150 m/sec)
Material damping (tan8)
0.1
Poisson's ratio (v)
0.25
The Piles:
End bearing piles - 8 soft wood piles
Unit weight (y)
48 Ib/ft3 (7542 N/m3 )
Pile length (I)
35 ft (10.668 m)
Effective radius (ra)
0.4166 ft (0.127 m)
Cross-sectional area (A)
0.5454 ft2 (0.05067 m2)
0.02366 ft4 (2.043x10-4 rn")
s,
1.728x108 Iblft2 (8.27x109 N/m2 )
Pile eccentricity (x-)
4 ft (1.219 m)
125
a) Single Pile
v =1/3
Ep/Gs
~
250
10.66
Pile slenderness ratio
=II R = 0.127
= 84
Vertical Motion: Using Eqs . 3.1a,
Parameters fv1• fV2 follow from Fig. 3.6 for end bearing piles in homogeneous soil as:
fV 1 = 0.058 fY2 =0.097 I
then
= 8.276 X 10
k
\'
9
x
0.0506 (0.058)
0.127 = 1.915x1 08 N/m
=1.31x1 07 Ib/ft
9
= 8.276 x 10
c v
x 0.0506 (0.097)
150
= 2.716x10 5 N/m/sec
= 1.85x104 Iblftlsec
126
Figure 3.19: Pile Supported Machine Foundation
r----------l !
1
,
I
I
--r-!
-
i
I
~ '.3 8 1rn
,
1.25 ft
I
--i::
I
I
I
I
J: t , ) ._ -
{ 16
3'
_r~x(t) ; Ze
/
~
I
N
rri
/1I1
"
I
u J
I
J
/
E:
-r/
I()
~
1
,
0 • -:;;-' ,'""1. :I 1.22m
,
"
: fl
1 /
/
/
,
/
I
W ty
.
-4-
;
/
"' r.. v.
4 fr
.. I •
)
i '-l
:.22 m
a. 3 m ,
4 ft
~
I f!
---:
)
I
/
L2 5 !t
I
-@l ,
.
E
tt 1
,
4 .5 ft
b : 4. 87 m
/ /
1'":1 ,' 0. ::8 I
1.37 rn
4 .5 tt
4.5ft
r/
I
I,
1.3 7m
:.3 '1m
I
y J
J
:4'1 5
I
i
- -~if; '....
.. C\I
l
I I
J
I
~
u,
Yc : 1.4 smj
Py{t ) I
I I!
..-.
I
-
-
I
I
I
Ye
~
7/
I
J
-$-
i
__
4'
TJ
, ~
- o/~ a.38 m
I- · 1-
1..
1.37m
~I·
1. 3 7 m
b " 4 .B77m (16
fT\
· ~7 · ~"'
J.37 m
,a.38m _
:
1
f l)
.- '-'-----...,
127
Coupled horizontal and rocking motion: Assume fixed head piles. Using Table 3.1 for Ep IG s = 250 and homogeneous soil profile the parameters are obtained for v = 1/3 by interpolation . This gives for a single pile: 9
_ 8.276 x 10 x 2.043 xI 0-4 (0.0692) (0.127r -
=5.716x10 7 N/m = 3.911x106 Ibltt c - EpI It
-
R2~.
1: 112
9
8.276xl0 x2.043xl0--4 (0.1636) = (0.127Y x 150
= 1.143x1 05 N/m/sec =7.824x1 03 Iblftlsec = 8.276xl0
9x2.043xl0-4
(0.5261)
(0.127)
=7x10 6 N m/rad = 1.2x1061b ftlrad 9
_ 8.276 x 10 x 2.043 x 10-4 (0.334) 150 ;:: 3.765x10 3 n rn/rad/sec
= 654.0 Ib ftlrad/sec
;:: 10.48x10 7 x (-0.1323)
=- 1.387x10 7 N/rad =- 3.114x10 61b/rad =
9
8.276xl0 x2.043xl0-4 (-0.1841) 0.127 x 150
=-16.34x10 3 N/rad/sec = - 3.668x10 3lb/rad/sec 128
Stiffness and damping constants for the pile group:
a) Pile Interaction Neglected Using formulae 3.2 to 3.6, the following data are obtained for the individual directions: Vertical:
= 8 x 1.915x1 08 = 15.32x1 08 N/m = 10.48x1 0 7 Ib/tt r
Cuu
=""" L.J C
u =8x2.716x10 5 = 21.73x10 5 N/m/sec=1.487x10 5 Ib/(ttsec)
r
Horizontal:
I. k;
k vv
:=
C vv
== """ L.J C v =8 X 1.143x105 = 9.147x105 N/m/sec =6.259x10 4lblftJsec
=8
X
5.761x10 7 = 45.72x10 7 N/m = 3.129x10 7 1b/ft
r
r
Rocking:
r
= 8 [7x10 3 + 1.915x1 05 x 1.22 2 + 5.716x1 04 x 1.4482 - 2(-1.387x1 04 ) x 1.448] = 3.616x1 0 9 N m/rad = 6.195x1 0 8 Ib ftJrad
r = 8 [3.767 + 271.6 x 1.22 2 + 114.31
X
1.448 2 - 2(-16.34) x 1.448]
=5.56x10 6 N m/rad/sec = 9.526x10 5 Ib ftJrad/sec
129
Coupling:
r
=8 x (-1.387X10
4
) -
(8 x 5.716x10 4 x 1.448)
= -7.731x10 8 N/rad = -1.735x108 Ib/rad
r = 8 (-16.34X10 3 ) -8 x 1.143x10 5 x 1.448
= -1 .455x1 06 N/rad/sec = -3.266x105 Ib/rad/sec
Torsional Vibration (see Figs. 3.9)
For slender piles
kil l GIl
can be neglected. Then from Eqs. 3.6
r
=4 X 5.716x10 7 [(1.22 2 + 0.68582)+(1.222 +2.952 )] = 1.756x1 0 9 n m/rad = 3x1 08 Ib ftfrad
r
= 4 x 114.34 x 7.68
=3.5126x1 06 N m/rad/sec = 6.018x1 05 Ib ft/rad/sec
130
b) Pile Interaction Considered
As an approximation, static interaction coefficients are used.
Vertical Stiffness:
Using formula 3.11
r
Gave - 1 P -- G
Coefficients a y are obtained from Fig. 3.14 taking A = 300 , p = 1 and interpolating for lid
= 42. Taking an inner pile (2) as reference, the interaction coefficients are: Pile
(sid)
(In sid)
a
1
5.4
1.68
0.209
2
0
--
1
3
5.4
1.68
0.209
4
10.8
2.37
0.134
5
11
2.39
0.134
6
9.6
2.26
0.155
7
11
2.39
0.134
8
14.44
2.67
0.109
y
Reduction in stiffness = 1/L o; = 0.479
kw = 0.479
X
15.32x10 8 = 7.31x108 N/m =5.021x10 7 Ib/ft
131
For damping, Cvv
~
L c, ~ 1.487x105 Ib/fUsec
Coupled horizontal and rocking motion: A correction is made only for the most important stiffness, kuu. For the ratio 1
(
E ~
J7 :::: (229)7" :::: 2.1733 1
the interaction coefficients, evaluated by means of Eq. 3.9a for the direction X, are listed below: Pile
S (ft)
1
4.5
2
~
auf
4.97
90
0.1209
--
--
--
1
3
4.5
4.97
90
0.1209
4
9
9.94
90
0.06
5
9.2
10.16
30
0.104
6
8
8.83
0
0.136
7
9.2
10.16
30
0.104
8
11.3
12.48
48
0.065
1
~+(~r
132
Thus, ~
Cl.uf = 1.71 and
k'uu
=26 .90x1 07 N/m =1.841x1 071b/ft
c'uu = 6.259x104 Ib/ft/sec
Example 2: Evaluation of Groups Impedance Using Dynamic Interaction Factors a) Approximate approach using Eq. 3.16 Evaluate the vertical impedance function for the pile group given in Fig. 3.19 using dynamic interaction coefficients from Fig. 3 .16 for frequency (co = 87.26
S·1
and V s = 150
rn/s. The dynamic interaction factor o: =Cl.1 +
iCt.2
is defined on p. 3.25.
a = dO) = 2 x 0.127 x 87.26 = 0.148 o
r.
150
Single pile properties from p.3.35 are kv
= 1.915 x 108 N/m and c, = 2 .712 x 105 N/m/s
Pile No .2 is chosen as reference pile. The factors c , obtained by interpolation considering even the far piles are given in the following Table A.
133
Table A Pile
SId
a
1
5.4
0.04 - 0.26t
2
0
0.984 - 0.122t
3
5.4
0.04 - 0.26t
4
10.8
-0.128 - 0.12t
5
11.0
-0.13-0.12t
6
9.6
-0.10 - 0.13t
7
11.0
-0.13-0.12t
8
14.44
-0.20 - Ot
Notice that even the contributions from the far piles are significant.
J. = .[v = dynamic jlexibilty coefficient v
J;
static jlexibilty coefficient
The dynamic flexibility of a single pile f
=1/Kv
where
=1.915x1 08 + t 87 .26 x 2.712x1 05 N/m 1.915xlo 8 -i2.366xl0 7
= 5.14x10-9 - 6.35x10-9 mIN
134
+ == .Jv
1" :::: .J v
1
9
1.915xl0
8 ::::
5.222 x 10- m / N
5.14 x 10 -9 - i 6.35 x 10-1 5.222 x 10- 9
K"G =
n
I
f
k
In + aI ' .
=n
0
k( ') +ba ') - i a-
==
2
0.984 _ i 0.122
b ')
a +b-
J
J
j=1
substituting 8
if + l:alj =
0.376 - i 1.132
j=2 8
KG = 8x1.915x10 x 0.376+i1.132. v 0.376 - i 1.132 0.376 + i 1.132 = 4.05x10 B + t12.14x10 B = 4.05x1 08 + 1.4x1 07 rot N/m Thus, the group vertical stiffness and damping constants are
b) Accurate evaluation of group stiffness and damping using the interaction matrix, Eq.
3.15. All the complex interaction coefficients are assembled in the interaction matrix [a] shown in Table B. The group stiffness is obtained from the sum of all elements
in the
matrix [c] by Eq.3.15, giving
135
;:;: 6.143x10 8 + L13.139x108
=6.143x1 0
8
+ 1.506x1 07 (OL N/m
The true stiffness and damping is from here
The approximate values are 4.05 x 108 for stiffness and 1.4 x 107 for damping. The approximate approach avoids matrix inversion but considerably underestimates stiffness in this case. Examples of the effects of dynamic pile interaction on harmonic response of foundations are shown in Figs . 3.20 and 3.21.
Figure 3.20: Horizontal and Rocking Response of Foundation by Fig 3.19 to Harmonic Horizontal Excitation.
136
Figure 3.21: Comparison of Coupled Response with that of One OOF System under Horizontal Excitation
~
w o
~ ...i
=> - ,
::: j
I fi
1
-J
~ ~ ~ ~ i
-1
0
{ ' '\
II" I
.\
II
~ ~l ~--. : : J: C
1
!r ii !,~
, I '\
,~l
:< ~
I
~ ~i o
:::
~
Nl <:> •
G
Col.¥led :-buo11, No L'1t.eractic'A'1
I
I
.1
\ 11
i t - SDJF, No rnt.e::=-action
~l
~ I~ \
J/ I
.
\
eoupled
\
/JM ~ \
I
,
3C·
!,' ~\ so
.
v.()t ion
I::1te;.ac::.ion
SrY"'.~
Tnt,:>..- cci,~ __ 5~_ ~,
'
~"
~~'- w
eo
;20
: :N
l Bli
e,
2!C
2H
137
4
RESPONSE OF FOUNDATIONS AND
STRUCTURES TO HARMONIC EXCITATION
4. RESPONSE TO HARMONIC LOADS
Harmonic loading is shown in Fig. 1.2. It represents the basic case of periodic loading and can be described as
pet) = P coset
(4.1)
where P force amplitude and co excitation frequency.
For reasons of mathematical
convenience, an auxiliary imag inary component may be added making the excitation force complex. Denoting the imaginary unit i is described as the complex harmonic force
pet)
=p ei ~)l= P (cosot +
/sincot)
(4.2)
analogous to a complex number (Fig . 4.1) z= a + b
=r (coso +isin
(4.3)
Figure 4.1: Representation of a Complex Number
in which
is the absolute value, also caned magnitude, modulus or distance, and
139
,
~
SIn 'f/
=
where
~
~
b
b
or tan If' = 2 2 va + b a I
= angle of the complex number (argument)
4.1 RESPONSE OF RIGID FOUNDATIONS IN ONE DEGREE OF FREEDOM The governing equation of the motion is obtained by adding the excitation force to the right side of the equation of free vibration. With viscous damping the governing equation for motion v(t) is:
mv
+ ell + kv = Pei(J)(
(4.4)
Because the excitation force is complex, the response is also complex. The lmaginary part of the solution will be labeled by
j
and can be deleted if only the real part of the
excitation force is of interest. The particular integral that gives the steady-state solution is: (4.5)
where Vc is a complex amplitude, Vc =
V1+iv2.
Substitution of Eq. 4.5 in Eq. 4.4 yields
vc{-ulm + kec + k) = P From this,the complex amplitude of the response is
v =k 1 = a(iev)P - mar? + IOJC C
(4.6)
in which
, )
a (lOJ =
1
2.
k - mOJ + uo c
140
(, J I
{~
-f
,
II J..J
[ 1-1
' ' /.
', 0'1
I
is known as the admittance (or transfer function) of the system. The reciprocal of a(ico) defines the impedance of the system .
. )= k
For a massless body, m = 0 and the impedance takes the form of complex stiffness defined for various systems in Chapter 2. The admittance is the ratio v« I P while the
= k-mco 2
impedance is P I vc . Note a
and b =
-Q)C
and multiply the nominator and
denominator of Eq. 4.6 by (a+ib); the complex amplitude becomes
=
V c
(P) X a + ib a - ib a + ib 1
=
i
a + ib P = e ¢ p 2 2 a +b r
in which
b a
tan¢=-=
-we ? k mor' r
Upon substitution of v c into Eq. 4.5, the particular solution is
(4 .7)
If the real vibration amplitude is denoted by
17' \
I
I
I
V=
p
~(k - m (
-
2
.: )
r
oJ
+ OJ2 C 2
-_1
the real response, described by the real part of Eq. 4.7, is
141
v(t) :::: V cos (O)t+~)
(4.8b)
The stiffness and damping constants k and c often depend on frequency, k k(co), c
>
=
c(w), as in the case of foundations exposed to excitation with a wide frequency
range. Then , the response has to be calculated from Eq. 4.8 . However, if k and care frequency independent or can be considered as such in the frequency range of interest, Eq. 4.8 can be rearranged into a more convenient form. By taking k in front of the radical and denoting
co;
kim
=
and
the real amplitude of the response can be rewritten as:
p
v
1
= --r========== = Vs' & k
(4.9)
~-
l. ' " ~ I-o rr Lt
in which
r
<
,
,
~
Vst
:;:;
-
-~
. ~
P/k :;:; the static displacement and c :;:; dynamic amplification (or
magnification) factor,
.
f
•
1
&
... . - 1.......
1
= ----r========= (4.10)
This factor is equal to the ratio of the amplitude of the response to the static displacement. The phase shift
142
(4.11)
The particular solution describing the steady-state response to the real excitation force given by Eq . 4.1 is v(t) = v cos (cot + ¢)
(4.12)
The complete solution is obtained by adding the complementary solution of the homogeneous equation [P(t) = 0] of Eq. 4.4 , which was found before, and becomes v(t) = v cos (cot + ¢) + Voe-at sin (co'ot + 4'0)
(4.13)
in which Vo and ¢o are integration constants depending on initial conditions .
The
complementary solution describes transient motion, which usually dies out due to damping , and is of little importance .
In applications involving random vibrations somewhat is different notation is often used.
The particular integral (4.5) is written as:
vet) =
~.
k-s mco
+Z())C
e i CV f
= P H«()))e icvt k
(4.14)
where
(4.15)
in which lHI =
£
is called the modulus of the complex admittance (or frequency) function,
H(co). (The terminology varies.) The particular solution is then written as
]43
v(t) = p IH(w)1 k
ei(OJt+¢)
and its real part is:
p
vet) = -IH(m)1 cos(mt + ¢) k
which is the same as Eq. 4.12 because
IH(w)J:::~ .
Identical results are obtained if the viscous damping
cv
is rewritten as kocv in Eq. 4.4.
The variation in the amplification factor e, Eq. 4.10 and phase shift
~,
Eq. 4.11, with
frequency is shown in Fig. 4.2a. The response starts from s = 1, i.e. from static displacement and, depending on the magnitude of damping, builds up into a response peak centred around the resonant frequency co =
Wo
at which the dynamic amplification
factor, given by Eq. 4.10, reduces to
1 2D
E=
(4.16)
The amplification factor at resonance depends only on damping and becomes infinite if damping vanishes . For small values of damping, the resonance amplification is great and the response far exceeds the static response. The phase shift ranges from 0 to 180 degrees and at resonance is -90 degrees (-71: I 2) for all values of damping. Above resonance the amplitudes diminish and approach zero as
(L)
approaches infinity.
]44
Figure 4.2; Dimensionless Response to Harmonic Loads (dynamic amplification
"
factor and phase shift)
.L
( J {..
', t
a) constant force excitation
,~
' .1 1.....
\)
)'
r 1 (
c.
o
l ' r P(t )
-- P
o C O S "lt
"
.' I
-
' /
(A r
3.0
4 .0
~.o
b) quadratic excitation
P ( t)
_. , -_ me ~~ ..... a;, 2 .-.. _C )S u.•. t ,
' 1--~--IlJl-\tI-;-----I
;.
z.c
J
\"'
I
"
.!!
r_9Q9
o -
-
_
. " ,
...
i
I
0
3_0
4 .0
~. O
J QJ
E ' ::>
IE
II
1.0 u
o
Quadratic excitation.
The above formulae and Fig. 4.2a were derived for an
excitation force whose amplitude P is constant. In many practical cases, the amplitude
145
- I G
I
of the excitation force is not constant but depends on the square of frequency. This is so with excitation stemming from centrifugal forces of unbalanced rotating masses, unbalances of reciprocating mechanisms, harmonic ground motion or vortex shedding. An unbalanced mass me rotating with an eccentricity e and circular velocity (J,) produces a centrifugal force
(4.17) The horizontal component of the centrifugal force is
(4.18) which is a harmonic force with frequency dependent amplitude. Substitution of this amplitude P =mee(J,)2 into Eq. 4.10 gives the response amplitude
V=
(4.19)
in which the reduced eccentricity p = mee I m and the dynamic amplification factor of quadratic excitation is
(4.20)
The variation in the dynamic amplification factor s' with frequency is shown in Fig. 4.2b. The response amplitude starts from zero, grows to the resonant amplitude
1 2D
v=p at (J,)
(4.21)
=(J,) 0 and then asymptotically approaches p. Thus , the dynamic amplification e' 146
ranges from zero to 1/2D at resonance and finally to 1 at high frequencies. Examination of the response peaks reveals that the actual maximum of the amplification factor exceeds the value of 1/2D occurring at
00
= 000 and is
1
This maximum appears at frequency
with constant amplitude excitation and at frequency
1
with quadratic excitation. Thus the true peak may appear below or above the undamped resonant frequency co
=CUo depending on the type of excitation (Fig . 4.3). For small
damping this difference becomes insignificant.
147
Figure 4.3: Response Curves for Constant Amplitude Exc'tation and Quadratic Excitation
q u adz e c i c fo rce
constant force
cirn l
w
General Relation Between Complex Amplitude and Real Amplitude In general, complex motion can be described by
where the complex amplitude is
v« =V1 + ivz According to Eq. 4.3, this amplitude is also
in which the real amplitude
(4.22)
and
(4.23)
Thus
148
and the real part of this motion is vet) = v cos (cot + ¢) The relation between complex and real amplitude is useful in many degrees of freedom. The theory of one degree of freedom can be used for vertical translation and rotation about the vertical axis (torsion); very approximately, it is also applicable to horizontal translation of very flat footings and rocking of tall footings about base axis. In cases involving rotation, mass moment of inertia, I, replaces mass, rn, in the above formulae and displacement, v, is replaced by the pertinent rotation. The basic formulae for mass moments of inertia are given below. The formulae are for rectangular bodies and cylindrical bodies; m is the mass of the body.
Figure 4.4: Mass Moments of Inertia
Mass Moments of Inertia
o[
--- ---r-----Ix-X
-~- --- --~-- -x'
V
I
m
2
x = 12
I Xl
=
1
( a +b
x
+
mx
2
)
2
.Z I
I
--:- j-X I
,.,.
--
__'-_
x
-:_-~-_.:- - ,I .
d
Z
= m
I
----X'
I
I
x Xl
=
2 d 8
=
m 2
R
2
2 2 d h + m(16 12)
=
I
X
+ mx
2
h-1
149
Examples of Respons e of Footing With one Degree of Freedom
The theory outlined above can be used to analyze the response of shallow foundations and pile foundations. The needed stiffness and damping constants can be evaluated using the approaches described in Chapters 2 and 3. As an example, the response amplitudes are examined for the machine foundation shown in Fig. 4.5. Using Eq. 4.8a, the vertical and torsional amplitudes are evaluated for different types of foundation and quadratic excitation described by Eq. 4.17. The response curves established are shown in Figs. 4.6 to 4.8. The amplitudes are given in a dimensionless form which actually corresponds to the dynamic amplification factor,
E',
defined by Eqs. 4.20 and 4.10. For quadratic excitation , the
dimensionless amplitudes of vertical and torsional response may be defined as
in which r = the arm of the horizontal force with regard to CG.
Frequency independent stiffness and damping constants were assumed . A detailed
numerical example is given at the end of this chapter.
150
Figure 4.5: Machine Foundation Used with Piles and Without Piles in Examples (1ft
=O.3048m)
Figure 4.6: Vertical Response of (A) Pile Foundation, (B) Embedded Pile
Foundation, (C) Shallow Foundation, and (D) Embedded Shallow Foundation (Bx
= m I p Rx3
= 5.81,
Novak, 1974a)
5
r'
~
....
I
J.
I'"
,
, ... I ~'
.
'. 1r
JI-' .
_ ......:.._--:!':---:-.t._... 20
60
80
I
100
• •
':20
.J........-. .. ,..,J
140
rGO
f R E ~ U EN C Y
'I. ~
I'
1.,1 l" ~'"
lj
(
' .... f
~
151
)
r
.
'J 1-·
r·
igure 4.7: Vertical Response of a (1 ) Embedded Pile Foundation with
,J
V' ,o.
,J
Jc
Pa~bolic
Soil Profile, and (2) Embedded Pile Foundation with Homogeneous Soil Profile and (3) Shallow Foundation Without Embeddment
"-
.' I
L, - r: _..
.J,
L0
u ..J .... ,
o
.1J
::::;,
I
!
_
]
J
o !
0=. cn-i
(, '
2:
,.
CI:
I
-ioo
ft1
I
2S Ci
FREJUEN Cr lR80. /SEC .
Figure 4.8 : Torsional Response of (1) Pile Foundation and (2) Shallow Foundation
152
- t. .
4.2 EVALUATION OF THE EFFECTS OF VIBRATION Using the above formula, the intensity of the vibration can be predicted; the effects of the vibration can then be evaluated. This evaluation includes the dynamic stresses in the structure, foundation and soil as well as settlement of the soil, physiological effects and effects on the operation of equipment. Physiological effects depend displacement amplitude.
essentially on vibration velocity
rather than
The velocity amplitude is calculated as vs» in which v
displacement amplitude and co
= frequency.
=
For frequency in Hz, the physiological
effects may be assessed using Fig. 4.9. Additional data on human susceptibility are given in Chapter 5 and in Richart et al. (1970). Some allowable amplitudes, established with regard to machine operation, are suggested in Table 4.1. The noise levels to be expected are indicated in Table 4.2.
4.2.1 Transmissibility Damping is favourable because it reduces resonant amplitudes but it has the undesirable effect of increasing the forces transmitted into the foundation of the system and its vicinity. To illustrate this point, consider a general case of harmonic vibration vet) in one degree of freedom resulting from excitation by a harmonic force. pet) = P coseot The steady-state response of the system is vet) ::
V
cos (rot + 4»
(4.24 )
in which
153
Figure 4.9: Human Susceptibilty to Vibration (after Reiher and Meister; amplitudes in inches and mm)(1in = 25.4 mm)
( ;r.r..;
( in)
,
i!
i !I I. I :I
:
I
I
:I III'
I
!
I
I
I
I
! i !I
I
II·III :I I
I !
!
I
I
0. :: 5
I
I
!
l' , I III i I·
1· :
I
t
I . I I! ::! • I .: I
i I
I "
!
I I
0 ,0 0 1
i
~_-:----t'-"'-:_-'l.c---;'
O,OOOl L - --------.:.-------.:.- , I iI
E lI ~
l-
-
I I
:...t: . _ _'
l,;
I ~ :' I I ! I!I I I i !I ~ I
I ;
:
I
-
: I
I I
;IIII
I i : [ :
~
0,0 0 001 1.yO
II
0
'
-
-'
I
-I
..
'N '·!II
~
•
0
•
: !
i I I
I,
I!.
I
!
\
I! . ' . I ; II
J :
I
.I
I
I
I
.
I
: I
I
I
•
: J
"
i
I
I i
I
i
I
0 . 0 (8 ] 5
1 00
10 F'r oq u c ri c y , C . ?S.
1
( II : )
154
Table 4.1 Limit vibration amplitudes for machine foundations (in mm)
1) Reciprocating engins with crank drive (diesel engins, piston compressors, fram e saws etc .) 1) ton 2) Hammers: foundations Anvil 2) ton anv il 3) ton heads 3) Crushers & Mill s 4) Turbornachinery 1) RPM ~ 3,000 2) RPM .~ 1,500 3) RPM ~ 750 1,000 -3000 RPM < 100 KW 5) Electric Motors 1,500 RPM> 100 KW 1,000 RPM> 100 KW 750 RPM 600 RPM 500 RPM 6) Factory Machines Lathe, drilling machines Shaper, milling machines Grinders , precise lathe 7) Paper Machines (according to make & component)
0.20 1.2 1.0 2.0 0.30 0.025 = 1 mil 0.050 0.100 0.050 0.090 0.10 0.12 0.16 0.20 0.01 0.003 0.008 to 0.025
Table 4.2 Noise Levels of Typical Machines That Range Above 90 dB Machine Hydraulic pres s Pneumatic press Wing bar drop press Swagger Shell press Mechanical power press Header Drop hammer Automatic screw machine Circular saw Ball mill Grinder
Average dB level 130 130 128 108 - 118 98 - 112 98 - 110 101 - 105 99 -101 93 - 100 100 99 80-95
155
p Va = - &
(4.25)
k
where k is stiffness and n is the dimensionless dynamic amplification factor which depends on the natural frequency w 0 and damping ratio D according to the equation
1
c = -r===========
(4.26)
The total force transmitted into the supporting medium results from both the restoring force and the damping force and is
F(t)
=
k vet) + c v(t)
(4.27)
With the motion given by Eq. 4.24, the expression for transmitted force becomes
F(t)
=
k V COS(OJt + rjJ) - c vOJ sin(OJ t + rjJ)
This is a sum of two harmonic motions having different amplitudes, the same frequency and a phase difference of 90 degrees. Thus, the transmitted force is also harmonic and its amplitude is
(428) Upon substituting for the amplitude from Eq. 4.25 and realizing that
the amplitude of the transmitted force is obtained as
156
(4.29)
With constant excitation amplitude P, the transmitted force varies with frequency and damping as shown in Fig. 4.10 in which the dimensionless ratio FJP, called transmissibility, is plotted . Damping reduces the transmitted force in the resonance region but increases it for frequencies w>
.J2 COo.
Figure 4.10: Transmissibility
3t-------:~....I....---+-----+------==-l
2 t-------ti:.......+"'+-i---+-----+----l
1
e::----.~~-_+_--_____ll----_i
i:""' .
1
2
3
With excitation stemming from the rotation of unbalanced masses , the force amplitude
157
is proportional to the square of frequency (4.30) where me and e are the unbalanced mass and its eccentricity, respectively . P can be rewritten as
P
mee =--m()) In
in which p
=:
2 = p k( ca J2
(4.31 )
())o
mse I m is equal to the amplitude v as
product pk is the restoring force as
0)
---7
00,
0)
---7
co and m
=:
k I ulo
The
This force can be used to normalize the
transmitted force. This normalized transmitted force F/(pk) is shown in Fig. 4.11. The increase in the transmitted force due to damping for co >
J2
Wo
is quite dramatic.
Hence, for low tuned foundations, high damping is unfavourable. The ratio F/P is the same as that shown in Fig. 4.10. Figure 4.11: Normalized Transmitted Force with Quadratic Excitation
I D=O
, V
~ , "'\.~\ ,' j
0.5
/
\\
3t-----+,&.f---';t----:-f'---"'7"----""7"-l
\1
'.
\.
ll-_*~-=-------------I
158
4.3 COUPLED RESPONSE OF RIGID FOUNDATIONS IN TWO DEGREES OF FREEDOM
Rigid foundations are constructed as rigid or hollow blocks . Their motion in space is described by three translations and three rotations and consequently they have six degrees of freedom. The six components of the motion are, in general. coupled. However, there is usually at least one plane of symmetry and this reduces the coupling between individual components of the motion.
Two vertical planes of symmetry
decouple the six degrees of freedom into four independent motions: vertical translation, torsion around the vertical axis, and two coupled motions in the vertical planes of symmetry; the latter motions are composed of horizontal translation (sliding) and rotation (rocking). The coupled motion in the vertical plane represents an important case because it results from excitation by moments and horizontal forces acting in the vertical plane. The motion is treated most conveniently if the centre of gravity of the footing and the machine considered together is taken as the reference point (origin of coordinates). Then, the horizontal sliding u (t) and rocking
\jI
(t) describe the coupled motion as
indicated in Fig. 4.12 in which the positive directions of the two components are indicated . For the analysis, inertia forces, stiffness constants, and damping constants must be established first.
159
Figure 4.12: Notations for Coupled Motion
x.o
~
b -..
,Y,v a
4.3.1 Mass, Stiffness and Damping Inertia forces are due to the mass of the footing-machine system and its mass moment of inertia about the axis Z passing through the centre of gravity. The mass of the system is m = m1 + m2 if
rn- is the mass of the footing and m2 the mass of the
machine. No additional mass to account for soil inertia is needed because this effect is taken care of by the variation of stiffness constants with frequency and the generation of geometric (radiation) damping.
In a massless medium, these two factors would not
occur. The mass moment of inertia is calculated in a standard way. If the footing is of a simple rectangular shape with the dimensions shown in Fig. 4-12, the mass moment of inertia of the footing-machine system is:
(4.32)
In this formula, the mass moment of inertia of the machine about its own axis is neglected because it is usually small compared to that of the footing.
160
Figure 4.13: Generation of Stiffness Constants
I J
r---_..
-..
1-.. ,k1{;if;, ................ 'J
II
_.._ / ~ __ T\Tk"ttu,, /
. t.
vJ,e !'
~t_
~~
-;I '~ '
I
R~
-...:;:
jv
-k u
I
. . . . . -t-t-
L
JJ c l
VJJ=I
J
/ -/
-......~R= ku Yc
b)
a)
Stiffness constants are defined as external forces to be applied at the centre of gravity in order to produce a unit displacement at a time with all the other displacements being zero. When the centre of gravity lies above the level of the base, two external forces (stiffnesses) are needed to produce a sole unit displacement. The unit translation calls for the horizontal force kuu and the moment requires the moment k w and the force principle,
ku'V
=
kljlu ,
k'l'u
kUlI'
(Fig . 4.13a) while the unit rotation
(Fig. 4.13b). Because of Maxwell's reciprocity
These stiffness constants are described as the stiffness constants
for translation and rotation at the centre of the base of the footing , transformed to the new reference point , the centre of gravity, CG. If the stiffness constants referred to the centre of the base are ku and k1jl' the stiffness constants referred to CG are: kuu = k u
(4.32a) (4.32b) (4.32c)
in which Yc > 0 if CG lies above the level for which ku and k., is are defined (this is the base in the cases considered in Figs. 4.12 and 4.13). Equations 4.32 are evident from
161
the geometry indicated in Fig. 4 .13 in which the reactive forces generated in the medium under the base due to unit displacement of the CG are also shown. surface foundations the constants
For
ku, k, are given by Eqs. 2.20 to 2.22 . For embedded
foundations , the resultant expressions are described by Eqs. 2.26. For pile foundations, Eqs. 3.3 to 3.5 apply. If the footing is supported by an elastic layer of cork, rubber or other material whose Young's modulus is E, shear modulus G, and thickness d. the stiffness constants of the base are
k, = GA I d
(4.33a)
=El z I d
(4.33b)
kv
In which A = base area and Iz = second moment of base area about the axis parallel to z. These constants are to be substituted into Eqs. 4.32 . The damping constants are evaluated in the same manner. Thus , the formulae for damping are obtained from those for stiffness by replacing constants k by c in Eqs. 4.32. The resultant expressions are given in Chapters 2 and 3.
4.3.2 Governing Equations of Coupled Motion The coupled motion can be caused by a horizontal excitation force, P(t) and a moment in the vertical plane, M(t),
where
00
P(t) = P cos rot
(4.34a)
M(t) = M cos rot
(4.34b)
= circular frequency of excitation and P = the force amplitude; the moment
amplitude , M, derives from the horizontal force and possibly from an independent excitation moment , Me, and is
162
(4.34c)
M = pYe + Me
in which Ye
=the vertical distance between the horizontal force and the centre of gravity
of the machine-footing system. With the mass, stiffness and damping constants established, the governing equations of the coupled motion, composed of the horizontal translation, u(t), and the rotation in the vertical plane, \jf(t), can be written by expressing the conditions of dynamic equilibrium of the foundation in translation and rotation.
Applying Newton's
second law and recalling the basic definitions of the stiffness and damping constants, the governing equations of the coupled motion are (4.35a) (4.35b) in which the dots indicate differentiation with respect to time, kUl v = kljlu, the sake of brevity, u(t) = u and ~J(t)
CUIjI
=
~u
and for
=\!J.
The governing equations, Eqs. 4.35,can be rewritten in matrix form as
[rn]{ii}+[C]{Ll}+[k]{u} = {PCt)}
(4.36a)
in which the diagonal mass matrix, the displacement vector and the force vector are
[m] =[
} {P(t)} = { P(t ) } ~ 0]1 ' {u(t)} = {U(t) If/(t) , M(t)
(4.36b)
and the stiffness and damping matrices are
[k] =
lk
uu
k tf/l~
J
ku'l/ [c] = k ' '1/'1/
[CUll C'1/U
(4.36c)
163
4.3.3 Solution of Equations of Coupled
otion
The governing equations, Eqs. 4.35 or 4.36a,of the coupled motion can be solved using two approaches: the direct solution and modal analysis. Both methods lead to closed form formulae and are easy to use. Direct Solution
The direct solution is mathematically accurate and is suitable with stiffness and damping constants which are frequency dependent or independent. For mathematical convenience, the harmonic excitation described by Eqs. 4.34 may be complemented by imaginary components iP and iM to yield pet) = P (cos rot + isin rot) = P exp (kot)
(4.37a)
M(t) = M (cos rot + isin cot) = M exp (ioit)
(4.37b)
With this complex excitation, the particular solutions to Eq. 4.36a are also complex and can be written as
{u
U(t ) } ;:::: c } exp(i to t) { lY(t) lYe in which
Ue
(4.38)
and \lie are complex displacement amplitudes. Substitution of Eqs. 4.38 into
EqsA.36a yields two algebraic equations for these complex amplitudes:
P= (kuu -
rnco' + i (U Curl ~c + (kulf + i (U CuvJ¥tc
These equations are readily solved using Kramer's rule. Introduce the auxiliary constants
164
(4.40)
Then, the complex vibration amplitudes are from Eqs . 4 .39,
(4.41 a)
(4.41b)
Separating the real and imag inary parts
(4,42a)
IJf
'l' c
. = nr + iu/ =M '1'1 '1' 2
/3& 1 1 2 &1
+j3& . /3& -/3& 2 2 + zM 2 [ 1 2
2 2 2
+ &2
&1
+ &2
(4.42b)
165
As in
Eq. 4.22, the true (real) vibration amp litudes u and
\jf
are:
(4.43a)
(4.43b)
When the motion is excited by a moment alone, P =
a and special, simpler expressions
for the real amplitudes result:
u=M
(4.44a)
If/ = M
(4.44b)
The phase shifts between the excitation forces and the response follow from Eq. 4.23 as
(4.45a)
(4.45b)
Dropping the imaginary components of the response labelled by i, the real motion of the centre of gravity is: u(t) = U cos (rot + ¢>u)
(4.46 a)
'V(t) ::: 0/ cos (Ot + ¢>u)
(4.46b)
From Eqs. 4.43 or 4.44 the response amplitudes are readily evaluated. Beredugo and
166
Novak (1972) formulated this closed form solution. For very high frequencies it may be advantageous to divide all constants a,
p and
2
s in Eq. 4.40 by co or co to avoid very
large numbers. As in the case of uncoupled modes, dimensionless amplitudes
(4.47) may be introduced to facilitate the presentation and analysis of the response to forces whose amplitudes are constant. This is the case of force amplitudes independent of frequency or force amplitudes evaluated for a certain operating frequency. With excitation due to unbalanced forces of rotating or reciprocating machines, the force and moment amplitudes are proportional to the square of frequency as described by Eq. 4.17. If the excitation is caused by an unbalanced rotating mass me acting at a height Ye above the centre of gravity, then
in which e = rotating mass eccentricity; the ratio M / P = Yeo The dimensionless vibration amplitudes are, in the case of frequency variable excitation,
m Au =u
(4.48)
me ' e
The uncoupled modes of vibration in one degree of freedom are special cases of the solution described . It may be noted that an alternative direct calculation may be formulated in which the complex amplitudes u, and
\jIc
are separated into their real and imaginary parts
beforehand. This approach leads to four simultaneous equations with real coefficients;
167
however, the computation requires more time and a closed form solution woul d be inconvenient.
From the motion of the centre of gravity, the horizontal and vertical
components of the motion experienced by the surface of the footing can be determined. The upper edge of the footing experiences vertical amplitude
Ve
and horizontal
amplitude u, that are:
Ve
= If! ~ , U e = U
+ (b - y JIJI
In the last formula , the phase difference between u and
(4.49)
\jf
is neglected and a,b are the
dimensions of the footing (Fig. 4.12). 4.3.4 Examples of Coupled Response
Examples of the coupled response calculated from Eqs. 4.43 are shown in Figs. 4.14 to 4.17. The foundation is the one shown in Fig. 4.5, the excitation is quadratic and the footing is founded either directly on soil or on piles. Frequency independent stiffness and damping constants are assumed. Figs. 4.14 and 4.15 show comparisons between pile foundations and shallow foundations.
As can be seen, pile foundations provide less damping than shallow
foundations . Fig. 4 .16 shows the response of the shallow foundation calculated for different soil shear wave velocities .
For this more heavily damped embedded
foundation, the second resonance region often is not marked.
Fig. 4.17 shows the
effect of soil stiffness on the response of pile foundation . As can be seen , the variation of resonance amplitudes with soil stiffness follows different patterns for shallow and pile foundations . Similar parametric studies can be conducted for various foundation conditions described in Chapters 2 and 3.
168
Figure 4.14: Horizontal Component of Coupled Footing Resposne to Horizontal Load. ( (Bx
= m IpR3x = 5.81, B'V = II pR\, = 3.46; (+) = modal analysis)
r£
t
I
A
I
I
I
-
I
•• ••• •• ••• ••• • •••.• •• .1. •.. •
I
--- --
.....
I
'Wz 20
,
40 FREQUENCY w (RAD / S )
( '0\. "t
.
';
169
Figure 4 .15: Rocking Component of Coup ed Footing Response to Horizontal Load.
'1.
~i
,
I L I
.,.
I
s '" f>,
IV
>-< , Cl
IE
2 :
Ii
? <{
I..L! 0
L
::::>
i
l
--l n,
I
::E <{
z, ~
I i :
«I
~
c
I
l-
0
a::
i
.......A'.:..-!=
0
0
...:-_
20
40
_ ... __..l '
60
...L-_ _---li.-._ _-"
.cc
80
:2 0
-'-_ _---'
140
160
FREQUENCY w ( RAD / S )
I
·
c
I
'0 i .J ~ ,
~ (
,
170
Figure 4.16: Horizontal and Rocking Components of Coupled Response of Embedded Foundation for Different Shear Wave Velocities of Soil (e IR = O.4
J
granular soil)
o
to
:c
150 m/sec 225 m/sec 300 m/sec
mjsec
mjsec 300 rn/sec
.....
tL.
:s::o
a: . ...J
....
a:;
. z: o
N -0
Q;.
dl'"1
-.: Q
w N
o
0
0
200
100
0
.
300
(RAD {ANS)
FREQUENCY
t
JJO
he 1./
.""
(
1.
~, / .' 0
"
'I'
r
r
, , L: . ( .
(;
)
r
,
VI
f
/;/
t -. . e (
/"
(. I~
~ .)
