Non-Linear Time History Analysis of Cable Stayed Bridges

THE PROJECT ENTITLED

NON-LINEAR TIME HISTORY ANALYSIS OF CABLE STAYED BRIDGES Submitted to the DEPARTMENT OF APPLIED MECHANICS

In partial fulfillment of the requirements for the award of the degree of

BACHELOR OF TECHNOLOGY IN CIVIL ENGINEERING

Submitted by

Guided by

Prakash Agarwal (U06CE029) (U06CE029 ) Shobhit Bhatnagar (U06CE043) C. Uma (U06CE056) (U06CE056)

Dr. G. R. Vesmawala Prof. A. J. Shah

DEPARTMENT OF APPLIED MECHANICS S. V. NATIONAL INSTITUTE OF TECHNOLOGY TECHNOLOGY Surat – 395007 GUJARAT

May 2010

EXAMINER’S CERTIFICATE OF APPROVAL

The seminar entitled “NON-LINEAR TIME HISTORY ANALYSIS OF CABLE STAYED

BRIDGES” submitted by Prakash Agarwal, Shobhit Bhatnagar and C.Uma in partial fulfillment of the requirement for the award of the degree in “Bachelor of Technology in

Civil Engineering” of the Sardar Vallabhbhai National Institute of Technology; Surat is hereby approved for the award of the degree.

EXAMINERS:

1.

2.

ABSTRACT

The concept of cable-stayed bridges dates back to the seventeenth century. Due to their aesthetic appearance, efficient utilization of material, and availability of new construction technologies, cable-stayed bridges have gained much popularity in the last few decades. After successful construction of the Sutong Bridge, a number of bridges of this type have been proposed and are under construction, which calls for extensive research work in this field. Nowadays, very long span cable-stayed bridges are being built and the ambition is to further increase the span length using shallower and slender girders. In order to achieve this, accurate procedures need to be developed which can lead to a thorough understanding and a realistic prediction of the b ridge’s structural response under different load conditions.

In the present study, an attempt has been made to analyze the seismic response of cablestayed bridges with single pylon and two equal side spans. This study has made an effort to analyze the effect of both static and dynamic loadings on cable-stayed bridges and corresponding response of the bridge with variations in span length, pylon height and pylon shape. Comparison of static analysis results have been made for different configuration of bridges - their mode shapes, time period, frequency, pylon top deflection, maximum deck deflection; and longitudinal reaction, lateral reaction and longitudinal moment at pylon bottom. Time history analysis results have been investigated for different configuration of bridges under the effects of three earthquakes response spectrum (Bhuj, El Centro and Uttarkashi) - axial forces in stay cables, deck deflections and stress diagrams at maximum peak ground acceleration of the above mentioned earthquakes.

ACKNOWLEDGEMENT

The project work would not have been possible without the valuable guidance of Dr. G R Vesmavala and Prof. A J Shah of Applied Mechanics Department, SVNIT, Su rat. We, hereby take this opportunity to express our deep sense of gratitude and indebtedness to them. We are also thankful to Mr. Kiran Joshi, M. Tech II, Applied Mechanics Department whose kind cooperation ensured the successful and timely completion of this work. We are also thankful to our Head of Department Dr A K Desai for providing us this opportunity of enlightening ourselves with the technology of cable-stayed bridges.

CONTENTS

Chapter no. Topic 1. Introduction Background Historical Development Basic Theory of Cable Stayed Bridges 1.3.1. Arrangement of stay cables 1.3.2. Position of cables in space 1.3.3. Types of tower 1.3.4. Deck 1.3.5. Main girders and trusses Notable Cable Stayed Bridges

Page no. 1 1 1 3 4 4 5 5 6 6

2.

Literature review 2.1. Concept of Cable Stayed Bridges 2.2. Static Analysis of Cable Sta yed Bridges 2.3. Dynamic Behaviour of Cable Stayed Bridges 2.4. Non-Linear Analysis of Cable Stayed Bridges 2.5. Seismic Analysis of Cable Stayed Bridges 2.6. Need of Study 2.7. Scope of Work

9 9 9 10 10 11 12 12

3.