)
·r
"
<,
, c.-'
,J(
-
-
n
I
1"1 ' r
If'I ~' .-
.~
r
-,,~
f
(
't ..
I
t,
mRO 1!=INS)
j
I
(..",
~
'_:7
SOC
400
t I'J - . ,
I
{/
300
FFlEOUENCI
I
&'1.-< r(
L~
f~~,- 'i)
e
200
f
t
1
I
I
100
400
-I
I )
171
(Ie
Figure 4.17: Effect of Soil Stiffness on Horizontal Comp onent of Coupled Response of Pile Foundation "
• \ _. ,
1\
o Lfl (\J
I'
GRANULAR SOIL
A
1-1 1 : ---
-
V s
225 m/sec
vs -'"
150 m/sec
o
('
I
.. ..
'
\
I
»:
v5
= 75 m/sec
I
o
J :'/(Ij~: '
,.--- 75 m/sec
.
UJ
O ~~-.......- _ - r
~. 0
100 .0
;'
.'
l.<
/ ; - 150 m/sec
/-----Z
·t\.. . .,
- ·1 '_
/
'
'-
. ' (.
225. m/sec
--r
r - -_ _, - - _ " " - -_ _
200 .0 300 .0 FRE QUE NC Y lRADI ANSJ
.-_----,
400.0
172
4.3.5 Modal Analysis of Coupled Response When the stiffness and damping parameters of the foundation are presumed to be frequency independent, a different approach to coupled response , known as modal analysis, is very suitable. The principal advantages of this approach are that it yields natural frequencies and damping ratios in addition to amplitudes and that the algebra involved is very simple. The method is described in more detail in Novak (1974a,b). First, the natural frequencies and modes of free vibration are calculated. These follow from the solution of the eigenvalue problem . The equations for the natural frequencies and modes follow from EqsA.39 by putting the damping coefficients Cuu, cIjIO/ and
cu~
as
well as P and M equal to zero. This yields, in terms of real amplitudes,
k uu [
. 2 mea
kU lf/
k if/'ll
kUlf/ 2]{U}={O } -ICO Ij/
The two natural undamped frequencies
(01
and
(02
(4.50)
are found from the condition that the
determinant of the coefficients must be equal to zero, which yields : J
2
{tJ
1,2
1 k If'If' 1 krill vv k;If/ =- - llu + -k - J + - (- - -k - J2 +-2( m 14m I ml
With these two natural frequencies wJ
0=
(4.51)
1,2) the two vibration modes (eigenvectors)
are, from Eqs. 4.50,
u.
a . =_ ) = )
with j
UE).
r
-k UIf/ k _ mOJ 2 j
(4.52)
IIlI
=1 or 2. (These equations provide a quick check of (OJ. With correct values of (01,2
173
both equations give the same results.) The two modes represent rotations abou t two two different points, as shown in Fig. 4.18.
Figure 4.18: Modes of Free Vibration of Rigid Footing
... .
--0"::..::-0-- ._. ; I !-- -- -1j -I
,I,
"' __ "t' l
I
,
, ,
I
f
I
I
~',I, f 't" l I
I I
ii
1
Ii
o
1st mode
2 n d Mode
Assume excitation by harmonic horizontal force P(t) and moment M(t) P(t)
= P sin rot
M(t)
= M sin rot = (PYe + Me) sin rot
(4.53)
The generalized force amplitudes, producing response in one mode each (Fig . 4.19), are: Pj = P
Uj
+ M 'Vi
(4.54 )
and the generalized masses (4.55)
in which subscript j = 1,2 denotes the mode and the corresponding frequency . The modal coordinates
Uj
and 'Vj and can be chosen to an arbitrary scale; e.g.1 and thus
174
from Eq. 4 .52,
Uj
= aj
Figure 4.19: Modal Superposition of Footing Response, (Excitation Proportional to co
2 )
_I Cll
1
LoU 0
:::> l -l
0..
!
~
I
q~uz
l
::t I
-l
a 0
~
w
0
w
i ,J ~ 90"
The two modal damping ratios pertinent to the vibration modes are, from Eq. 21 in Novak (1974b),
(4.56)
in which damping constants c are given in Chapters 2 and 3.Then,the footing translation and rocking are: 2
u(t)
=
"q .u . sin(wt+¢J.) ~ J J
(4.57a)
j =[
175
:2
¥/(t) = Lqj¥/j sin(mt +
(4.57b)
j~l
is in which the amplitudes of the generalized coordinates
(4.58)
and the phase shifts
lV.w .w ¢j = -arctan 0/ J_ ~'J. = -arctan J
2Dj {t}/
1m.
(.!
With respect to the phase difference between and \jf
(4.59)
1- ca/ /
translation u and rocking
])2
J
/ CI) .
.J
~1
and
~2
the true amplitudes of
are the vector sum of the two modal components, i.e.,
Eqs. 4.60 give results usually very close to those obtained by direct calculation from Eqs, 4.43, particularly if the damping is small and the resonance peak is well pro
nounced. This can be seen from Figs. 4.20 to 4.22. The difference seen is due to the inaccuracy of the modal damping evaluation but is not very significant because the dynamic amplification is quite small when the error is large. The greatest advantage of modal analysis occurs when calculating the maximum amplitude in the first or second resonant peaks of the coupled motion.
When the
damping of the resonating mode is not too large and the response curve shows a peak, the contribution of the nonresonant modal component to the resonant amplitude can be
176
neglected in most cases .
It is small and the phase difference between the two
components is close to 90 degrees (see Fig. 4.19). Then, the resonant amplitudes of the coupled motion at resonance j
U=1 or 2) are approximately
r»,
(4.61a)
2D.M.w~ ) ) )
P) ·lf) · lfr,j - 2.D .M .w·2 -
)
)
(4.61b)
)
Figure 4.20: Sliding Component of Coupled Response Computed Directly (exactly) and by Means of Modal Analysis (for foundation shown in Figure 4.5)
EXAC T ' AN A l y S I S MODAL ANALYS I S, £ O .. 4.6 02 E O.
4. 6 3a
...J
«
I
4
2'
o N
! W, . !) I = 5 .9 % I .
I
I
w2 , O
I
20
2 :
I :,
.,~
I
40
60 80 10 0 FREOU ::NCY w ( R A D I S )
12 G
177
Figure 4.21 : Rocking Component of Coupled Response Computed Oirectly (exactly) and by Means of Modal Analysis
-
EXACT ANALYS IS
.••.. ••••• MOO A LAN A L Y S IS, E~ .
+
4 . 6 Cb
EO. 4..63b
J-.._---'
_ _l - - - ' - - - - - - - _ - L _ _-L-_ _
ol,.-~=--L---..L_L.--
o
20
40
60 80 FREQUENCY w
i OO
12 0
140
160
( R ;:D ! S~
The resonant amplitudes calculated from these very simple equations are shown as crosses (+) in Figs. 4.14 and 4.15. The agreement between the accurate and approximate values is quite good in this case. The agreement can be further improved by the vector addition of the non-resonating modal generalized coordinate obtainable from Eq. 4.58. With the omission of damping, allowable because of large differences between
0)1
and
0)2,
the amplitudes of the non-resonating mode k at frequency
O)j are
(4.62a)
178
(4.62b)
Because the phase shift between the resonating and non-resonating modes is close to 90 degrees, the resonant amplitudes are approximately '")
u" j
U ;,j
'J
+ U ,~,k (4.63a)
I-D ~ J
v: .+ 'fI !
')
r ,J
n ,k
I-DJ~ in which k
(4.63b)
*j.
The inclusion of the damping in the denominator yields the approximate value of the maximum resonan t amplitude instead of the somewhat smaller amplitude at frequency (OJ. (The maximum amplitude does not appear exactly at (OJ as can be seen in Fig. 4.3). Equations 4.63 yield an even better estimate of the resonant amplitudes than Eqs. 4.61 . The amplitudes calculated from Eqs. 4.63 are shown in Figs. 4.20 and 4.21 where the complete response curves, obtained directly and by means of modal analysis , are also plotted . The agreement between the two approaches is very good .
179
Figure 4.22 Comparison of Direct Analysis with
odal Analysis for Embedded
Foundation c; . -, '-~
i
i I I
i
!
1\
Ii
EX'-I'='l
k..'l.041 y s l. .s
1'\oda-l A. n alysu
e; i
. ...,
. ,
~
~ 1
a; ,..-.
:::0 : '; ' ; ' • • \1 :' , . "
f
... ~.' -i
..,..; :
"
o
~- 1 1
:=:c::: ! --'
I
~(~ l I
!, Cl
u . ·.;
SC.
Q
, : :2 0 . 'J
~ P.~ ·C J:n c '"
.
~gc . o
~
, ;< ';0. 0
I
300. Q
9 C; C ; M,\ 5 / ~ C C l
180
4.4 MULTI·DEGREE-CF·FREEDOM SYSTEMS
When the foundation has fewer than two planes of symmetry, the response is coupled in three or more degrees of freedom . The governing equations have the form of Eq. 4.36. The displacements a re sought as u(t)
= Uc exp (kot),
The solution is readily
obtained using a suitable computer program such as DYNA5 (Novak et al., 1999). Examples of the response are given in Novak and Sachs (1973).
Some other
approaches are described in Arya et al. (1979).
181
NUMERICAL EXAMPLE: EVALUATION
O~
THE RESPONSE OJ= A ACHINE
FOUNDATION TO HARMONIC EXCITATION The foundation shown in Fig. 4.23 is analyzed as a shallow foundation and as a pile foundation. These data are given below:
The Machine: Weight:
2000 Ib (88.96x10 3 N)
Heiqht of horizontal excitation
12 ft (3.657 m)
The Footing:
R.C. density:
150 Iblft (23.57x1 0 3 N/m 3 )
Dimensions: a = 10 ft (3.048 m) b
= 16 ft (4.87 m)
c = 8 ft (2.44 m) height of centroid of system (Yc)
4.75 ft (1.448 m)
Masses: from 1 and 2
Total mass of the system (m)
6583 slug (9.60x104kg)
Mass moment of inertia about
117490 slug .ft2
axis Z, (lz)
(1 .598x105 kg .m 2 )
182
Exciting Forces:
Excitation forces occur due to rotor unbalances and act in the vertical direction Y , the horizontal direction X and as a moment about axis Z and are: Px (t)
=mae ro2 cos cot
P y (t)
=mee 0/ sin cot
Mz (t)
=Px (t) Ye,
Ye
=7.25ft = 2.20m
where me = mass of rotor , e = rotor eccentricity and co = frequency of rotation
The results will be shown in dimensionless form so mae will not be given. The foundation is the same as the one used in the examples in Chapters 2 and 3. The stiffness and damping constants are evaluated for the shallow foundation on pages 2.42 to 2.46 and on pages 3.33 to 3.38 for the pile foundation.
The response will be calculated for the following vibration modes: 1.
Vertical mode
2.
Coupled horizontal and rocking vibration
3.
Torsional mode.
183
Figure 4.2~
r------ I
---l I I
I
I
/
T,
I
I
I
I
Ye
~
X;; ~
E '1"
~
N
11
V
'/ /
, I
1.381 m
• I.
I
1.25 f r
1.37m
4.5 ft
.~ .
I w .!.
, y
1.
I
1.37 m
:.37 m
tt
4.5
!I
1 1 ////// ///
'/ /
/
I
4
~
J.o.~8 ! rn
0.3m\
I"If!. •
1.25ft
4 .5ft
1.22m 4ft
--1
(i6~t ) ..._._. __
b" 4 .87rn
(
.
'w ·w·
I
"$I, -
2.
~ (t)
ill \J../
3'
4'
I.
7 7 C\}
~ ;..
-$: •
1.37rn
.~..
'J
-07-,
1.37 m
I
1.37 m )fl'",
~I
°
r0 0
,
I
a.38m
I-
E
YJ
I-~-
1fJ
I
----~Ze I
, °T :::
..... ,
,.,
~I -o/~
E r<> 0
0.38 m fa:_ .~
b = 4.877m!!6 fl)
184
CASE (I) - SHALLOW FOUNDATION Shallow foundation overlying a deep homogeneous soil layer (halfspace) with no embedment.
Natural Frequencies and Damping Ratios
a) Soil Material Damping Neglected: Vertical Vibration: kw = 4.074x1 08 N/m
{O o =:
C w :::
5.68x10 6 N/m/sec
8
{k =:
4.074 xl0 =:65 .14rad /sec 9.603 X 104
~;;;
D =~ = c = 5.68 X 10 = 0.454 Le.45.4% 4 8x9.603 cer 2.Jkm 2-J4.074x10 x10 6
Torsional Vibration: 9
k'1 11 = 1.789x1 0 N m/rad
Cr,T] = 4.39x1 06 N rn/rad/sec
Mass moment about Y (Eq. 4.32, Fig . 4.4)
(00
= 82.3 rad/sec
D = 0.10
Coupled Horizontal and Rocking Vibration: k'l"V
= 1.668x1 09N m / rad
kuu = 3.683x10 8 N / m ku'V
= -S.34x1 08 N m / rad
CW'i'=
1.1Sx10 7Nm/rad/sec
cuu = 3.18x10 6 N / m / sec C UIjI
= -4.61x1 06 N m / rad / sec
185
The natural frequencies of the coupled motion follow from
(1) 1 ;::
41.3 rad / sec
(02
Eq. 4.51 as
=112.3 rad / sec
The modes (Eq. 4.52) are a- ::: 2.61, a2 ::: -0.638 Modal damping ratios (Eq, 4.56) are 0 1
;::
0.135, O2 = 0.299
b) Soil Material Damping Included: tan8 ;:: 2B ;:: 0.1 Use formulae (2.18). Since in these expressions
CD
is variable, we can set
CD
=COo
Vertical Vibration:
;:: 3.708x1 0 8 N f m = 2.537x1 07 1b / ft
;:: 6.306x10 6 N / m / sec
>
4.315x10 51b / ft / sec
Torsional Vibration:
k~1]
= k q l1 - 2fJ c1]7]OJ o = 1.789 x l 0 9 - 0.1 x 4.39 x l 0 6 x 82.3
=1.752x109 N m / rad ;:: 3x108 Ib ft / rad
= 6.563x10 6 N m / rad / sec
>
1.124x10
61b
ft / rad / sec
186
Coupled Motion:
For the first mode ro ::::
W1,
for the second mode
(0 ::::
W 2
For the more important first mode R
6
k'/Ill =k /Ill -2j3c /Iii OJ l = 3.683x10 -O.lx3.18xl0 x 41.3 :::: 3.551x108 N 1m:::: 2.43x1 0 7 Ib 1ft
c'
till
=
c + 213 k 1111
I~ U
I OJ1 = 3.18 x 106 + 0.1 x 3.683 x 10 8 -.;- 41.3
= 4 .071x1 06 N I m I sec::; 2.786x1 05 Ib / ft / sec
k'~~ =k~~ -2j3c ~~ OJ1 =1.668xl0 9 -O.l x1.15xl0 7 x41.3 =1 .620><109 N m / rad ::; 2.776x10 8 Ib ft I rad
c'
'If~
=c
'If~
+ 2j3 k 'lflii IOJ 1=1.15 xl0 7 +O.1x1.668xl0 9-.;-41.3
=1.554x10 7 N m / rad / sec
k'. =k 1I~
u~
-2j3c
u~
OJl
>
2.662x1061b ft / rad / sec
=-5.34xl0 8-O.1 x(-4.611 xl0 6)x41.3
::; -5.149x1 08N/rad::; -1.156x1 08 Ib / rad
:::: -5 .904x1 06 N I rad I sec
>
-1 .325x1 06 Ib I rad / sec
Natural frequencies and damping ratios with material damping included:
For comparison the values obtained with material damping neglected are shown in brackets .
187
V ertical Motion:
kw
=3.708x10 8 N / m
COo
fk = 62.14 rad I sec (65.14) = V;;
c o = .jk;;; 2 km
Cw
=6.306x1 06 N/ m / sec
C ll ll
=6.563x106 N m I rad / sec
= 0.528 (0.454)
Torsional Vibration: kill!
=1.752x1 09 N m / rad
COo =
fE = 81.47 rad / sec (82.3)
VIII
D = 0.1776 (0.10)
Coupled Horizontal and Rocking Vibration: 001=
a1
40.75 rad / sec (41.3)
=2.631 (2 .61)
0 1= 0.188 (0.135)
CO2
= 110.34 rad / sec (112.3)
a2
=- 0.632 (-0.638)
D2 = 0.563 (0.30)
Response to Harmonic Loading The Vertical and Torsional Amplitudes: the amplitudes follow from Eq . 4.9. With Py
=rn,e co 2 and k =molo, the amplitude is, as in Eq. 4.19,
188
Expressing the amplitudes in a dimensionless form (Eqs. 4.23)
The response curves are shown in Figs. 4.24 and 4.25 .
Coupled Motion:
Using the direct analysis, the horizontal and rocking amplitudes follow from Eqs. 4.43 .
These are
shown
in Fig.
4.26
as dimensionless
horizontal
amplitudes
and
dimensionless rocking amplitudes defined by Eq. 4.48,
Figure 4.24: Effect of Soil Material Damping on Vertical Amplitude of Shallow Foundation : 1) soil material damping neglected and (2) soil material damping included w
o
0
I-
'"
::::J
•
....J
a:.
{l)
u
L...
-~ ~ - -- • .<.
0
a: ..: ur >
5'0
i
90
f RE2 UENCi iRRD. / SC: C. I
189
Figure 4.25: Effect of Soil Material Damping on Torsional Amplitude of Shallow Foundation: (1) soil material damping neglected and (2) soil material damping
included
w "]
a
0
= .: ~
(0
a: G ......
'"
.
N
o
6'C
9'0
. ... '-'
f tH:~Q U ==' r~c r :~ ;:;:; .
~C: '"
.ec
I S :.C. J
190
Figure 4.26: Horizontal and Rocking Components of Coupled Response to Horizontal Loads: (1) material damping neglected and (2) material damping included
c on
w o
:::> l
CJ
~
.,J
n, :L
a:
c
-.J
~
a:
'
. -;
.....
a::
o
:c:
I
50
I
i
I
90
lZO
I SO
.ac
FR:' QUf.NCY lA RD . IS EC. )
j
~o
I
90
120
I ~O
18 0
i
210
f RE.QUE NC Y [RAD. I SE C. )
J 91
CASE II - PILE FOUNDATION
Summary of Results for Pile Foundation
1) Vertical Vibration:
a) Pile-soil-pile interaction neglected:
kw :;:; 15.32x10 8 N 1m
5
Cw = 21 .732 X 10 N
I m I sec
0=0.0896
b) Pile interaction included:
k'w
=7.310x10 8 N I m
0)'0=
c'w::::: 10AOx10 5 N I m I sec
87.13
0=0 .0618
2) Coupled Response:
a) Pile interaction neglected:
7
kuu = 45.728x10 N I m
klll\!,
k UIjI
9
3.616x 10 N m I rad
=-7.73Jx1 08 N I rad
C UU
5
= 9.1472x10 N I m I sec
cljI'jI =5.5606x10 N m I rad I sec 6
C UIjI =
-1A55x1 06 N I rad I sec
b) Pile interaction considered: 7
k'uu = 26.90x10 N I m
5
c'uu = 5.38x10 N I m I sec
k'lVlI' == ~1jI
192
c'UII'
=
ClJ\V
Response After obtaining the stiffness and damping constants for both cases, the response to harmonic loads is obtained in the same way as in the case of shallow foundation . The results are plotted in Fig. 4.27 for the vertical response and in Fig. 4.28 for the coupled response to horizontal excitation.
Figure 4.27: Vertical Response of Pile Foundation: (1) pile interaction neglected and (2) pile interaction included (2)
w
Cl ::::J
c
(1 )
~
l :J CL
k n; ..J
LI)
...·
a: U
l-
a: W >
t:)
·
m
'------
II)
-· °0
sa
so
.
90
I
I
120
lS0
===== leo
210
FREQUENCY rARo. /SEC. )
193
Figure 4.28: Horizontal and Rocking Response of Foundation
by Fig
~ .19 to
Harmonic and Horizontal Excitation
J
r
•••• • •.(;.• • ,~
4.4 RESPONSE OF RIGID FOUNDATIONS IN 6 DOF 4.4.1 Governing Equations
When the rigid foundation is of general shape , the response is in six degrees-of freedom, three translations and three rotations, all of them, possibly, coupled . These directions are indicated and labelled in Fig. 4.29. The stiffness and damping constants of the footing are best established for the elastic centre of the base (C.B.) first and then transferred to the centre of gravity (C.G.) of the footing-machine system , analogously to the coupled constants of a 2 DOF system by Eqs. 4.32a, band c. Thus, the individual stiffness constants kij and damping constants
Cij,
or the impedance functions
194
refer to the centre of gravity and strictly satisfy the basic definition according to which
K ij
is the external force to be applied in the nodal direction i when there is a sole unit vibration amplitude to occur in direction j. The positive directions for forces and displacements are indicated by arrows in Fig. 4.29 . For embedded foundations, details on the coupled impedance functions can be found in Novak and Sachs (1973). For shorter writing, describe the stiffness and damping properties in terms of impedance functions Kij. Then, the typical governing equations in the directions X and \V, being conditions of dynamic equilibrium of forces and moments in the two directions
respectively, can be written using Newton's second law as (4.64a)
where dots indicate differentiation with respect to time. Figure 4.29: Notations and Sign Convention for Rigid Footing
,;
Pz.
/'
/'
/'
P.
f----- •
2 Y,v
I I
I ~-L.:.r-:-+-----...J[-"::'--+-~
X, u
z,w
195
Similar equations can be written in the other four directions. In Eqs. 4.64, li = mass moment of inertia in direction i and Dij = products of inertia. The effect of the latter is usually small unless the asymmetry of the footing is very large.
Listing all the
displacements and rotations in a vector,
{u} = [u v
W
S
ljI
SY
(4.65a)
and the excitation forces in the loading vector,
{P(t)} = lpx(t)
Py(t)
Pz(t)
/v(.(t)
M y(t)
Al z(t)J (4.65b)
the impedance matrix can be readily written and the governing equations expressed in the standard matrix form, i.e.
[Tn J{U} + [KJ{u} =
{pCt)}
(4.66)
The mass matrix is diagonal when D jj l = O.
4.4.2 Free Vibration Undamped free vibretion In the case of free vibration, {P(t)} = 0 in Eq. 4.66 . When damping is neglected,
Cij
= 0 and [K] = [k] . Assume that the stiffness matrix is frequency independent. Then, the particular solution for all displacement in Eq. 4.65a is {u(t)} = (u) sin cot, where (u) lists displacement amplitudes and co is the unknown natural frequency. Substituting the particular solution into Eq. 4 .66 yields
([k] -
0) 2
[mJXu } = {a}
(4.67)
Eq. 4.67 represents the classical eigenvalue problem whose solution yields six natural
]96
frequencies and vibration modes.
These are best determined using a suitable
subroutine such as IINROOT" in the IBM Scientific Subroutine Package or the subroutine "GVCSP" in the IMSL Mathematical Subroutines Package. If the stiffness and damping matrices are taken as frequency dependent, the eigenvalue problem becomes a nonlinear one and its solution is more difficult. The natural frequencies can be more easily identified from the response curves of the undamped or lightly damped system to harmonic excitation .
Damped free vibretion If damping is considered and the impedance matrix is constant (frequency independent), the free vibration analysis leads to a nonclassical eigenvalue problem . Its solution, carried out in terms of complex eigenvalues and modes, yields six damped natural frequencies and associated modes, which feature phase shifts between individual motion components, and six modal damping ratios. The analysis can be carried out using a suitable subroutine such as 'IRGG" in the EISPACK package or "GVCCG" in the IMSL package.
Details on the complex eigenvalue problem can be
found in Novak and EI Hifnawy (1983) or elsewhere.
4.4.3 Response to Harmonic Loads If the excitation forces are harmonic with frequency
{P(t)}
CD,
they can be written as
= {P]}e iM = {P} (cos CO t + i sin (0 t)
(4.68)
The particular solution to Eq. 4.66 is
{u(t)} = {u}e;(lj{ 197
where {u}
=vector of complex amplitudes.
Substituting into Eq . 4.66 yields
[[K]-m 2[m]]{u} = p
(4.69)
This is a system of linear algebraic equations for the complex amplitudes that can be solved using the IMSL package or any other. Alternatively, both the complex impedance functions and the amplitudes can be split into their real and imaginary parts, i.e.
[K] = [k] + i OJ [ C]
,{u} =
{u] } + i {u2 }
(4.70)
Substituting Eq. 4.70 into Eq. 4.69 and realizing that both the real part and the imaginary part of the latter equation must vanish, two coupled equations for the real (ul) and (U2 ) are obtained, which can be written as
(4.71)
The dimension of the problem is doubled to 12 x 12 but all is real. Consequently, the solution is easily obtained by Gaussian elimination or any of the basic subroutines available such as "SIMQII" in the IBM package. In either situation , after {U1} and {uz} have been established, the real amplitudes and phase shifts follow analogously to Eqs. 4.22 and 4.23. An example of the coupled response in 6 OOF is shown in Fig. 4.30 . The figure shows the vertical response and rotation about the horizontal axis X for an irregularly shaped, large compressor foundation exposed to harmonic unbalanced forces and moments. Notice that all six possible resonances need not be discernible. (The vertical response is shown for the edge of the footing.)
198
Figure 4.30: Response of an Irregular Compressor Foundation in 6 OOF in Vertical Translation (Z) and Rotation about Horizontal Axis X. (30m layer of clay overlying bedrock)
U
LlJ (f)
"
Do
<0 cr.. Ul ~
. o
o o
C"
o
~
>(
....,
o.......
0 ~
•
...
f
.....: a
f
L:.L: I
x
0 co
:1
o
)
----.--/ o -1----, . - . I, ---,..'-'-----.;:=--.- I' I I '=> J 5 10 IS 20 2 $ 30
r" 35'r--"10, ~is"I
Fr:EOUEl-IC )'
I
I
50
55
i l l
66
6S
]C
• R:A.O . ISEC · .
199
o
CI
o ...,
,..... 1:
z;
0 0
.
CD
~
0 0
Q
~ ,g
X ..."
,::,
.... -e, ;
CJ
.J
<""I
;::
~
~ I)
..:
(I:
I· I r-J 0
'" r'
c
c: 0-
Q
'f
5
I
10
15
20
FREQUUlC Y
-------_._----- . ._ - - _._.-.. .. _.._---- -_._
l-
X~
46 m
_._- ---j'
I
if j -
Y
- - ---&
Z
200
4.4.4 Effect of Symmetry When the foundation with the machine has two vertical planes of symmetry, many of the off-diagonal elements of the impedance matrix vanish, decoupling it as shown in Eq. 4.72.
KUI/
s; [K] =
KI/f/1 K IN
K,.;
(4.72)
~
«.; Thus, the response decouples into two coupled motions in 2 OOF, each comprising a horizontal translation and rotation in one vertical plane, a vertical motion in 1 OoF and a rotation about the vertical axis. Such a situation often exists or may be assumed to be approximately valid simplifying the analysis. Oecoupling into six 1 OOF systems could only be valid for a very thin doubly symmetrical mat, not a practical foundation. (If the foundation is very thin it is flexible, not rigid as assumed here.)
4.4.5 DYNA5 Computer Program The calculation of the footing response in 600F for any type of loading and foundation can be conveniently carried out using the code OYNA5 (Novak et aJ. 1999).
4.5 STRUCTURES ON FLEXIBLE FOUNDATION In many practical situations, a flexible structure is supported by rigid footings or mats . This is the case with frame foundations for turbine generators or paper machines, buildings, towers and many other structures. Two basic configurations may occur as
201
diagrammatically depicted in Fig.
4.31.
Each column
of the building has its own footing
(a) or all flexible columns rest on one common rigid mat (b).
Case (a) is typical of
towers and chimneys and frame structures on individual spread footings.
Figure 4.31: Structure Foundations in the Form of (a) individual spread footings and (b) common mat
G
:l
h
7
a)
b)
Consider horizontal response of a structure comprising horizontal translations u, and rotations
\j! i,
of individual lumped masses . The foundation stiffness and damping
can be described by the same 2 x 2 impedance matrix as used in the analysis of rigid footings in par . 4.2 .2 and included in Eqs . 4.35 and appearing as the first submatrix in Eq .4.72. This foundation impedance matrix can be introduced into the governing equations of the structure and the governing equation written in the standard form
[m]{ii}+[K]{u}
=
{pet)}
(4.73)
For individual spread footings, each supporting one column, the mass matrix lists all the lumped masses including the footing and the global impedance (stiffness) matrix is obtained by superimposing the footing impedance matrix on the structure
202
impedance matrix giving
l
[K] = f[K s ]
l
(4.74)
[K fJ]J
For a frame on a common mat, schematically represented in Fig. 4.32, a somewhat different procedure can be followed. Figure 4.32: Building on Mat Foundation
G: o
. i
. . :: " -
j
!
I
i
, o
I\( ;-I
For each of the floors (masses m, to mn) one condition of horizontal equilibrium is written giving n equations. If the structure is a shear building or if the structure matrix was condensed eliminating floor rotations, the total number of degrees of freedom is (n+2) where n
= the number of storeys
and the other two DOFs represent the footing
203
translation,
Ub
and rotation.w . The remaining two governing equations are obtained
by
writing the equilibrium condition of the building as a whole in the horizontal direction and rotation,
respectively .
Then ,
denoting
the
mass
matrices
and
floor
relative
displacements
the governing equation of the building response
[m]
{m}
{mh}
n
{m }T
It
m«: Lm i I
{mh}T
I
" Lmjh j I
rl ub
Lmihi
" 2 1+ Lmjh J
'1/
I
[Kl {O}T
{O}
{O}
{u}
KUI/
«;
Ub
{O}T
KIf/lI
K/{/V/
1/1
= {pet)}
(4.75)
The matrix [K] lists all the stiffness and damping constants of the structure and {OJ is the null vector. The total mass matrix is not diagonal, however. The mass moment of inertia includes all floors and the mat is
Matrices [m] and [K]s are matrices the structure would feature in case of a rigid
204
foundation. The foundation submatrix 2 x 2 is clearly separated from the rest. Notice the relative displacements
Uj
in the vector {u} are measured relative to the displaced and
rotated axis of the building. {pet)} is the vector of the horizontal loads on the building. When the motion of the structure results from horizontal ground motion ug(t) , the acceleration ii 8 (t) is added to
itb
with the only result being that the right side of Eq.
4.75 becomes n
{P}
= -[ {m}
LmihilT iig(t) 1
More details on this analysis can be found in Novak and EI Hifnawy (1983). The effect of soil-structure interaction on seismic response of buildings resting on various types of both shallow and deep foundations is examined in Novak and EI Hifnawy (1984) . The inclusion of piles into the analysis of a frame structure supported by a flexible mat is discussed in Aboul-Ella and Novak (1980). A large amount of data on soil-structure interaction is available in the proceedings of conferences on earthquake engineering.
4.6 DESIGN CHECKLIST FOR MACHINE FOUNDATIONDS After the response of the proposed foundation is predicted from the dynamic analysis, it is checked against certain design requirements including : 1. the usual check of bearing capacity and settlement, and structural strength of the foundation under static loads.
205
-----For piled foundations, the maximum load for any pile
2. the maximum bearing pressure (static + dynamic) should be less than 75% of the allowable pressure of the soil.
(static + dynamic) should be less than 75% of the design capacity of the pile . 3. comparison to tolerance for dynamic behaviour which includes a) maximum vibration amplitudes, see figure; b) maximum velocity (co x displacement amplitude) and acceleration (ol x displacement amplitude), see fiqure ; c) maximum magnification factor, should be less than 1.5 at resonance; d) possible resonance conditions, the operating frequency of the machine should not be within ± 20% of the resonance frequency (damped or undamped). 4. Consideration of possible fatigue failure in the machine components and connections. 5. Consideration of environmental requirements and physiological effects on workers. Sometimes, a factor of safety (FS) could be used to account for the relative importance of the machine to overall plant operation. The predicted amplitude is multiplied by a service factor (safety factor) FS (1.5-2) to obtain an effective vibration amplitude. The following figures and tables may be used to check the compliance of the vibration amplitudes with different design requirements. 1. Figure 4.33 shows dynamic response limits in terms of limiting "single amplitude" vibration at any frequency.
The figure has 5 zones of sensitivity shown by persons
(standing and subjected to vertical vibration). 2. Figure 4.34 may be used to establish permissible horizontal vibration amplitudes for rotating machinery. 3. Figure 4.35 shows the response spectra for the structures; displacement, velocity, and acceleration vs frequency . The lines labelled "Rausch" are the same as the limits
206
for safe operation of machines and foundations , (Fig. 4.33). The cross-hatched area gives possible structural damage which may be caused by steady state vibrations . 4. Figure 4.36 shows the vibration standards for high-speed machines. 5. Figure 4.37 shows the vibration limits for foundations supporting turbomachinery. 6. Table 4.3 gives suggested limits of peak velocities for various categories of operation.
207
Figure 4.33 General limits of vertical vibration ampli udes for a particular frequency
0.0001 100 200
SOO 1000 2000 SOOO 10,00
FREQUENCY. + o
CPM
Steady state vibrations Steady state vibrations
Due to blasting
.Shaded line represents limits for safe operation of machines and foundations (not for satisfactory operation) . • Dotted lines are limits associated with blasting . Do not apply to steady state vibration. 208
j.
1: :. ..{/' ~
0
I
\
I,
J . ,
(UC
t.
,
"
~·lt-lC"~\'o J tv [- _~ j:l
" Vt ('j ~ j
Figure 4.34 Vibration performance of rotating machines
-/
,
I
..-......,
~ /:~ '~'
\"E V
Ie
(
I "~
~.,
.>
I
j '
I t
",
C,. t·1
,I
r
0.10
':~
\-0
~" ~(;)
O
...a-= :::»
•4
•I li•.z:.. •z: -!C a •-> ..... III
0·01
0
III
E
.. 0
III
g
:t
C
L
JI
.• 4
..J
-... •z: • •z:
0.001
..J
A.
~
..J
t! Z
0
N
It
-=
A
~
III
•
0
%
0.0001 100
'REQUENCY •
E D C B A
10,000
1000
Dangerous, shut it down immediately Failure is ncar, Correct very quickly. Faulty, correct quickly. . ' i ' \ . Minor faults. No faults, typical of new equipment.
(
CP.
'C'
(,
r:
209
Figure 4.35 Response spectra for allowable vibration at facility
Frequency.
epa
210
Figure 4.36 Vibration standards of high-speed machines
6O ...........Q...--I----I--.........- - - t - - - - t
4000
SPEED, RPM \
,
.
-'
(
211
Figure 4.37 Turbomachinery bearing vibration limits
256 0 0 0
...
.-;
<,
'28
?'>
~
"1(
...
UJ
0
-
&t
~.(
~
~
E
....J
Q..
~
32
-e
S; UJ
~
16
UJ
U
:s
0..
UJ
8
0
Z 0
4
~
E
~
a:l
1--1
0
2
>
0
320
5
L
( \ '
\
\
r
212
Table 4.3 General machinery-vibration-severity data
Horizontal Peak Velocity
Machine Operation
(in/sec) < 0.005
Extremely smooth
0.005-0.010
Very smooth
0.010-0.020
Smooth
0.020-0.040
Very good
0.04-0.080
Good
0.080-0.160
Fair
0.160-0.315
Slightly rough
0.315-0.630
Rough
>0.630
Very rough
After Baxter and Bernhard (1967)
REFERENCES Aboul -Ella, F. and Novak, M. (1980) - "Dynamic Response of Pile Supported Frame Foundations," Journal of Engineering Mechanics, Vol. 106, No. EM6, December, pp. 1215-1232. Arya , S.C., O'Neill , M.W. and Pincus, G. (1979) - "Design of Structures and Foundations for Vibrating Machines," Gulf Publishing Company, Book Division, Houston, Texas, p. 191. Baxter, R. L. and Bernhard, D. L. (1967) . "Vibration Tolerances for Industry", ASME Paper 67-PEM-14, Plant Engineering and Maintenance Conference, Detroit , MI, April.
213
Beredugo, Y.a. and Novak, N. (1972) - "Coupled Horizontal and Rocking Vibration of Embedded Footings," Canadian Geotechnical Journal, Vol. 9, No.4, pp. 477-97. Novak, M. (1974a) - "Dynamic Stiffness and Damping of Piles," Canadian Geotechnical Journal, Vol, II, pp. 574-598 . Novak , M. (1974b) - "Effect of SoH on Structural Response to Wind and Earthquake," International Journal of Earthquake Engineering and Structural Dynamics, Vol. 3, No.1 , pp.7996. Novak, M. and Beredugo, Y.O. (1972) - "Vertical Vibration of Embedded Footings," Journal of the Soil Mechanics and Foundations Division, ASCE , SM12, December, pp. 1291-1310. Novak, M. and EI Hifnawy, L. (1983) - "Effect of Soil-Structure Interaction on Damping of Structures," Journal of Earthquake Engineering and Structural Dynamics , Vol. 11, pp . 595-621. Novak, M. and EI Hifnawy, L. (1984) - "Effect of Foundation Flexibility on Dynamic Behaviour of Buildings," Proc. 8th World Conference on Earthquake Engineering , Vol, 111, San Francisco, pp. 721-728. Novak, M. and Sachs, K. (1973) - "Torsional and Coupled Vibrations of Embedded Footings," International Journal of Earthquake Engineering and Structural Dynamics, Vol. 2, No. 11, 33. Novak, M., EI Naggar, M. H., Sheta, M., EI-Hifnawy, L., El-Marsafawi, H., and Ramadan , 0 ., 1999. DYNA5 a computer program for calculation of foundation response to dynamic loads. Geotechnical Research Centre, The University of Western Ontario, London , Ontario. Richart, F.E., Hall, J.R. and Woods, R.D. (1970) - "Vibrations of Soils and Foundations," Prentice-Hall, lnc., Englewood Cliffs, U.S.A. Urlich, C.M. and Kuhlemeyer, R.L. (1973) - "Coupled Rocking and Lateral Vibrations of Embedded Footings," Canadian Geotechnical Journal, 10, pp. 145-160.
214
5
FOUNDAnONSFORSHOCKPRODUaNG
MACHINES
5 FOUNDATIONS FOR SHOCK-PRODUCING MACHINES
Shock producing machines generate dynamic effects which essentially differ from those of rotating and reciprocating machines and the design of their foundations, therefore, requires special consideration.
5.1 Introduction
Many types of machines produce transient dynamic forces that are quite short in duration and can be characterized as pulses or shocks. Typical machines producing this type of load are forging hammers, presses, crushers and mills. The forces generated by the operation of these machines are often very powerful and can result in many undesirable effects such as large settlement of the foundation, cracking of the foundation , local crushing of concrete and vibration. Excessive vibration may impair the operation of the facility and the health of the workers, cause damage to the frame of the machine and expose the vicinity to unacceptable shaking transmitted through the ground. Some machines operate with fast repeating shocks and consequently, the effects of vibration may be aggravated by resonant amplification of amplitudes such as is the case with rotating or reciprocating machines. The objective of the foundation design is to alleviate these hazards and secure optimum operation of the facility. Hammers are most typical of the shock-producing machines and therefore this report is limited to them. This is not a serious limitation, however, because the design and analysis of the other shock producing machines follow criteria that are in many respects similar to those applied to hammers.
213
5.2 Ty pes of Hammers and Hammer Foundations
There are many types of hammers. According to their function, they can be divided into forging hammers (proper) and hammers for die stamping. Forging hammers work free material into the desired shape while die stamping hammers shape the material using a mould or matrix. According to their mode of operation , hammers can be classified as drop, steam and pneumatic, although other systems are also used. More details on the various types can be found in Major (1962) Because of the powerful blows generated, hammers are mounted on block foundations of reinforced concrete separated from the floor and other foundations . The basic elements of the hammer foundation system are the frame, head (tup), anvil and foundation block (Fig.5.1). The frame of forging hammers is separated from the anvil. In die stamping hammers, the frame is usually connected to the anvil to give the system rigidity and precision of blows.
Figure 5.1: Schematic of Forging Hammer and its Foundation
214
The forging action of hammers is generated by the impact of the falling head against the anvil , which is a massive steel block. The head is allowed to fall freely or in order to obtain greater forging power, its velocity is enhanced using steam or compressed air. The size of the hammer can be judged by the weight of the head, which ranges from a few hundred pounds to several tons . The intensity (energy) of the blows can be expressed as a product of the head weight and the height of the drop or the equivalent height of the drop. Only a part of the impact energy is dissipated through plastic deformation of the material being forged and conversion into heat. The remaining energy must be dissipated in the foundation and soil.
Different foundation arrangements are used to
this end. In small hammers, the anvil is sometimes mounted directly on the foundation (Fig.5.2a). This is done for the sake of simplicity and hard shocks . The main drawback of this arrangement is that the concrete under the anvil suffers from the shocks and, depending on the hammer type , also from high temperature. Repairs often may be necessary. To reduce the stress in the concrete and shock transmission into the frame, viscoelastic suspension of the anvil is usually provided (Fig. 5.2b). This may have the form of a pad of hard industrial felt, a layer of hardwood or, with very powerful hammers, a set of special isolation elements such as coil springs and dampers.
Such a
suspension reduces the impact of the anvil on the foundation by prolonging the path of the anvil and by energy dissipation through hysteresis and plastic deformation .
2J5
J:igure 5.2: Types of Foundation Arrangement
/
r
GA P
/
\
SPR lNGS -TROUGH
(c)
( d)
The foundation block is most often cast directly on soil as indicated in Figs - 5.2a and b. When the bearing capacity of soil is not sufficient or undesirable settlement is anticipated, the block may be installed on piles. When the transmission of vibration and shock forces in the vicinity and adjoining facilities is of concern, a softer mounting for the foundation may be desirable. This can be achieved by supporting the block on a pad of viscoelastic material such as cork or rubber (Fig-5.2c) or on vibration isolating elements such as rubber blocks or steel springs possibly combined with dampers (Fig. 5.2d) . A trough, which adds to the cost, is needed to protect these elements. The material of the pads must be able to resist fatigue as well as moist environment due to condensation and must have a long lifetime. Rubber pads should
216
have grooves or holes to allow lateral expansion because the Poisson's ratio of rubber is 0.5. Slabs of solid rubber are quite incompressible. The gap around the footing, which rests on a pad (Fig-5.2c), may be filled with a suitable soft material which allows the block to vibrate freely but prevents blockage of the gap by debris.
Figure 5.3: Suspended Footing Blocks
~
~
. .' 4
D ..
-
a)
..
4.
•
~
-
II"
"
d.
...
_
. .... ..
..
.. #
-. -.
~.
_
...
b)
With springs, the space around the footing must be wide enough to provide access for installation, inspection and replacement. This is necessary because springs sometimes crack. However, access space is not always available. For reasons of easy access and convenience, the springs are sometimes positioned higher up and the footing is suspended on hangers or cantilevers (Figs. 5.3a,b). Such a design is more complicated and costly. Careful reinforcement of the cantilevers for shear is necessary. The soft suspension of the footing block on pads or isolators is particularly efficient on stiff soils. It increases the vibration amplitude of the block but reduces the force transmitted into the soil. Additional damping, if provided, is very useful because it reduces the vibration amplitude. The inertial block is sometimes deleted and the anvil
217
suspen ded directly on isolators (GERB). More complicated foundations are sometimes designed to protect the frame of the hammer from shocks which can hinder the operation and cause fatigue cracks . To interrupt the flow of shock waves into the frame, additional joints with viscoelastic pads or elements (Fig. 5.4) separate the upper part of the block from the rest. Klein and Crockett (1953) describe the example shown in Fig. 5.4b .
Figure 5.4: Schematic of Foundation with Additional Joints to Protect Hammer Frame: (a) outline and (b) prototype
, - . • !' ·· -·-. , -, .~~ . . · -·.- -.· . . - --- . --.. I'~ - , . , . .- . . .. , · . - . - . / i -: :-. ''·r'x ·x , . . , .. ~ > . ·· . - - ·· - - . - - i(
..........
c;'"
".
".
.-
,""' ,\
r
~
". ' ".
/~ 7 ~ ~.
/"
'~ '
a)
b)
5.3 Des ign Criteria
The hammer foundation must be designed so as to facilitate efficient operation of the hammer without failure and cause minimum disturbance to the environment. This general objective may be achieved if the vibration amplitude, settlement, physiological effects, and all stresses remain within acceptable limits. In addition, resonance should be avoided with high-speed hammers.
218
5.3.1 Vibration Amplitudes There are no unique limits on the vibration amplitude unless specified by the manufacturer or codes. In the absence of such specifications, the allowable level of vibration amplitude is estimated on the basis of experience and physiological effects with some accommodation for the fact that larger hammers produce greater shocks and usually larger amplitudes. A few values of maximum allowable amplitudes are suggested for guidance in Table 5.1.
Table 5 .1: Maximum Allowable Amplitudes for Hammer
Foundations
For foundations built on soils susceptible to settlement, such as saturated sands, smaller amplitudes are desirable. On the other hand, larger amplitudes may sometimes be admitted for large hammers provided they satisfy the criteria for physiological effects and settlement. Amplitudes larger than about 0.16 in (4 mm) can impair the operation of the hammer, however.
5.3.2 Physiological Effects of Vibration Physiological effects depend on vibration velocity and acceleration rattler than displacement, but vary with the type of vibration and the sensitivity of individuals. The
2]9
velocity may be considered a criterion in the moderate frequency range typical of hammers. The amplitude of vibration velocity can be calculated approximately as (5.1) In which
Vrn =::
the maximal (peak) displacement and roo ::: the natural circular frequency
of the foundation. Various authors (see Richart et. al. 1970) have collected many data on human perceptibility. The data given by the German Code DIN4025 and shown in Table 2 are useful in that they provide an indication of perceptibility of vibration as well as the effect of vibration on work. The data are shown as a function of the physiological factor K calculated as r
K =O.80vm
oJ'!(
>
-'
r.A. "
I
~'.
(5.2a)
,OJ I
for vertical vibration and (5.2b) for horizontal vibration; V m is peak vibration velocity in mm/s (1 inch
>
25.4 mm) . For
machines operating intermittently, the effect on work may be one category lower than that given by the calculated value of K.
l" I
(\
(
.,
;' l (.'
rI
10---
\
(
"
'. r 1
. "t-
-
_I
- ,
[-
II
rjr}
,
"
I
\ !-
r
~/ l
<
r
~\
1
0"
v-
,.
0 J\
\
220
Table 5.2: Physiological Effects of Vibration
D1N4025 (German Code)
K (m/s)
Classification
Work
0.1
Threshold value, vibration
Not affected
just perceptible 0.1 - 0.3
Just perceptible , scarcely
Not affected
unpleasant, easily bearable 0.3 - 1.0
Easily noticeable,
Still not affected
moderately unpleasant if lasting an hour, bearable 1.0 - 3.0
Strongly noticeable, very
Affected but possible
unpleasant if lasting over an hour, still tolerable 3.0 - 10
Unpleasant, can be
Considerably affected , still
tolerated for one hour, not
possible
tolerable for more than one hour 10- 30
Very unpleasant, not
Barely possible
tolerable more than ten minutes 30 -100 Over 100
Intolerable
Impossible Impossible
221
Another physiological effect is noise. Hammers are noisy and have a level of noise of about 100 db (decibel) but some presses are even noisier.
5.3.3 Stresses Stresses in all parts of the foundation have to remain within allowable limits. This includes compressive stresses in the pad and the concrete underlying it; bending, shear and punching shear stresses in the block, and finally, the stresses in the soil or piles as well as in the vibration absorbers , if used. Dynamic stress is repetitive and can cause fatigue . This effect can be accounted for by lowering the static allowable load or by multiplying the dynamic stress by a fatigue factor.
A fatigue factor
J.!
=3
is often
recommended for all parts of the system. Steel springs are also subject to fatigue, particularly when they have initial cracks.
To eliminate this possibility, the springs should be X-rayed before they are
installed. Temperature effects can contribute to the decay of the pad under the anvil and the underlying concrete because the temperature of the anvil can rise to as high as 1000C (2120F) with hot forging.
'- I . 5.3.4 Foundation Settlement
I
c r
t •
I
1-' I
•
"' \
Settlement can be a serious problem with hammer foundations. Where the soil is unreliable in this respect, piles or absorbers should be considered (Figs .5.2c, d and 5.3).
Piles limit the settlement by transmitting the loads to deeper strata while
absorbers reduce the settlement by reducing the forces transmitted to the soil.
222
5.3.5 Mass of the Foundation Block The adequacy of the foundation mass and dimensions is best proven by detailed analysis of stresses and amplitudes . This is particularly true for the more complex foundations.
Nevertheless, some guidelines have been suggested for the preliminary
choice of the weight of the foundation block. Assuming that the anvil weight is about twenty times the weight of the head, Go, the weight of the block. G, can be estimated using the formula (Rausch, 1950)
(5.3) in which Co
= the
maximum velocity of the head and c,
:=
reference velocity taken as
1.8.37 ft/s (5.6 m/s). Smafler masses can be used if the response is limited by special measures such as shock absorbers. As for the general layout of the foundation, it is desirable that the centreline of the anvil and the centroid of the base area lie on the vertical line passing through the centre of gravity of the footing with the hammer. Misalignment and eccentric blows can result in tilting of the foundation and differential settlement
5.3.6 Vibration Effects on Environment Vibration propagates from the footing into the surroundings in the form of various types of waves. At greater distances, surface waves (Rayleigh waves) usually prevail. The vertical amplitude of the ground motion, vr , at a distance x from the vertical axis of the foundation can be evaluated approximately as
223
'1 GJ
Y
t'.
6 I l,l
' ,'
,I
Vr
- .ff
r:
-va -e -a(r-roJ r
~ \
in which v» = the footing amplitude, ro
",I
fc-~ ,
\
.
iJ """.J. {~ _
"I
/
"'--"
~
.l
<,
, "
(5.4)
,
,
=the distance of the footing
edge from the vertical
axis of the foundation (Fig. 5.5) and a = empirical coefficient accounting for the effect of soil hysteresis (viscosity) . Experiments indicate the values of a range from 0 to 0.15 fr1 (0 to 0.05 rn") or more for r in ft or m, respectively.
Figure 5.5: Ground Motion Attenuation with Distance
, I
_,lo ( ---
}
I
r
'c
I
l
<, . \ l~_
,
•
,
,
/ / 7 / 7 7 7
.
~_ro_
_
.-:.... r
_ I
I
The propagation of surface waves is characterized by soil particle motion, which is a retrograde ellipse whose vertical axis is greater than its horizontal axis. This pattern is obtained from theory even when the original disturbance acts only vertically as in the case of a hammer foundation.
However, the physical properties of the soil medium
differ considerably from the ideal properties assumed in the elastic half-space theory. Consequently, the motion actually observed usually differs from the theoretical pattern. In particular, the horizontal amplitudes are often greater than the vertical ones . Thus, it may be assumed for practical calculations that the ground motion in the vicinity of the
224
'
foundation has vertical amplitude predicted by Eq.5.4 and horizontal amplitude of about the same magnitude. Then, the response of a structure located near the hammer foundation can be predicted using the methods of structural dynamics. The effect of ground motion may be augmented by dynamic amplification of the structural response.
5.4 STIFFNESS AND DAMPING CONSTANTS OF THE SYSTEM
The prediction of the response of the hammer foundation requires the description of the stiffness and damping of the foundation and the pad under the anvil. Foundation on Soil Stiffness and damping of foundations supported on soil can be evaluated using the approaches described in Lecture 2. For a rectangular base, the equivalent radius can be established as
(5.5)
in which a and b are the width and length of the base respectively. The vertical stiffness, k, and damping, c, of a foundation are defined as forces associated with a unit amplitude and unit vibration velocity, respectively.
For an
embedded foundation, stiffness and damping constants can be evaluated approximately by Eqs. 2.24 as
(5.6a)
225
(5.6b)
Here, C 1 and
C2 are dimensionless parameters related to the stiffness and damping ,
respectively, derived from the medium under the base (the elastic half-space) or a stratum . S1 and
S'2 are constants related respectively to the stiffness and damping
derived from the reactions of the layer lying above the level of the base (Fig.5.6) and acting on the vertical sides of the footing . G = soil shear modulus and p = mass density of the half-space while Gs ,
ps
are the shear modulus and mass density of the side layer
(backfill), respectively. Mass density is unit weight , y, divided by gravity acceleration, g. Finally, I =.embedment depth. Figure 5.6: Embedded Foundation
p
--r---.,I
. s. ,.
H
. . t .. ,5,',2 . I
J
, '
"'(:7,
. .. . £7
_0
L
t
--.'
:
t
o
•
£7
0
SEPARATION '
•
"
0
£7
' pO '...
-
.. _
o
P
0
•
0"
#
..
0
",' ..
~
t:J
"
t~ Gs • Ps . ' , . • . .. ~ . l
~.
(7
.
"BACKFILL : -' .'
:, ' "
D-
0 - · • •
,
~
r
~
~
~..
.-
'2
226
The parameters C and S depend on the dimensionless frequency. However, the analysis can be simplified if the frequency dependent parameters are replaced by suitably chosen frequency independent constants.
For the range of dimensionless
frequencies typical of hammers (0.5 to 1.5), such constants can be taken from Table 2.1. Adjustments of the theoretical values are desirable . To be on the safe side, it appears advisable to divide the theoretical values of Cy2 shown in Table 2.1 by a factor of about two . The second correction involves embedment effects. The theory indicates that embedment provides a significant source of geometric damping and contributes also to stiffness. However, with the heavy vibration typical of hammers, the soil may separate from the footing sides and a gap may occur-as indicated in Fig. 5.6. This gap is likely to develop close to the surface where the confining pressure is not sufficient to.maintain the bond between the soil and the foundation. The separation may be accounted for approximately by considering an effective embedment depth , I, smaller than the actual embedment depth, L. The effective depth, of course , depends oh conditions. The best bond is obtained when the block is cast directly into the excavation . Another way of accounting for footing separation is to assume a weakened zone around the footing (Novak and Sheta, 1980) When the footing is cast in forms and then backfilled, the backfill shear modulus and density are usually lower than the original values. accurately, the ratios
ps
Unless established more
Ip = 0.75 and Gs I G = 0.5 may be adequate.
227
Soil
eterlel Damping Foundation stiffness and damping are also affected by soil material damping .
The material damping of soil is hysteretic and independent of frequency.
It is con
veniently described using the complex shear modulus
G* = G + iG' = G (1 + itan8) rrr':
in which i = -..j-l , tanf
= G' / G with
(5.7)
e = the loss angle and G' the imaginary part of the
complex soil modulus . Another measure of material damping is the damping ratio
p=
1/2 tane. Material damping can be incorporated using the correspondence principle of viscoelasticity. In the sense of this principle, the shear modulus, G, in Eqs. 5.6 has to be replaced by the complex shear modulus defined by Eq. 5.7. After some manipulation , the stiffness and damping constants including material damping become
kh
=
k - tan 6' C OJ
(5.8a)
c;
=
C+ tan 8 k/ OJ
(5.8b)
in which k and c are evaluated from Eqs. 5.6 without regard to material damping. As Eqs. 5.8 suggests, material damping reduces the stiffness but increases the total damping. However, with a half-space, i.e. a deep layer under the footing, these effects on vertical vibration are small and can be neglected. With shallow layers, the incorporation of material damping is important because the geometric damping is quite small, as can be seen in Fig. 2.12, or may not materialize at all if the first natural frequency of the hammer foundation is lower than the first natural frequency of the soil layer . Thus, the evaluation of the damping provided by
228
shallow layers should commence with a comparison of these two natural frequencies. The procedure is described in Lecture 2.
Soil Properties The magnitude of material damping depends on the type of soil and increases with strain.
It ranges from 0.05 to 0.20 with a typical value of tano being about 0.1.
Shear modulus of soil (or shear wave velocity) depends on the type of soil, the level of strain and confining pressure. The shear wave velocity of most soils ranges from 300 to 1,000 ftIs (90 to 300 rn/s) and should be established by field or laboratory experiments.
Because of the strong effects of confining pressure and strain , the shear modulus entering the calculations should be established for a representative reference position. Such positions are suggested in Fig. 5.6. With deep embedment, the effect of confining pressure variation with depth can also be accounted for by dividing the depth I into sections and using different Gs in each of them .
Pile Foundations Vertical stiffness and damping of a foundation supported by a group of piles can be evaluated using the procedures described in Lecture 3.
Block Suspension by Pads and Absorbers Pads
When the foundation block is suspended by a pad of viscoelastic material (Fig.5 .2c), the vertical stiffness constant of the block is
229
kp in which E,
(5.9 a)
=Young's
modulus of the pad, Ap
=area of the pad and d = its thickness.
The damping constant, calculated in terms of the complex modulus as in the case of a shallow soil layer (Eq..5.8b), is
(5.9b)
where 8p = the loss angle of the pad material and
(00 ::::
the natural frequency of the
block calculated with kp. A variety of reinforced cork, rubber and other materials are available with a broad choice of Young's moduli for each of them. For cork, Young's modulus and material damping vary with the make and static stress (load); typical values may be about Ep = 2700 psi
tan8p
>
19162 kPa
= 0.05
However, the effectiveness of the pad does not derive from its properties alone but depends also on the stiffness of the soil supporting the trough . The two elastic media, the pad and the soil, act in series and have a total flexibility of
1 1 F=-+
kp
(5.10)
k
in which the soil stiffness k can be calculated from Eq. 5.6a using the dimensions of the trough. The joint stiffness of the two media is kt
= 1 / F which yields
230
(5.11)
This equation indicates that the pad is effective and worth the expense only if the soil is much stiffer than the pad, i.e. when k » kp .
Absorbers. Absorbers can be installed as shown in Figs. 5.2d and 5.3. Rubber elements or steel springs combined with dampers are used. They can be selected from available lines or made to order (GERB) . The total stiffness and damping of a set of absorbers is the sum of the individual contributions. Absorbers provide the softest suspension and greatest reduction in forces transmitted into the vicinity. Their effectiveness also depends on the stiffness of the soil as expressed by Eq. 5.11.
Suspension of the Anvil The anvil can rest on a pad of hard industrial felt, Wood or special resilient plates . The Young's modulus of felt varies with the make and typical values may start at about E == 11000 psi = 75 .85 MPa The strength of the felt is approximately equal to one tenth of E. The Young's modulus of hardwood across the grain is about E == 0.70 to 1.50 x 10 psi == 490 to 1034 MPa and about ten times more in the direction parallel to the grain.
The stiffness and
231
damping constants follow from Eqs, 5.9.
5.5 MATHEMATICAL MODELS OF HAMMER FOUNDATIONS Hammer foundations are modeled as lumped mass systems.
The number of
degrees of freedom (independent displacements) depends on the foundation type and on whether the blow of the head acts along the centreline of the system or eccentric ally. The mathematical models which can be used are shown in Fig. 5.7. In many foundations, the anvil rests on an elastic pad as indicated in Fig. 5.2b. Then, a two-mass model shown in Fig. 5.7c is adequate. With e = 0, this model has two degrees of freedom and is most often used in hammer foundation design. Most practical cases can be analyzed on the basis of the models shown in Figs. 5.7a and c and therefore, further discussion focuses on these models.
For a given model whose
properties have been established as outlined above, the response depends on the magnitude and nature of the impact- forces.
232
Figure 5.7: Mathematical Models of Hammer Foundations (OOF
=degrees of
freedom)
1
J
HEAD
~.
~I
I
mo
~ e p
,P Tv
ANV~Ll71~7Xh j ,kn i
oor (a
Tv
mt)f
rn
ex
o
!
I
BLOCK
orn
~
u
k //
3 DOF ( b)
1
I
~
HEAD
mme
me
"T v,
ANVIL
Tv} 1<1
T
V2
BLOCK
me k2
TV3
TROUGH 2
DOF (c )
k;s
3 OaF (d )
233
5.6 IMPACT FORCES
The energy of the impact is determined by the weight of the head, which is given, and its impact velocity . The foundation response to the impact also depends on the time history of the force resulting from the impact.
Impact Velocity of the Head
For gravity hammers, the head falls freely. The maximum velocity just before the impact is
(5.12)
in which g
=gravity acceleration equal to 32.2 fUs 2 =9.81 m/s2 , ho =the drop height and
n = correction factor <1, characterizing the efficiency of the drop. With steam hammers this factor stems primarily from the resistance of the exhaust steam. For well adjusted hammers, n should be close to unity. Power hammers are the prevalent variety and are all double-acting. They utilize the steam or compressed air not only to lift the head but also to accelerate its fall. The impact velocity of such hammers is
CO
= n 2g hs
Go (1+PSJ
(5.13)
in which h, = the length of the stroke, p the mean pressure on the piston (in psi or kPa) , S = the area of the piston and Go the weight of the hammer head . The correction factor of these hammers is lower with the average quoted value being about 0.65 .
234
Time History of the Impact Force
The hammer head moving with impact velocity Co has a momentum moCo if ma = Go Ig is the mass of the head. During the impact with the anvil, part of this momentum is reinvested in the rebound of the head and the rest is transferred to the anvil in the form of a pulse. This pulse is a transient force , P(t), of short duration, tp • The time history of the pulse and its duration depend on the conditions of forging and are to a high degree random; little is known about them. However, the total power of the pulse follows from the theorem of conservation of momentum. The duration of the pulse, tp , is very short, in the order of 0.01 or 0.02 sand is usually much shorter than the fundamental period of the foundation, T. It can be shown (Novak, 1983) that the foundation response decreases as the pulse duration increases and that for ratios of tplT lower than about 0.1, the peak response is practically independent of pulse duration and equal to that obtained with an infinitely short pulse. Even for durations tp
= 0.2T or so, the
peak response is only slightly less
than the maximum . With real pulse duration of 0.02s, the ratio 0.2 1 0.02 = 10 which implies that for natural frequencies smaller than 10 cps (Hz), the infinitely short pulse yields a satisfactory prediction. For frequencies higher than this limit, the infinitely short pulse overestimates the real response and its assumption is, therefore, conservative. Thus, it appears possible to predict the response using the assumption of an infinitely short pulse. When the pulse is very short, it expires before the system starts moving and the resultant motion is free vibration triggered by initial velocity .
235
5.7 RESPONSE OF ONE
ASS FOUNDATIO S
When the anvil is rigidly mounted (Fig.5.2a) and the hammer blow does not act eccentrically, the one-degree-of-freedom model shown in Fig.5.7a is sufficient to analyze the response. The response is obtained as a solution of the governing differential equation that expresses the dynamic equilibrium of inertia, damping and restoring forces. With the notation of Fig. 5.7a, the governing equation of the response v= v(t) is
mv+cv+kv- 0 in which m
= the
(5.14)
mass of the foundation with the anvil and frame; c and k are the
stiffness and damping constants evaluated e.g. from Eqs. 5.6. Finally,
v== dv/dt and t = time. From the elementary theory corresponding to initial velocity of the system,
v= d2 V Idt2 and
of vibration , the solution to Eq. 5.14
C, can be written as
,..
vet) in which
=
C
-e UJ'o
-Dro
f
0
•
SIn UJot
(5.15)
C is the initial velocity,
(5.16) is the undamped natural frequency,
(5.17) is the damped natural frequency and finally,
236
D=
c 2.Jkm
is the damping ratio. With small damping, (i)'o =0 CUo The initial velocity,
C,
can be obtained from the consideration of the collision
between the head whose mass is rn, and the foundation having mass m. Because the pulse resulting from this collision is presumed to be infinitesimally short, the restoring and damping forces have not been activated during the collision.
Consequently, the
collision is governed by the relations valid for two free bodies. The impact velocity of the head, Co , follows from Eqs. 5.12 or 5.13 while the velocity of mass m is zero at the beginning of the collision . Conservation of momentum requires (5.18) in which c'o and
C are the
unknown velocities of the head and foundation, respectively,
after the collision. For these two unknowns, Eq. 5.18 is not sufficient and one more equation is required. This equation is obtained by introducing the coefficient of restitution, k., defined by Newton as the ratio of relative velocity after the impact to relative velocity before the impact, i.e.
c- C'o (5.19) From Eqs. 5.18 and 5.19, the initial velocity of the foundation is:
(5.20)
The coefficient kr depends on the material of the bodies and ranges from 0 for plastic
237
collisions to 1 for perfectly elastic collisions. For hammers, the lowest k, occurs when forging nonferrous materials for which it is close to zero.
For hot forging, kr is about
0.25 but increases as the material gets colder. For cold forging, k, is about 0.5 (Barkan , 1962). Thus, kr = 0.5 may represent the adequate mean value for design purposes. Occasionally, the impact occurs with the sample absent, yielding the highest k, and hardest shock . With the velocity
Ccalculated
from Eq. 5.20 and substituted into Eq. 5.15, the
complete response is determined. A few examples of the response are plotted in Figs. 5.8 and 5.9. A foundation without an anvil pad modeled according to Fig. 5.7a is assumed to be supported either by soil (Eqs. 5.6) or by eight timber piles (Eq. 3.11) with all other conditions being the same.
The pile supported foundation exhibits smaller
peak amplitudes, a higher natural frequency and smaller damping than the soil supported (shallow) foundation (Fig. 5.8). If the piles were used in two different types of soil, the stiffer soil would reduce the response amplitudes and increase the natural frequency even more (Fig . 5.9). The maximum (peak) displacement occurs at a time, tm r for which dv/dt = O. This condition yields
1 .JI-D 2 t = -arctan - - -
m ID (v
(5.21)
o
The peak displacement,
v
r
follows from Eq.5.21 for time t m substituted. When the
damping is not very large, tm ;; X T and the peak displacement becomes
"
"
C
v=-e 0)0
... 7!- D 2
(5.22)
238
Figure 5.8: Response of Hammer Foundation Supported (A) directly by soil and (8) by eight piles
"
I
... (
,r ... . IoP ' I,
SOlL
3
A
0.30
o.~o
;;,
"'.J
o
,
Figure 5.9: Response of Pile Supported Hammer Foundations for Two Types of Soil: (A) shear wave velocity V s and (b) shear wave velocity 2Vs
...~
D. '0
.... Z
I
QJ
'" u '" a: -'
a'>
o
"0 o
239
The maximum force transmitted into the ground is
(5.23) and the peak dynamic stress on the soil is
(J
= FlAb where
An the base area . With
piles, the maximum dynamic load on one pile is, on average, F (1) = FIn
(5.24)
where n = the number of piles . If group interaction effects are significant, the load is not distributed evenly and Eq. 5.24 gives the average load. The effect of dynamic stresses on the soil or piles is to be evaluated with respect to fatigue. When the anvil is mounted on an elastic pad, a two mass system should be used to analyze the response .
5.8 RESPONSE OF TWO MASS FOUNDATIONS When the anvil rests on an elastic pad as shown in Figs. 5.2b,c, d or Fig. 5.3 , a hammer foundation should be considered as a two mass system as shown in Fig.5.7c and reproduced in Fig. 5.10a. In this model, rru is the mass of the anvil and m2 the mass of the pad under the footing. k 1 is the stiffness constant of the pad under the anvil calculated by Eq.5.9a and kz is the stiffness of the soil, piles or any other support of the footing block . At first, damping is neglected.
240
Figure 5.10: Hammer foundation as a two masses system and its vibration modes c)
b)
a) ,
¢m
HEAD
o
t
ANVIL kl
BLOCK
mZ
MODE 2
MOOE 1 W\ r - - - - -.,
vI
-+v2
R
I
Wz r il V
r
I
rr::
--,
I
1-- --1 ~
~
l:~~-~~~J ~2'
7;~; Ul/U~77 7/~~/ ::' ~-,t 7
Undamped Vibration If the duration of the collision between the hammerhead (mass ma) and anvil is short relative to the natural periods of the system, the impact of the head will be followed by free vibration of both masses, V1(t) and V2(t). The governing equations of motion follow from Newton's second law in the sense that the product of each mass and its acceleration must be equal to the sum of all forces acting on the mass and thus
m; -
dZv.., (t) -.., =-kzv.., -kt(vt-v..,) dr
With the notation
(5.25) the governing equations of the motion become
241
(5.26)
These are two coupled , homogeneous differential equations of the second order with constant coefficients . Hence, complete solutions for the unknown displacements V1(t) and V2(t) can be written as sums of two independent particular solutions. Because the system is undamped and linear, particular solutions can be expected to be harmonic with an unknown frequency. (OJ , as in the case of free undamped vibration in one degree of freedom, and written as (5.27)
Substitution of Eqs. 5.27 into Eqs. 5.26 gives (5.28a)
(5.28b) These are algebraic homogeneous equations for constants V1 and V2. When the stiffness constants are frequency independent, the solution of Eqs. 5.28 represents the "eigenvalue problem" . A nontrivial solution exists only if the determinant of the coeffi cients of Eqs . 5.24,~ vanishes, i.e. 1
~
=O. This condition
yields two natural frequencies
(529)
242
With the two frequencies (OJ
a= 1,2) substituted into Eqs . 5-28 one at a time, two ratios
of displacements V1 I V2 can be calculated . The ratios represent the undamped vibration modes and are from Eq. 5.28a and b, respectively
for j = 1 or 2
(5.30)
(This relation can be used to check the correctness of (01 , and ~) To distinguish the vibration amplitudes of the two modes, double subscripts are introduced. The first subscript identifies the amplitudes of mass m1 or m2; the second subscript indicates the frequency and mode with which the amplitude Vu is associated . The two ratios, Eq.5 .30, characterize the vibration modes shown in Figs.5.10b and c. In the first mode, the two masses vibrate in phase; in the second mode, the two masses vibrate in antiphase. Because the two natural frequencies are different, Eqs. 5.27 give two different particular solutions and the complete solution to Eq. 5.26 can be written as
(5.31)
Amplitudes Vij are determined from initial conditions which are V1(0)
= 0 , V2(0) =0
(5.32a) (5.32b)
v = dVj(t) I dt
and the initial velocity
c follows from
the basic formula for collision, Eq.
5.20, as
(5.33)
243
where mo, Co are the mass and impact velocity of the hammer head respectively and k, is the coefficient of collision.
Applying the initial conditions and denoting the natural
frequency of the anvil in case of a rigidly supported foundation block
(5.34) the four amplitudes determining the solution described by Eqs. 5.31 are
(5.35a)
(5.35b)
for the anvil and
(5.35c)
(5.35d)
for the foundation. The motion of each mass has two harmonic components whose amplitudes are in the ratios given by Eq. 5.30 or Eqs. 5.35. For a foundation with a rather stiff anvil pad , an example of the response of both the anvil and the foundation is shown in Fig. 5.11a. The anvil oscillates about the instantaneous position of the foundation block. For a softer anvil pad , the contribution of the second vibration mode is much more significant as can be seen in Fig. 5.12a. (The calculation is given later in an example)
244
Figure 5.11: Response of Two Mass Hammer Foundations with Stiff Anvil Pads: (a) undamped and (b) damped (Dl
=10%, D2 = 5%) (amplitude in inches)
10
I.
FOurH.lATtON
5
~
.
~
:50
zo
~
100
~
~
-e;
~5
{a) - 10
..... 5
-
~
~ o
~
20
(b)
60
100
In( l-lE-ll
""'-5
Figure 5.12: Response of Two Mass Hammer Foundations with Elastic Anvil Pad: (a) undamped and (b) damped (D1 =51% and D2
=7.47%); (amplitude in inches)
10
.
5
~
~ ~o ~
"
~
-5
(e) -10
:r.
5
~o ..... ~
-5
(b)
245
Because
(;)2
is usually much greater than
(;)1
the peak displacement of the foundation is
approximately 5,36) and the amplitude of the total motion of the anvil (5.37) However, the stress in the pad does not depend on the absolute magnitude of the motion but only on the relative displacement between the anvil and the block. As Fig. 5.12a- suggests, the relative peak displacement of the anvil is (5.38) The amplitudes
Vij
are given by Eqs. 5.35, The peak displacements given by Eqs. 5.36
to 5.38, represent the upper bound because damping, neglected thus far, reduces the amplitudes as is shown in the next paragraph.
Damped Vibration In the two mass model, damping is defined by the constants
C1
and
C2
(Fig.
5.10a) . It could be introduced into the governing equations of the motion but the accurate solution gets more complicated. It is more convenient to predict the damped response approximately using the notions of modal analysis and modal damping outlined for structures by Novak (1974b) . The foundation response comprises the two vibration modes shown in Fig. 5.10. The damping ratio associated with vibration in each of these modes can be evaluated by means of an energy consideration if it is assumed that the damped mode is
246
approximately the same as the undamped mode. This consideration yields the damping ratio associated with the foundation vibration in the jth mode (Novak, 1983)
(5.39)
in which the generalized mass of mode j (5.40) where j =1, 2. In Eq. 5.39, the damping constant of the anvil, 5.9 and the damping constant of the foundation ,
C2,
C1,
is obtained from Eqs,
from Eqs. 5.6b. The frequencies
appearing in these equations are the natural frequencies
(:) 1
5.29. Thus, for each natural frequency (mode), one set of
and
C1, C2
(:)2
calculated from Eq.
may be necessary due
to converting a constant hysteretic damping to equivalent viscous damping. amplitudes
V1j
and
V2j
The
are the. undamped amplitudes given by Eqs . 5.35. Alternatively,
arbitrary modal amplitudes complying with Eq. 5.30 may be used in Eq. 5.39; in this case, one amplitude can be chosen for each mode, e.g . V1j = 1, and the other calculated using the ratio aj. If the frequency CO2 and
V22
«
V12
»(:)1
which can be the case with a hard anvil pad, then
V11 ~ V21
and
Consequently, Eq. 5.39 simplifies to approximate expressions for modal damping ratios,
(5.41)
247
in which
~p = the damping ratio of the pad.
Realizing that Eqs.5.31 represent the superposition of vibration modes, the damped vibration of the anvil and the foundation block can be written as
(5.42)
in which
V1j
and
V2j
are the undamped amplitudes established from Eqs . 5.35 and the
damping ratios Dj are given by either Eq. 5.39 or Eq. 5.41. The damped natural frequencies
An example of the damped oscillation described by Eq. 5.42 is plotted in Figs. 5.11b and 5.12b . The damped response shown pertains to the same foundation as were used to exemplify the undamped response. The only difference is the inclusion of damping . Even the modest damping incorporated smooths the response quickly, eliminating the second harmonic component in the case of a stiff anvil pad but not with a soft pad. The peak displacements are approximately
and occur at time
1 ~1-D12 t m, =-arctan--- D CUj
(5.44a)
1
where
248
(5.44b) The peak relative displacement of the anvil, determining the stress in the anvil pad, is approximately
--~~;2 -\
~(VI2 + h20~
(5.45)
The total peak force transmitted into the ground (or piles) comprises the restoring force and the damping force and, with respect to the 90 degree phase shift between them is approximately
,
LFF _.tr~"
)
2 v }
~k 2
~+
V2j
/ v(
<:'(
( , . \ \ r .v
( '
( ,
, ) 2
CzOJ}
~ ~l
,~
(5.46)
/
--
in which
9S -\l ~' ,
are the modal contributions to V2 obtained from Eq. 5.42 or 5.43; thus, V2
:=
V21 + V22 . The force acting on the pad is (5.47)
More Complicated Systems.
The above approach can readily be extended to include more complicated systems shown in Fig. 5.7d .
Complex Eigenvalue Approach
The damping ratios, Dj obtained from the energy consideration can be verified by introducing the damping constants,
c
into the governing equations of the motion, Eq.
5.26, and solving the complex eigenvalue problem. The modal damping ratios calcu
249
lated using this mathematically accurate procedure are practically identical
with those
calculated from Eq. 5.39. This agreement indicates that the energy consideration yields the same damping as the complex eigenvalue approach. (More details on the complex eigenvalue approach may be found in Novak and EI Hifnawy, 1983.)
5.9 PRELIMINARY TWO-STEP ANALYSIS For a preliminary check of the design, a very simple two-step approach can be used . This approach is based on the assumption that the response of a two mass foundation can be evaluated approximately as a sequence of two collisions: the collision between the head and the anvil and the collision between the anvil and the footing block. After each collision, the amplitudes of the anvil and foundation block are established using the formulae for one degree of freedom. The underlying assumption is that the duration of the collisions is short compared to the natural periods involved . The impact of the head yields the initial velocity of the anvil given by Eq. 5.33,
(5.48)
and the peak, undamped displacement of the anvil, as in Eq. 5.22 "
va = -
" C
OJ a
(5.49)
in which the frequency of the anvil
(5.50)
The second collision results in the initial velocity of the foundation
250
(5.51 )
in which the collision coefficient kr reflects the behavior of the pad and may be taken as approximately 0.6. The undamped amplitude of the foundation is
(5.52)
in which the frequency of the foundation
(5.53)
In this approach, damping is usually ignored but could be included as in one degree of freedom using Eq. 5.41 for damping and Eq. 5.15 for amplitudes. The stress in the soil and the anvil pad is obtained using Eqs. 5.23 and 5.47 , usually with the omission of damping. This simple approach, proposed by Rausch
(1950), gives a reasonable estimate of the response .
Mass of Hammer Head. In the above analysis, mass rn. is the mass of the anvil. Depending on the type of hammer, restitution coefficient and mode of operation mass ma can remain in contact with the anvil; this is more likely to occur at very low restitution coefficients . In such a case, mass rn- should be replaced by (ma + rn.) in all the above formulae. The practical difference is not great since rna « rn, « m2.
251
5.10 ECCENTRICITY OF THE IMPACT If the blow of the hammer head acts with an eccentricity. e, as indicated in Fig. lb, the blow produces an initial angular velocity of the anvil, \jf. in addition to its initial I'
velocity, C . Conservation of momentum requires (5.54a) for the vertical translation and (5.54b) for rotation where I is the mass moment of inertia of the anvil. The third equation needed can be written using Newton's definition of the restitution coefficient,
kr.
In this
case, this definition includes the contribution of the angular velocity to the relative velocity after the collision,
C+ e\jf - c'a,
yielding
(5.54c)
From Eqs. 5.54 the initial velocity of the anvil becomes
(5.55)
and the initial angular velocity of the anvil
(5.56)
in which the square of the radius of gyration of the anvil i2 1
= 1/ rn-. For a centric blow e 252
= 0, Eq. 5.55 reduces to Eq. 5.33 and 'Jf = O. For a one mass foundation (Fig. 7a), mass
rn- would be replaced by the total mass m in the above formulae. If only-vertical motion and rotation are considered, i e. two degrees of freedom , the analysis of a one mass system is mathematically almost identical with the analysis of vertical vibration of the two mass system outlined above. With more degrees of freedom, the analysis is more complicated and is conducted most efficiently in terms of the complex eigenvalue approach (Novak and EI Hifnawy, 1983). An example of an eccentrically arranged two mass foundation is shown in Fig. 5.13. The foundation is the same as the one used in the numerical example later herein, except for the eccentricity of the anvil. Horizontal translations U1, U2 and rocking 'Jf1, 'Jf2 have to be considered in addition to the vertical translations
V1, V2.
Consequently, the
two mass foundation has six degrees of freedom. The response , obtained by means of the complex eigenvalue analysis as described in Novak and EI Hifnawy (1983) , is shown in Fig. 5.14. Depending on conditions, the horizontal translation and rocking can be quite significant.
253
Figure 5.13: Two Mass Foundation with Eccentrically Mounted Anvil
/
T
00
It)r
roN
o
Nit)
14:T~z
~L
Vz
- - -;/7777777/ /7"7'"',." "''6 .56 .~L.
J
•
FT Lrn)
.- -'--
----'--
Ir - -'"
(2.00)
' - - - -
I
" '"
6.56 (2.0 0 )
~"' " -, I
3 .28
I
( 1.00 )
16 .40
( 5.00 )
-- ----_..- -
254
Figure 5.14: Three Components of Response for the eccentric foundation shown in Fig 5.13 (amplitudes in inches, rocking in radians, time in seconds)
.
o
.
U)
C\l l
-
W
fj"0-+....L...._~ h------+-----.'~---'i P-rd~~ =;;::-::t=::::;;;:"\I17. l (
-< • U'>
J
.
4u.
{a) VERTICAL
HORIZONTAL
.
= N I. W •
_U'>
.
U)
I
. I c ) ROCKING
255
5.11 RESPONSE FOR KNOWN IMPACT FORCE If the time history of the impact force, P(t), is known, the response can be predicted accurately using those methods of structural dynamics that are suitable for transient loads . The following methods are well-suited to this end: The Duhanel integral in combination with modal analysis and numerical integration , fast Fourier transform , and direct integration of the governing differential equations such as the Wilson method.
e
All these methods presume the use of the computer and are described in
Clough and Penzien (1975).
A special Fourier analysis is described in Lysmer and
Richart (1966) . The complex eigenvalue approach is particularly efficient as described by EI Hifnawy and Novak (1984). All these methods work very well.
However, their practical usefulness in the
hammer foundation analysis is limited by the lack of reliable information on the time history of the impulse force , its inevitable randomness and variability. Finally, because the initial velocity approach is bound to give the upper bound estimate, it seems quite sufficient to use this method for design purposes .
5.12 STRUCTURAL DESIGN The moments and shear forces generated in the foundation block also must be evaluated. The static forces and moments follow from the equilibrium between the gravity loads and soil resistance as in usual foundation design. The dynamic forces are complex but can be approximately evaluated from the
256
equilibrium between the impact force passing through the pad, Fa, and the inertia forces , Pi, acting on the mass of the footing and distributed in proportion to its mass (Fig. 5.15). For the geometry depicted in Fig. 5.15 , the equilibrium is
The forces Pi and F I 2 result in shear forces and moments in the planes I and II which can be readily evaluated . Special attention should be paid to shear reinforcement in the cross section I and horizontal reinforcement under the anvil. The latter reinforcement is needed because of the horizontal tensile forces resulting from the concentrated pressure of the anvil. When the shear stress is high, the shear reinforcement is sometimes designed for inertia forces acting not only up but also down in anticipation of stress reversal. The basic reinforcement is shown in Fig. 5.16.
257
Figure 5.15: Inertia Forces Acting on Block
ri
Figure 5.16: Reinforcement of Foundation
258
5.13 EXAMPLE: HAMMER FOUNDATION The response plotted in Fig.. 5.12 was calculated for a hammer foundation shown in Fig. 5.17 and characterized by these data :
1) Hammer Weight of head (tup, Go)
3000 lbs (1361 kg)
Weight of anvil (G 1)
60000 lbs (27216 kg)
Weight of frame mounted on
the block (G F)
50000 lbs (22680 kg)
Impact velocity of head (eo)
21.33 ft Is (6.5 m I s)
Frequency of blows
40 to 90 c/min
Coefficient of restitution (k.)