Validation Study and Comparison of 2D and 3D analysis 3.1. Introduction 3.2. The 2-Dimensional Model 3.3. The 3-Dimensional Model 3.4. Analysis Results 3.5. Validation

14 14 14 15 17 20

4.

Modeling of Cable Stayed Bridges 4.1. Introduction 4.2. Material Properties 4.2.1. Concrete 4.2.2. Steel 4.2.3 Steel Tendon (for cable) 4.3. Modeling of Deck 4.4. Modeling of Girder 4.4.1. Horizontal Beam 4.4.2. Edge Beam 4.4.3. External Brace 4.4.4. Internal Brace 4.5. Modeling of Pylon 4.5.1. Pylon Bottom 4.5.2. Pylon Interior 4.5.3. Pylon Top 4.6. Cable Section

21 21 22 22 22 22 22 22 23 23 23 23 24 24 24 24 24

4.7. Support Conditions 4.8. Time History Analysis 4.9. Time History Functions Used 4.9.1. The Bhuj Earthquake 4.9.2. The El Centro Earthquake 4.9.3. The Uttarkashi Earthquake

25 25 26 26 26 27

Results and Discussion Discussio n

28

5.1 Introduction 5.2 Non-Linear Non-Linear Static Analysis 5.2.1 Time Period vs Mode Number Graphs 5.2.2 Frequency vs Mode Number Graphs 5.2.3 Pylon Top Deflection Deflecti on vs Pylon Height 5.2.4 Deck Deflection vs Pylon Height 5.2.5 Longitudinal Longitudinal Reaction Forces at Pylon Bottom 5.2.6 Lateral Reaction Forces at Pylon Bottom 5.2.7 Longitudinal Longitudinal Moment at Pylon Bottom 5.3 Non Linear Dynamic (Time History) Analysis 5.3.1 Maximum Deck Deflection 5.3.2 Axial Forces in Cables 5.3.3 Shell Stresses at Deck

28 28 29 30 31 32 33 34 35 36 36 39 41

6.

Conclusions Conclusio ns

45

7.

References

46

5.

LIST OF FIGURES

Figure No.

Topic

1.1

Stay ropes on Egyptian sailing ships

1.2

Cable Arrangement Systems

1.3

Space Positions of Cables

1.4

Different Shapes of Pylon

1.5

Types of Main Girders

3.1

Layout of 2D Model

3.2

Pylon Layout

3.3

Layout of 3D Model

3.4

Deflection of 2D Model

3.5

Deflection of 3D Model

4.1

Schematic diagram representing various sections sections of the bridge b ridge model

4.2

The 3D CSB model in SAP2000

5.1

Time Period vs Mode Number Graphs for A and H-shaped Pylon Models

5.2

Frequency vs Mode Number Graphs for A and H-shaped P ylon Models Models

5.3

Pylon Top Deflections for A and H-shaped Pylon Models

5.4

Maximum Deck Deflections for A and H-shaped Pylon Models

5.5

Longitudinal Reaction at Pylon Bottom for A and H-shaped H-shaped Pyl P ylon on Models

5.6

Lateral Reaction at Pyl P ylon on Bottom for A and H-shaped Pylon Models

5.7

Longitudinal Moment at Pylon Bottom for A and H-shaped Pylon Models Maximum Deck Deflection for A-Pylon Models under Different Load Time

5.8

Histories

Maximum Deck Deflection for H-Pylon Models under Different Load Time 5.9

Histories Axial Forces in Cables for A-Pylon Models under Different Load Time

5.10

Histories Axial Forces in Cables for H-Pylon Models under Different Load Time

5.11

Histories

5.12

Deck Stresses for A-shaped pylon of Height = Span/2

5.13


5.14


5.15

Deck Stresses for H-shaped pylon of Height = Span/2

5.16


5.17


LIST OF TABLES

Table No.