0.5
2. Soil (a deep layer of coarse sand with gravel)
Shear wave velocity (vs )
500 ft Is (152.4 m Is)
Unit weight (y)
120 Ib/ft 3 (1922 kg/m 3 )
Poisson's ratio
0.25
Material damping (tano)
0.10
Depth to bedrock
100 ft (30.48 m)
Allowable stress
6000 Ib I ft2 (0.287 MPa)
3. Backfill Shear modulus (G s )
0.5G
259
Unit weight (Ys)
0.75y
Poisson's ratio
0.25
4. Pad Under Anvil (hard felt)
Thickness (d)
6 in (0.1524 m)
Young's modulus (Ep)
15000 psi (103,43 MPa)
Pad dimensions
6.56 ft x 4.92 ft (2m x L5m)
Area (Ap)
32.28 ft2 (3.00m2)
Material damping (tanop)
0.10
Allowable stress
500 psi (3,45 MPa)
5.Foundation Block (reinforced concrete)
Unit weight (y)
150 Ib / ft3 (2400 kg 1m 3 )
Base dimensions
16,4ftx 13.12 ft (5m X 4m)
Base area (A2)
215.57 ft2 (20m 2)
Total depth
8.2 ft (2.5m)
260
Figure 5.17: Hammer Foundation
/
l(")
.
-,-ro -
0
0
N
in
a)
N
,
r
St tiT .1
PA D
0'
0
N
m
~
l.O
-
'-"
FT (rri)
I~
4 .92 {1.50}
r------
I
·i•
6.56 (2.00)
16.40 (5 .00 )
~
I
I-
4 .92
{l.50}
1 ~I
ot[)
-N '<;1"'':::
ol.O
-N
.q: ....:.
261
Solution MASSES Mass of the anvil: rn- ::: G1 I g::: 60000 132.2 = 1863.35 Ib2 1ft Weight of the block: Gb
=150 (16.4 x 13.12 x 8.2 - 6.56 x 4.92 x 3.28) =2487801bs
Check of the weight using Eq. 5.3:
Gb
j2 =75Go(~J2 =75X3000(21.33 =3030001b 18.37 ) Cr
The weight of the block seems adequate. Total foundation weight G t ::: Go + G1 + Gf + Gb = 3.0 + 60.0 + 50.0 + 248 .78 :: 361.78 Mass of the foundation block: mb:::
G b Ig
=248780 132.2 ::: 7726 .0 Ib2 1ft
Mass of the frame: mf::: 50000/32.2 ::: 1552.8 Ib2 1ft Mass of the block with the frame: m2 :::
mb + m, ::: 7726 .0 +1552 .8 = 9278 .8 Ib2 1ft
STIFFNESS AND DAMPING CONSTANTS Stiffness of the anvil pad (Eq. 5.9a):
k1 ::: kp ::: Ep Ap I d ::: 15000 x 144 x 32.281 0.5 = 1.395 x 1081b 1ft Stiffness of the embedded foundation (Eq. 5.6a)
262
Equivalent radius (Eq. 5.5)
Yo
=
~~ = -!16.4x13.12/1r = 8.28ft
Soil shear modulus
G = v; xp = 500 2 x120/32.2 = 9.317xl0 51b/ ft2 Embedment: Assume separation s = 3 ft (Fig. 5.6) Effective embedment 1 = 8.20 - 3.0 = 5.20 ft Embedment ratio Ilro
= 5.20 I 8.28
= 0.63
Stiffness parameters from Table 3 C, = 5.2, S1 = 2.7
Stiffness constant of the foundation (Eq. 5.6a)
= 9.317x1 05 x 8.28 (5.2+0 .5 x 0.63 x 2.7)
= 4.668x1 07 Ib I ft
Damping of the foundation (Eq. 5.6b)
From Table 3 the theoretical constant C 2
=5.0. With the correction factor taken for
radiation damping as 0.5, the corrected value C 2 = 2.5.
Constant S
2
= 6.7.
263
The foundation damping coefficient (Eq. 5.6b)
= 8.28 2.J120/32.2 x 9.317 x 105 (2.5 + 6.7 x 0.63)0.75 x 0.5) = 6.496x105 Ib / ft
PRELIMINARY TWO-STEP ANALYSIS
Stress in the anvil pad:
Consider collision between the head and the anvil. Initial velocity of the anvil (Eq. 5.48)
3000
= (1+0.5) [3000 + 60000 ] 21.33
= 1.52 ft / s Natural -frequency of the anvil with rigid soil (Eq. 5.50)
m~ =~= 1.395x10
».
8
=7.4865x10 4
1863.35
The approximate value of the maximum relative displacement of the anvil (Eq. 5.49)
Va =
c= OJ a
1.52 =5.555x10-3 }1=0.067 in 273.61 264
Dynamic force on the pad
Dynamic stress on the pad 5
a-
= p
Fp
=
Ap
With fatigue factor !-t
a-~
:=
7.75 X 10 = 2.40 X 10 4 lb / ft2
32.28 .
3
= 3 X 166.72 = 500.15 psi
Static stress on the pad
- - ~-
a -
60000
- 32.28/
Ap
_ 12 91 . - . pSI
/ 144
Total peak stress on the pad
Up =U p +a-~ =12 ..91+500.125=513.06psiOK
Foundation block amplitudes and stress on soil
Response of the block is considered as a collision between the anvil and the-block with
kr = 0.6.
Initial velocity of the block (Eq. 5.51)
= (1 + 0.6)
1863.35 1.52 = 0.4067 ft / S
1863.35 + 9278.8 .
265
Natural frequency of the foundation (Eq. 5.53) 7
k2
(m1 + m 2 )
-
4.668x10 = 64.73s-1 1863.35 + 9278.8
Peak amplitude of the block (Eq. 5.52)
6283 x l 0 - 3 fit = 0 .07 54 in
v" 2 = -(;2 = 0.4067 =. lUI 64.73
This exceeds the recommended limit of 0.05 in (Table 5.1) but may be reduced due to damping . Dynamic force on soil
Dynamic stress on soil 5
8 2 = F2 = 2.933x10 = 1361.11b/ ft2 Az
215.17
With a fatigue factor ~
:=
3,
8~ =3x1363.1 =4089.4lb/ ft
2
Static stress on soil
(]' =~= 361780 =1681.4Ib/ fl2 A2
215.17
Total stress on soil
266
== (52 +
(52
0-; == 1681.4 + 4089.4 == 5770.81b / ft2
< 6000 lb Iff OK
All stresses and amplitudes appear to be adequate. The design need not be changed at this stage and the more detailed analysis can be started.
DETAILED ANALYSIS Natural frequencies: The stiffness constants of the coupled system (eq. 5.25) are k 11 :: k 1 :: 1.395x 10 8 lb 1ft
:: 1.8618x1 08 Ib I ft Masses: rn, ::
1863.35 Ib S2 1ft
m2::
9278.80 Ib S2 1ft
Natural frequencies (Eq. 5.29) are
267
Undamped Amplitudes (Eqs. 5.35) Initial anvil velocity
c = 1.52 ft I s, and w; = 7.4865x104 s' 2 "
V11
= 0.0520 in
V12
= 0.0494 in
V21
= 0.0494 in
V22
= -0.0105 in
?
The peak undamped displacement of the anvil (Eq . 5.37) is
'Or
= VII + VI2 = 0.0520 + 0.0494 = 0.1014 in
and for the foundation (Eq. 5.36)
V2 = V21 + IV22 I= 0.0491 + 0.0105 = 0.0596 in ,.., 0.06 in These values exceed the recommended limits listed in Table 5.1 but the actual values will be reduced due to damping . With the undamped amplitudes available , the complete response can be calculated using Eqs.5.31. The undamped response is plotted in Fig.5.12a. However, the response is affected by damping.
Damping Ratios The damping constant of the foundation C2
= 7.4708x10 5 Ib I ft, for j = 1,2
The damping of the anvil is (Eq. 5.9b) Clj
C11
=tan8p k1 I Wj, j = 1,2
=0.10 x 1.395x10 8 I 64.41 =2.165x10 5Ib 1ft 268
C12
=0.10 X 1.395x108 /301.30 =0.4630x10 5 Ib 1ft
Generalized mass (Eq. 5.41) M 1 = 1863.35
X
0.052 2 + 9278.8 X 0.0491 2
M 2 = 1863.35
X
0.0494 2 + 9278.8
X
=27.408
0.01052 = 5.570
Damping Ratios (Eq. 5.39) 01 = (2 x 64.41 x 27.408r 1[2.1658 (0.0520 - 0.0491)2 + 6.496
X
0.0491
2]
5
x 10
= 0.44 = 44% 02 = (2 x 301.3 x5.57)-1[0.4630 (0.0494 - 0.0105)2 + 6.496 X 0.0105 2] x 10
5
::: 0.070 = 7% Check of the modal damping using the approximate Eqs. 5.41:
D, and with C1
D2
=
5
6.496 x 10 = 0.45 7 2.J4.668 xl 0 x 11142.15
= 45%
= C12
=
5
0.463 x 10 = 0.0454 = 4.540/0
8 2J1.3596xl0 x1863.35
With the damping established the total response can be calculated from Eq.5.42. For both the anvil and the foundation , the damped response is plotted in Fig.5-12b. It can be seen that damping has a significant effect on the magnitude and character of the response. The peak displacements can be established from Fig. 5.2b or calculated separately. The character of the response plotted in Fig. 5.12 differs from that shown in Figure 5.11. The difference is primarily due to the difference in the stiffness of the anvil
269
pad.
Figure
5.11 represents the response typical of a stiff pad white
~ ig .
5.12
corresponds to a moderately soft pad. With the soft pad, the contribution of the second mode is greater and the time history of the response is more irregular. The peak displacements, if not established from the time history, have to be established considering both vibration modes for the footing as well as the anvil.
Amplitudes of damped vibration The damped frequency
The peak displacement of the foundation occurs at the time (Eq. 5.44a)
1 ~1- D I _. -arctan
2
t
= m
f
(VI
1 tan ---arc 57.916
DI
.J1- 0.5106
2
0.4425
0 019 S =.
The peak displacement of the foundation is (Eq. 5.43) "-
V 2 == 0.0491 exp(-0.5106 x 64.41 x 0.0187) sin(55.38 x 0.0187) +
0.0105 exp(-0.074 x 301.3 x 0.0187) '"
V2
=0.0228 + 0.0069 =0.0297 ~ 0.03 in.
<0.05 in. OK
Without damping the amplitude was 0.06 in. The peak displacement of the anvil
270
"-
Vj
= 0.0228 x 0.052/ 0.491 + 0.0069 x 0.0494 /0.0105
"
VI = 0.0241 + 0.0325 = 0.0566 ~ 0.06 in.
<0.08 in. OK
Without damping the amplitude was 0.10 in. With damping included, the amplitudes are within acceptable limits. The peak relative displacement of the anvil taken as approximately equal to the first amplitude of the anvil response in the second mode would be (Eq. 5.45)
Va
=
(O.0494+0.010S)e
~ O. 074 2
=O.OS3in.
This amplitude of the relative displacement of the anvil is smaller than 0.08 in allowable according to Table 5.1 . This maximum is also smaller than 0.067 in obtained in the pre liminary analysis. The force transmitted to the ground by the first harmonic component is (Eq.5 .46)
F?
-
= 0.03
12
~(4 .668x107)2+(6.485xI05x57.916)2 ·
.
= 1.627 x 105 Ib This is less than the value of 2.933x10 5 found in the preliminary analysis. The contribution from damping is significant. Comparison of these results with those obtained using the approximate, preliminary analysis indicates that the preliminary analysis gives very good estimates . Thus, the preliminary calculation of stresses need not be repeated.
27]
REFERENCES Barkan , D.D. (1962). "Dynamics of Bases and Foundations ," McGraw-Hili Book Co., lnc., Chapter 5, pp. 185-241. Clough, R.W. and Penzien, J. (1975) . "Dynamics of Structures," McGraw-Hili, p. 634. EI Hifnawy, L. and Novak, M. (1984). "Response of Hammer Foundations to Pulse Loading," Int. J. Soil Dynamics & Earthquake Engrg., Vol. 3, No.3, pp. 124@132 . GERB, "Vibration-isolation Systems," Published by GERB, 1000 Berlin 51, P. 91. Klein, A.M. and Crockett, J.H.A. (1953). "Design and Construction of a Fully Vibration Controlled Forging Hammer Foundation ," Journal of the American Concrete Institute, January, pp. 421-444. Lysmer, J. and Richart, F.E. (1966) . "Dynamic Response of Footings to Vertical Loading," Journal of Soil Mechanics and Foundation Division, ASCE , Vol. 92, SMP, pp. 65-91. Major, A. (1962). "Vibration Analysis and Design of Foundations for Machines and Turbines," Collet's Holdings Limited, London, Chapters XII and XIII, pp. 221-269. Novak , M. (1983). "Foundations for Shock-Producing Machines ," Canadian Geotechnical Journal, Vol. 20, No.1, pp. 141-158. Novak, M. and Sheta, M. (1980). "Approximate Approach to Contact Problems of Piles," Proc. of Geotechnical Engineering Division ASCE National Convention "Dynamic Response of Pile Foundations : Analytical Aspects ," October, Florida, pp. 53-79. Novak, M. and EI Hifnawy, L. (1983). "Vibration of Hammer Foundations ," International Journal of Soil Dynamics and Earthquake Engineering , Vol. 2, No.1, pp. 43-53 . Prakash, S. (1981) . "Soil Dynamics ," McGraw-Hill Book Co., p. 426. Rausch , E. (1950). "Maschinen Fundamente ," VDI-Verlag, Dusseldorf, (in German), Chapter 6, pp. 107-232. Richart, F.E., Hall, J.R. and Woods, RD. (1970). "Vibrations of Soils and Foundations," Prentice-Hall, lnc., p. 414. Srinivasulu, P. and Vaidyanathan, C.V. (1976). "Handbook of Machine Foundations," Tata McGraw-Hili Pub!. Co. Ltd., New Delhi, Chapter 4, pp . 103-134 .
272
6.1 DAMAGE AND DIS TURBANCE
Vibration damage and disturbance can be categorized into the following main groups.
6.1.1 Operational Disturbance in the Technical Process Disturbances in technical processes due to vibration may cause deterioration in product quality and/or a reduction in production capacity. The clients will reject lower quality products, which translates into economic losses and may ultimately lead to a total loss of the market.
6.1.2 Damage to Machinery Damage to machinery can result in stoppages that may cause large economic losses.
The extent of the damage depends on the duration of the stoppage, the
importance of the machine for the whole operation and on the size and output of the industrial plant. The disruption of operations could result in daily losses in the order of hundreds of thousands of dollars in a manufacturing facility, a mining operation, a large sawmill, a large paper making machine or a few millions of dollars in a nuclear power plant.
6.1.3 Personal Discomfort Human susceptibility to vibration is related to vibration velocity. As the vibration velocity increases, the level of discomfort increases. The working environment could be annoying or even painful at higher vibration levels. Consequently, the work is affected
274
and in some situations becomes impossible. Unpleasant working environment results in reduced productivity and sometimes health hazards for the workers.
6.1.4 Damage to the Ground, Foundations and Buildings Some foundations are built on soils susceptible to settlement, such as saturated sands, which can be a serious problem. Large settlements and/or differential settlement may lead to malfunctioning of the machine and/or increased wear of machine components. These conditions may result in lower quality products an/or increased maintenance cost. Dynamic stresses due to machine vibrations are repetitive in nature and can cause fatigue. This can affect the foundation elements and/or the buildings . Vibration induced cracks could cost a lot of money over the years in repairing concrete structures and foundations. Operation disruptions and production losses could also accompany these repair costs.
6.1.5 Disturbance Outside the Industrial Area Vibration disturbance outside the industrial area can result in costly and drawn out disputes with people living nearby and with the environmental health authorities. It may also affect the quality and productivity of nearby precise operations in the same facility.
6.2 PROBLEM ASSESSMENT AND EVALUA TION The fundamental principle in analyzing a vibration problem is to define the vibration system concerned and to find the combination of measures that will produce
275
the desired result at the lowest cost. The main approach is to define and evaluate the relationship between force and motion. The approach is simple but the number of parameters is large and they often difficult to determine.
6.2.1 Evaluation of System Parameters
The dynamic forcing function is not generally known . As discussed previously, different machines produce different types of dynamic loading , and often the manufacturer does not define these loads. Also, the material properties in existing installations are often insufficiently known. This means that even if vibration amplitudes (displacement, velocity or acceleration) can be measured, it is generally extremely difficult to find theoretical solutions.
It is therefore necessary to employ a practical
procedure consisting of a suitable combination of measurements and calculations, the general method being adapted to the problem encountered in each individual case . The method can be illustrated with the aid of the vibration model of one degree of freedom . The complex relationship between mass, stiffness, damping and the motion of the system shows that the task of finding suitable remedial measures in conjunction with harmful vibrations is not amenable to an unambiguous solution. Figure 6.1 shows the components of the simplified one degree of freedom model.
ill
k Figure 6.1
276
The governing equation of motion for that simple dynamic system is given by
kw+ cw+ mw = pet)
(6.1)
It can be seen easily that the dynamic response of this system can be altered by the modification of one or more terms in the equation of motion of the model , i.e. by the modification of: elastic forces (spring force, kw) damping forces (dash pot force, cw) inertia forces (mass inertia force, mw) disturbing forces (dynamic forcing function, P(t)) Each one of the above modifications can be induced in different ways, depending on the kind of machine, foundation and substructure. For instance; 1.
Increasing the surface area and/or the mass of the foundation can alter the spring force and the damping force . This alters the dynamic properties of the substructure (improving the soil or increasing the number of piles), connecting the foundation to some other foundation or improving the stiffness of the foundation block using grouting and/or post-tensioning. The dynamic stiffness is primarily important in the case of undertuned foundations.
2.
Increasing or decreasing the mass of the foundation block can alter the mass inertia forces. This applies primarily to overtuned foundations.
3.
The disturbing force can be altered by balancing rotating machines, balancing mass forces in machines with crankshafts mechanisms, changing the speed of revolution, etc.
It must be noted that the task is complicated by the fact that alteration of one parameter
277
also causes a change of other parameters in an unfavourable direction,
which can
reduce the overall effect of a primary remedial measure.
6.2.2 Procedure for Evaluation of System Parameters The main steps followed to evaluate the system parameters are outlined herein. 1.
Evaluate the soil dynamic properties. If measured properties were not available from soil investigations, then empirical correlation relationships can be used to estimate the dynamic shear modulus (or shear wave velocity) and soil material damping ratio .
2.
The spring constants for vertical, horizontal and rocking motion of the foundation can be estimated using the dynamic soil properties and the techniques described in Chapters 2 and 3. Alternatively, the measured natural frequency (from vibration measurements) and the estimated mass of the foundation plus the machine can be used to evaluate the spring constant.
As an example, the equivalent spring
constant, kw, in the vertical direction , can be evaluated by
(6.2) where m 3.
=mass of foundation plus machine , and f =measured natural frequency.
Evaluate the dynamic forcing function. This step depends on the type of machine in question. For centrifugal and reciprocating machines, the unbalanced force can be estimated from the dynamic balancing process. For impact producing machines or any other machinery that cause transient or random force, the force is backfigured from vibration measurements. In this procedure, the displacement, velocity or acceleration (normally velocity or acceleration) is measured. The
278
measured time history is digitized and using a recurrence type numerical integration and or differentiation procedure, other vibration quantities are obtained (e.g. if velocity time history is measured , then displacement and acceleration time histories are obtained using integration and differentiation of the velocity time history, respectively) . The dynamic forcing function is then evaluated using the single degree of freedom equation, given by Eq. 6. 1. 4.
In order to verify the accuracy of the developed parameters, the obtained forcing function is used to compute the response time history for the same system. The comparison between the measured and computed values should verify the evaluated parameters.
6.3 REMEDIAL PRINCIPLES Various measures can be taken to reduce harmful vibrations. These measures can be categorized in three main groups: 1.
Measures for reducing vibration at the vibration source. These measures rna be divided into the following subgroups: i)
Measures to reduce the disturbing force : correct the alignment and perform dynamic balancing for the machinery, change the speed of the machine.
ii)
Measures to change (reconstruct) the foundation characteristics: active vibration isolation (springs and dampers), modify the size of the foundation block (alter the mass).
iii)
Measures to change the dynamic parameters of the substructure:
279
soil improvement, pile grouting, addition of piles. 2.
Measures for preventing the propagation of vibration: these measures include the provision of expansion joints, the construction of trenches, and the construction of sheet pile walls (or other obstacles in the ground) surrounding the source of disturbance.
3.
Measures for alleviating the harmful effects of vibration on building structures, other machines and components: the objective of these measures is to modify the dynamic characteristics of the affected building. These measures comprise alteration of the mass, damping and stiffness of the bUilding components, analogous to the measures applied to the machine foundations.
6.4 SOURCES OF ERROR Experience gained from cases of damage shows that it is very rare to find a clear explanation in conjunction with vibration damage in industries and power plants . The interaction between different sources of error often necessitates long and complicated investigations and leads to lengthy disputes between the parties involved. Causes of damage and sources of error can be classified as follows: 1.
Faults during procurement.
2.
Faults in design .
3.
Faulty workmanship (structural work, machine installation).
4.
Faults during completion of the project and commissioning.
Because of the complexity of the problem, it is difficult to give general recommendations as to the way that the work should be executed. In principle, it is
280
desirable for the supplier of machinery to specify all dynamic forces and for the structural engineer to state how the different requirements have been complied with. The following summary of the causes of documented cases of damage is therefore given by way of illustration.
6.4.1 Procurement It is very rare for dynamic conditions and requirements related to them to be specified in conjunction with the procurement of: - The services of consultants for design - Contractors - Machinery and equipment - Control during implementation of the project - Checking at the time of commisioning and during the guarantee period . Experience shows at the same time that, in the cases when vibration specifications were drawn up on the basis of actual conditions, the common types of vibration damage or disturbance did not occur. The conclusion that can be drawn from this is that it is the lack of appropriate dynamic specifications in conjunction with procurement which is the fundamental cause of the many defects that result in vibration damage. The absence of regulations and specifications relating to vibrations in documents is one of the reasons for inadequate interaction between the parties involved. These difficulties are reinforced by complicated channels of communication, particularly in cases where machinery is delivered from abroad . It is therefore essential that liability in
281
the event of defects should be regulated by some for m of vibration clause in the tender and contract. The importance of the form of the form of tender, with regard to the number of parties involved in the tendering process etc. must be pointed out. insurance matters should be sorted out between the parties in good time before the contract is finished.
6.4.2 Design In many cases, the primary cause of defects is lack of consideration of dynamic loading or propagation of vibrations. This also applies when new machines are installed in existing industrial plants or where existing machines are repositioned . This failure to take vibrations into consideration is due to various factors , such as: Inadequate data from the machinery supplier. Failure to carry out the appropriate geotechnical investigations. Lack of knowledge, lack of design recommendations, etc. The following faults may occur during planning, design and sizing of the foundation: i)
Choice of the wrong loading model.
ii)
Choice of the wrong analytical model.
iii)
Inadequate or faulty calculations.
iv)
Faulty design of the machine foundation.
v)
Inadequate supporting foundation.
vi)
Choice of the wrong construction material or the wrong grade .
vii)
Choice of the wrong structure and/or defective construction .
282
I
_'. l ev
' I
6.4.3 Implementation of the Project The wrong choice of construction procedures and/or materials, earthworks, and installation of machinery may result in defects. The sources for errors during the implementation include: - The choice of unsuitable construction materials, method of compaction, etc. - The choice of the wrong piling method. - Various kinds of negligence in conjunction with excavation and/or concreting - Defective installation of prefabricated units (tightening or locking of nuts,etc.) - Underfilling with grout (material and/or workmanship) - Anchorage and, if applicable, preloading of retention bolts .
6.4.4 Commissioning the Project Errors made during commissioning the project may result in vibration damage. These errors include: failure to carry out a vibration inspection in conjunction with commissioning Failure to document the relevant vibration conditions Failure to carry out a monitoring survey of the plant at the end of the guarantee period . Inadequate handing -over procedure and division of responsibility between the project and operational organizations.
I r
I
i f
,
~(
A ')
~
V
I
" I
I I
283
'
1
(~
6.4.5 Other Factors Among the other factors that give rise to vibration problems , is the changes in time, or in dynamic parameters and/or the properties of different objects, I.e. the elements in the machine-foundation-supporting foundation-environment system that are of the greatest importance. These changes may, for instance, occur due to: The variation of the ground water table. The effects of vibrations on the properties of the soil, construction materials and isolator materials. The effect of machine wear. Monitoring and regular inspection of foundations subjected to dynamic loading are therefore important. Detection and planned repair of incipient vibration damage is often a precondition for the prevention of costly repairs and expensive breakdowns.
284
D SIGN OF FOUNDATION FOR DYNAMIC LOADS PART II SEISMIC LOADS By: Ayman Shama P.E. Ph.D. PARSONS 100 Broadway New York, NY Ayman.shama@parsons. com
1
BASIC GEOTECHNICAL
EARTHQUAKE
ENGINEERING
PRINCIPLES
2
1.. BASIC GEOTECHNICAL EARTHQUAKE
ENGINEERING PRINCIPLES
1.1 Seismology and Earthquakes According to the simple plate-tectonic theory, the Earth is formed of several layers as shown in Figure1-1 that have very different physical and chemical properties. The crust, which is the outer layer, consists of several, irregularly shaped plates (Tectonic plates) that slide over, under and past each other. An earthquake is the vibration of the Earth's surface that follows the release of energy inside the Earth's crust where the plates meet as a sudden movement of the earth. The crust may first bend and may break if the stress exceeds its strength . Seismic waves are generated during the process of breaking.
1.1.1 Description of Earthquake Location The location of an earthquake can be defined in terms of several parameters. The point from which the wave first emanates as shown in Figure 1-2 is called the earthquake focus or hypocenter, and the point on the ground surface directly above the focus is called the earthquake epicenter. The focal depth of an earthquake is the depth from the Earth's surface to the focus. The location of an earthquake is commonly described by the geographic position of its epicenter and by its focal depth. The distance on the ground surface between a site and the epicenter is known as the epicentral distance, and the distance between a site and the focus is called the focal distance also known as the hypocentral distance. Earthquakes with focal depths from the surface to about 70 kilometers are classified as shallow. Earthquakes with focal depths from 70 to 300 kilometers are classified as intermediate. The focus of deep earthquakes may reach depths of more than 700 kilometers. It should be noted that the focuses of most earthquakes are concentrated in the crust and upper mantle.
1.2 Types of Seismic Waves The two general types of waves producing the shaking during an earthquake are the body waves, which travel through the Earth and the surface waves, which travel along the Earth's surface.
1.2.1 Body Waves Body waves travel through the interior of the Earth. They follow curved paths because of the varying density and composition of the Earth's interior. Body waves transmit the preliminary shaking of a seismic event but have little damaging effect. They are divided into two types: primary (P-waves) and secondary (S-waves}.
3
.,.----(rust
Figure 1-1. Structure of Earth
Epicenter
Ground Surface
Focus or hypocenter
Figure 1-2. Description of Earthq uake Location.
4
1.2.1.1 P waves The first kind of body wave is the P wave or primary wave . This is the fastest kind of seismic wave. P waves generally travel twice as fast as S waves and can travel through any type of material. They travel as shown in Figure 1-3 in a manner as it spreads out; it alternately pushes and pulls the rock in the direction of propagation. P waves are generally felt by humans as a bang or thump. The speed of P waves is determined in terms of the elastic properties of the material they travel through as:
v = [M p
Vp
(1.1)
where pthe mass density of the material and M is is the constrained modulus defined according to the theory of elasticity as: (1.2)
where E is Young's modulus and u is Poisson's ratio. Speed of P waves ranges from 5 km/s to 7 km/s in typical earth's crust.
1.2.1.2 S waves They are also known as secondary, shear, or transverse waves, cause shear deformations as they travel through a material. S waves do not travel as rapidly through the Earth's crust and mantle as do primary waves; they can travel only through solids. During their travel , they displace the ground perpendicularly to the direction of propagation, alternately to one side and then the other as shown in Figure 1-4 so that the motion of an individual soil particle is perpendicular to the direction of S wave travel. The direction of particle movement is used to divide S waves into two components , SV (vertical plane movement) and SH (horizontal plane movement) . The speed of S waves is determined as:
!;-
I V. _
~- ,
=1* 1
(1.3)
·7
where G is the shear modulus of the soil medium, also referred as the initial or maximum value of the shear modulus when the soil deposit is not yet affected by the cyclic earthquake loading. Speed of S waves ranges from 3 km/s to 4 km/s in typical Earth's crust. Seismologists determine approximately the distance from a location to the origin of a seismic wave by taking the difference of arrival time from the P wave to the S wave in seconds and multiply by 8 kilometers per second.
5
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Surface waves result from the interaction between body waves and surficial layers of the earth . They travel more slowly than body waves. They usually have the strongest vibrations and probably cause most of the damage done by earthquakes. There are two types of surface waves : Rayleigh waves and Love waves.
1.2.2.1 Rayleigh waves: They are produced by interaction of P and SV, so they involve both vertical and horizontal particle motion. They are similar, in some respect, to the waves produced by a stone thrown into a lake. The particle motion of a Rayleigh wave is elliptical as shown in Figure 1-5. Also, they are of main interest for tunnel engineers because of their capacity to induce large axial strain under certain conditions. Speed of Rayleigh waves ranges from 2 krn/s to 4.2 km/s in typical Earth's crust
1.2.2.2 Love waves These waves result from the interaction of SH waves with soft layers of the crust surface and have no vertical component. Hence, they cause horizontal shearing of the ground as shown in Figure 1-6. They are usually slightly faster than Rayleigh waves. They are largest at the surface and decrease in amplitude with depth . Speed of Love waves ranges from 2 km/s to 4.4 km/s in typical Earth's crust. r .
1.3 Quantification of Earthquakes
(.
I
The severity of an earthquake can be expressed irlc several ways. The magnitude of an earthquake, usually expressed by the Richter scale, is a measure of the amplitude of the seismic waves. The scale is logarithmic so that a recording of 7, for example, indicates a disturbance with ground motion 10 times as large as a recording of 6. Earthquakes with a Richter value of 6 or more are commonly considered major. The moment magnitude of an earthquake , which is commonly used nowadays, is a measure of the amount of energy released. This scale assigns a magnitude to the earthquake in accordance with its seismic moment as:
M
=:
w
UogMo) -10.7 1.5
(1.4)
where Mw is the moment magnitude, and Mo (dyn-cm) is the seismic moment, a parameter directly related to the size of the earthquake source and quantified as: Mo
=:
JlAD
(1 .5)
where Jl is the rupture stress of the material along the fault, A is the rupture area, and D is the average amount of slip. The seismic moment is a measure of the work done by the earthquake .
7
Rayleigh Wave
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Figure 1-5. Rayleigh wave Propagation
Love Wave
PA
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f.
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Figure 1-6. Love wave Propagation
The intensity, as expressed by the Modified Mercalli scale, is a subjective measure of damage to works of man, and of human reaction to the seismic event. The Modified Mercalli Scale expresses the intensity of an earthquake's effects at a certain location in values ranging from I to XU. The most commonly used adaptation covers the range of intensity from the condition of "l -- Not felt except by a very few under especially favorable conditions," to "XII -- Damage total". Evaluation of earthquake intenslty can be made only after eyewitness reports and results of field investigations are studied and interpreted.
1.4 Sources of Seismic Hazard 1.4.1 Plate Boundaries The tectonic plate boundaries as shown in Figure 1-7 are classified into three basic types. Their characteristics affect the nature of earthquake that occurs along them. 1.4.1.1 Spreading Ridge Boundaries: In certain areas, plates move away from each other and hot, molten rock rises and cools adding new material to fill the gap between the spreading plates. Most spreading zones are found in oceans; for example, the North American and Eurasian plates are spreading apart along the mid-Atlantic ridge. Spreading zones usually have earthquakes at shallow depths (within 30 kilometers of the surface). 1.4.1.2 Subduction zone Boundaries: Sometimes one plate overrides, or subducts , another, pushing it downward into the mantle layer below the crust. An example of a subduction-zone plate boundary is found along the northwest coast of the United States, western Canada , and southern Alaska and the Aleutian Islands. Subduction zones are characterized by deep earthquakes . lo ',-rd (c'; c', - ~ Cl ,.,' ,:""'o-€ / 1.4.1.3 Transform fault Boundaries: ,""
-J
Transform faults are found where plates slide past one another without creating new material or overriding each other. The San Andreas fault, along the coast of California and northwestern Mexico is an example of a transform-fault plate boundary. Earthquakes at transform faults tend to happen at shallow depths and form fairly straight linear patterns. 1.4.2 Shallow Crustal Sources
r
ShaHow faults may exist in the form of fractures in the earth's crust in which the rock on one side of the fracture has measurable movement in relation to the rock on the other side. They may extend from the ground surface to depths of several tens of kilometers. The presence of a shallow fault does not necessarily mean that an earthquake can be expected. On the other hand, an inactive fault in an area does not guarantee that this area is immune of earthquakes.
9
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na nhcuakes
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ANTARcnc PLATE
...
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n.
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Voicanic centers (hot spots )
:
zone
..l..6...A. Subduction
FIGURE 1-7. Major Tectonic Plates
Piate boundary
Ridge axis
Direction of plate movemem
ANTARCTIC PLATE
Transform fault
Bouvet Is,
9
The Seattle fault, Puget Sound fault, and Tacoma Fault in Western United States are some examples of the shallow crustal sources. 1.4.3 Intraplate Sources Earthquakes can also occur within plates, although plate-boundary earthquakes are much more common . As the subducting plate overrides another , stresses and physical changes in the subducting plate may produce large devastating earthquakes. The New Madrid earthquakes of 1811-1812 and the 1886 Charleston earthquake occurred within the North American plate. Other examples are the 1949 Olympia and 1965 Seattle-Tacoma events. 1.4.4 Other Sou rces In addition to rupture of the rock at plate boundaries and faults, there are other sources of seismic activity that produce smaller earthquakes . These sources include shallow earthquakes associated with volcanic activities, seismic vibrations produced by detonation of chemical or nuclear explosives, and '. J r/'(/.. ~ reservoir induced earthquakes.~ e 'r r~ c. '1 .UJ . ( i
/0
1.5 Characteristics of Earthquake Ground Motions 1.5.1 Ground motion measurement Ground motions during earthquakes are usually measured by instruments called accelerometers. An accelerometer is an electronic transducer, which produce an output voltage proportional to acceleration. The relationship between the time and acceleration through the duration of an earthquake is called an accelerogram. Accelerometers can measure the three orthogonal components of ground accelerations (two in the horizontal direction and one in the vertical direction). These instruments may be located on free-field or mounted in structures . Ground motion accelerograms are usually corrected to remove any errors associated with digitization . The process of digitization, correction, and processing of accelerograms is usually carried out by the United States Geological Survey (USGS), or by California Strong Motion Instrumentation Program (CSMIP). A typical corrected accelerogram and the integrated velocity and displacement for the Northridge earthquake recorded at the ground level of Fire Station 108 in Los Angeles California are shown in Figure 1-8. . 1.5.2 Ground Motion Characteristics
Earthquake ground motions are usually characterized by its peak values, duration, frequency content, Arias intensity, and root-mean-square of acceleration. These parameters control the extent of damage a structure may undergo during an earthquake. These parameters are briefly discussed below: 1.5.2.1 Peak Ground motion Values
Because of their natural relationship to inertial forces, peak horizontal accelerations (PHGA) are the most commonly measure of the amplitude of ground motions. Peak vertical ground acceleration (PVGA), Peak horizontal
II
ground velocity (PHGV), and peak horizontal ground displacement (PHGD) are also used in some engineering applications. As an example, PHGD is more significant than PHGA in the analyses of some structures such as tunnels, and underground pipelines. Peak ground motion values are influenced by a number of factors such as: the earthquake magnitude, distance from the source, local soil conditions, style of faulting, and the variation in geology along the travel path. Peak ground motion values such as PHGA are related to these parameters through attenuation relationships. Attenuation relationships are usually based on statistical analyses of recorded data. Since they are primarily dependent on source conditions and fault mechanism, different relationships were developed for different zones in western United States to reflect different natures of seismic sources. On the other hand, few attenuation relationships, which are based on theoretical models, are available for eastern and central United States due to the little number of recorded motions. 6
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Figure 1-8. A typical corrected accelerogram and the integrated velocity and displacement
1.5.2.2 Frequency Content The seismic behavior of a structure is usually controlled by the frequency content of the ground motion. The response of the structure is amplified the most when the frequency content of the motion and the natural frequencies of the structure are close to each other. The frequency content of a record can be investigated by converting the motion from a time domain to a frequency domain through a Fourier transform. Fourier amplitudes and power spectra, which are based on this transformation, are usually used to characterize the frequency content. The Fourier amplitude spectrum is defined as the square root of the sum of the squares of the real and imaginary parts of the Fourier transform. A broad spectrum such as the one shown in Figure 1-9 implies that the motion contains a wide range of frequencies that produces an irreqular time history like the one shown in Figure (1-8). On the other hand, a narrow spectrum indicates that the motion has a dominant frequency. The power spectrum assumes a ground as a stochastic process. It illustrates how the variance of a record is distributed with frequency . It is also used as an input of the excitation in random vibration analysis of structures . The response spectrum is another mean to demonstrate the frequency content of ground motions. As shown in Figure 1-10, It expresses the maximum response of a single-degree-of freedom system to a certain ground motion as a function of the natural frequency and damping ratio of the system. While used frequently in structural analysis, response spectra are not widely used in geotechnical engineering. Their primarily applications are in the selection of time histories for input to site response analysis or for the selection of seismic response coefficients for simple methods of seismic design.
1.5.2.3 Duration The duration of earthquake ground motion has a significant role on the seismic damage of structures. The number of load reversals during an earthquake is responsible for the material strength degradation of certain structures. It is also responsible for the liquefaction of soil during earthquakes. Duration is usually determined as the strong portion of the accelerogram, and there are different procedures for evaluating such portion. The bracketed duration as shown in Figure 1-11 (Bolt 1969) is defined as the time interval between the first and last acceleration peaks greater than a specified value usually taken as 0.059. Since there is no standard method for determining the duration, the selection of a procedure will depend mainly on the purpose of the intended application.
13
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0
20
25
Figure 1-9. Fourier Amplitude Spectrum of an Earthquake
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14
0.6 - , - - - - - - - - - - - - - - - ---- .......,
0.4
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10
Bracketed Duration
=If).~
20
40
30
sec
50
TIME {sec)
Figure 1-11- Bracketed duration of an earthquake
1.5.2.4 RMS Acceleration The root- mean-square acceleration is a single parameter that includes the effects of the amplitude and frequency content of a ground motion record. 1t is defined as the square root of the square of the acceleration integrated over the duration of the motion and divided by the duration: (1.6)
where RMSA a is the .rrns of the acceleration time history, a (t) is the acceleration time history and tf is the duration of the ground motion. The RMSAa can be viewed as average acceleration for over the duration of the time history. It is also directly related to the energy content of the motion .
1.5.2.5 Arias Intensity The Arias intensity is directly proportional to the square of the acceleration integrated over the duration of the time history: It
IA =~ J[a(t)Ydt 2g o
(1.7)
where 9 is the acceleration of gravity and tf is the duration of the ground motion. Arias (1970) demonstrated that this integral is a measure of the total energy of the acceleroqram.
15
1.6 Vibration Theory Applied to Foundation 1.6.1 Basic Concepts Dynamics is the branch of science , which considers forces and displacements that vary with time. Dynamic loads such as earthquakes produce time dependent displacements of the system called dynamic response, which is usually oscillatory. There are two types of mathematical models for the structural representation of vibrating systems , namely distributed mass and lumped mass models. In distributed mass models, the mass is distributed through the system, while in lumped mass models the mass is concentrated into a number of points and the structural elements between the lumped masses are considered as massless. Figure 1-12 illustrates examples of distributed and Jumped mass models. The number of dynamic degrees of freedom of the system is the number of independent variables required to define the displaced position of is masses , One lumped mass has three possible translations and three possible rotations representing six degrees of freedom. A system whose position is described by a single variable is known as a single degree of freedom (SDOF) system. A SDOF system consists of a rigid mass, rn, connected in parallel to a spring of stiffness, k, and a dashpot of viscous damping coefficient, c, and subjected to some external dynamic load, Q (t) as shown in Figure 1-13. -
The external dynamic force is resisted by the inertia and damping forces in . addition to the spring force . The equation of motion can be expressed as:
.
~
'
mii t)+cu(t)+ ku(t);= Q(t) _
(1 .8)
I
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' . where u(t), u(t) ,andu(t) are acceleration, velocity and displacement respectively . L ' 'w ~ t- 1.6.1.1 Free Damped vibration of SDOF system: Under free vibration the natural frequency of the system is defined as: ?l 'I., -=:
- "(.
(1.9)
:J
The viscous damping of the system c is a measure of the energy dissipated in a cycle of vibration . There is a critical value of damping cer, below which the system will be in the state of oscillatory motion . Mathematically , the critical damping coefficient is expressed in terms of the system mass and natural frequency as: c
C(
= 2m(i) = 2.Jkm = 2k/(i) /
(1.10)
16
aJ
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Figure 1-12. (a) Distributed models, and (b) lumped mass models
acn
Figure 1-13. SOOF system subjected to extemal load
u
Figure 1-14. The effects ofviscous damping on free vibration
17
The damping ratio is the ratio of the system's damping to the critical damping:
s=~=_c_ ccr 2mcu
(1 .11)
If the damping ratio is less than one, the system is defined as under-damped. The motion of an under-damped system is oscillatory but not periodic, the amplitude of vibration is not constant during the motion but degreases for successive cycles. Nevertheless, the oscillation occurs at equal intervals of time as shown in Figure 1-14. If the damping ratio is more than one the system is defined as over-damped , and , for a ratio equals one, the system is defined as critically damped . The resulting motion for both systems is not oscillatory but decays exponentially with time to zero . The logarithmic decrement is a useful tool that free vibration provides for determining the damping ratio . According to Figure 1-15, the logarithmic decrement is defined as:
S=
In(~J = un + 1
(1.12)
2ns
~1- (/
Rearranging allows the damping ratio to be determined from the logarithmic decrement as:
~
=
S/2n
[1 + (S/2nY
(1.13)
I" s
For small values of 8, can be taken as 8/2n , and if the decay is slow, it may be easier to compare amplitudes of several cycles instead of successive amplitudes. In this case the logarithmic decrement is evaluated as:
O~~ln(u~:J
(1.14)
For example, suppose the oscillation in free vibration decayed from amplitude of 0.36 in. to 0.05 in. in 22 cycles. Then the logarithmic decrement would be:
8 = 1/ 22In(.36/ .05) = 0.0897,
this gives
~
= 0.0143.
1.6.2 Foundation Vibrations Soil-structure-interaction is the principle of determining the dynamic response of the structure interacting with the soil foundation under seismic waves or any other kind of dynamic loads. This response may be computed using rigorous methods such as finite element modeling or by establishing the dynamic properties (impedance functions) of the foundations for each degree of freedom . As shown in Figure 1-16 the vibration of foundations is characterized by six degrees of freedom: (dynamic) displacements along the axes x, y, and z; and
18
Figure 1-15. Determination of damping ratio from logarithmic decrement
Figure 1-16. Foundation block with its six degrees of freedom
19
(dynamic) rotations around the same axes. We consider separately the response of the foundation for each degree of freedom, hence the SDOF model is utilized for the calculation of the impedance functions, as will be shown in chapter 2, in terms of foundation geometry, soil properties, and soil damping. The energy of vibration in soils is dissipated through two mechanisms: the inelastic deformation of soil and the effect of propagation of the reflected seismic waves away from the structure. Inelastic deformation of soil is considered as material or hysteretic damping and expressed in terms of the energy dissipated hysteretically by the slippage of grains with respect to each other. The effect of wave propagation is close to viscous in character and is referred as geometric or radiation damping
1.7 Measurement of Soil Dynamic Properties The shear wave velocity of the soil medium is directly related to the shear modulus through equation (1 .3). The stress-strain response of soil due to the cyclic earthquake loading is commonly characterized by a hysteretic loop. Therefore the actual behavior can be simulated as that of a hysteretic system described through the tangent-at-the-origin (initial) shear modulus, Go also known as the maximum shear modulus G max, and a damping ratio ~o . Therefore it is clear that the initial shear modulus Gmax , or the corresponding shear wave velocity Vsmax :::: ~Gmax I p is the most significant soil parameter influencing the response of foundations under dynamic loads.
1.7.1 Standard Penetration Test (SPT) The standard penetration test (SPT) is the traditional in situ test in Geotechnical engineering, and also used in a number of Geotechnical earthquake engineering applications. In the SPT test a standard split spoon sampler (Figure 1-17) is driven into the soil at the bottom of a borehole by repeated blows (usually 30 to 40 blows per minute) of a 140-lb hammer released from a 30 in. height. The sampler is usually driven 18 in.; the number of blows N required to achieve the last 12 in. of penetration is considered as the standard penetration resistance. Since different hammer designs have evolved and they vary considerably, it has become common to correct the N value. to an energy ratio of 60%. The required corrections are included in the following formula (Youd and Idriss 1996): (1-15)
( J ,
~
where (N]) 60 is the corrected standard penetration resistance; N is the measured standard penetration resistance ; C E is the correction factor for hammer energy ratio, Cs. is a correction factor of borehole diameter; CR is the correction factor for samplers with or without liners; and CN is an overburden correction factor calculated as:
I
~'
j
\ ,
I
I
20
Figure 1-17. SPT sampler (1-16a) where cr~ is the effective vertical stress in (US) ton/tt or (1-16b)
The effective vertical stress is calculated as: (1-17)
where Yt is the total unit weight of soil, Yw is the unit weight of water, d is the depth to sample, and dw is the depth to ground-water level. The initial tangent shear modulus Gmax is related to the corrected standard penetration resistance (Nr) 60 for sand using the following empirical formula adopted by FEMA 356 (Seed ~t al 1985): G max :: 20,OOO(N] )601/3 ~cr~
\ _ - ('.
L.'
,j
" (
(1-18)
where Gmax and o'; are both calculated in Ib/tt.
21
Table 1-1 Corrections to SPT (Youd and Idriss , 1997) Factor
Equipment Variable
Term
Correction
Safety Hammer Donut Hammer -
/
Borehole Diameter
Rod Length
Sampling Method
)
65 }ai1~mm . / ( ~mm
0.60 to 1.17
CE
Energy Ratio G..
0.45 to 1.00 .[,'
I
-e-
CB
•
\.. l
r -' :,::;v "' ''' : I
1.0
1.05
200 mm
1.15
3 to 4 mm
0.75
4to 6 mm
0.85
6 to 10 mm
CR
0.95
10 to 30 mm
1.0
>30 mm
1.0
Standard Sampler Sampler without liners
Cs
1.0
1.2
1.7.2 Cone Penetration Test (CPT) In a CPT test, the cone penetrometer (Figure 1-18) is pushed into the ground at a standard velocity of 2 cm/s (0.8 in/s) and data is recorded at regular intervals (typically 2 or 5 ern) during penetration . The standard cone penetrometer has a conical tip of 10 cm 2 (1 .55 in 2 ) area, which is located below a cylindrical friction sleeve of 150 cm 2 (23.3 in2 ) surface area. The tip as well as the friction sleeve are connected to load cells to record the tip resistance qc and the sleeve friction resistance fs during penetration, and the friction ratio FR defined as FR =fJ qc(Figure 1-19). The initial tangent shear modulus Gmax is related to the penetrometer tip stress using the following empirical formulas (Kramer 1996). For sand:
22
G max -- 1634(q c )0.250 (a rv )0.375
(1-19)
For clay:
G max: -- 406(q c )0.695 e -1.1 30
(1-20)
G max , qc, and a~, in equations 1-19 and 1-20 are calculated in kPa.
1.7.3 The Cross-hole Seismic Survey This method determines the variation with depth of in-situ low-strain shear wave velocity V smax • As shown in Figure 1-20, the cross-hole method is based on generating shear waves in a borehole and measures their arrival times at the same elevation in neighboring boreholes. The wave velocity is calculated from the travel times and the spacing between the boreholes. The initial tangent shear modulus is calculated from the measured low-strain shear wave velocity Vsmax using equation 1-3 as: (1-21 ) in which, p is the mass density of the soil and g is the acceleration due to gravity. For successful results of a cross-hole test , there should be at least two boreholes, which are spaced about 3 to 5m (10 to 15 ft) apart. Also, the source must be rich in shear wave generation. The SPT can offer a good inexpensive solution . Moreover, the receivers must be in good contact with the surrounding soil.
1.7.4 The Seismic Down-hole Survey This method offers an economical alternative to the cross-hole test as it needs as shown in Figure 1-21 one borehole, inside which the receiver can be moved to different depths, while the source remains at the surface, 2 to 5 m (6 to 15 ft) away. Alternatively, the test can be done by fixing an array of multiple receivers at predetermined depths against the walls of the borehole. Travel times of body waves (S or P) between the source at surface and the receivers are recorded for various depths and a plot of travel time versus depth can be generated, from which V smax or V pmax are then computed at the same depths as the slope of the travel time curve at that depth .
1.7.5 The Seismic Cone Penetration Test This test as shown in Figure 1-22 is a combination of the down-hole and cone penetration tests. The cone penetrometer is modified by mounting a velocity seismometer inside it, just above the friction sleeve. At different stages during penetration of the cone penetrometer, penetration is paused to generate impulses at the ground surface. Travel time-depth curves can be generated and interpreted the same way as the down-hole test..
23
s
...
s
I
7
Ii
~
I
!
:s
1
J
1 I
I
. ' -'
I
: .
. ,..
.. ...
i I
.~
;;It
----*
35.6
rnm
• I
~
1 Conir.:lll pl"lil'1T (10 CIIl') ~ l.noo cell J S tJ.~iJ ) ga . ~~~ 4" Friction sleeve ( 151) .;.oi ·)
5 AdjuRtment r ing I'J W~te tp~ lXJ r bushi ng 7 Cable S Connection I'r'ilh rods
Figure 1-18. CPT penetrometer
6 4 2 0
o 10
---
20
a. a
30
:::: .c Q)
40
50
Friction ratio (%)
Bearing resistance (tons/ft)
Friction
resistance
(tons/ft)
I !
100 200 300 400 500 .
o
2 468
.
··1···......;..,.....J,• .•.. . .
. ~=====~.!C;• • .• . . .• .. . . ' . .•.
··
.
· · ·····T ··· · · · ·~ · ·· · ··
_ __ • _ • • ;
. o w • w• • •
.:. . . . .
:
:
·
. ' 0 ... .
•
..
. r:. .···. .
.
. ~ . ':' '' A '' •• · :
.
. ...:
e O ;
..
=
• ••
- . - ~ ----_
... . .
60
Figure 1-19. Results of cone penetration sounding
24
A significant advantage of this method is that with a single sounding test , one can obtain information for the stratigraphy of the site, the initial tangent shear modulus of different layers, as well as static strength parameters. A limitation of this method is that it may not be adequate for some types of soils containing coarse gravels
1-8 The Design Spectrum The design spectrum should satisfy certain requirements because it is intended for the design of new structures or seismic assessment of existing structures to withstand potential earthquakes . Therefore, it should in general sense be representative of ground motions recorded at the site during past earthquakes or at other sites under similar conditions . The design response spectrum is usually based on statistical analysis of the response spectra for the ensemble of ground motions for a specific site. Different codes have developed procedures to construct such design spectra from ground motion parameters. One such procedure of FEMA-356 and the LRFD Guidelines for Seismic Design of Highway Bridges is outlined herein as an example 1. From the U.S. Geological Survey web site (http://earthquake.usgs.gov/) determine the 0.2-second and 1-second spectral accelerations Ss and S1 . These values can be obtained by submitting the latitude and longitude, or the zip code of the site of interest. These are spectral accelerations on rock outcrop (class B). 2. Classify the site according to the average shear wave velocity in the upper 30 m. Site class A is defined as hard rock with average shear wave velocity greater than 1500 m/sec (5000 ftlsec) . Site class B is defined as rock with average shear wave velocity that ranges from 750 to 1500 m/sec (2500 to 5000 ftlsec) . Site class C is defined as very dense soil and soft rock with average shear velocity that ranges from 360 to 760 m/sec (1200 to 2500 ftlsec). Site class D is defined as stiff soil with average shear wave velocity that ranges from 180 to 360 m/sec (600 to 1200 ftlsec) . Site class E is defined as soft clay with average shear wave velocity that is less than 180 m/sec (600 ftlsec) 3. Determine the site coefficients Fa and Fv for the short period spectral acceleration and the 1-second period spectral acceleration respectively. These values are displayed as function of the site class as shown in Tables 1-2 and 1-3. 4. Calculate the design earthquake response spectral acceleration at short periods, SOs :: Fa Ss, and at 1 second period, S01 :: Fv S,. 5. Determine the periods Ts and To required for plotting the design response spectrum, where Ts :: S01/S0s , and To :: 0.2 Ts .
25
-- -Lm (l3ftl
/"m (13fl)
[mpQct
+ TfQnsduc~
I
...
"J I
I _
~ J'
- --
r
~
~
I'" ~ 1
I
....
,
.
~
)
-
I
Figure 1-20. Seismic cross-hole test
26
._-----~ ----------_.~.~--
._-- _.•.__ .._- .._
_-_..
,.
~-
Figure 1-21. The Seismic down-hole test
V. (m/s)
o
z
~
lQl
I5ll 100 1'iO
[bar)
o"'--<::_.........,,........._~_
,..
' "'9:J.'ft;,w:"W7~~~
~~s~
5
~
Il.
w
c
•••
..
/
E e
c
s _
Soi.",ic CPT
-
-
-
...,.--
---I::>
._ - - - - -_."._ - -_ .._._..
_ .~_._--
Figure 1-22. The Seismic cone penetration test
27
.5v .: '7-
,
. cD
~
..
"
.(.,
l
.s, \
\. .f
6. For periods less than or equal to To, the design spectral acceleration, Sa, shall be defined by: Sos To
Sa = 0.60-T + 0.4080s
(1-22)
Note that for T ;;; 0 seconds, Sa shall be equal to the effective peak ground acceleration. 7. For periods greater than or equal to To and less than or equal to Ts. the design spectral acceleration shall be defined by: Sa =Sos
(1-23)
8. For periods greater than Ts , the design response spectra! acceleration, Sa, shall be defines by:
S = 8 01 a
T
(1-24)
The steps involved in the development of the design spectrum are displayed in Figure 1-23.
1.9 Site Response Analysis The local soil profile at a project site can have a pronounced effect on the earthquake ground motions, and subsequently on the response of the structure to the earthquake. The nonlinear behavior of soils under strong earthquake loading is a highly complicated problem. Generally soil is a nonlinear, anisotropic material. Despite this complex behavior, isotropic elastic models either linear or nonlinear have been used in the past for practical considerations. Methods of evaluating the effect of local soil conditions on ground response during earthquakes are based on the assumption that the main responses in a soil deposit are caused by the upward propagation of shear waves from the underlying rock formation. The most commonly used model to represent the soil behavior in seismic analysis is the equivalent linear model (Seed and Idriss , 1970). According to this model, the appropriate equivalent linear shear modulus G as shown in Figure (1 24) is the secant modulus, which is less than the initial tangent shear modulus Gmax.
28
Table 1-2. Values of Fa for different values of spectral acceleration (LRFD guidelines for seismic design of highway bridges 2004) Spectral Acceleration at Short Periods Site Class Ss::;; 0.25 g
S5
=0.50 g s, =0.75 9 s, =100 9
Ss ~ 1.25 g
A
0.8
0.8
0.8
0.8
0.8
B
1.0
1.0
1.0
1.0
1.0
C
1.2
1.2
1.1
1.0
1.0
D
1.6
1.4
1.2
1.1
1.0
E
2.5
1.7
1.2
0.9
0.9
Table 1-3. Values of Fv for different values of spectral acceleration (LRFD guidelines for seismic design of highway bridges 2004) Spectral Acceleration at Short Periods Site Class 51::;; 0.10 g
51
= 0.20 9
51
=0.30 9
51 = 0.40 9
51 ~ 0.50 9
A
0.8
0.8
0.8
0.8
0.8
B
1.0
1.0
1.0
1.0
1.0
C
1.7
1.6
1.5
1.4
1.3
0
2.4
2.0
1.8
1.6
1.5
E
3.5
3.2
2.8
2.4
2.4
29
I
I I I " " ' -~- -- ~ - - ._- -
SOS == FaS s
S01 == FvS 1 TO
==O.2Ts
s . . S01
T
SOS
PGA
~
I
:
I
:
!! ,' '
-
._- - j-
---
. \ \
I \
:
I So1 - ~ -I -··· T
I_l: I
"
~,, ;
-. i ._ _
-- - -- -_. _ . . . .~:=--_
TO 02 Ts
1.0
Figure 1-23. Example of a Design Spectrum
30
Meanwhile, the area of hysteresis loop has expanded, indicating an increased dissipation of energy resulting from sliding at particle contacts, hence, the equivalent linear hysteretic damping ratio ~ is larger than~o . Therefore, the bigger the cyclic shear strain, the smaller the equivalent shear modulus G and the larger the equivalent damping ratio ~ as illustrated in Figure (1-25). Two different levels of site-specific seismic site response analysis are available. In levell, the simplified methods recommended by codes are usually followed. As an example , FEMA-356 established six classes of sites for seismic depending on their shear wave velocities. According to this classification sites range from hard rock (class A) to peats and organic clays (class F). Table 1-1 illustrates the recommended values for the effective shear modulus to account for the non linear behavior of soils for different sites and peak ground accelerations. The recommended LRFD guidelines for the seismic design of highway bridges (2004) also recommends for regions of low-to-moderate seismicity (PGA<0.3g), a value of G=0.5 G max while for regions of moderate-to-high Seismicity (PGA> 0.5g), a value of G = 0.25 Gmax is recommended. Both FEMA-356 and LRFD guidelines request level II, which is a dynamic site response analysis , for organic soils (class F). Dynamic site response analysis is also requested for major projects and critical facilities when global time history analysis is mandated to establish the ground motions at the foundation levels. I
Typically, a one dimensional soil column that extends from the ground surface to bedrock is used to model the soil profile. Two dimensional soil profiles may also be used in special cases such as basins. The soil layers in a one dimensional model are characterized by their unit weighs, maximum shear wave velocities and by relationships defining the nonlinear shear stress-strain relationships of the soils such as the one shown in Figure 1-24. The computer program SHAKE originally developed by Schnabel et al. (1972) and updated by ldriss and Sun (1992)under the name SHAKE91, is the most commonly used computer program for one dimensional equivalent-linear seismic site response analysis . The program requires a set of properties (shear modulus, damping and total unit weight) to be assigned to each sublayer of the soil deposit. The analysis is conducted using these properties and the shear strains induced in each sublayer is calculated. The shear modulus and damping values for each sublayer are then modified according to the relationship relating these two properties to shear strain. The analysis is conducted iteratively until strain-compatible modulus and damping values are reached.
1.9 Liquefaction of Soils Liquefaction occurs during earthquakes due to loss of strength of soil, which may occur in sandy soils as a result of an increase in pore pressure. This phenomenon can take place in loose and saturated sands.
31
G-eao-GIIIO.Il Mol'c.ooic toadin9 Q,jry.
Figure 1-24. Equivalent linear representation of the soil hysteretic cyclic stress strain behavior
O.S l---- --
x:
0.6
_i__
-~-+---.
.. f---+--
--I-- .- -
--+-----+-~.----------,P__----+-----l 1 5 i! t
o E
~ C)
~
1 0.41--- - - 1..- - -
c: .&
-j- ---;f--'\'-.;-------'f------ -
O.2t-----~+_--++_---+__'"
0001
0 .01 Shear SJroin -
- -
-:--
10
eo
0
5
O.i %
Figure 1-25 Typical shear modulus and damping relationships used in equivalent linear soil
32
Table 1-4. Effective shear Modulus Ratio (G/G max) after FEMA-356
Effective Peak Acceleration 0.40 50S SITE CLAS5 0.40 50S = 0
0.40 50S = 0.1 0.40 50S = 0.4 0.40 5DS
A
1.00
1.00
1.00
1.00
B
1.00
1.00
0.95
0.90
C
1.00
0.95
0.75
0.60
0
1.00
0.90
0.50
0.10
E
1.00
0.60
0.05
*
F
*
*
*
*
*Site-specific geotechnical investigation analyses shan be performed
and
dynamic
=0.8
site response
33
The increase in pore pressure causes a reduction in the shear strength, which in certain cases may be totally lost. In such cases soil will behave like a viscous fluid, and in some earthquakes liquefaction appeared in the form of sand fountains . Structures founded on liquefiable sands may settle, tilt, or even overturn during earthquakes due to loss of bearing capacity as a consequence of soil liquefaction. Examples of failures of structures due to liquefaction during the 1964 Niigata Japan earthquake and the 1999 Izmit Turkey earthquake are illustrated in Figures 1-26 and 1-27. 1.9.1 How Liquefaction Build-up during Earthquakes
Seismic shear waves during its passage through different soil layers will tend to compact loose saturated sand deposits, and thus decrease its volume . If these deposits cannot drain rapidly, there will be a gradual increase of pore water pressure and decrease of the effective stress of soil with increasing the ground shaking. Since the shear strength of cohesionless soils is directly proportional to the effective stress, liquefaction will occur at the point when the pore water pressure becomes equal to the total overburden pressure . At this stage there will not be any shear strength for the soil, and it will tend to boil like a fluid. 1.9.2 Liquefaction Potential Evaluation
The LRFD guidelines for seismic design of highway bridges (2004) recommend that no evaluation of liquefaction hazard potential at a site be done if the following conditions occur: • The distance from the ground surface to existing or potential ground water level is more than 15 m. • Bedrock underlies the site. • The soil is clayey with particle size < 0.005 mm greater than 15% A simplified procedure for the evaluation of liquefaction potential was originally developed by Seed and Idriss (1971, 1982) with subsequent refinements by Seed et al. (1985, and 1990). The procedure compares earthquake cyclic stress ratio (CSR) at a certain depth of a cohesionless stratum to the cyclic resistance ratio (CRR), which is defined as the cyclic stress ratio required inducing liquefaction for that given depth. This evaluation procedure uses correlation between the liquefaction characteristics of soils and field tests such as the Standard Penetration Test. The procedure involves the following basic steps: • Determine the cyclic stress ratio (CSR). During an earthquake, the soils will be subject to cyclic shear stresses induced by the ground shaking. The average cyclic stress ratio (CSR) may be estimated by the following formula: CSR==
'r ~v ==0 .65 (amax)(cr~Jrd Go
g
(1-25)
Go
34
Figure 1-26. Failure of the Kawagishi-cho apartment buildings following the 1964
Niigata earthquake due to soil liquefaction (courtesy of EERC, Univ. of California)
Figure 1-26. Failure of a building following the 1999 Izmit Turkey earthquake (courtesy of EERC, Univ. of California)
35
o0
0.1
02 03 0.4 05
0.6 0.7 0.8
0.9 1.0
3 (0) 6(20)
-
::= '-' E
'" ..r:: -..
0..
Average values Mean values of rd
9(30)
12(40) 15(50)
. . .. . .
tU
0
18(60)
21 (70)
24(80)
27(90) 30 (l 00)
.. . ..... . . . .
. . . ..
...
. . '
l--...;.....L...---J.~.a...;...~:.......;....L_...o...-..;...a...----JI.-.-""""'--..I
Figure 1-27. Stress reduction coefficient rd versus depth curves (youd and ldrlss , 1997)
36
Where amax is the maximum acceleration at the ground surface;
1.9.3 Liquefaction Induced Settlement A significant consequence of liquefaction is the volumetric strain caused by the excess pore-water pressure generated in sands by the cyclic ground shaking. The volumetric strain results in settlement, which could lead to collapse or partial collapse of a structure particularly if there is a pronounced differential settlement between adjacent structural elements. The post-liquefaction volumetric strain can be estimated from the chart depicted in Figure 1-30 after Tokimatsu and Seed (1987). Knowing the strain caused by the liquefaction, the ground surface settlement may be estimated by multiplying the thickness of each layer by the strain.
1.9.3 Post-Liquefaction Lateral Spreading One of the consequences of liquefaction is the degradation in undrained shear strength of soil, which may lead to lateral spreading. If there is differential lateral Spreading under a structure, there could be sufficient tensile stresses developed in the structure that it could be literally torn apart. Lateral spreading can have disastrous consequences on lifelines. Figures 1-31 and 1-32 show examples of lateral sp read, following the 1995 Kobe, Japan earthquake , and the 1994 Northridge earthquake. The magnitude of the lateral spread hazard can be assessed us ing an approximate procedure (Barlett and Youd, 1992) that was based on regression analyses on a large database of lateral spread case histories from past earthquakes. They proposed two statistically independent equations , one for areas near steep banks with a free face , the other for ground slope areas with gently sloping terrain.
37
0.6
. 2'
ur. I
I
Percenr Fines =35
0.5
,
15
I
I
I I I I
,
I
I
,
I
-
~
, I
.Jl
~
10
0.3
. lZ
.~
'"
~
;,
~
&'b ·q101) . ~'O'
~,.nl ' J.. 'ii> ;l,",~
1.1'0<1 ' (h~ 0.1 {7 ,,"°1/'
~~ o
o
t
,
t
1
,
I
t
3S percem lines, respectively
1
I
j/ V ~. '
20
~o~.. o
JO•
,' ---/.... CRR curves for 5,15. and
I
J
,"a.
IQ
IQ· "10·
C O.2
I
....tQ
I
I
,
60-·
.;!
U
, "~
I I
,
,,---1
t
4~
I
I
I I ~o
I
I
I
r
0.4
tl
s, 5
I
r .~
~~ !l
I
"Y»J @
~.A~o
/
~] iJ:
II
'0
S
FINESCOl\'TENT:?; 5%
2
O'-'tYJ
Modifted Chi nese Code Propos>! felay eerueru =5';;; ) @
"')0 .
Ma . aI
No
U quefoaion l.iq:lOn l.KEfac1,00
•
Pan • A.merian data
AdjllStmenl
•
Recomrreeded Jap;m=~ra By Workshop Chinese data
30
0 A
A
20
10
Q
g
40
CorrectedBlow Count, (Nl)6Q
Figure 1-28. Curve Recommended for Determining CRR from SPT Data (Youd and Idriss, 1997) 4j
I
4
angeof recommen I MSFfromNC Workshop
3.5 3
25
1.S
.... . ~
o 5.0
6.0
7.0
8.0
9.0
Earthquake Magnitude. Mw Figure 1-29 Magnitude scaling factors for the SPT data (youd and Idriss, 1997)
38
0.6 Volumetric strain (%)
0.5
1054 3
2
~
0.5
· · ·:0.2 .... .. .. ..• • 0.1 t.: ....... . . ........ . .....'......... .... . . . .. ... ..»...» . . . .. ...... . . .. ...... ·•···• •
II
0.4
a:
1
.
•
0.3
#
#
0.2
.9
#
•
•
."
.' .. .'
0.1
o
•
•
,.~ ~
o
10
20
30
40
50
(N1)SO Figure 1-30. Estimation of post-liquefaction volumetric strain from SPT data and cyclic stress ratio for saturated clean sands and rnaqnitude > 7.5 (Tokimatsu and Seed, 1987)
39
Figure 1-31. lateral spread following the 1995 Kobe, Japan earthquake in Tempoyama Park Osaka (courtesy of EERC, Univ. of California)
Figure 1-32. Lateral spreading in Granada Hills at Rinaldi St. (Granada Hills, California) following the 1994 Northridge earthquake (courtesy of EERC, Univ. of California)
40
For free-face conditions: Log OH = - 16.3658 + 1.1782 M - O.9275Log R - 0.0133 R + 0.6572 Log W + 0.3483Log T15 + 4.5270 Log (100 - F15) -0.9224 050 15 (1-26) For ground slope conditions: Log OH = - 15.7870 + 1.1782 M - 0.9275 Log R - 0.0133 R + 0.4293 Log S + 0.3483 Log T 15 + 4.5270 Log (100 - F15) - 0.9224 050 15 (1-27) where, OH is the estimated lateral ground displacement in meters; M is the moment magnitude of the earthquake; R is the horizontal distance from the seismic energy source, in kilometers; W is the ratio of the height (H) of the free face to the distance (L) from the base of the free face to the point in question, in percent; T15 is the cumulative thickness of saturated granular layers with corrected blow counts,(N1)60, less than is, in meters; F15 is the average fines content (fraction of sediment sample passing a No. 200 sieve) for granular [ayers included in T 15 in percent; 050,5 is the average mean grain size in granular layers included in T 15 , in mm; and S is the ground slope , in percent. The LRFO guidelines for the seismic design of highway bridges (2004) recommends using this approach only for screening of the potential for lateral spreading , as the uncertainty associated with this method is generally assumed to be too large. Alternatively, more rigorous methods such as the Newmark sliding block analysis can be used to assess the potential of post-liquefaction lateral spreading at a site. I
1.9.3 Post-liquefaction Flow Failures Flow failures are the most catastrophic form of ground failure that may take place when liquefaction occurs in areas of significant ground slope. Flow failure may be triggered when farge zones of soil become liquefied or blocks of unliquefied soils flow over a layer of liquefied soils. Flow slides can develop where the slopes are generally greater than six percent.
1.9.4 Mitigation of liquefaction Hazard Mitigation of liquefaction potential can be established either by site modification methods or by structural design methods. Site modification methods include but not limited to: • Excavation of the site and replacement of liquefiable soils, which is applicable only to small projects due to the expenses of excavation and soil replacements . • Oensification of in-situ soils through advanced compaction methods such as vibroflotation. This principle involves lowering a machine into the ground to compact loose soils by simultaneous vibration and saturation. As the machine vibrates, water is pumped in faster than it can be absorbed by the soil. Combined action of vibration and water saturation
41
rearranges loose sand grains into a more compact state. After the machine reaches the required depth of compaction, granular material, usually sand, is added from the ground surface to fill the void space created by the vibrator. A compacted radial zone of granular material is created. • In-situ improvements of soils by using additives such as the stone column technique. The stone column technique, also know as vibro-replacement, is a ground improvement process where vertical columns of compacted aggregate are formed through the soils to be improved. These columns result in considerable vertical load carrying capacity and improved shear resistance in the soil mass. Stone columns are installed with specialized vibratory machines. The vibrator first penetrates to the required depth by vibration and air or water jetting or by vibration alone. Gravel is then added at the tip of the vibrator and progressive rising and repenetration of the vibrator results in the gravel being pushed into the surrounding soil. The soil-column matrix results in an overall mass having a high shear strength and a low compressibility • Grouting or chemical stabilization. These methods can improve the shear resistance of the soils by injection of chemicals into the voids. Common applications are jet grouting and deep soil mixing. Designing for liquefaction may be accomplished by the use of deep foundations which are usually supported by the soil or rock below the potentially liquefiable soil layers. These designs would need to account for additional forces that would develop because of potential settlement of the upper soils that could occur due to Iiquefaction.
1.10 References Arias, A (1970) " A measure of earthquake intensity, ~ Seismic Design for Nuclear Power Plants, MIT Press, Cambridge, Massachusetts , pp. 438-483 Bolt, B.A. (1969) "Duration of strong motion, "Proceedings of the 4th World Conference on Earthquake Engineering, Santiago, Chile, pp. 1304-1315. Bartlett, S.F. and Youd, T.L. (1992). "Empirical analysis of horizontal ground
displacements generated by liquefaction-induced lateral spread, "Technical
Report NCEER-92-0021 , National Center for Earthquake Engineering Research,
Buffalo, New York .
FEMA (2000). "Prestandard and commentary for the seismic rehabilitation of
buildings", FEMA-356, Federal Emergency Management. Washington, D.C.
ldriss, I.M. and Sun, J.I. (1992). "SHAKE91: a computer program for conducting equivalent linear seismic response analyses of horizontally layered soil deposits , "User's Guide, University of California, Davis, 13pp.
42
Kramer, S.L. (1996), "Geotechnical Earthquake Engineering," Prentice-Hen, Inc ., Upper Saddle River, New Jersey, 653 pp. MCEERJATC (2003) "Recommended LRFD guidelines for seismic design of highway bridges", MCEERIATC 49, Applied Technology Council and Multidisciplinary Center for Earthquake Engineering Research. Schnabel, P.R, Lysmer, J., and Seed, H.B. (1972). "SHAKE: computer program for conducting equivalent linear seismic response analyses of horizontally layered sites," Report EERC 72-12, Earthquake Engineering Research Center, University of California Berkeley. Seed, H.B. and Idriss, I.M. (1970)."Soil moduli and damping factors for dynamic response analyses," Report EERC 70-10, Earthquake Engineering Research Center, University of California Berkeley. Seed, H.B. and Idriss, I.M. (1971). "Simplified procedure for evaluating soil liquefaction potential," Journal of Soil Mechanics and Foundations Division, ASCE, Vo1.107, NO.SM9, pp.1249-1274. Seed, H.B., Tokimatsu, K., Harder, L.F., and Chung, RM. (1985). "Influence of SPT procedures in soil liquefaction resistance evaluations," Journal of Geotechnical Engineering, ASCE, Vol. 112, No.11, pp.1016-1032. Seed, R.B. and Harder, L.F. (1990). "SPT-based analysis of cyclic pore pressure generation and undrained residual strength," Proceedings. H.B. Bolton Seed Memorial Symposium, University of California Berkley, Vol. 2, pp.351-376 . Tokimatsu , K. and Seed, H.B. (1987) Evaluation of settlements in sand due to earthquake shaking," Journal of Geotechnical Engineering , ASCE, Vol. 113, No.8, pp.861-878. Youd, T.L. and Idriss I.M (1997) Proceedings of the NCEER Workshop on evaluation of liquefaction resistance of soils. Report NCEER 97-22, National Center for Earthquake Engineering Research, Buffalo, New York.
43
2
SEISMIC DESIGN OF
SHALLOW
FOUNDATIONS
45
2-SEISMIC DESIGN OF SHALLOW FOUNDATIONS 2.1 General Shallow foundations are usually suitable for sites of rock and firm soils. The stability of these foundations under seismic loads can be evaluated using a pseudo-static bearing capacity procedure. The applied loads for this analysis can be taken directly from the results of a global dynamic response analysis of the structure with the soil-foundation-interaction SFI effects represented in the structural model.
2.2 SFI Representation in Global Structural Models SFSI effects can be incorporated into global structural models by means of two methods, the foundation dynamic impedance function method, and the Winkler spring model method. The dynamic impedance function method is adequate if the seismic foundation loads are not expected to exceed twice the ultimate foundation capacities (FEMA 2000). The Winkler spring model approach is more applicable for life safety performance-based design, where it is essential to represent the nonlinear force displacement relationships of the soil-foundation system. As illustrated in Figure 2-1 , the dynamic impedance model is an uncoupled single node model that represents the foundation element. On the other hand, the Winkler approach can capture more accurately the theoretical plastic capacity of the soil-foundation system. The non-linear spring constant for this approach are usually established through non-linear static pushover analyses of local models of the soil-foundation system using general-purpose finite element programs such as ABAQUS or ADINA, or by using a Geotechnical soil structure interaction programs such as FLAC OR SASSI. An upper and lower bound approach to evaluating the foundation stiffness is often used because of the uncertainties in the soil properties and the static loads on the foundations. As a general rule of thumb, a factor of 4 rs taken between the upper and lower bound (ATC-1996). The procedure is to make a best estimate of foundation stiffness and multiply and divide by 2 to establish the upper and lower bounds, respectively.
22. 1 The Dynamic Impedance Approach This approach is based on earlier studies on machine foundation vibrations, in which , it is assumed that the response of rigid foundations excited by harmonic external forces can be characterized by the impedance or dynamic stiffness matrix for the foundation. The impedance matrix depends on the frequency of excitation, the geometry of foundation and the properties of the underlying soil deposit. The evaluation of the impedance functions for a foundation with an arbitrary shape has been solved mathematically using a mixed boundary-value problem approach or discrete variational problem approach. Both approaches are mathematically rigorous methods. In another approach the problem has been approximately solved by defining an equivalent circular base. The impedance function of a foundation is a frequency dependent complex expression, where its
46
real part represents the elastic stiffness (spring constant) of the soil-foundation system and its imaginary part represents both the material and radiation damping in the soil-foundation system. At small strain levels typically material damping ratio ~ associated with foundation response is on the order of 2% to 5%. Radiation damping is close to viscous damping behavior. and is frequency dependent. Considering the range of frequencies and amplitUdes in earthquake ground motions compared to machine foundations, it is reasonably to ignore the frequency dependence of the stiffness as well as the damping parameters. There are two methods for evaluating the dynamic impedance functions for a shallow foundation that are commonly used in practice. The first is based on the approximate solution for a circular footing rigidly connected to the surface of isotropic homogeneous elastic half-space. adopted by FHWA (FHWA-1995). which provides the static stiffness constants for each degree of freedom. The second approach is based on the more rigorous mathematical approaches. Evaluation of the stiffness coefficients using the equivalent circular footing is carried out in four steps as follows: Step 1: Determine the equivalent radius for each degree of freedom, which is the radius of a circular footing with the same area as the rectangular footing as shown in Figure 2-2:
=Rv =~BLl1t ro =Rh =~BL/1C
(2-1-a)
ro
r0
-R _[16(B)(L)3]1/4 r1 -
-
_R
ro
(2-1-b)
-
---=--z....:.....:.-
31t
_ [16(B)3(L) 31t
]1/4
r2 -
2 2>]1I4
_[16(B +L r0 -R t - ---=---'61t
(2-1-c)
(2-1-d)
(2-1-e)
Step 2: Calculate the stiffness coefficients for the transformed circular footing for vertical translation ksv , horizontal (sliding) translations ksh1 and ksh2 , rocking, kr , and torsion, k t
k sv
_ 4GR v
-
(2-2-a)
1-u
8GR k sh1 = k Sh2 = h 2-u
(2-2-b)
47
p
Foundation forces
Uncoupled stiffnesses
Winkler spring model
Figure 2-1. Analysis models for shallow foundations
x
/ : RECTANGULAR FOOTlNG
I-
-~
I
I
I
J
I
I
I I I I
~------if--"""'y
2l
co N
--t--EQU1VALENT
CIRCULAR FOOTING
Figure 2-2. Calculation of equivalent radius of rectangular footing
48
(2-2-c)
where, G and v are the dynamic shear modulus and Poisson's ratio for the soil foundation system. Step 3: Multiply each of the stiffness coefficients values obtained in step 2 by the appropriate shape correction factor C1 from figure 2-3 (Lam and Martin 1986). This figure provides the shape factors for different aspect ratios LIB for the foundation . Step 4: Multiply the values obtained from step 3 by the embedment factor ~ using Figure 2-4 for values of D/R :5:0.5, and Figure 2-5 for D/R > 0.5 (Lam and Martin 1986). D in these figures is the footing thickness . The second approach for calculating the impedance functions for the soil foundation system is based on the results of rigorous formulations (Gazetas 1991). This approach is adopted by FEMA-356. Using Figures 2-6 and 2-7, a two-step calculation process is required. First, the stiffness terms are calculated for a foundation at the surface. Then, an embedment correction factor is calculated for each stiffness term. The stiffness of the embedded foundation is the product of these two terms. According to Gazetas , the height of effective sidewall contact, d, in Figure 2-7 should be taken as the average height of the sidewall that is in good contact with the surrounding soil.
2.3 Dynamic Bearing Capacity of Shallow Foundations The general vertical soil bearing stress capacity of a shallow footing is: 1
qull
= en, Sc + yDNqSq +2yBNySy
(2-3)
In this expression : C = cohesion property of the soil
=
bearing capacity factor depending on angle of internal friction, ¢' Nc , Nq ,N.( and evaluated as:
N = eittan
q
(2-4a)
= (N q -1) cot (cj»
(2-4b)
N y = 2(N q +l)tan
(2-4c)
Nc
Sc, Sq Sr = footing t
shape factors (see Table 2-1), y = soil total unit weight.
49
1 .20..,.------------------------~__.
.« ,
u tlJ ri
L.
0
Z
t-
O
<:
I....
1.10
W
a...
-c
~ 1,
1,OOH--r-1r-T-,.....,.....,.....,r-T--.--T-T"lIT-.---.--....,iT-.---l-i'""'1r-r-....,...,r-r""T""'i o 1.5 2.0 2.5 .3.0 J5 4.0 lIS
Figure 2-3. Shape factor a for rectangular foundations 2.5
-T""---....--- ---....------------:----....
2,0- .
~~~~ J-f .
~
•
.
,
•
~
. ,
R_
.
•
. :
t ·
.
!
Z
w ::2' o w
co ~
w
0.5 O/R
Figure 2·4. Embedment factors for foundations with D/R :5,'; 0.5
50
EMBEDMENT FACTOR ~
TRANSlA1100AL (\lERTlCAL AND HORI ZONTAL)
-----.-.-.-....-.-.-
TORSIONAl Afo'D ROTATION.AL.
IS
Figure 2-5. Embedment factors for foundations with D/R > 0.5
51
Degree of Freedom
Stiffness of Foundation at Surface
Translation along x-axis
Translation along y-axis
K
y,su"
GB [..,
=
2 _. v -').
4(L)0.65 + 0 4b. -\- 0.8J B . B
Translation along z-axis
Rocking about x-axis
«:
s ur =
Rocking about y-axis
7~:[0.4(~) + 0.1 ] 3
(L)2 .4 + 0.034J B
K
=
GB [0.47
KZZ, s ur
=
3 L GB [0.53(B ) 2.45 + 0.51J
.r .v. SUI'
Torsion about z-axis
1 -- v
Figure 2·6. Gazetas approach for foundation stiffness step-1 (FEMA-2004)
52
Degree of Freedom
Correction Factor for Embedment
Translation along x-axis
Translation along y-axis
Translation along z-axis
Py
=
~z
Px :0=
Rocking about x-axis r:l.
I-'xx
=
[1 + 2\
~~ + 2.6 Z)].[1 + O.32(d(~~ L)y / 3]
d[
.2~J 1 +251 + -2Bd(d)-O . B D L
Rocking about y-axis
d 1.) +3 .7(~1.9( d)-O'<1 ( JO.6[_ 1) D J
~yy =1+1.4L Torsion about z-axis
~z:
d
=
B ( B)(d)O.9
1 + 2.6 1 + L
=height of effective sidewall contact (may be less than total foundation height)
h = depth to centroid of effective sidewall contact
For each degree of freedom, calculate
Kemb = ~
«;
Figure 2-7. Gazetas approach for foundation stiffness step-2 (FEMA-2004)
53
/
'~QJ. ' f
'."
- I ,
. :(., .
r ,,-"
J (/
c'
"
r
~
-.
/
'
,r \
I
;r-
,,- p ..
:..
'
J
. ',
o = minimum distance from ground
level to the bottom of the footing
B = width of footing
-.
T.able 2 1 B eanng Shape Fac tors Bearing Shape Factors Shape of Footing
Strip Rectangle
Circle or square
Cohesion
Surcharge
Density
(c;
(q
(r
1.0
1.0
1.0
B Nq 1+- L Nc
B 1+-tan$
1-0.4
1+ tan $
0.6
L
N 1+-3.
Nc
B
L
Earthquakes will induce moments and horizontal loads in addition to the traditional vertical loads applied to a shallow foundation. To represent the combined effect of the seismic forces and moments a resultant load that may have to be inclined and applied eccentrically can be applied in lieu of the seismic vertical forces, seismic horizontal forces, and seismic moments. Therefore, a procedure is established to account for load inclination , and load eccentricity of footing. Through this procedure, the general bearing capacity equation of shallow footing is adjusted to account for these effects.
2.3.1. Evaluation of the Dynamic Bearing Capacity using equivalent static methods This procedure is carried out in three steps as follows : Step 1: compute the seismic vertical loads , seismic horizontal loads, and seismic moments imposed to the footing. These seismic loads and moments can be taken directly from the results of a global dynamic response analysis of the structure with the soil-foundation-interaction SFI effects represented in the structural model. For each direction , these forces are then combined into a single resultant force with an inclination angle ~ with respect to the vertical and eccentricities, eb and elfor both lateral and longitudinal directions. Step 2: calculate the equivalent dimensions for the footing to account for the load eccentricity, which is caused by the seismic moments applied to the foundation in both directions. The vertical load can be transferred to an eccentric position defined by eb =Mt/Q and el =M/Q, where Q is the central vertical load due to seismic load plus other service loads; Mb and M;: : seismic moments about the short and long axes of the footing; and eb and el eccentricities of the load Q
=
54
about the centroid of the footing in the direction of the short and long axes respectively. It is known from basic principles of strength of materials that if the eccentricity in one direction is less than 1/6 of the foundation's length in that direction, the footing is in compression throughout. As eccentricity exceeds this value, a loss of contact occurs. The concept of effective width was introduced by Meyerhof (1953) who proposed that at the ultimate bearing capacity of the foundation , it could be assumed that the contact pressure is identical to that for a centrally loaded foundation but of reduced width. Highter and Anders (1985) who complemented Meyerhof earlier work and provided design charts for four cases of a footing subjected to two-way eccentricity depending on the magnitudes of el/Land eb/B :
Casel: el/L ~ 116 and et/B ~ 1/6. The effective area for this condition is shown in Figure 2-8, where: (2-5a)
A' =..!..B1L1 2
in which:
B, =B(1.5- 3:
b
Effective
L = L(1.5 _ 3e, I 1
\.
area
:
(2-5c)
L }
The effective length L', is the larger of the two dimensions, that is B1 or
L 1. So, the effective width is B'::; A'/L' Case 1/ : O
The effective area for this case is shown in Figure 2-9a, where:
(2-6a)
~
I
I
i
I
L1
j
L
I
1
I(
B
..I
Figure 2·8. Effective area for two-way eccentricity case 1
in which, the magnitudes of L 1 and L2 can be determined from Figure 2-9b . The effective width is
A' B' = - - - - - - - - L1orL2 (whichever is larger)
(2-6b)
The effective length is: L'
=L1 or L2
(whichever is larger)
(2-6c)
55
Effective area
B
L
- --+~"'~r/?f--
I I
,
1
•
(a) O.5~--.......------.----.........- -.........--..,
e!JB= 0.167 0.1
O.3~r+-+-~~~~~~~:---rO'~--4 0.06
0.04
0.11----+---
For
Figure 2-9. Effective area for two-way eccentricity case 2
56
Case 11/: elll < 1/6 and O
N
=
2.(B 1 + B 2)L 2
(2-7a)
The effective width is : B,=A' L
(2-7b)
Effective length is quantified as:
L' =L
(2-7c)
The magnitudes of 81 and 82 can be determined from Figure 2-10b.
Case IV: eJL < 1/6 and
area for this case, which can be quantified as: A'
= L2B +2.(B + B 2)(L - L2) 2
(2-8a)
The effective width is:
A'
B'=-
(2-8b)
L
The effective length is:
L' = L
(2-8c)
Step 3: adjust the bearing capacity equation for inclination and eccentricity. Load inclination factors are added to equation 2-6 to count for load inclination as follows: =
qU11
en, ~cAc +yDNq~qAq + ~ y8'
Ny~yA y
(2-9)
where ACT Aq , and A.{ are load inclination factors calculated as (Vesic 1970, Hunt 1986):
A_[1- V +8'L'ccot~ H ]m
(2-10a)
q -
A "I
H
V +B'L'ccot~
A = /" c
]m+1
=[1------
q
1-A q
N c tan~
(for c and ~soil)
(2-10b)
(2-10c)
57
L
Effective area
B·--~
I...
(a) 0.5 r-o----r------,---,---r------,
edt-> ,-.......-+--0.1671----+-----1
0.1
•
0.08 O.31-H-+-~~~~~~~~-O.06----l
0.04 0.02
POo .. ~
0\
~
';)
0.1 eill- =
0.4
For """""IIf~~~~:-+-""'" obtaining BzlB 0.6 0.8
BtIB, BzlB
-_.-
.. ..
- - ---~)---------------_....
Figure 2-10. Effective area for two-way eccentricity case 3
58
.....---B--~ )or 1
L
~~ffl?:t---+-.Effective
area
(a)
0.20 For 0 btai taimng B 2/B \0
.-l "'1'
O.-l~
o
o
.
0.02 = eJ.1L
0.2
0.4
For obtaining L 2/L 0.6 0.8
1.0
B 2/B, LVL
(b) Figure 2-11. Effective area for two-way eccentricity case 4
59
A =1C
mH B'L'cN C
(for c soil)
(2-10d)
where H :::: horizontal component of load, V = vertical component of load, and m is an exponential factor relating B/L or LIB ratios as follows: For load on short side: (2+B/L) mb = -7-(1-+-B-/L-+)
(2-10e)
For load on long side: ml =
(2+L/B) (1+L/B)
2.3.2. Evaluation of the Dynamic Bearing Capacity using modified bearing capacity factors Prakash (1981) proposed new bearing capacity factors that are functions of the angle of internal friction of soil 4> and the eccentricity eb of the footing, and expressed in terms of the ratio of eccentricity eb to width B of the footing. The loss of contact of the footing width with increases in eccentricity was accounted for while evaluating the new bearing capacity factors henceforth called dynamic bearing capacity factors . The dynamic bearing capacity factors Nc , Nq • N, have been plotted in Figures 2-12 through 2-14, to be used in lieu of the calculated values using equation 2-4. Prakash (1981) also proposed the following values for the dynamic bearing shape factors to be used with the dynamic bearing capacity factors:
Sy
=l+(~ -O .68)(~)-(~ -0.43 )(~ r
Sq = 1.0 for all shapes of footings
(2.11a) (2.11b)
c;,c :::: 1.2 for square footings
= 1.0 for strip footings (LIB ~ 8)
(2.11c)
2.3.3 FEMA-356 Procedure for dynamic bearing capacity Soil yield, uplift, and rocking of shallow rigid foundations under seismic-induced moment loading can reduce the ductility demand in a structure. Accordingly, FEMA-356 allowed rotational yield to occur under earthquake loading. In the absence of moment loading, the vertical load capacity of a rectangular footing QUit is given by: QUit
= qultBL
(2-12)
where L is the length of footing. If seismic moment loading is present in addition to the vertical load, FEMA-356 assumes, based on earlier studies (Bartlett 1976) that the ultimate moment capacity, Me, is dependent on the ratio of the vertical
60
load to the vertical capacity of the soil. This theory assumes a rigid footing and that the contact stresses are proportional to vertical displacement and remain elastic until the vertical capacity is reached. Hence, a factor of safety, FOS, can be defined as FOS= qUlt/q • where q is the vertical contact stress according to the vertical load on footing. If the factor of safety is greater than 2, uplift will occur prior to plastic yielding of the soil. On the other hand, if FOS is less than or equal to 2 the soil at the toe yields v\before uplift starts. r
II I I
.e/S c 0.4
0.3
o.r
0.2
0.0
'E" ~
c:
:; 20 I-,f-I.~~.,L+----+-----=---+---l----+---!----l----L--:------l
9 " 4(f
'"
'-c" c:
10l-l--Jr+-I--I+--+----+---+--
o
Q
10
.. 20
40
- t - ---:-----1----t
50
50
70
e/B
Nc
0.0
94.83
D.l 0.2
. 66.60 S4.45
0.3
36.30
0.4
18,15
80
90
10~'
Cohesion factor No
Figure 2-12 . Bearing capacity Factor Nc vs. ~ and e/B (After Prakash 1981)
0-2
" 0.1
0.0
-s
.~
!i~
30l--+--A-7I'~A",.e.=---_+----l----+-.....,._-r__--t--_+_-______j
~
:: st:
~ 201-J-,fN.~--_+--_+_--+--+---1_--t---L_:;:;_-_j
.,
C,
c:
e/B 0.0 0.1 0.2
<:
N~
81.27 10 56.09 45,18 0.3 30.1a 0.4 15.06 01L...----.l.--....L---'--...,-J~-~:__-_+.:_~___:_f:_-__:::::__-~ . 40 ' 50 90 70 80 30 ·0 10 20 Surchel'9s factor N ~
Figure 2-13. Bearing capacity Factor Nq vs. ~ and e/B (After Prakash 1981)
61
etB '"
40
I 0.4
0.3
=
, 0.0
,.,---
.~
U
I
0.1'
0.2
,........ V ~ ~ /" »> :E .. !~ / -e
.2
I
! ,,
30
~
.1:. "Q)
c
:;: 20
o
V
I1l ~
i
c
to o
o
.
I 10
20
30
40
50
60
70
80
4J=40-
-
90
etB
Ny
0.0 0,1
0,2 0.3
115.80 71.80 41.60 18.50
QA
4.62 100
110
Weight factor Nl'
Figure 2-14. Bearing capacity Factor Nl' vs. 4> and e/B (After Prakash 1981) On the basis of these assumptions, the moment capacity of a rectangular footing can be expressed as:
M QL(l-_l_J =
c
» .....1;.
~
..
- .
2
(2-13)
FOS
where Q = the vertical loading on footing and L is the length of footing in the direction of bending. I
2.3.4 Richards method for dynamic bearing capacity This procedure was developed by Richards et aJ (1993) and is the only method that directly correlates the dynamic bearing capacity of the foundation to the earthquake parameters . This theory assumes that seismic settlements of shallow foundations on granular soils that are not attributable to liquefaction can be explained in terms of seismic degradation of bearing capacity. This procedure as shown in Figure 2-15 uses a simplified Coulomb's failure mechanism extended to dynam ic earthquake situation. According to this theory, the dynamic bearing capacity for a strip foundation can be evaluated as: (2-14)
62
B
P /' /'
~
H::BtonpA
Y/'(OrOndfl
-----
~---r----
/'
H/tonpp
-I
L
Figure 2·15. Simplified Coulomb's failure surface for dynamic bearing capacity evaluation .
where, qUE is the dynamic ultimate bearing capacity; NeE, NOE , and NyE are dynamic bearing capacity factors evaluated in terms of the angle of internal friction of soil, and the earthquake acceleration ratio. Figure 2-16 shows the variations of the dynamic bearing capacity factors as a ratio of the static values determined by equation 2-7 with earthquake acceleration ratio tan 8 for different values of soil friction angle ~, where: tan 8
=-.JSL I-k
(2-15)
v
where, kh is the horizontal coefficient of acceleration due to the earthquake, and k" is the vertical coefficient of acceleration due to the earthquake. The seismic settlement of sand soils according to this theory can be evaluated as:
y2 k*
SEq
= O.174- i
Ag A
tanPA
(2-22)
in which, V = Peak or pseudo-velocity response spectrum (m/sec): A = acceleration coefficient for the design earthquake ; g = acceleration due to gravity rn/sec"); k~ settlement; and PA (9.81
= critical acceleration coefficient for incipient foundation = the inclination angle of the failure surface's active wedge
region with respect to the horizontal as shown in Figure 2-15. The values of k~ and PA can be obtained from Figures 2-17 and 2-18 respectively .
63
NcEINeS 1.0
'.0
0.8
0.8
'.0
0.8
0.4 fone =
~ l-kr
(cJ
(b)
(0)
Figure 2~16. Seismic to static bearing capacity ratios
3.0
<...
/.0
Q
U
tl: ~
, , I
o0
0.1
.....<:J o V'l
/.0
,(~ ~ l~~ ' , ',
, , ,, , , ,
0.2
0.3
1',,~~),~ ~~~ ,,.,.,, 0.1
0 .4 0
a2
------ :< - --- -
. ~ 4.0
0.3
0 .4
4.0
"0 Vi 3.0
3.0
2.0r
2.0
I
~o
o
~o
(c) ~ .. 30o [
o
1
I
I
,
J
0.1
t
I . ,
r
rJ2
1
z
F
t,
I ,
Q.3
f
!
I !
of . . .
0.4 0
( d)¢: 40o I
t
0 ./
1<; - -
I
•
•
,
I!
0.2
J
)
_
•
J (
a3
t
.
I
I
0.4
Figure 2~17. Critical Acceleration k~ for incipient foundation settlement
64
2.0 r----r----r----r----,------r---..--,
1.5
0.5
o
0./
0.2
0.3
0.4
0.5
0.6
Horizootot Acceleration Coefficient. kh
Figure 2-18. Tangent of active Coulomb Wedge Angle PA
2.3 Example1 Calculate the stiffness coefficients for a strip foundation for the purposes of seismic analysis. The width of the foundation is 5m and the length is 20 m. The distance from ground level to the bottom of the foundation is 5 m and the thickness Is 1 m. The estimated static load on the footing was 35000 Kn. The soil is a deep gravelly sand deposit with angle of internal friction 38°, and dry unit Borings were taken and groundwater was not weight equals16 kNfm 3 . encountered. The standard penetration resistance of the layer has been determined to be 21 blows per 300 mm using a standard sampler, a drill rig with a safety hammer efficiency of 60%, a drill rod of length 10m, and a borehole of diameter 127 mm (5 inches). The effective peak acceleration for the site was determined to be 0.35 g. Use the approximate solution for a circular footing resting on the surface of elastic half-space .
65
5o/ufion
DATA: Foundation width
B:= S'm
Foundation lenqth
l- := 20·m
Depth from ground surface
1/ := ? ·m
foundation thickness
d:= l -m
Unit weight
,
of soil
Poisson ratio for the soil medium
Ang le
iKN'
y := ;6 :? m v := 0,:-70'
of internal friction
~ := '58deCl
Effective peak acceleration
r
~a:;:::
Measured penetration resistance from SPT test
O~(
.:/'?
N;= 21
STEP 1: Determine correction factors to SPT biowcount Reffering to Tabl e 1-1 the correction factors are : Overburden pressure Effective vertical stress at the foundation level
cr;=
y.t/
r . -nl" r ,/lI l ( e rVL1i..:::; ,....l,/j !-"
-
'1 \ r
/
c = 80kPa Overburden correction factor
tWa
CN := 9,18 · ---;;CN
=
( eo.uatim! - 16b)
1.09'5
66
/j
Energy ratio:
Ce
/i'
Borehole diameter:
C~ :=
.•
Rod length: /
, / Sarnpling method
:=
1.0
f.o (,R := !.o
C5
:=
1.0
, since safety hammer is 6010 efficient , since diameter is 5 in. (127 mm)
.slnce rod length is 10m
.since standard sampler used
Corrected standard penetration resistance
N60 = 22.962
( equaJ:,ict11 -
I?)
STEP 2: Evaluate shear modulus f rom the information of the 3PT test
I Maxlrnu rn (initiaI) shear rnodu Ius
Um." := 200ClO.N60' .j
r.
{Ama)'..
.p, ( e"""t im! - IBJ
(J,
p 5T
= u: :,'.2 C~ /'./ IVI~AP a
For effective peak ground accel eration of 0.4 and site class C Table 1-1
Effective shear modulus ratior
Rf
:=
O,BO
Effective shear rndulus
u = B9002MPa STEP 3: Evaluation of the equiivalent radiUS for each de gree of fre edom
Translational modes
l
~
r? ,_ _
"'1 .-
-\ ( eCju3tl' Cf1 2_ - 'ta;
1t
67
Rocking modes
\0.28
-z;
'£. ~ I --"
I?
Iv ·v·> i '
I
?Jt)
'''n .- [
transverse axis parallel to width
~ I?
. ,_
IY 2 ·-
longitudinal axis parallel to length
Rr2 =
i6 -B--" l [ 5·Jt
\O.2So I
( eqJaLion2 - ld)
)
8.071m
Torsional mode
Rr. = 15.782 m STEP 4 : Calculate the stiffness coefficients for the transforrned circular footing
assume axis x is parallel to the length. and yaxis is parallel to the width
4·G,·i(v ~v:=-
Vertical translation
I-v
K~;v
I
Translation along transverse axis
I
m
Kshi :=
~h i
Translation along longitudinal axis
'1 : ' MN = ?.o9 x ,0
B'~'Rn
2-v
5 MN = 2A:?? x 10 111
~h2 := ~hJ
68
8 -.:A.R,.? k,., ,; :=
Rocking about transverse axis
'
( eo..uatll7.12 - 2()
2
3 ·( I - v ) -rn
6 MN krl = 1.0756 x 10 m
8 ·(..1f. ·R "r2'
~2 :=
Rocking about longitud inal axis
2
3·( I - v )
( eCitJati0'l2 - 2c)
1Jl
? MN Kr2 = 1.92 x 10 - ' m
1(.7
I.
l uT5ion
KJ,
:=
v
16 f. v - .(..1. 3
2 m
( eo.u8tim2 - 2d)
/,. MN
kt = 1.24:? x 10° -" m
STEP 5: Evaluation of th e shape f actors a. for each mode The shape fa ctors are ca lculated from Figure 2- 3 for LIB = 4 a v :=
1,13
a t:=
1.1 1
ahl :=
1.1 7?
v.. r1 .= i ,IBe,.»
N
;
a r 2 := i.16
a h2 := 1.06
STEP 6: EvaluatioD of t he embedment factors
•
Qf or
each mode
The embedment fa ctors are calculat ed from Figure 2-4 for d=lm and R values f or each mode
, .
I
vertical
d
ratiO:=
Rv
h-t " Pv ;= r., Of)·' '7./ 1'/
from figure 2-4
f rans!atiCf'al
ratio = 0 .117
rabo :=
d RJ(i
from fig ure 2-4
J3h := 1.2? 69
Rocking about transverse axis
d
rat io
ratio =
~ rI :=
from figure 2-4
0.062 1.08
d
rali o ._ .
Rocking about longitudinal axis
t(r2
from figure 2-4
ratio
J3 r2
:= i.2'-3
rat io
= 0 .015
~ t :=
from figure 2-4
1.26
STEP 7: Evaluate the ad justed st iffn ess co efficients based on shape and embedment f actors
_
Vertical
7 MN
.
Kva = 5.8i9 x;O
rn
Tanslational a long transverse axis I
K5hla
~
..-::\{)
= ./ ,:/ \/ x
10 -:.c../ MN j
m
Tra nslational along longitudinal axis
Rockiriq about transverse axis
L
j
._
"na ·-
kr j .Nv.. n!. Ct
,
t-' :'":
70
Rocking about ionqitudlnal axis
k.ta
z;
'=
N'N
1,8:?2 x lOCI -' m
2.4 Example2 Following seismic response analysis for the problem in example 1, the following estimates of the peak dynamic forces acting on the foundation are as follows: Vertical force = 6500 kN Horizontal force in the direction of the longitudinal axis of the footing
=12500 kN
Horizontal force in the direction of the transverse axis of the footing = 14000 kN Moment about the longitudinal axis Moment about the transverse axis
=42000 kNm
=83000 kNm
If the estimated static load on the footing was 35000 kN, evaluate the dynamic bearing capacity of the foundation
5 01ufion
DATA: Foundation width
B:= 5>-r;.
Foundation 1611gth
[.,:=
Depth from groulld surface
i7 := 7-m
foundation thickness
d := l-m
Unit weight of soil
'Y
20·m
kN := :6 7
rn Angle of internal frjction Vertical static force
~ :=
58deC]
V5 := ?:70 00·kN
71
Step 1: Evaluate seismic f orc es
VD:= 6e700·kN
Vertical se ismic force Horizontal seismic force in the transverse direction
Hi := 14000 -kN
Horizontal seism ic force in the longitudinal direction
H~ := j 2::?OOlN
Moment about the t r ansverse axis
Mf := B?OOO·kN·m
pI\- := 42000-kN·m
Moment about the longitudinal axis
Step 2: Calcu late equiva lent dimensions to account for load eccentricity
M..I
Eccentricity in the longitudina l direction
el := -
-
Vs+ Vr; Ecc entricity in t he longItudinal direction
et:= -
-
Rat io of eccentricity in the longitudinal direction to t he length
Ratio
of eccentricity in th e transverse
= !.0 12 m
e.c
V5 + Vr; e iII!
"=
ell = 0 .1
-
' . L. e~
dire ction to the width cit:= -
B
ett :::; 0 .202
since ei/L < 116 and O
of B1/B
BIB:= O,B::?
ratio of 52/5
( f iOiJrel - lOb)
B2B := 0,21
(\. f'lopreI - lOb' i/) I
Distance 51 in Figure 10 -a
BI := BH3·B
f?1= 4.2e7 m
Dist ance 51 in Figur e 1O-a
172 := B2B·B
B2 = l.OSm
Effective area Effe ctive width
Effective length
f\; := 05 ·( [:>1 + B2)·~ A;. Bd
I~
:= -
L.
l d := ~
=
2
e7? m
Bd ;::: 2,6 '? m ~d
= 20 m 72
Step 3 : Evalu ate t he dynamic bearing capacity inclu ding incli nat ion and ecce nt ricity Consider case of horizonta l load acting on t ho transverse direc t ion Bearing capacity f actors e lt · tal (
Nq ._
~)
.( tan ( "17
.deC! + .? .(j)) ) 2
Nq
t~ t . 2· ( l~ q + I) -tan ( ep)
=
Q 'I ," 48 .7 J)
Ny = 78 .024 ( eqJat.icn
2-
"1-)
Shap e f actors
(,y
= 0.9 ( fable 2 - I)
Inclinat ion factors
'- '
L.
'l1
:=
B
+
l
1.8
111
f?
1+
L.
'}.. 0. :=
The bea ring stress capacit y
Clult
2,70 1 x 10
?
!:.Pa
The ultimate load The factor
of safety is t hen given by:
f5 .-
fS
=
5.269
73
The bearing stress capacity
%It
= 5.12 x 10 7 kPa
The ultimate load
The factor of safety ls then given by:
P5 := ----;::::::=======
f5 = 4.5140'
Step 4: Check slid ing resistance Interface friction angle between soil and concr ete 4
The friction capac ity
Rf = L957 x ;0 .kN
The factor of safety is then given by:
kf f5 :=-
HI
f5 = 1.582
74
2.5
eferences
ATC (1996) Seismic evaluation and retrofit of concrete buildings volume 1," A TC 40, Applied Technology Council, Redwood City, California. II
Bartlett, P.E., (1976) , "Foundation rocking on a clay soil, " Report no. 154, M.E. thesis, University of Auckland, Auckland, New Zealand . FEMA (2000) . "Prestandard and commentary for the seismic rehabilitation of buildings", FEMA-356 , Federal Emergency Management. Washington, D.C. FHWA (1995) "Seismic retrofitting manual for highway bridges," Federal Highway Administration. Washington, DC. Gazetas, G. (1991). Foundation vibrations, Chapter 15 in Foundation Engineering Handbook, 2nd edition, H.-Y. Fang, ed., Van Nostrand Reinhold, New York, pp.553-593. Highter, W.H., and Anders, J.C. (1985) . "Dimensioning footings subjected to eccentric loads," Journal of Geotechnical Engineering, ASCE, Vol. 111, No.8, pp.659-665. Hunt, R.E. (1986), "Geotechnical Engineering Techniques and Practices," McGraw Hill, inc ., New Jersey, 729 pp. Lam, I.P. and Martin, G.R. (1986), "Seismic Design of Highway Bridge Foundations," Report No. FHWAIRD-86-102, U.S. Department of Transportation, Federal Highway Administration, McLean , Virginia , 167 p. Meyerhof, G.G. (1953), "The bearing capacity of foundations under eccentric loads, u Proceedings :rd International Conference on Soil Mechanics and Foundation Engineering, Vol. 1 ppA40-445.
Prakash, S. (1981), "Soil dynamics," McGraw Hill, Inc., New Jersey, 426 pp.
Richards, R., Jr., Elms, D.G., and Budhu, M. (1993). "Seismic bearing capacity
and settlements of foundations," Journal of Geotechnical Engineering, ASCE,
Vol. 119, NoA, pp.662-674.
Vesic, A.S. (1973)" Analysis of Ultimate Loads of Shallow Foundations, Chapter
3 in Foundation Engineering Handbook, 1st edition, H.-Y. Fang, ed., Van Nostrand Reinhold, New York, pp.553-593 .
75
3
SEISMIC EVALUATION
OF
PILE FOUNDA TIONS
76
3-SEISMIC EVALUATION OF PILE FOUNDATIONS
3.1 General Pile foundations generally consist of pile groups connected to a pile cap with pile diameters usually less than or equal to 24 inches . Pile foundations are preferred under the following conditions: • • •
The upper soil layers are weak or susceptible to liquefaction; Excessive scour is likely to occur; or Future excavation is planned in the vicinity of the structure .
Essentially, the seismic response of piles requires consideration of six degrees of freedom ; that is, three translational components and three rotational components. The lateral soil reactions are usually mobilized along the top 5 to 10 pile diameters . The axial soil resistances, however, develop at greater depths . Hence, the axial and lateral capacities of piles are considered to be uncoupled. In general, seismic design of pile foundations is a three-step process , which includes determination of the seismic demands, evaluation of the pile foundation capacities, and finally comparing demands to capacities and assessing the seismic response of the foundation . For a proper determination of the seismic demand, it is imperative to estimate precisely the foundation stiffness to be included in the overall structural model for determination of the demands. The seismic response of a pile group foundation depends on the response of individual piles under both lateral and axial loads and on the lateral response of the pile cap. It is conventional to compare the lateral displacements under seismic loads to acceptable levels of displacements dictated by the design criteria of the project. In assessing the vertical response of pile foundations under vertical seismic loads, the pile loads are compared to the compressive capacities of the piles and any potential uplift.
3.2 PiIe Foundation Stiffness Solis are inherently nonlinear, starting from incredibly minute load levels. Lateral loads on piles are resisted by the surrounding soil. Therefore, piles exhibit non linear load-deflection characteristics. This behavior is represented assuming the pile as a beam supported on Winkler springs that are characterized by non-linear p-y curves for lateral loading , or t-z and q-z for vertical loading. These curves characterize the latera! soil resistance per unit length of pile as a function of the displacement. These relationships are generally developed on the basis of semi empirical curves , which reflect the nonlinear resistance of the local soil surrounding the pile at a certain depth. The two most commonly used p-y models are those proposed by Matlock (1970) for soft clay and by Reese et al. (1974) for sand. The most commonly used t-z and q-z models are those developed by McVay at al. (1989). The general form of a single pile stiffness matrix can take the form:
77
-K
0
«;
0
0
Kzz
0
0
0
0
s:
0
0
-KByx
Ko,y 0
0
0
Key
0
0
0
0
0
0
s;
K .", 0
0
0
0
s;
0
0
0
0
.~oy
(3-1 )
in which, Kxx, Kyy are the lateral stiffnesses; Kzz is the axial stiffness; and Kxeyand Kyox are corresponding coupled stiffnesses between shear and overturning moment. There are two methods for modeling the behavior of pile groups for seismic response studies, the soil-pile stiffness as will be explained in the following subsections. 3.2.1 Coupled Pile Foundation Stiffness Matrix In this method, a quasi dynamic analysis for the pile group is conducted by applying loading (either as forces or displacements) at the interface node between the superstructure and foundation model using linearized properties for the soils. Linearized properties for a single pile can be achieved by assuming secant foundation stiffness at 0.5 to 0.65 of peak deflection (Lam et al., 1998). A stiffness matrix can be obtained by prescribing a unit deformation vector for each degree of the six degrees of freedom, while keeping the other five degrees zero. The resultant force vector corresponding to each unit deformation vector can be used to form the corresponding column vector in the stiffness matrix. The stiffness matrix must be positive definite otherwise, numerical problems may be expected when the stiffness matrix is implemented in the overall structural model. One way to ensure that the stiffness matrix is positive definite is to invert it and check that the diagonal elements in the inverted (compliance) matrix are positive values . Programs such as LPILE (Reese et at., 1997) and FLPIER (Hoit and McVay, 1996) may be used to establish the foundation stlffness matrix or the nonlinear load deformation characteristics of the pile group. 3.2.2 Simplified Procedure for Pile Group Stiffness Matrix This simplified method involves five basic steps (Lam et al. r 1991) as follows: 1. Determine the stiffness coefficient of a single pile under lateral loadinq, 2. Determine the stiffness coefficient of a single pile under axial loading. 3. Superimpose the stiffness of individual piles to obtain the pile group stiffness. 4. Solve for the stiffness contribution of the pile cap. 5. Superimpose the stiffness of the pile cap to the pile group.
78
Details of the each step are described herein. Step 1: Single Pile under Lateral Load
Lam and Martin (1986) came to a realization that lateral load-deflection characteristics representtng the overall stiffness of the soil-pile system are dominated by the elastic pile stiffness over the nonlinear soil behavior. Moreover, the localized zone of influence is limited to the upper five to ten pile diameters. Hence, they concluded that linear solutions may be adequate for pile stiffness evaluations. They developed single-layer pile-head stiffness design charts for lateral loading as presented in Figures 3-1 to 3-4 for the no embedment case and in Figures 3-5 to 3-7 for the embedded pile case. These charts are applicable for piles up to 24 inches. Two parameters are required to define the soil-pile system: the pile bending stiffness, EI, and the coefficient of variation f of soil reaction modulus Es with depth. The coefficient f has units of force/unit volume. Recommendations for the values of f in sand have been published (Terzaghi, 1955; O'Neill and Murchison, 1983) and shown in Figure 3-8. Values for piles in clay soils are given in Figure 3-9 (Lam et al., 1991). Examination of these charts yields the following observations : 1. Average value of soil conditions for the upper five pile diameters should be used when reading values from these charts. 2. Embedment effects on stiffness are larger for slender piles and tend to reduce for stiffer piles. For a depth increase from 0 to 3 m (0 to 10ft), the stiffness is likely to increase by almost 150% for rigid piles and 350% for slender piles. 3. The effect of the embedment depth is greater in dense sands than for slightly compact sand. 4. The lateral stiffness reduces by changing the boundary condition of the pile-to-cap connection from fully restrained to partially restrained condition. Hence, a realistic representation of the pile-head connection is very significant and often of the same importance of the selection of soil parameters. Step 2: Single Pile under Axial Load
A graphical procedure that gives the axial stiffness coefficient and includes soil layering and slippage along the sides of the pile was developed by Lam and Martin (1986) and consists of the following steps: Compute the axial soil load displacements relationships by assuming a rigid pile condition. This step can be done by calculating the ultimate pile capacity using conventional procedures (Figures 3-9 and 3-10) for skin friction and end-bearing capacities for the different soil layers.
79
:: 1071--§De~~.~gl
-
-
z
, 10 00
~
6 -
=
: -
-
Coeff. of Variation of Soil Reaction Modulus with Depth, f (1bf1O 3 ) I
10 10
10 11
!
1111l1l
I
I I JI l I !
10 ' 2
Bending Stiffness, EI (LB-IN 2)
I I ./
K- 1.07SS-E-1 T3
T= (~al )1/5
Figure 3-1.Lateral pile-head stiffness for fixed-head condition (Lam and Martin 1986).
80
l//:
~V
......
s :.0
»: ' /
~0
~
~
10
~~
9
"//
1/
~/, V
~v v
~v
c: 0
:I
~
10 8 //
V
~
~R'"
Kr-
<,
'/
V// /v
'iii
a::
./
....0< h
~
~
~ ~
~ r;; ~~
1010
~ .,..
fa CD c: !E CJ)
'/
v
r-,
r-, {=2OO f-100
" <,
~1:;60
"f -10
r-, "'=5
1:;0.5 ~ ~':;1
1=0.1
'/
h
!/'/'/
1/
Coeff. of Variation of Soil Reaction Modulus wtlh Deoth. f (IbJIn 3 )
Bending StIffness, El (1b-ln 2 )
=Ka-li+ ~a"e Mt :::Ks6"5+K e·e Pt
,
J 1
K = 1.499"E-1
T3
I
I I I I r
T= (Ef- ·
yl!
Figure 3-2. Rotational pile-head stiffness (Lam and Martin 1986) .
81
./':
~ ~ ~:-/'
~ 1/
./
J
./
/ h l<:/
......:: /./
v ./
I~
1.....-:
~
t?
~
~
/v ~ L::: "'v V
:1.0'
-7
.,,
V/ V/
"",?";
.><;,.
y,. X
»:
>< "., ?
"
~
'./
1/
/ / ' ".
') P<;-.."
t>
V'
~"
t'-.. -, r-...
»:
"7' ./
/'
17
7
t>
-: -:
V
.,, A ... t =200 I;::: 150 100
,=
<,
.r-; <, vV'
><
~~
A.
./ ./
~ ::::
"x
x
~ ~;...; v
~
'" <>
~
1=80 1=60 '"
f~40
1=20 r- f= 10 ~ 1= 5 t-.. f = 1 r-..
1=0.5 f = 0..1
eoefl. of Varia on of SoH Ree.ction Modulus with Depth, f {lblln S } T [TTTTIT I T 1 111111
1010
10' 1
1012
BendingStiffness, EI (lb-ln2 )
I I /
(
•I
!--......IiI
,
1
J t I
I
Figure 3-3 , Cross-coupling pile-head stiffness (Lam and Martin 1986).
82
c
I:;:::~
--:::. r;::; v
L:::
~ 10 6 "d c
8
"C
as
~
(I)
J:I
$
It:
...
.---:__
v
V 10 5
(I)
-I
-::::---
-:: ,.,-
,I-'
»>
G)
a;
....-
-'"
/'
i--'
I--'
V V I--'
V 4 10
V
\:12( \f\
v
v
.......-
V
V
~
V'\
,......
'\
...,)t'\\ \
\l\\\f\
I--"
V
v
tZ
as...
...----: v
......-
"'"
V
,g
~ en
V
---v
v
----
~
l\\
f\ f=200 1\ f= 150 f=100 I' f= 80 1\ f= 60 1\ f;;;;4O
1\\
1\ f= 5
I\.
v
, 1\
V-
I--"
~~~;ijg~~~*mt~~~mJ~l\~\.\ : V
....-
\.
f;;;;20 f= 10
~-+--H--H++++--t-+-+-H-+l+il---+-+-H-t+tH--"n\1\
f;;;; 1 f r::s 0.5
f = 0.1 Coeff. of Variation of Soil Reaction
-+---I---t-+++1I-+tt---f--I--H-+-Hf+---L---J.......I..-JL....I..Lu.J----'1 \
Modulus with Depth. (lbJin 3 ) 3 10 -+--+--+-++++trH---+--+-i-+++t+t--r--r-T"T'TTTr'lr--"""T"""""T""T""TT'T'TT1 I I I 111111 I I I I I III 13 11 12 109 10 10 10 1010
Bending Stiffness, EI (Ib-in 2) Free head Pile Stiffness /
J
I I
I
r
1
I 1
I
Figure 3-4. Lateral pile-head stiffness for free-head condition (Lam and Martin 1986).
83
10 7 : Embedment I
--
--.
t: ~ .0
-
I I
I I
a
-
ee
I
- - - - - 51
-
Co
I
-
.
- ·10'
~
- 10 6 ~
::
CD
-
--
I
--
~
CD .~
~ 10
5
CD
c:
.,
CiS
-
+:
0
~
...
,," ...
""
v
v
_.....
~
".
I
~
....
......
10 3
»>
.
...
e- :_
~
=
....
... 1
... ............ ""
...
/
...
~--'"
F
....
1-/ v~
~
CI)
~
~
--'" "...
.....
~
V
-'......
...
...........
ct) tI)
m c 10 4
".
... I ....
f=100_~
'C
co
...
"...
-
......
-
.....
..
Coeff. of Variation 01 Soil reaction MOdulus with D9eth, f llb/in 3 )
""
I
10
10
10
I I I I1III
11
10
I
-
J I I 1III
12
Bending Stiffness, EI (Ib-in 2 )
PIle H9w:l at Gradll Lswl
Embedded pae Head
Figure 3-5. Lateral embedded pile-head stiffness for fixed-head cond ition (Lam and Martin 1986)
84
I
= :
I
I
I I III
Embedment 0'
~
- - -
-
-
c:
.~
- - 5' ----- 10'
/?
"'C
-
m 1010
IA
a:
V
~~
C :.;:;..
k?
v?ft< ~
~
~;.
~.~~
10 9
E
v ".
:.;:::0
~~
(J)
m
E 0 a::
V .1,,1
~
~~
tf' III
~,
" .... f 100 <,
Q)
0
,.
i-'..~
v:
A.
I
c::
~
,.,
.0
(J) (J)
~
l.t?
10 8
h
v' !{/ I/.
%
1"I.-
1/,tP
vii'~
v~
........
v.~7
.....
b)
~tv
<,
['I'
I'
r-.
'~
f = 10 f=1
1= ~ tottt-
t-
f =0.1 ~ 1= r-
I-
/~
~'l
t-
~
I-
,,9 ~
Coeff. of Variation of Soil Reaction Modulus wilh Depth, F Oblin 3 ) J I I 1'111
I
10 10
1011
I
I
I
~
I II I '
1012
Bending Stiffness J EI (lb-in 2 )
r r-
/77
Pile Head at GradeL.ab61
Embedment Deplh 118
~
. ,(I
711~
u
1
Embedded Pile Head
Figure 3-6. Embedded pile-head rotational stiffness (Lam and Martin 1986).
85
l
-_
:
I
I
IIIII
Embedment
0' - ---- 5' ----10'
-
-
-
! ~
!
"'
.... f= 1 <,
f= 0.1
CosH. of Variation of SoU Reactlon Modulus with Deeth. f 1Ib/in 3 ) I
I I II
1Il1
Bending Stiffness, EI (Ib-in 2
T T T TTTlT
)
Pt :::Kl).D+KM·e M l = K~e·D+ Ke·a
Pile Head at Gracre level
Figure
3~7 .
Embedded PUe Head
Embedded pile cross-coupling pile-head stiffness (Lam and Martin 1986).
86
Friction Angle, ~
aoo
~
VERY VERY LOOSE MEDIUM DENSE DENSE LOOSE 100 -+-...;;;.;;.=;.....a....-------'--------'-----r--<-----;
Recommended by Terzaghl. 1955
(AfterO'Nen andMurchison, 1983) _
80
~
-+-------L.------L..----t------+-.----;
c:
s
:;.
t
~
~
c:
60
Q)
'u
:E CD o
o
~ -t------r----r-----t-'!SJr~
U
e
~ ::::J
CIJ
20 -+-----t-----+-r--"""'1:I'''''''"------t------j
O-+----f-----+----+------+-----;
o
20
40
60
80
100
Relative Density, Dr (Percent) Figure 3-8. Recommended coefficient of variation in subgrade modulus (f) with depth of cohesion less soils.
87
Blowcount (blOWs! ft)
o
2 4
8
so
15 STIFF
VERYBTIFF
HARD
6)'100 c:
13
-! ~
...en
vary Soft
80
CD
~ eo C)
.c
:J C/)
.5
c: 40 -1----..---
~ ttl
'C
~
15
20-+--
=z::
8 o
o
1
2 3 Cohesion (ksf)
4
5
Figure 3-8. Recommended coefficient of variation in subgrade modulus (f) with depth of cohesive soils.
88
Two pile load displacement curves can be developed based on the published skin-friction and end-bearing pile displacement relationships . The following two forms can be used as follows:
F = Fmax(2~~cr -~)
(for friction curve)
(3-2)
(for tip resistance curve)
(3-3)
Where, Fmax and Q max are the ultimate skin friction and point resistance of the pile; Z is the displacement at any loading stage; Zcp and Zcf are the ultimate displacement of the pile corresponding to the ultimate point resistance and ultimate skin friction resistance can be evaluated (Vesic 1977) as follows: (3-4) where C p is an empirical coefficient that ranges from 0.02 to 0.05 the pile diameter; L is the embedded length of pile; and qo is the ultimate point stress (3-5)
where 0 is the diameter of the pile; and C, is an empirical coefficient evaluated as: (3-6) ii
The resulting load-displacement curve can then be obtained by summing the skin friction and end bearing capacities at each axial displacement. This curve represents the state of an axially-rigid pile and is a lower bound on the actual pile displacements.
iii
Calculate the flexible pile load displacement curve from the rigid pile solution . This can be achieved by adding an additional component of displacement at each load level Q to determine the axial displacement at the pile head due to the compression of the pile under axial load but neglecting the surrounding soil. This displacement is given by:
89
t
Q."#"J' total up ward capaciry
Thickness WL. Angle of Shearing Dens,ty Resistance Zone pi negligible resistance
:.if increasing resistance
Zol'lt!
Total
LJ
'V
'0
G. (1
Ll
11
$1
LJ
length. L etc,
Zolle of constant
etc.
resis(MC e
H
L1
B. Diameter
t- I
Downward CaPJtcitv
Qcap (- J == P, N q At +
I:
F di Pi
i= I
Where P, = Effective vert. stress at tip
"i
a ,'i
L;,
t _J
r, = L
Lz
'Yi
S
P
@
L() +- 20B
i=O
= Bearing capacity factor A, = Bearing area at. tip
N
tan
~
F Ji:;; Effective horiz. stress factor for downward load (Coefficient of skin Iriction) PI';:: Effective vert. stress at depth i i
Pi
8, = a l ==
=L
'Lj
rj
5 P @ Lv';' 20 B
j~O
Friction angle between pile/pier and soil at depth i
Surface area of pile/pier pet" unit length
t-I
Upward Capacity
Qcap(+)
=
L
F ui Pi tan 0i as
Lt
i= I
=
Where F u,' Effective horiz.. stress factor for uaward load " other parameters as for downward capacity
Figure 3-9. Pile capacity for granular soils (NAVFAC, 1986)
90
t
Q
,.~ . ; . total upward capacity
+ Q""""~J '
total downward capa city
Thickness
Zone of negligible resistance
WL. Density
Cohesion
Ll)
'Yl ;
C ;l
L:
Y.
c,
L}
Tmal length,
L
Zone ofconstant resistance
etc.
etc.
H
L,
B. Diameter
l-l
Downward Capacity
Qcap (-) =
Ct
N c At +
l:
C ai
as
4
i=1 Where c
I
= Cohesion strength of soil
N•. = Bearing capacity factor
9.0 for depths greater than 4B
A J :; Bearing area at tip
,.. = Cohesion strength of soil at depth i a. = Surface area of pile/pier per unit length
C
Upward Capacity
Qcap (+) ~
l-l
I
eai
as i;
:'= 1
Where parameters are as for downward capacity
Figure 3-10. Pile capacity for cohesive soils (NAVFAC, 1986)
91
Table 3-1. Pile bearing capacity factor Nq (NAVFAC 1986) Angle of Shearing Resistance for soil;
Placement Dr;",'en Pil~
Drihed ?ier
26
:18
so
~1
52
-5~
S4
5$
10
1,5
21
24
29
35
/32
50
5
1
I
s
12
10
I
14
~
(degrees)
5$
~7
ss
59
62
77 1 86
120
I
21
17
30
25
1115
!
43
38
40+
60
n
I
Table 3-2. Friction angle, 0 (NAVFAC 1986) s
Pile/pier Materf31 Steel
20
concrete
0 .75
~
T1mbel"
0-75
<)
Table 3-3. Pile horizontal stress factors, Fdi and Fui(NAVFAC 1986) Pile/Pier TYpe
Driven H·pile Drive
straight
Downward
upward
Fe!i
FuJ
low
high
lOW
high
0.5
10
0.3
0.5
1.0
1.5
0.6
2.0
1 .0
0 .9
0-3
prismatic pile
I
Dri ve tapered pile
1.5
Driven i erteo p-re
0.4
I
or ruec Pier
0 .7
l
I
1.0 1 .3
I
0 .5
04
Table 3-4. Pile (NAVFAC 1986) Plfe Material
consistency of Soft
I
COhesion, ct
Ad11es1on, C3
low
high
luw
very sort
a
250
0
250
SOft
2S0
500
2 S0
480
high
limber and
Mec: . sti ff
sao
1000
118C
Concrete
Stiff
'1 000
200ll
750
I
9$0
very St!ff
:l0C'Q
4000
950
I 1
1300
verv sost
0
250
0
Soft:
250
SOO
:150
460
Med. Stiff
500
1000
460
7CC
Stiff
1COO
2000
7CO
720
very St iff
2000
4000
no
750
Ste el
I
I
!
750
250
92
o = QL C
(3-7)
AE
where Q = axial load; L = length; and AE =axial rigidity of pile. The flexible pile solution can then be obtained by adding Dc from equation 3-6 to the rigid-pile load displacement curve at pile loads that correspond to the rigid pile load displacement curve. This curve is an upper bound on the actual pile displacements. iv
The actual solution is bounded by the rigid and flexible load-displacement solutions derived in steps ii and iii and depends on the nature of the soil pile system . Hence, the actual axial load displacement curve can be obtained by averaging the curves from steps ii and iii for the rigid and the flexible pile solutions.
v
The actual pile stiffness can be evaluated by determining a value for the pile secant stiffness from the load displacement curve obtained in step iv over the range of expected displacements. The pile secant stiffness is used as the equivalent axial stiffness coefficients in the pile stiffness matrix. Expected displacements can be obtained from the range of expected axial loads which would be in the range of 50 to 70 percent of the ultimate pile capacity.
The steps involved In this procedure are illustrated in an example shown in Figure 3-9. In this example the equivalent stiffness was taken at an expected load of 70% (195 kip) of the ultimate pile load (278 kips). The corresponding displacement was determined to be 0.125 in. It should be noted that the range of displacements can also be set by the design engineer according to the project's design criteria. The recommended LRFD guidelines for the seismic design of highway bridges recommend the following simple equation for the determination of the axial stiffness of the pile: \
Kv
=
E A
1.25-PL
I ·:
\,
L
~
-
c'
J.',
\
r
~
(
(3-8)
-/
This equation and the computer solution for the stiffness of this problem were posted on Figure 3-11 for comparison with the above procedure. It can be shown that the above solution represent a best estimate for stiffness calculation. Equation 3-7 will always yield a stiffer pile in the axial direction . FLPIER software was used to determine an equivalent stiffness for the same axial load level used above .
93
300
r--~--------.r--------,----------,---------,
STEP 3
-
200
-
~
~
-._
-
-
-_.
_
-
. :
-
- - - ,-,,- - - --- - - -- - --, - - - - -
--
J
150 - '
.. --- --., ---~ ~ - --------j . ---- ---.. --
100 ..
, ,, _,"_j_... __"_._\__... ";~~:= curve
15
~
-
,
STEP 4
~
/ : -
STEP 5 : :J ~ • _
.'"
" .••• _ -l _
~'
_
' .0
•.
'• •
-. ,K
. '"
' 0.
' 0.
.. . . ¥
•• •
Rigid pile solution - - Rexible pile solution Correct solution ........AxiaJ pile stiffnes
LRFD-Sa..UTfOO
-
cavPUTER S(1lJT1O',1
oe----------;---------;-----------! o
0.5
0.25
0.75
Displacement (in)
Figure 3-11. Steps involved in determining the pile axial stiffness coefficient
94
Step 3: Pile Group Stiffness
The stiffness of the single pile can be used to establish the pile group stiffness matrix. If the pile group consists of vertical piles, the stiffness summation procedure is relatively straight forward. The stiffness for the translational displacement terms (the two horizontal and the vertical displacements) and the cross-coupling terms can be obtained by multiplying the corresponding stiffness components of an Individual pile by the number of piles. It is worth mentioning that a unit rotation at the pile cap wilt introduce translational displacements and corresponding forces at each pile head (e.g. vertical forces for rocking rotation and lateral pile forces for torsional rotation), which will work together among the piles and will result in an additional moment reaction on the overall pile group . ln general, the axial stiffness of the piles will dominate the rotational stiffness of the group. Therefore, the rotational stiffness terms require consideration of this additional stiffness component. The following equation (Lam et ai, 1992) can be used to develop the rotational terms of a pile group: N
KRG
= N K RP + L.K&1S~
n=1
In which, KRG and KRP are the rotational stiffness of the pile group and an individual pile respectively; N is the number of piles in the pile group; Kon is the translational stiffness coefficient of an individual pile (axial for group rocking stiffness and lateral for group torsional stiffness); and Sn is the distance between the nth pile and the axis of rotation. STEP 4: Stiffness Contribution of the Pile Cap
In addition to the component of soil resistance acting on piles, a pile foundation may experience additional resistances due to soil acting on the pile cap. These resistances include: (i) passive pressure acting on the front face of the pile cap; (ii) side shears acting on the two vertical side surfaces of the pile cap; and (iii) base shear acting on the bottom of the pile cap. Because of potential interaction between the pile cap and the supporting piles, shear forces at the bottom of the pile cap and the two side surfaces can be ignored . Therefore, the passive pressure soil resistance on the vertical pile cap face is the only component that can be added to the stiffness and resistance obtained from the pile members. This is contingent on stable level ground conditions . The pile cap stiffness can be estimated as the ultimate soil capacity divided by an estimated displacement to mobilize this capacity . Centrifuge tests (Gadre , 1997) showed that the deflection level to reach the ultimate pile cap capacity occurs at about 0.02 times the embedment depth. This equivalent linear secant stiffness can be added to the stiffness of the piles. Methods of estimating the ultimate passive pressure capacity of a pile cap are illustrated in Figure 3-12. 95
SolI Parametam
Effective UnitWT 'Y
Shear Strength
Cohesion c.
I
Friction an Ie
Awrage Passive Pressure H
(J
m
on Pile Cap at Mid Depth zm
L Passive PressUrB 0
Overburden
on Pile Cap
Pf888U1V
Effective ~~ = Y'Z
Recommended M81hod for Passive Preaaure Capacity (I) For Frictional SoD (41 Onty): Average Passive Pressure Capacity = K p '1 I Z m Kp Based on Caquot & Kerisel (1948) for Intmface FrictionAngle ~ ;:;;; 0.6' (II)
For COhesive Son (c Only): Based on Rankine Pressure Theory
Average Passive Pressure capacity =
(III) For c and , Soils: Average Passive pressure Capacity =
'1 • Zm + 2 c
Kp 1 'Z m + 2c Tan{45° + 4112)
Total Force capacity on Pile cap Per Unn Width
Average Passive Pressure Capacity x Thickness of Cap (H)
=
Figure 3-12. Passive pressure capacity of pile cap (Lam et al 1998)
96
Step 5: Superimpose the Stiffness of the Pile Cap to the Pile Group
The resultant pile cap stiffness obtained from step 4 can be added to the diagonal latera! translational stiffness coefficients in the pile group stiffness matrix for the total pile group~pile cap stiffness matrix. 3.2.3 Pile Group Effects The above procedure does not account for group effects which relate to the influence of the adjacent piles in affecting the soil support characteristics. Full scale tests by a number of investigators demonstrate that the lateral capacity of a pile in a pile group may be less than that of a single pile due to the interaction between closely spaced piles in the group (group efficiency). As the pile spacing reduced, the reduction in lateral capacity becomes more pronounced. In general, pile spacing of less than three to five pile diameters are necessary before the effects of pile interaction becomes significant in practical terms. Type and strength of soil, number of piles, and loading level are other factors that may affect the efficiency and lateral stiffness of the pile. Moreover, in addition to group effect, gapping and potential cyclic degradation were also subject of many investigations (e.g., Brown et aI., 1987, McVay et aI., 1995). It has been shown that a concept based on p-multiplier applied on the standard static loading p-y curves can work reasonably to account for pile group and cyclic degradation effects.
3.3 Solved Example A portion of a viaduct is to be supported on pile-supported bents. Each bent will consist of two columns, each 1 m in diameter, transferring the load to a cast-in place pile cap on top of the foundation piles. A static load of 2500 kN including the pile cap was calculated. Six 12-m HP 12 X 84 steel piles were selected for the foundation. The soil properties as well as pile configuration are shown in Figure 3-11. If the bridge is to be built across a large flood plain, calculate the stiffness matrix of the pile foundation.
5o/ufion Uee ~if7P!ified procedure for pile qroup ~f ifffle7? ~inq/e
pilelafera! 51iffflC5~
tiP 12X 84- Jeplh Depfh 01 which ibe evdudlon of lafera! =>fjffne~~ lror: chari
Jr := 510 .f7f7 0 := ' ·Jr
o = 1,'5'5
f7
en
1---<::>,
""/14""'·""-:-:""' _ /fIW-'i'<''-:':''' i/N,M 711.i'''''llk &''''''''; ~ .---l-
I
0"A~T~;;;;-ffAV~-
I
I
3.°
x
~m
1.
~ rl
lI
I '
E
": A M
I
I I __L
t
r
i.
II" r ; i, r ••_.
i I
2.0m
-,;-Y
_t__
• '
GWT
Loose , fine , to medium sand $:: 33 ' '( :: 18.5 kNI m3
8.5m
:
-
-.. .
I
.-' I I.
<:>
+ , -+-----,--
-- . -.
i O,75 m
I
0.3'"
,(1')1I:
: [ -1
--f--:-:·---r.
J• _._._. . ,
!---+
im I
t . to-15 m I
!
1
t
. _.l--... . _
'_..
Dark brown dense sand ;; 38 0
y= 19 kNI m3
-{
~
3.0m
NOT TO SCALE
4.0",
Section A-A
Figure 3-10. Pile foundation configuration for example 1.
98
,
fra~/afional 5Iiff~?
in local y~y [ron liqare 5~'5 (or(.-;1(; Ib/ir6 and e(1bedr7enl ~
'
M
/(yy :;::
~ Ibf IJ ·10 '- . In
82 (f
Rolafional5fiffoc?? aboul x-x [ron Iiao«: 5.-0 (or ("'1(; Ib/ir6 and e(1bedr?enT = 82 (I /(ex
kN
= (;1191 (1. rad
Croo» couplirq 51if(ne~ [ron [tqore 5~1
(or (=1(; Ib/ir6 and e(1bedr7enl .-; 8.2 (I
8endinq 5Iiff~~ abod local asi» y-y
EIyy
fran?lafional 51if(oc?? Inlocal x-x [ron [iqore 5~'5 (or ("",1(; Ib/lr6 and e(1bedr?enT 82(1
/(xx :=
;:: 1/121
kN ·(1
2
~ Ibf ID ·10 .
In
P'
R0Iaflonal51Iffnc:>:> abcoi y-y [ron [iqore 5-(; (or (~20 Ib/ir6 and e(1bedr?enT 82 (f d
/(e
r
:= 2,/
·10
8 , /bf ·m· rad
99
t>endinq 5liffoc55 aboul local asi» y-y
EIyy = /7727 kN ·f?
Trandaflonal 511ffoc55 in local y-y [ron [iqure 5-5 (or (,,:>20 Ib/lrO and ef?bedf'lenf ~ 82 (f
RoIaflonal 5liffoc5~
y-y [ron fiqure 5-6 for (=20 Ib/irO and err!bedf'lenf ~ 82 fI about
I(xx.
I(ey :=
I(ey
Cro~~ couplirq 5Itffoc~~
[ron [iqore 5-7
;=
~
z
1M
/D ·/0 . in
1M
B
2.1 ·10 ·in· rad
= 25727
I(xey := 52
kN (1-
rad
6
·/0 ·Ibf
(or (~2O Ib/irO and ef?bedf'1enf .,:: 82 ff I(xey =
11-254- kN
5irl{//e pile I1xjal51i!(OC~5 Calwlafion of pilevllif?ale capacity Dafa: DepfhalSWT
Thickne::o of
(Ir~1 sobrerqed layer
Hz:= 8:5·f?
100
rhidne~~
H;.
of second ~bf?Crqed layer
:= :!J of?
kN
"'I 0= 18'5 . j
Unit weiqh! of fir:d layer
'
•
;.
f?
5ubr?erqed unl! wdqh! of flr~! layer
tAl
8. 1 ;.
1/x b --
/'7
Unl! weiqhf of second layer
12
kN
;=;c;; o~ {'1
5ubf"lerqed unif weiqhf of second layer
'-I := 0'5 ' /'7
Lenqfhof pile above6WT
Lenqfh of pile In~ide flr~1 layer
l. 2 := 8.'5 -n
Lenqfh of pile in~ide eecond layer
5kin [ridton faclor
Fd
tiP eedion deplh
dp
HP sedlon tlonqe width
Wf :=
f;earinq area al lip
f;earinq capacily faclor Fddlon anqle bdween pile and ~il
:=
:=
10
12.28 -ln
IZ,ZC;;'5
-in
WI;:::
051
I\f
0./
=
/'7
2 f'1
N(/ :=86
o := 2-0 -deq 101
Erredive
~fre:x>
of fop of
eMIr GV'lr
Effedive ~re::>::> of fop of second
layer
r Z := r, + Yl:vb .nz r z = 12~.4-;;
kFa
Effedive ~rex> of pile lip
f'1 := f' Z f', =
+ y Z~b
1~/.o2
-tJ;J Pt =157 I..n,,; - --
---'
kFa
r oin! Rc:>i~af)Ce
op
= 1;;/~,4- kN
GirOJrJ. lire:a// rt
Dala for cdcddionof ('1(Zi.irvn di::>p/acerlCrJf::>: crlbedded depfh of pile
enpatca poin! rc>i~ooxcodficial
enpcricaJ::>kin ltkiion codliciat
102
Ullirnfe: poin! ~re:!!>!}
Ullrrnfe: dl!!>p1accncrl cor{c>pandlrq 10u/Ilrde: paIn! t=blarcc
UI/if'lafe: d/~p1ace:f'lCfIf
corrc~pandirq 10:>Kin Iridian rc~idaocc:
Zd = 02
rJ/'I
P oin!(e:~bl(jr= OJNe:
I
o co ""= 0 p .[.-!...-'I~ Z cp)
coo "
x := coo
o 'I
0
z :=00 .l rr: .. lo.rJ/'I
.00'5
101
0'5'5
2.51
./0'5
2-'74
2'5'5
:571
XJ'5
4-20
1!/0'5
682
(0)
:;130'5 '71:;
«)
'7.%'7
y :=coo
20
1'500 r -
-
-,-
-
-
J:;I'5 J:;I'5 )
, - - - - - - ,-
-
--,
1000
O L-
o
---'---
-...L-
- --'-
- -'
20
103
5kfn [tidion asve
ZI ;:= 0 ,0'5 om 0.2 of'7/')
coo
x
::= coa
y
;:= coa
0
0
00CY'5
164
O~'
4!/0
./0'5
'521
2'5'5
%4
50'5
%4
tzo»
564
!/BO'5
564
9509
%4
'I
::=
(0 )
(I )
20
%4- )
5kin Frfdfon!? c~;:;fafKX C UNe I
t
I
-
100
-
I
obfaincd by!:JJmirK} fhe: din (ridion anderJ bearfrK} copoalte» aI oxh axial di!>P/accncrtl,
Riqid pile: XJ!vlion b
(0 ) X
:= cor
y ;:= cor
0
0
0.00'5
21/
D5'5
01!/
./0'5
81'5
2'5'5
9W
50'5
981
15~
IU'5
~80'5
1'5!/8
9509
1819
cor ::=
20
'I
1819 } 104
R'iqid pile ~olufion if:> obfained
by xf1f1inq fhe din [ricilon and end bearlnq
capocliie» af each axial dif:>place:f'7enf.
x := cor
cor :=
(0)
y:= cor
(1)
')
0
0
0 ,00'5
211
,0 '5'5
f,15
./0'5
81'5
,2'5'5
960
50'5
984
150'5
124-'5
5.80'5
1'558
95f,9
1819
20
1819 )
Riqid Pile: 5olufion
2000
1'500 ~
~ ~
j
_Y_l000
~
~
'500
Oi------..L.---......L.----L---~
o
10
20
x Di:?pJocet?CfJ1 (f'lf'i)
105
Flexible pileXJ!ulion /~ ochieYcd by:!Vf'7I'?/rq ibepilehead di~placer?Cnf of eachloadleYd 10 lheriqid pile::dulion Area of pileeedion 0pi! := (0 Z11 6i5 8/'5
9GO 984 12+'5 /'558
/8i9
/819 ) ·iN
106
/'dl.loJ pile=!ufion b oMaiocd by o/eroqln; lhe fJaiMe: and tiqld pile !X>!u/io~
Ac ;= v;:=
z
:~
o '\
o
k(O) (I)
I1c
0.'5/6
211
1::;2'5
61;;
1,61
8/'5
207
960
2.16
984
;;66
/21-'5
6.1
1'5;;8
12.9
/819
2;;.5'5 /819 )
I1dl.ld :301u/ion
2OO0r------r -----.- ---.---~-__,
1'500 '"'
~
8 -.l
I:l
y v
--/000 z
~ '500
6raphlcd ~iffnc~ :>oIu/ion i~ oblaincd by dderrJlninq a vdoe for lhe pile: xcad diffrt=" (or lhe odod =/ufion al10X of fheullinak oxjdpile: capacity
Di:>placenad cotcopondirq10 107. of lhe vllimle: copociiv
L\ := 2.62
·rJrJ
/Uiolpile: ::Nfocx> f1
LRFD :>o/ufion(e:qualion ;;-8)
107
Pile group ~/jf[rx;~~
n :=G
50/
Dldance 10 kax/~ of roidion
:=
0:1 ·(7
DI:;-fance fa Yraxi» of rofalion
"XX6 := n·"xx
xN
KXX6
= 10'507G.I (7
r. '('(6
=
kN 1'5/GI4-Y5 ("J
('1
KXeY6 :: 8540'5.8G
xl\!
KYeX6 = 221'521.4-4- kN
KeX6 = 18829'50.GI
KeY6 = 4078908.1'5
r. eZ6 = 2'5/4-:5G,4-'5
kN ('1.
rad
kN rad
('1.
kN ('1.
rod
108
:5 focz fhe bridqc i~ fo be buill oaoso a krqe flood plaIn corrlr/buflon of fk f'OX'~ re!>idance of Ik pile cop10 Ik lafad ~iff~ CQflrof befaken ido = rr.:iderafion
Tbado«: Ik ::J'rffne= nalrixcan be o:pr~d o»
K . '
- 1882.9'50.6 1
0
0
0
0
0
0
0
/882C;X;.6/
0
0
0
0
0
4018908.1'5
0
0
0
0
0
10"5076.1
0
0
0
1'516/4'.1'5
0
0
0
~0/2-66~,12
0
22.1'52./,4-4
- /882.9 '50.6 1
0
0 2 2./'521.#
1
2.'514-%.4-'5 )
3.4 References Brown, D., Reese , L.• and O'Niell, M.(1987), "Cyclic lateral loading of a large scale pile," Journal of Geotechnical Engineering, ASCE , Vol. 113, No.11. Gadre, A. (1997), "lateral responseof pile-cap foundation systems and seat-type bridge abutments in dry sand, Ph.d. Dissertation, Rensselaer Polytechnic Institute. Hoit, M.L and McVay , M.C., (1996), FLPIER User's Manual, University of Florida, Gainsville. Florida . Lam, I.P. and Martin , G .R. (1986), "Seismic Design of Highway Bridge Foundations," Report No. FHWAIRD-86-102, U.S. Department of Transportation, Federal Highway Administration, McLean, Virginia, 167 p. Lam, LP., Kapuskar, M., and Chaudhuri,D . (1998) "Modeling of pile footings and drilled shafts for seismic design" Technical Report MCEER-98-0018, Multidisciplinary Center for Earthquake Engineering Research, Buffalo , New York. Lam , I.P. Martin , GR. , and Imbsen, R. (1991) , "Modeling bridge foundations for seismic design and retrofitting," Transportation Research Record 1290. LPILE (1995), "Program lPILE Plus, Versiuon 2.0" Ensoft Inc., Austin , Texas, Matlock, H.(1970), "Correlations for design of laterally loaded piles in soft clay", Offshore Technology Conference, Vol. 1, Houston , pp.579-594.
:fId
McVay, M.C., O'Brien, M., Townsend, F.C., Bloomquist, D.G., AND Caliendo, J.A. (1998), "Numerical analysis of vertically loaded pile qrcups ," ASCE Foundation Engineering Congress, Northwestern University, Illinois, pp.675-690.
109
McVay, M.C., Casper, R. and Shang, T.(1995),"Lateral response of three-row groups in loose to dense sands at 3D and 50 Pile Spacing," Journal of Geotechnical Engineering, ASCE, Vol. 121, NO.5. NAVFAC, (1986), "Foundations & Earth Structures, " Naval Facilities Engineering Command, Design Manual 7.02. O'Neill , M.W. and Murchison, J.M . (1983), "An evaluation of p-y relationships in sands", Report No. PRAC 82-41-1 to THE American Petroleum Institute Terzaghi, K. (1955), "Evaluation of coefficients Geotechnique, vol. 5, No.4, pp.297-326.
of subgrade
reaction",
Vesic, A.S. (1977}, " Design of pile foundations", Transportation Research Board, National Research Council, Washington, D.C.
110
-
--
4-RETAINING WALLS UNDER SEISMIC LOADS 4.1 General During the past earthquakes, gravity earth retaining walls have suffered considerable damage which ranged from negligibly small deformations to disastrous collapses. Performance of retaining walls during past earthquakes has revealed the fact that the damage is much more pronounced jf the wall is extending below the water level. According to Seed and Whitman (1970), failures in walls extending below water level may have resulted from a combination of increased lateral pressure behind the walls, a reduction in water pressure on the outside of the wall and a loss of strength due to liquefaction. As an example, extensive failure of quay walls during the 1960 Chilean earthquake and the 1964 Niigata earthquake in Japan have been attributed to backfill liquefaction. Fewer cases were reported for walls constructed above the water level. Few cases of minor movements of bridge abutments were reported during both the San Fernando and Alaska earthquakes. This section will focus on two of the most commonly retaining walls used in construction, gravity retaining walls and cantilever retaining walls. Under static conditions, these walls will sustain body forces related to the mass of the wall, soil pressures, and any external forces. Equilibrium of these forces is mandated for a proper design of the retaining wall. During an earthquake. however, inertial forces and changes in soil strength may breach equilibrium and cause unfavorable deformation of the wall. Failure in sliding, tilting, or bending mode may occur when excessive permanent deformations take place. Gravity walls usually fail by rigid~body mechanisms such as sliding, which occur when the lateral pressures on the back of the wall produce a thrust that exceeds the available sliding resistance on the base of the wall. Cantilever walls are SUbject to the same failure modes as gravity walls and also to flexural failure modes. If the bending moments required for equilibrium exceeds the flexural strength of the wall, flexural failure may occur. The seismic performance of retaining walls is usually evaluated using pseudo~ static methods, where the transient dynamic pressures induced by the earthquake are added to the initial gravitational static forces exerted on the wall before the occurrence of the earthquake. Hence, a brief overview of the static earth pressure is presented in the following section.
4.2 Static Pressures on Retaining Walls Static earth pressures on retaining walls are strongly affected by the movements of both the wall and soil. Minimum active earth pressures are mobilized when the wall moves away from the soil behind it, and this movement is sufficient enough to activate the soil strength behind the wall. On the other hand, the maximum passive earth pressures develop as a result of movement of the wall towards the soil. A number of simplified approaches are available for the computation of static
112
loads on retaining walls. The most commonly methods used in practice are outlined below.
4.2.1 Rankine Theory According to this theory, the pressure at a point on the back of a retaining wall can be expressed as: PA =
KAO'~ -2C~KA
(4-1)
where O'~ is the vertical effective stress at the point of interest, c is the cohesive strength of the soil, and KA is the coefficient of minimum active earth pressure evaluated as;
K
A
= 1-sincP = tan 2(4S-
1+sincj)
~)
2
(4-2)
For the case of backfills inclined at angle ~ to the horizontal, the following equation can be used to compute 1<.4; K = A
cos~ coS~_~cos2~-coS2cj) 2 2 cos ~ + Jcos ~ - cos ~
(4-3)
For dry cohesionless soil backfill conditions, Rankine theory predicts triangular active pressure oriented parallel to the backfill surface. The active earth pressure resultant, Pa , acts at a point located H/3 above the base of a wall of height H with magnitude: .
1 2 PA =-KAyH
(4-4)
2
The wall pressures under maximum passive conditions are given by:
pp = KpO'~ + 2c~Kp
(4-S)
in which, Kp is the coefficient of maximum passive earth pressure evaluated as:
Kp -- 1+sin~ -- tan 1-sincj)
2(45 +-2cj))
(4-6)
For inclined backfills with angle ~ to the horizontal:
\
113
K = cosf3 cos f3 + ~~~2~=-~~~2
cos p-
~COS2 (3 -
COS
2
(4-7)
The passive earth pressure thrust, Pp , acts a point located H/3 above the base of a wall of height with magnitude:
1 2 Pp =-KpyH
(4-8)
2
Active and passive pressure distributions for various backfill strength characteristics are illustrated in Figure (4-1). It is important to note that the presence of water in the backfill behind a retaining wall influences the effective stresses and hence the lateral earth pressures acting on the wall. Therefore the hydrostatic stress due to the water must be added to the lateral earth pressure.
4.2.2 Coulomb Theory According to this theory, if the wall is allowed to deform enough to mobilize active or passive pressures, lateral earth pressure that develop at the back of the wall may be evaluated using a rigid plastic model to describe the soil behavior. Under minimum active earth pressure conditions, the active force on a wall such as the one shown in Figure 4-2a is obtained from the equilibrium of forces (Figure 4-2b). Force equilibrium is used to determine the magnitude of the soil thrust for both the active and passive conditions. For the critical failure surface, the active thrust on a wall retaining a cohesionless soil can be expressed using equation 4-4. in which the active earth pressure coefficient can be expressed as: (4-9)
in which 0 is the angle of interface friction between the wall and the soil can be determined using Table 4-1, and p and are as shown in Figure 4-2a.
a
For maximum passive conditions, equation 4-8 can be used to determine the passive thrust with the passive earth pressure coefficient calculated as: (4-10)
114
COHESIVE SOIL, NO
GRANUl.AR SOIL
FRICTtoN.AL RESISTANCE
ctMBINf.D COIifSION AND "RCnc»II
ACTIV£ PRESSURES
/ ..~O
~A1~ I.. 'OfF,r.'\\.
9.. ...
KA;TAN2(4S-~)
Zo :~<:/r
~:ICAYZ
O'A:i)'Z-ZC
PA:~yH2lt
',," )'11 2/2 -tCH + 2.,2
~~==X:
P"
Zg.i-f>TAH (45+~f2.)
OA :1'7 1I1N2.(45~M)-2C'DiN(~-~> PA : (~)TAH2(4S-.I2H()oIWIl(.45"'4li: +2C2/y
Ptt-$$IVE PfU$l)flES ,.........2C
J- ,2
< ) l(
""
Kp:DH2(4$"~)
a'p'KpYZ
ppfl(p1'tt2/z
.... "V
~·1'Z.2C
Pp'
TYH2 +ZCH
O"p:;.rz TAN! (45+#Zh2CTAN(45~) pp.;( ~I TAN2(4!S+~)+2CH'aN
(45+eW2)
Figure 4-1. Rankine active and Passive earth Pressure Distributions For Backfills with Various Combinations of Frictional and Cohesive Strength (NAVFAC 1982).
115
~
F
(a)
(b)
Figure 4-2. Wedge of Soil and assumed Failure Surface for Calculation of
Coulomb Active Earth Pressure Coefficient
116
Table 4-1. Typical Interface Friction Angles (NAVFAC 1982) ,<'<""
_-.
-,
~.,d,.
....... __
.. _..." .. ,".".WH,W'.-"N ,mm//M'''''''.-...... ''
~H"~·.,
INTERFACE MATERIALS Mass concrete against
Clean sound rock Clean gravel, gravel-sand mixtures, coarse sand 1 Clean fine to medium sand, silty medium to coarse
sand, silty or clayey gravel Clean fine sand, silly or clayey fine to medium sand " Fine sandy silt, nonplastic sill Very stiff and hard residual or preconsolidaled clay
Medium stiff and stiff ciav and siltv clav
Steel sheet piles against Clean gravel, gravel-sand mixtures, well graded rock fill with spalls Clean sand, silty sand-gravel mixture, single size hard rock fill Silty sand, gravel or sand mixed with silt or clay Fine sandy silt nonplastic silt Formed concrete or Clean gravel, gravel-sand mixtures. concrete sheet piling against well graded rock fill with spalls
Clean sand, silty sand-gravel mixture. single size hard rock fill Silty sand, gravel or sand mixed with sin or clay Fine sandv silt, nonolastic silt
Dressed soft rock on
Dressed hard rock on Dressed hard rock on Masonry on Steel on
Dressed soft rock Dressed soft rock Dressed hard rock wood (cross grain) steel at sheet Dile interlocks
~.....
. ""M'-'
,.",_>
INTERFACE FRICTION ANGlEfD
35
29-31
24-29
19-24
17-19
22-26
17-19
22
17
14
11
22-26
17-22
17
14
35
33
29
26
17
117
4.2.3 Caquot-Kerisel Chart Solutions Caquot and Kerisel developed charts, which provide values for the coefficients of earth pressure modified for the adhesion angle 5 to conform to log-spiral surfaces and are particularly useful in finding values for the passive coefficient of earth pressure Kp for analysis of flexible retaining structures. Values of KA and Kp for various values of <\> and (3, are presented in Figures 4-3 and 4-4 for the cases of sloping wall and sloping backfill. These values are used in the Rankine equations 4-4 and 4-8. The horizontal component of earth pressure Ph can be found from KA cos 0 and Pv = KA sin o. It is important to note that the curves in the figures are for the case of 5 1 ~ =-1.0. The reduction factor R for other values is also displayed in the figures. In general, the Caquot-Kerisel solution, also known as the logarithmic spiral method, gives active earth pressure coefficients KA that are considered to be slightly more accurate than those given by Rankine or Coulomb theory with a small negligibly difference. However, the passive earth pressure coefficients given by this method are considerably more accurate than those given by Rankine or Coulomb theory. It is also worth mentioning that Rankine theory greatly under-predicts actual passive pressures and is seldom used for that purpose. Coulomb theory over-predicts passive pressures, with an error, which increases as the interface angle increases. Therefore it is recommended that Coulomb theory not to be used to evaluate passive earth pressure when 0 > <\>/2. 4.3 Seismic Pressures on Retaining Walls
4.3.1 Mononobe-Okabe Method for Cohesion/ess Soils The most popular method for evaluating the seismic pressures on retaining walls is the Mononobe-Okabe (M-O) method. This method is pseudo-static and based conceptually on the Coulomb method. In the M-O method, pseudo-static accelerations are applied to a Coulomb active (or passive) wedge. The pseudo static soil thrust is then evaluated from force equilibrium of the wedge. It is important to note that this theory is limited to cohesionless dry soil. Also, a constant value for the angle of wall friction, 0, must be assumed. The forces acting on an active wedge in a dry cohesionless backfill are shown in Figure 4-5 (A similar method can be used for passive pressures). As shown, pseudo-static forces that relate the mass of wedge to the coefficients of horizontal and vertical seismic accelerations are also added to the forces that exist under static conditions. The Mononobe-Okabe (M-O) equation for the active earth pressure coefficient for seismic loading can be expressed as: (4-11)
118
II
Tr
it"
,
j
i
I
I!
iii
i.
V II
~
, .,; ~
i
i
&
4
~
i
i 21
I
I til
.;
i
i
A"". ' -
!!!.6,,~
If .51-~ONE ..""".
III
~ A ij .3 ~
i
.21
~B
'0
g
II
...
'I
I [
~
...... ....... _ _ ,.
•
..........
---.......
""-
..... ........
.... __J . ~
1""'110.
I. I 1 I f
~
I I . J ... 1 10! ,
I I I J
J
~
i""""oo.' ..... -
~
.t
! I I I I
•.
r-'
~~'
__
F~.d:
I
~
.~~
[:').:T~'li·-ir I ~=I.e=-~ - -
Figure 4-3. Active and Passive Earth Pressure Coefficients Using Logaritmic Spiral Method for the Case of a Sloping Wall (NAVFAC 1982)
119
JEOOCTlON FACTOR (It) OF ftlfl VARIOUS RATIOS OF
9O,01---l-.....-' 8OC,'-4---I-
I(p
-81+
7OOt-+-++-4-t+--I-A--I-l fOO I---!-----+. 5QO t--t-·-;-'-+-+HYh~.{.A'
40.0 1---+--:"'...-1
30.0 I
I
,~~
4..' I ;;( I
V::+- I
j
!
Y.J
M ~"-,6
2.0
iJ!~~~ftftrrt-HH14Jll{J/
"
'0
i
... ,
n,
I
I
0
I
«
I
I
,
20
!
i
,
I
J
30
J
,
At«;l..£ Of: 1N1ERNAL FRtCTtON.". tBRE£S
,
I
I
40
I
«I
45
Figure 4-4. Active and Passive Earth Pressure Coefficients Using Logaritmic
Spiral Method for the Case of a Sloping Backfill (NAVFAC 1982)
120
tv
-
.....
::J
o
~
o
"'0
(l)
ao"
.2:
-
in which:
'If =
tan-1[~] 1-k
{4-12}
v
and:
ktJ
and kv are coefficients of horizontal and vertical accelerations, which
may be assumed according to FEMA-356 equal to Sos/2.5 and 5osl4
respectively where 50s is the short period spectral acceleration adjusted
for site class as explained in section 1-9; ,
4j) is the internal angle of friction of the backfill;
~ is the backfill inclination;
ois the wall/soil interface friction angle;
9 is the wall batter.
The total active thrust can be evaluated in a form analogous to that developed for static condition, that is, PAE
= ~KAE'YH2(1-ky)
(4-13)
where, PAE is the active thrust and includes the static component in addition to the dynamic increment, 'Y is the total unit weight of the backfill, and H is the wall height. In some cases the base of the retaining wall is embedded to some depth within the foundation soil. Effects of emebedment include the development of passive restraint against sliding. In order for the passive restraint to be mobilized some deformations must be developed. For the passive seismic limit state:
PpE
=
~ KpE'YH2(1- ky)
(4-14)
where KpE is expressed as:
(4-15)
It should be noted that the value of KpE calculated using equation 4-15 increases considerably with increasing the wall/soil friction angle (), Hence the value of this angle should be selected carefully to avoid unconservative values for KpE.
122
4.3.2 Prakash Method for C and "soils Prakash (1981) provided a general solution, which was also based on the Coulomb method for determination of total (static plus dynamic) active earth pressures for C~~ soils. The uniform surcharge effects and only the horizontal inertia forces resulting from the earthquake are included in this method as shown in Figure 4~6. q/unit area
I~'
'I
I
r
I
Jc
He
H
b
Figure 4-6. Forces acting on a wall retaining c~~ soil and subjected to a seismic load The active thrust that includes the static component in addition to the dynamic increment can be evaluated according to this method as:
PAE =yH 2Nar +qHNaq -CHNac
(4~16)
In which y is the total unit weight of the backfill; H is the height of retaining wall free from cracks; q is a unit surcharge per unit area; C is the cohesion of the soil; and Nay, Naq , and Nae are earth pressure coefficients can be calculated as follows: N = (n r +1/2)(tana+tanO)+n/tanO [cos(a+cjl)+khsin(a+c\l)] ~ ~na+O+cjl+~
(4~17a)
123
N
= [(n r +1)(tana.+tanO)][cos(a,+
N
(4-17b)
sin(a.+O+.p+ 0)
aq
= cos(a.+O+.p)secO+coscj>seca.
(4-17c)
sin(a. + e+ ell + 0)
ac
Where, nr = HdH is the ratio of the height of the retaining wall with cracks to the height free from cracks. Other variables are defined as illustrated in Figure 4-6. It is important to note that the earth pressure coefficients are expressed in terms of the angle of inclination of the failure surface a, which makes the solution indeterminate. Hence, a number of potential failure surfaces must be analyzed to determine the critical failure surface. Location of the failure surface, according to the upper-bound limit analysis, will be such that the least amount of resisting force will bring it about (Le., the failure surface is the one that produces the greatest active thrust or the smallest passive thrust). Application of this method and comparison with the Mononobe-Okabe method are illustrated by an example. Example 1: Let us assume a retaining wall of height Hi = 6m inclined at 10° with the vertical and retains soils with unit weight y= 1.732 T/m 3 C ::: 0.5 11m2 and
ttl :=6·,.,
Un# wek]frl of fhe badJill
., :=
Arrjle of In/erna frld/oln of badflll
, :=XJ·dt:4
Co~/on
C:=·~·z
of bacillI!
1.1:52-
.!.... ,.,~
T
f?
Irdlnaflon of wall wifh fhe velNcai
0:= 10.dec;
ttor/rorlal x/!>f'lic codflcienf
Kh:= 0.1
Verf/cal xl!>f'llc codflclerl
Kv:"'O.o
Irlerface anejle bdween !>OIl and wall
5 := 21- ·dt!:tJ (fab/e4--f).
f?;odfll! Irdlnaflon
Ii
:= O·deej
124
!1ononobe ~ Okabe t1e:fhod
'II
KItE
:=
kh "\ alan ( I-k ) v
'II
= ~.111
(co~ (.
;==
co::> (V
(e:quafion 4--12)
deq 9
).(co~ (9))2 .co~ (li + 9 + 'J1 ).[1 +
KItE
== 0.4-~8
F/IE
:== 0.'5
'II ))2
~in(a + ej))·~/n(, - p - V) ]~ (co~(li +9 +'II)·co::>(P -9)) (equafJon 4--11J
PItE
.KItE·y·t1/·(/-kv)
r
= 14-.2-64- -
(e:quafion4--I:?)
f?
Prakadrt1e:fhod 11~5U/'1e
n:=OO
fir!?! c~ of no len~ion crack
5e1e:d failure: 5Urface:~ 50 thai their inc/ina/ion anqle:!> ranqe: approxlnafely frof? to (45++°)
.0
a := 10 -dec; .1'5 -Je:q ..GO .Je:q B (a) := a + 9 +. + a
[Cn + ,~).(lanC a) + Ian (9)) + n2. .fan (9)].(CO~ (a + +) + kh'~in( a + 4
NolB ,a):= - - - - - - - - - - - - - - - - - - - - - - f>in(B( a))
(equafJon 4--/1aJ
Nac(B ,a)
co~( a
+ 9 + ej))·xcC9) + co~( +)·xc( a) (equa/ion4--11c)
:=
. (
~m
2
(
B a.
F/tE(a):=y·t1, .Na/B,a)
))
(equalion 4-~/G)
125
P/rC(a)::::
r
9.4 10.9
f1
12.1 13.1 13.8
F/t£(u)
1~
<..
14.2 13.7 12.7 10.6
"'"
7.1
a
It can be observed that in the absence of the soil cohesion C ,the surcharge q, and the vertical seismic inertia force, Prakash method conforms to the Mononobe-Okabe method. Consider now the M-O equation with the effect of vertical seismic coefficient:
V t:rlica! x/!}rJic cexfflclt:nf
kv :=006
Kh I '1/ := afan ( I -k ) v
'1/ =
6012
f(1rE :=
(CO!}
(equaf/on 4-~/2)
dt:r;
(~
9 '1/)}2
CO!} ('1/ )'(C05 (9))2 .CO!} (8 + 9 + '1/).[ 1+
f(Ae =0.4-0
f'IrE := O.~ ·f(IrE·r ./1 /.(1 - Kv)
~
+ fp'f}/n( -13 - '1/) ] (CO!} (8 + 9 + 'I' )·CO!} (13 - 9)) t:>/n(8
~
(t:quaf/on 4-~ //)
f'!Ie.
= 1~,'518
r
-
rJ
(equal/on 4-~/~)
It can be observed that kv when taken as one-half to two thirds the value of kt" affects PAE by less than 10% (almost 5% in this example). Seed and Whitman (1970) concluded that vertical accelerations can be ignored when the M-O method is used to estimate PAE for typical wall design which agrees with the assumptions of Prakash method. Consider now the effect of soil cohesion in the Prakash method:
126
z·e
Depfh of fefl!>ion cracJ:5
411
(
tic := -y-.fan 4-'5·det1 + z)
tfc=!f? (Fiqure4--f)
tic n'-- .- ti,-tic
1?afio of depfh of fefl!>ion crack fo fhe heleJhf wifh no fefl!>ion crack
n=02 =:>e:Ied failure 5Ur(ace5!XJ fhaf fheir inc/ina/Ion afk:Jle. rafk:Je appro4rJafdy (rof?
+0 to (45+,0)
a := fO .deeJ ,1'5 ·deeJ .. GO ·det:; B (ex) := ex + 9 + 4l + 8 2
[(n + ,'5).(fan( ex) + fan (e)) + n ·fan (O)].(C05 (a + 4l) + kh'5In( a + 4l)) NalB .0.) := - - - - - - - - - - - - - - - . ; . . - - - - - - - - - - . . . ; . . , 5In(B( a.)) (equaf/on 4--/1aJ
C05( a.
+ 9 + 4l)'xc (0) + C05( 4l)'xc (a.)
Nac(B ,a) :=
. ( 51n
(
B a
(equafion4--l1c)
))
P&(0.) := y.tf/" .Noy(B ,a) -C.(tf/-tfc)·Nac(B ,a) (equaflon4--16)
P,A.E(a) =
9.6 11.9
T
rJ
13.7 15.2 16.2
P/le(U)
16.7
,
16.7- ~
15.9 14.2 10.9
5.4
a
It can be observed that the M-O method underestimates the seismic pressures on the walls which retain cohesive soils behind them.
127
4.3.3 Application Point To determine the point of application of the total dynamic earth pressure, it is necessary to determine the distribution of earth pressure along the back of the wall. Experimental results suggest that the point of application act at a higher point than the H/3 of a wall of height H under dynamic loading conditions. The total active thrust PAE, as determined by equations 4-13 and 4-16, can be divided into a static component PA (equation 4-4), and a dynamic component, 8PAE: PAE = PA + aPAE
(4-18)
The static component is known to act at H/3 above the base of the wall. According to Seed and Whitman (1970), the dynamic component can be taken at approximately O.6H. Hence, the total active thrust will act at a height h above the base of the wall calculated as: h = PAH/3+aPAE (O.6H)
PAE
(4-19)
4.3.4 Effect of Saturation on Lateral Earth Pressure For saturated earth-fill, the saturated unit weight of the soil shall be adopted in equation 4-13. For submerged earth-fill, Matsuzawa et al. (1985) developed a procedure to modify the M-O method to account for the presence of pore-water within the backfill. According to this method the excess pore-water pressure in the backfill is represented by the pore pressure ratio, ru defined as :
r. = uexcess . u
(4-20)
0'3c
where, Uexcess is the excess pore water pressure due to liquefaction during the seismic event and O'~c is the effective confined stress. The active soil thrust acting on a wall can be quantified from equation 4-13 using the following modifications for the unit weight Y and the angle 'I' as follows:
Y =Ysub(1-ru)
(4-21)
'I' = tan-1[
(4-22)
Ysat kh ] Ysub(1-ru)(1-k v
The total thrust shall be calculated as the sum of the soil thrust and an equivalent hydrostatic thrust based on a fluid of unit weight Yeq
= 'Yw + ruYsub
(4-23)
128
For partially submerged backfills, the soil thrusts may be calculated using an average unit weight as follows: 'Yav = '}..2 'Ysat
+ (1- A,2}Yd
(4-24)
in which 1I.is the ratio of the height with saturated soil to the total height of the bacfill.
4.4 Seismic Bearing Capacity of Retaining Walls Seismic induced reduction of bearing capacity has been studied by Richards et aI., (1990), and Shi (1993). For simplicity, a "Coulomb-type" of failure mechanism was adopted in these studies within the foundation consisting of an active wedge directly beneath the retaining wall and a passive wedge that provides lateral restraint. According to this method, shear transfer between the footing and foundation soil is conveniently described by a shear transfer coefficient, n, where: (4-25) where, tV is the friction angle of the foundation soil, N is the sum of the vertical forces transmitted to the soil quantified with reference to Figure 4-7 as: N =PAE sin(ow + 0) +W - PPE sin{ow -( 2 )
(4-26)
F is the sum of the horizontal forces' quantified as:
F = PAE cos(ow +O)+khW -PpE cos{ow -82 ):S: Ntanof
(4-27)
The seismic bearing capacity according to this method can be expressed in terms of seismic bearing capacity factors and quantified as:
Pd = CNCE + qN qE + 21 '}'8·NyE
(4-28)
Where, q is the overburden due to depth of the footing; C is the cohesion of the foundation soil; '}' is the unit weight of the foundation soil; B' is equal to 8-2e, where B is the width of the footing and e is eccentricity computed as described below; 0 is the depth of embedment of the footing; and Pd is the seismic bearing pressure. The seismic to static bearing capacity factors (NqE/N qs , NeE/Nes, NYE/N"fS) are expressed in terms of the friction angle of the foundation soil.tV. seismic acceleration coefficient, ~, and shear transfer coefficient, n. These ratios are
129
PAl:
T h
\
T
t
Y c
D
1
1
&f~F N
I...
B
R
I
PpE
~l
Figure 4-7. Forces acting on a retaining wall during a seismic event
130
Phi = 35 degree
Phi = 30 degree
Phi = 40 degree
II)
0"
Z
W
0"
0.6
0.5
Z n=025 0=0.5 n=0.75 0.0' 0.0 0.2 0.4 0.6 0.8 I
I
,
I
J
1.0
0.01 02 0.4
0.6 0.8
1.0
0.0 02 0.4 0.6 0.8
1.0
kh
1.1 1.0
~.;:;oO..L2.0°_L_.1. 30° 4rf'
0.9
I 1=1
f= 0
l/l= 10°
0.8
(,) 0.71
~y,
Z -
~4>= 10°
".~
f/)
0.6 W (,) 0.5
\~
0.4
\
0
\2d'
20
~\
Z
\
~
\ \ 30°
0.3
'\
02
1=2
'40°
\30°
\
\.400
0.1
00' '0.0 0.2 0.4 0.6 0.8 1.0 t
•
'
,
,
0.0
~.2
0.4 0.6 0.8 1.0
0.0 0..2 0.4 0.6 0.8 1.0
kh (~)
NcE/Ncs
131
Phi
Phi = 30 degree
00' '0.0
, 0.2
• 0.4
,,"--','- I 0.6 0.8 1.0
=35 degree
0.0 0.2
Phi = 40 degree
0.0
0.2
0.4
0.6
0.8
1.0
(c) NYe/Nys
Figure 4-8 (continued). Ratio of seismic to static bearing capacity factors (Shi 1993).
132
displayed in Figure 4-8. The ratio for NcelN cs is presented in terms of the friction factor f F/Nkh n tan(~)/~, instead of n.
=
=
The effective stress must be used to compute q in the second term and the submerged unit weight must be used in the third term of equation 4-28, if the foundation is submerged above the base of the footing. If the foundation is submerged below the base of the footing an equivalent unit weight must be used in the third term of equation 4-28 as:
'Yeq
= 'Ysub(1-Z/B)+ y(Z/B)
(4-29)
where Z is the depth to the ground water surface below the base of the footing and B is the width of the footing. If Z is greater or equal to B, then 'Yeq 'Y.
I
=
The seismic bearing capacity is evaluated by comparing the seismic vertical force resultant at the base of the retaining wall to the seismic bearing capacity of the foundation soils computed with equ~tion 4-28 with eccentricity e computed as:
B Mnet N
e="2-
(4-30)
in which, N is the vertical force resultant determined using equation 4-26, and Moat is the net moment of forces about the toe of the wall (point C in Figure 4-17) calculated as MR-Mo1 where Mo is the overturning moment computed with reference to Figure 4-17 as: Mo = khWYc +PAE cos(5 w +8)h+PpE sin(ow -8 2 )(0/3)(tan82 )
(4-29)
and the resisting moment can be computed as: ~ =
WX c +PAE sin(Bw +8)(B-htan8)+PpE cos(Bw -82 )D/3
(4-30)
4.5 Seismic Stability of Retaining Walls The safety factor against seismic induced bearing capacity failure as: F:'C = PdB' (4-31) N The wall is considered stable under seismic induced loss of bearing capacity if the computed factor of safety F:'c for the peak acceleration is equal to or greater than one.
133
The safety factor against seismic overturning instability is quantified as: O.T _ MR
F5
(4-32)
--
MO
The wall is considered stable with respect to overturning if the computed factor of safety is equal to or greater than one. 4.6 Seismic Displacements of Retaining Walls Estimation of the permanent displacement of retaining walls is necessary for performance based seismic design. Richards and Elms (1979) method is used for the estimation' of this allowable permanent displacement. According to this method, the fevel of acceleration that is just large enough to cause the wall to slide on its base is defined as the yield acceleration defined as:
[t
.I.
ay = an'l'b -
PAECOS(O+9)-PAESin(o+9] W 9
(4-33)
where, ~b is the angle of internal friction of the soil beneath the wall's base. This method works with the M-O method for calculation of PAE. Hence, the solution of equation 4-33 must be obtained iteratively because the M-O method requires that ay be known. According to this method, the permanent displacement is quantified as: 2
dperm
3
=0.087 vmax ~max
(4-34)
ay
where,
Vmax is
the peak ground velocity. amax is the peak ground acceleration.
Example 2: Check the seismic stability of the reinforced concrete cantilever wall shown in Figure 4-9 for the maximum considered earthquake (2500 years return period). The wall is located near Memphis, Tennessee (350 3' latitude, _90 0 O' longitude). The site consists mainly of coarse sand of unit weight 18 kN/m3. The average initial shear modulus up to a depth of 100 meters is 180 MPa.
5olufion For a5i1e localion: ~O ~' Ialifude, 9000' IOl7t:Jifude, and a 2'500 year!>!??, fhe 5horl period 5pedral acceleralion of bedrock (rorl fhe U,5, 6eoloqical ?UNey !>ife (hHp:// earlhc,uake,u!>tJ!>.eJov/)
51> := 0,6/ '4
134
--
O.6m
-7!!!&/l1I§:,····· <1>=30°
r =18.00 kN/m3 7.5m
p
!
121
,
I 3'
B=6m
Figure 4~9. Geometry of the retaining wall Irveraqe ~ear f'?Odulu~
6 rlax := 180 ·I1Pa
kN
y:=18·
Un!f wek;hf of 50/1
~
f?
yc = 2;;,%;;
Un/f weicjhf of concrde
kN ~
f?
Irveraqe *ar wave vdocily
For f>fiff 5011 wlfh (:xci/on 1-8)
V,,:=
J 6~.q
rI ~
180 ro/f> <' V ~ <' ;;6~ I'll~ ~ife if> dG:Jf>/fied G:J D
5/fe codiden! 5ife adju!>fed !>pedral accderaf/on
F Q := 1,264 5D~ :=F o ·55
505
tforizonlal accderaf/on coefficien! Verlical accderaf/on coefficienl
tfelc;hf tf
V ~ = ;;1;;.1%
of rela/nine; wall
kh:= 2,~.q
50!>
(Tablel - 2)
=0.84-1
cj
kh = 0;:;»
kv:=OO tf := 1.'5 .f?
135
t1
Ef'lkdnenl depth
r:= ~.('I
Widfhofbax
f? := 6·f'I
Wall fhlckne~!}
T
Wldlhalloe
f?, := 01·f'I
~ax Ih/ck~!}
T,
/ft7tJle of IrJerna frldloln of backlitl
• := X)
Inc/ina/Ion of wall wilh fhe venied
:= 0.6·('1
:= 0.':5 '('1
·de:
9 := O.de:
aw := 0 ·da:;
Inferface at7tJle k/wa:n !>OIl and wall Irlefface at7tJle belween !XJil and foundallon
(conxNaflve Q!}~I)f'Iplion)
8 f := !JO ·deq
f?acJ!illlrd/naflon
fl
KA :::: O,!;!J!J
:= O·deq
(equal/on 4-J7)
kh ')
'" := alan
K~:::::
( I-kv )
(equallon 4-~/2)
'" :::: 18.114- da:; (CO!} (cjl - 9 _ '" ))2
eo~ (",).(eo~(e))2 ·eo~ (a w + 9 + 'II)'
~~-m~8-+-cjl~.-~~n(~+--~fl---",\)rr
[
(eo~ (0: + 9 + '" ),co~(fl _ 9))
1+
Klie :::: 0614 P ~ := 0'5 .KAe·y·t/2.(1 -kv )
(equallon4-~II)
Plfe = !JIOBI IN f'I
(equaflon 4--/~) 136
. (C05 Cit' + 9 _ 'I' ))2
f(PI:
.=-~[-.:.-.:...-!>in 8 w +' .!>in (, + ~ - 'I'
.
CO!> (
'I' ).(co!> (9))2
'co~ (8 w - 9 + '1').
1-
{co!> (8 _ 0 + 'I' ).co!> w
)] _
(~ _ OJ)
/(PE = Z~28
(eC/ualion4--I/)
PPt: := 0,'5 ·f(Pt:·y .11/ .(1 -kv) PPt: = 188,'581 IN f'J
(e:qualion4--I!'J)
CalQJlallon ofwdqhf!>
WI
:=
(tt- T,).(t? -t?,- T).y
WZ
:= (11- T,).T·yc
W.'
:=
khW1
l~
t?·T,·yc
W :=WI +WZ +W"'
(t? - t?,- T)
AI:= +t?/+T Z ItZ
:=
It)
n
.'5·T + t?,
~.
M
~'
:=.'5.t?
f;! :=
t?Z
¢
(I1-T,) Z
+ T,
t?/
T,
:=-
f;Z
Z
=
4 f'J
= 02'5 f?
Didance of re:xllartl ofverlical force5 frof'Jloe
X := WI ·1'11 + WZ. t1Z +
w"' .~
X =!'J,Z4-'5 f?
Didance of rt:Xllanf ofhorlzonfal force> frof? foe
Y:= ft?I·(WI + WZ) + t?2·W",]
W
Y = .'(:/52 f?
137
Applicaf/on poinf I!J( := /(At: -/(II
11 Kit";; +AJ(·06·11
h ---.- · - - /(IJE
h = ;;,414
IVJ
Check for dJdlng along fhe box But?rKJIion of drivlnej force!>
Fd
:= PIlE
F d = '568Bf7 kN
+kh'W
IVJ
8uf?f1aflon of rt:~/dinc;force:!>
IN
N:=W
N = 16/.8'54-
f'I
IN
F r := PrE + N·lan (ar)
F r = 628.444
I'?
Fr
F ador of :xidy OCJaln~f ~/idinc;
F8 ~ := F d
F8:>
::=
1./0'5
O/(
Check for overlurn1trJ 81abilizinej f'IOf'leri
11/
HI( := PrE'""3" + W·x
HI( = 2.661
x
10~ kN
.(1
f'I
Overlurniinc; f'IOf?enf
110 := FAt:.h+kh·W,Y Ho
=
~
f'I
2.004 x 10 kN· f?
F ador of t>afely aqain!!i overlurnlnc;
F8 o
:=
HI(
Ho
F8 0
=
1:!J28
nK 138
-14TJ
.
<;;heQ; :;x:i~f?ic bearit?4 capacifv n = 0,86~
PItC + kh' W -Pre
5hear fran!.'ler codficienf
W.fan ( ep)
n :=
5ei~f?ic beari(J(j capacify f acfor~
Nqe./Nq~
N qr := 02-'5
N'1:> := ""./an ( +) ·fan (
NqC
:=
(by inferpo/afion fiqure 4--8a)
r
~ .d,,9' + i
Nqr·Nq:>
N
:>
'1
NqC
Nye./Ny~
N yr :=
Ny:>
:= 2.(Nq:> + I).fan( ep)
NyC
:=
Ol),?'5 Ny:>
= 18,"f01
=
(e;quafion:2 - f
0)
4,6
(by inferpo/afionfiqure4-8c)
=
22.402
(equafion
2 - rb)
NyC =2J28
Nyr·N y:>
fJ I1K- 110 ecc := -Z - --N- ecc = ZJ~ f?
eccenfrlcify
fJo
:=
f;o = /,726
fJ - 2- ·ecc
f?
/
Pd := y.t1,.Nqc + 2 ·y·f;o·NyC
5ei~jc lif?iI fo bearinq pre~t:XJre
Pd = 28/.47'5 kN f?2.
139
l~
_11 _ _.L!
~1~
_~
. : _ _ L.- _ _ ,....._
J
(JI
.. ~
....,---r-r
~
_.1#
F aclor of 5afdy acjain5f Xi5f?ic induced bearinej capacify failure
F 5~ ._ Pd·f?D vC .-
F 5f?C = 0,6:58
COf?pu!ed fador of Mdy for xi!>f?ic beadnej capacily i5 le5!> fhan one. rherefore lo!>!> of beadncj capacify and filfinq of fhe wall i!> o.peded dUrinej fhe 2500 year earfh4uake. ttencel xi!>f?ic rdrofif i!> required. /(drofil ?fraf!!ZJY,' Place a lieback fhrouc;h fhe wal15fef? fo reduce drivinq I'1OMenf!> and Increax Xi5f?ic beadnej capacify.
I
1.0 m
tD'U'-1~~---'-"
- - _.._ _.... F,.cos 15'
~
T
r _______________ F.. sin15°
II
w
~
I
vJ /
1.'' ':' ': ' ' ~ · '0
1,
1
i
{
"-""
f
L-__.. ,...
I
PAE
""."".,,,,F""""""""
I! ,
!
16.5m
h
j
J
.
N
n:= 0.'5
a5!>Urte !>hear frander codficienl of fhe rdrofiHed wall
(3 := 20 ·deq
inclinafion anqle of lie-back rod
PItE + kh' W -PPE -n-W-fan(+)
Force in fiC'back rod
Flie:
:= - - - - - - - - - - n'5ln ((3)·fan (cr.) + co!> ((3)
Flie:
=
1"54.4-4-1 kN f?
140
=:>ei!>f'7ic beariQ:} capacity fador1> \
Nere/Nq~
N qr
:= 0,40
(by inferpolalion fit/ure 4-8a)
NqE := Nq(.Nq~
N q£: = 1.!J6
Nye/Ny~
N yr
:=
02-0
Ny£: := Nyr·Ny~
(by inferpolalion fiqure 4--8c)
NyE =4.48
11Kr := 11R + Flit: ·co!> (rJ)·(I1-I.f'/) + Flie·!>in(rJ)·!?,
l10r := NO
N
N := W + Flie·~in(rJ)
= 8/4,616 kN
f'7
eccenfricily
I1Rr- N OT
2 N
!?
ecc :=
ecc = 0,99> r?
!?o :=!? -2·ecc =:>ei!>f'1ic lif'1if 10 beariQ:} pre!>!>Ure
!?O == 4.021
f'7
1
Pd :== 1 .I1,.NqE + 2. '1 ·!?D·Ny£: Pd = J1j'59,591 kN
2
f'7
F ador of ~afdy acjain!:i ::>e:i!>r?ic induced beariQ:} capacify failure
Pd·!3Jo F5fX,:==
N
F5e;c
=
2,162
141
5-SEISMIC PERFORMANCE OF CAISSONS
5.1 General Caissons are very large concrete boxes that are excavated or sunk to a predetermined depth. They are used usually for the construction of bridge piers or other heavy waterfront structures, and they often become advantageous where water depths exceed 10 to 12 m. Caissons are divided into three major types: (1) open caissons, (2) box caissons -(or closed caissons), and (3) pneumatic caissons. Open caissons are concrete shafts with the top and bottom open during construction. This type is provided at the bottom as shown in Figure 5-1 with a cutting edge. After the caisson is sunk into place, soil from the inside of the shaft is removed through a number of openings by grab buckets until the bearing level is reached. Once the bearing stratum is reached, concrete is poured into the shaft, under water, to form a seal at the bottom. After the concrete has matured, the caisson is pumped dry and filled with concrete. This method does not guarantee thorough cleaning and inspection of the bottom. Box caissons as shown in Figure 5-2 are cast on land with spaces open for buoyancy. They are then transported to the construction site and gradually sunk by filling the inside with sand, ballast, or concrete. Pneumatic caissons are closed at top and open at bottom. Overburden materials are excavated by hand or machine from a working chamber while compressed air is used to keep water from entering the chamber. Penetration depth below water is limited to about 40 m (130 ft) as higher pressures are beyond human endurance. Despite of their higher cost as compared to the other two methods, this method of construction yields proper bearing stratum and concrete will be of adequate quality. A common feature of caissons produced by the three methods is that they are massive structures that respond to seismic loads in a primarily rocking mode about the base plus some translations.
143
lJ:I~r
Section at A-A
A
A
•L
~
i
_
..
'I
.,.
"
,"
·t ~
~
. !
_---.J
Water Level _ -=
~
....... , ....
, j
/)707/;:;:""
": 11/;<0 II;<0 " "
Soil Seal
Cutting Edge
Figure 5·1. Open Caisson
··D.Ll
Section at A-A
.•• _.~--' " ,!_.. -.
,.
'--'-'"1"'" -- t
it
A I
,_f
11;<0
71~1'
1" I:,
' ~A .J
j
tI
f~
Water Level
'J
n7)~)i~
.-.JI-;- -J 1 :'
Soil
Figure 5-2. Box Caisson 144
5.2 Modeling of Caissons for Seismic Loads The behavior of caissons under lateral seismic loads is essentially nonlinear. Geometric nonlinearity dominates this behavior due to rocking of the caisson and gapping at the soil-caisson interface. Soil material nonlinear behavior at the interface with caisson may contribute to the general nonlinear behavior but it is minor when compared to the geometric nonlinearity when the caisson starts to rock under high seismic drifts. Nevertheless, the material of the caisson will maintain its linear behavior during the course of the seismic event. The following steps are usually involved for modeling caissons in global models: 1- The global model, which includes the superstructure, shall also include a discretization scheme for the caisson-soil interface. This scheme shall be ' capable of capturing the physical behavior of caissons such as the gapping geometric nonlinearity associated with rocking response of the caisson under large seismic drifts. Example of such modeling technique is depicted in Figure 5-3. The caissons in the global model can be represented by a combination of three-dimensional elastic beam elements representing the spine of the caisson, constraints (rigid links), and spring elements with plastic material properties and gapping capability (Winkler elements). The nonlinear load displacement behavior of the interface elements of the global model shall be established using the results of analyses of a detailed 3-D finite element model of the caisson and supported soil, or other simplified procedure as explained in the next subsection. 2- The detailed 3-D finite element shall include constitutive relationships for the nonlinear behavior of the soil, and special interface elements that can capture gapping between the soil and the caisson at the base and side walls. This model will be referred to as the local model. 3- Nonlinear static pushover analyses shall be carried out to establish the soil-structure interaction behaviors for implementation in the global model. Pushover analyses of the local model shall be performed by applying a point load at the center of gravity of the rigid caisson for each mode of soil resistance. 4- The soil response results from the different pushover analyses of the local model in each direction shall be extracted and distributed to the soil spring elements at various nodal points in accordance with the discretization scheme of the global model. 5- The behavior of the caissons under seismic loads shall be assessed through nonlinear time history analysis of the global model. The performance of the caissons is evaluated by comparing the maximum drifts from the results of the time history analysis to the permissible levels according to the performance based design criteria of the project.
145
(a)
!, ~~umm\
Spring element with to simulate horizontal shear tractions at the base
--- ---+
Seismic Excitatio
......... ~-.
4:: 41.' roo I 4 1 -
\
\
j
\
!
T-r11111
.
// Spring element with gapping property to simulate normal contact pressure
Spring element with ~ apping property to simulate ~ assive pressure on side wall s
~rl
/
~~ ..J~
~,
seismic
~E
(-
xcitalion
.....,....,...,.
I I I i l l 1
Seismic Excitation
(b) ...... ......
~------
caisson
Rigid links 551 elements
Figure 5-3. Modeling of Caissons for Seismic Analysis
146
5.2 Seismic Performance Evaluation of Caissons using Simple Methods. 5.2.1 Theoretical evaluation of Capacity Since, caissons are massive rigid structures; it is likely that it will maintain its material linear behavior during earthquakes. Therefore the term capacity refers to the maximum drifts that the caisson can withstand without affecting the structural safety of the superstructure seriously. The capacity can be expressed in terms of moment~rotation or lateral load~displacement relationships. The theoretical formulation in this section is developed, for this purpose, for rocking about the caisson's longitudinal axis. An analogous derivation can be established for rocking about the transverse axis. Consider the caisson under forces and reactions in Figure 5-3a. For simplicity, a linear distribution of the contact stress acting at the soil footing interface, resulting in the triangular stress block shown in Figure 5-3a. In the following analysis, it will be assumed that onset of rocking is accompanied by maximum soil lateral earth pressure on the embedded sides of the front faces of the caisson. Brooms (1964) reported that the ultimate lateral earth pressure at failure can be taken in the range of two to three times the passive Rankine earth pressure. It is assumed here that maximum lateral earth pressure will reach a maximum value of two times the Rankine passive earth pressure when the portion of the base width in contact with soil reaches a value of half the width. Also, the active earth press'ure which may develop on the embedded sides of the back faces are assumed very small compared to the passive earth pressure. Eventually, at high rocking drifts loss of contact between the back walls and the surrounding soil may occur. Hence, 'soil reactions at the back faces were neglected. At any stage during rocking the bearing pressure, p, at the caisson's base and the associated deflection, A, can be quantified according to Figure 5-3b as:
o
0< X«A-L)
P(X) =
(5-1) 2W (X-A+L) L2B
(A-L)< X< A
where, W is the total buoyant dead load of the caisson, A is the width of the caisson, and L is the portion of the width in contact with soil.
147
00
~
-
"T1
::s
c.
C"
' -"
-
::s !fJ ID ::s
cr
U1
::s
co
3'
0
:E
< Ci)"
m ::s
"'U
Q)
--
co U1
0
"a
m
<0
ID
0,
:t§ C"c. S.co -, .,
O::s Cir U1
CO 0U1
o ID .,(')U1 -,
~ "T10
I
(J1
CD
ca' e:
x
TIt11 •: .F "....
"
," 0"
, 0"
'
'
. ..
." ..
1 I - - - - o I - - -...111"1011
1. o i 1.
< x
"
..... '"
.
•
....
:
,
. .....
,
.:. • ...:t't:.
.'
I-.---~li!<-- I - - - - - -......1
);>---
':~"': ~>":':'_'" .. ~l':":
:
.. .. ,. t
~~ ~~
..I
1
-<
'i=*
..Ilo....:..
'~':',: :.. ~
..
.~ ·1 .· . '_.'-t.. TF..:~· .:, 'to ..... ',., "
"
~.: j: . ::'
.::'
'IS
•
• I
'
".:
'.'
....
Ix
I
I
I
.J:,:., .....:.\::.. ~:::..
.'. :: ...
;·of';'';.":
;~~ .~. ~.
)CZJ"' ,.. '·1, " . :" . ". ~.
".~
~ •.•.
.
",:~
~ :'. .:
r 'QO',:. -
IX I
!i-.--CD
-<
2W
o< x
A(X) =-2-(x-A+L) L: Bk
< A
(5-2)
In which k is vertical modulus of subgrade reaction (unit pressure required to produce a unit deflection) associated with the rocking mode. By assuming that the caisson's base is free to rock on a surface of elastic half space, with no side pressure effects, and imposing conditions of onset of rocking ( i.e. L=A), the vertical modulus of subgrade reaction k is evaluated:
k
= 12~ay
(5-3)
AB
in which, KOx is the rocking stiffness of the base about the longitudinal axis evaluated as (Gazetas, 1991):
Key = ~~ [0.4(~)+O.1]
(54)
Where G and v are the soil's shear modulus and Poisson's ratio respectively. The overturning moment Mo =
Mo at any stage of rocking can be evaluated as:
2WA (1- 32) a + 6A. yh 3BK p
(5-5)
=
where a UA is the ratio of the portion of width under contact to the total width, y is the buoyant unit weight of soil, h is the embedded depth of the caisson, B is the length of the base, and Kp is the coefficient of Rankine passive earth pressure calculated according to equation 4-6 as: 1+sin+ =tan 2(45 ++) K = 1-sin+ 2 p
(5-6)
A. is a coefficient to determine the increase in the coefficient of passive earth pressure at different stages of loading determined as:
{~a
0.5 < ex. < 1 ex. <0.5
(5-7)
The rotation associated with the overturning moment is:
0y= A(A) = L
2W o,2A 2 Bk
(5-8)
Therefore, by varying the value of ex. from 0 to 1, a moment rotation relationship can be established for the caisson. The case of ex. = 1 represents the condition of
149
onset of rocking, while the case of a caisson.
=0
is equivalent to full non-stability of the
The primary mode of deformation is rocking about the caisson's base accompanied by translation of the portion of the base in contact with the soil. Thus, the resultant inertia load for this mode is located at a vertical distance Ho to the base, henceforth called the effective moment arm, which can be obtained by summing the moment of masses about the center of rotation at the base:
Ho ; ; ; ~4H" +A
2
(5-9)
12
Where, H is the total height of the caisson. At any stage during rocking the ba~e shear can be quantified as the sum of the ultimate force to induce sliding and an additional force required to produce rocking:
M Ho
Vx ;;;;;;J.lW-P+o
(5-10)
Where, P is the passive resistance at the front side and J.l is the interface friction angle between the caisson's base and the soil. The total displacement at the top of caisson is the sum of displacement due to rocking and displacement at the base:
L\x=
JJ.wK -P +H8 :xx
y
(5-11)
in which, Kxx is the stiffness of the base interacting with the soil in the transverse direction, evaluated according to the elastic half-space theory (Gazetas 1991) as:
GB [(8)0.65 +0.4(B) A +0.8 ]
Kxx ; ; ; 2-v 3.4 A
(5-12)
It is important to note that the uplift is the only source of nonlinearity in this method. as no soil nonlinearities are considered for the soil beneath the caisson base. This method can be used in lieu of the detailed local finite element modeling to develop the lateral load displacement relationships to be implemented in global models.
5.2.2 Capacity Evaluation Procedure On the basis of the theory presented above, the following steps summarize the procedure to evaluate the nonlinear load displacement relationship (pushover capacity) of the caisson: i.
For the considered caisson calculate the effective moment arm Ho using equation 5-9.
150
ii.
Calculate the half-space transverse and rocking stiffness coefficients, Kxx, and Key for the caisson base (equations 5-12 and 5-4).
iii.
Calculate the vertical modulus of subgrade reaction k using equation 5-3.
iv.
Assume a value for the ratio of the portion of width under contact to the total width a..
v.
Calculate the overturning moment Mo using equations 5-5 to 5-7.
vi.
Determine the rotation associated with the overturning moment from equation 5-8.
vii.
Determine the caisson's base shear from equation 5-10.
viii.
Calculate the total displacement at the top of caisson as the sum of the nonlinear displacement due to rocking and the linear displacement at the base from equation 5-11. Change the value of a. and resume steps from iv to viii.
ix.
5.2.3 Evaluation of Demand and Structural Performance Once the pushover curve has been established, the structural response can be displayed in the form of normalized spectral acceleration versus spectral displacement. This technique is known as the capacity spectrum method, which requires that both the demand response spectra and structural capacity (pushover curve) be plotted in the spectral acceleration versus spectral displacement domain. Spectra plotted in this format are known as Acceleration Displacement Response Spectra (ADRS) after Mahaney et aI., 1993. To convert a spectrum from the standard spectral acceleration Sa versus T format to the ADRS format, it is necessary to determine the value of the spectral displacement Sdi for each point on the curve Sail Ti . This can be done with the equation:
T=2
Sd' =_I_S .g I 41t2 al
(5-13)
If it is assumed that the caisson can be modeled as a rigid single-degree-of freedom system vibrating in a mode in the direction of the application of the pushover force, then any point on the capacity curve can be converted to the corresponding point Sail Sdi on the capacity spectrum using the equation: Sal
V;
=~
and
Sdi
= Ax
(5-14)
Through this method the performance of the structure is estimated graphically as the point where the capacity curve intersects with the elastic demand spectrum curve. The following example is given to illustrate the method.
5.3 Example A caisson foundation was designed to support a suspension bridge tower in Tacoma Washington. The dimensions are 24.4 x 39.6 m. the height is 74.5 m
151
with an embedment depth of 22.5 m. The soil is silty sand with unit weight of 19.6 Kn/m 3 . The total buoyant weight of the caisson was estimated as 1023 MN. The results of the cross-hole seismic survey test indicated that the average shear wave velocity of the soil foundation is estimated as 305 m/sec. Estimate the seismic performance of this caisson during a potential earthquake.
501ulion 51ep I: Develop Ihe
~/andard
denand ~pedrun:
Ux lhe nelhod explained in xdion 1,8 fo develop Ihe de!?iqn re!>ponx !>pedrun,
Fronlhe U5, 6eo/oqica/ xNey web !>ile Ihe 0,2 xcond 51» I-xeond ~pedra/ aeee/eraflon!> 5/, and peak qround acee/eralion P61t were delemined a~:
5:> "" /,21 q 5,
0:12- q
P'
PSIt
~ 0,'55
q
5ince fhe averaqe !>hear wave ve/ocily i!> !J0'5 M1!> Ihe !>ife i!> cla:?!?ified a!? ealeqory D, rhe !?ile eoefficienl:;:, 1'5 de/emined frof? r able!? (1'2-) and (I,!J) at>:
Fa"" /,0/6 Fv"" /''58 Calcu/ale Ihe de!?iqn earlhquakere!?ponx t>pedra/ acce/eralion al t>horl period 5D:::- a nd al'-xcond period, 501 501>~· /,0/6X 501 ""
1.2-lq ~ 1,25 q
1/58 X0:1% "" 0,6Gq
Delerf?ine Ihe: period!? r!? and r 0
r ~ 066/ /,25 ~ 0,'54 !? r 0 ~ O,2X 0,'54 - 0./08 :;:, d
Con!?lrud Ihe ~% danpinq de:5iqn !?pedfUf1 u!>inq equalion!> 1'2-2Ihrouqh /,24-, rhe de::.5iqn re!?pon!?C !?pedrun I!? depided in Fiqure '5'4-.
152
IA
I
I
1.2
~
v ~
0.8
l
0.6
~
l
~ O.~ 0.2
a 2-
I
0
~
4
Period (::ec)
FiCJurc ~~4-. 'X Darlpif7CJ De5iCJn R c!:>po~ :5pcdrurl for T ocorJa WO!>hlnc;fon 5fep 2: E::>faUidl fhe Pudrover Cu/V.e:
Follow fhe !:>fep::> in ::edion 5.2.210 develop fhe capacily cUNe:
rJAfA: f oundatiGfl width
A = 24.?B4m
Fcul1dati(ifJ lenqth
B=
~.624m
P01550115 ratio
v :=
a,??
5hear wave velOCIb1
V5:=
?o?~ 5
Y
Unlt wt of soil
= 19,6?6 kN
:?
m MalC.imUm shear
modulus
/'.
. Y
tAmax'= -.
v52
Umax. = 1.86:; x I08 pa
Cj
153
--(
~ffective
moment arm
Halfspace transverse stiffness
HO :=
J(4.H
Z
Z +A )
HO = 4?J,89Bm
12
(17 ,0,67 + 0.4·-[? + 0,8J
CtA [
kXX := - . ?A· 2-v A)
A
kxX = 6,129 x 106 kN m
Halfspace rocKinCl 5tlffne55
Ct.A'?.[ OA· key := -.I-v
([7'- O.J]. A)
+
I
rad
kOY = 1.246 x I09kN.~ rad
Coefficient of vertical 5ubgrade reaction
k. := 12.
Kay
A'? ·18
k = 2,604 x 104 kN '? m
The pU!5hover CUNe I!:> depided In Fiqure 5-5, II? dlown in fhe fiqure fhe re!5pon!5e fran 0 fo It ;!:> linear wilh un/lom dlf:>fribulion of file pre!:>?Ure on fhe cai!5!50d!:> box. 5lidlnc; !:>/arf!:> 10 occur of point It and con/lIVe!:> up 10 polnf ~J whree if I!:> occonpanled by rockinq frof'7 ~ fa C, J
154
BOJ,CX::O
.'"'"-'-'
. -:-1' -~.-,-~-
__ ._....• _ _._ - _---,-_
__ . "--,,.....•.. ,
.
-· ..
1
C
i
!
600,CX:O f
\ t'
:::;;z;» .........
r
3
v
~
4
~ 2
.CX::O
+t
I
o. o
0.2£'
j
I
I
0.7
0.17
1J1$p/ac.eJrent ( m:>
FiCJure ~~~. Exal"lp/e of a f'udlover CUNe for a Cai!:Jf>On. 5fep 5: E!:Jfab/if:>h fh~.._...r;;.qJ?Cl{;!.fy... 2f?t;d[(!.tL0u.Ne and Check Perfomonce the capacily !:Jpedrun CUNe if:> dif:>played in FiCJure ~-6, II i~ ~hONn fhaf in fhb reCJion fhe caif:>=>on f1ay experience ~one f:>/idinq under a pofenfial earfhquake. Neverfh/e~~J rocki!1CJ ('1oy nof occur. IA
!
!
;
,
1.2
11
\
-a
..
\
0.8 ;
v
-{'
0,6
o
V -----
....... k-,""'
Q-r
0.2
~:
,.-
I
/
'1/ ,._-_ ...
:
I
:
I
'"
·ww_·
I
-IJ~
1
~... ~~
,I
--- ;
- I
-
1
-----'r---..
,
1;
I
-S:-Q.
rq
("') ~~
Jb
r'\-+('}",e (' c~
I
_.._.---!
i
-
---=
,
;
..-1-._........_...................
)
.-_ ............................... ....• "
Fic) 12 Cvvt. Qh.
J
k ~ ~~ 10-- ~0oAkr kJ P(Z etA. I ('vI I 9'~ ,:;>.\
IAJ ~ XA~
v', S"lA.
155
, ~
5.4 References Gazetas, G. (1991). Foundation vibrations, Chapter 15 in Foundation Engineering Handbook, 2nd edition, H.-Y. Fang, ed., Van Nostrand Reinhold, New York, pp.553-593. Mahaney, J.A., Paret, T.F., Kehoe,B.E., and Freeman, S.A. (1993) "The capacity spectrum method for evaluating structural response during the Lorna Prieta earthquake, "National Earthquake Conference, Memphis.
156