Topic

1.1

Notable Cable Stayed Bridges

3.1

Sectional dimensions of the 2D model

3.3

First 10 mode shapes of the 2D model

3.4

First 10 mode shapes of the 3D model

3.5

Comparison of time periods of the 2D and 3D model

1. INTRODUCTION 1.1 Background

During the past decade cable-stayed bridges have found wide applications in large parts of the world. Wide and successful application of cable-stayed systems has been realized only recently, with the introduction of high-strength steel, orthotropic type decks, development of welding techniques and progress in structural analysis. The variety of forms and shapes of cable-stayed bridge intrigue even the most demanding architects as well as common citizens. Engineers have found them technically innovating and challenging. Modern cable-stayed bridges are at present considered to be the most interesting development in bridge design. The increasing popularity of these contemporary bridges among bridge engineers can be attributed to its appealing aesthetics, full and efficient utilization of structural materials, increased stiffness over suspension bridges, efficient and fast mode of construction and the relatively small size of their substructure.

Cable-stayed bridge construction differs from conventional suspension bridges since in the former the girder is supported by individual inclined cable members which are attached directly to the tower, rather than by vertical hangers which are supported by one member as in the case of cable suspended bridges. One of the main difficulties an engineer encounters when faced with the problem of designing a cable-stayed bridge is the lack of experience with this type of structure, predominantly due to its nonlinear behavior under normal design loads. As accurate measurements of seismic responses are scarce in designing these bridges; the need for accurate modeling techniques has arisen. The methods available to the designer for the study of the bridge’s dynamic behavior are the forced vibration test of the real structure, model testing and computer analysis. The latter approach is becoming increasingly popular since it offers the widest range of possible parametric studies.

1.2 Historical Development

Early nineteenth century gave rise to the concept of long span bridges using steel. Towards the end of nineteenth century, reinforced concrete was first used in bridges followed by composite

1

construction with steel and concrete, and pre-stress concrete being successfully used first in 1959. Mid twentieth century saw the revival of the cable-stayed bridge, which in concept, dates back to seventeenth century Venice but is generally credited to Loscher (1784) in the form of a complete timber bridge.

The history of stayed beam bridges indicates that the idea of supporting a beam by inclined ropes or chains hanging from a mast or tower has been known since ancient times. The Egyptians applied the idea for their sailing ships as shown in Fig 1.1. Redpath and Brown in England and Frenchman Poyet, early in the nineteenth century, designed bridges with steel wire cable and steel bar stays respectively. The first concrete structure to utilize cable stays was the Tempul aqueduct with the main span of 60 m in Spain in 1925. However, the first modern cable-stayed bridge with a steel deck, designed by F. Dischinger, a German engineer, was built in Sweden in 1955, with a main span of 183 m and fan type cable configuration supported on twin column bents.

The cable-stayed bridge is an innovative structure that is both old and new in concept. It is old in the sense that it has been evolving over a period of approximately four hundred years and new in that it’s a modern day implementation began in the 1950s in Germany and started to seriously attract the attention of bridge engineers in other parts of the world, as recently as 1970.

Fig. 1.1 Stay ropes on Egyptian sailing ships

2

It was very unfortunate that engineers have faced, for a long time, series of failures in early attempts in building of cable-stayed girder bridges. It is widely believed that those early failures were mainly caused by the lack of suitable high strength construction material, especially for cables that were hung loosely under dead load and did not provide effective support for the bridge girder under live load until the deflection of girder became extremely large, thus causing overstress of girder. In those days, efficient analytical tools were not available for such huge and complex structures. Now-a-days bridges of this type are entering a new era with main span length reaching 1000 m. This fact is due to the relatively small size of the substructures required, availability of very high strength materials, the development of efficient construction techniques and rapid progress in the analysis and design of these types of bridges.

1.3 Basic Theory of Cable Stayed Bridges

A cable-stayed bridge is a non-linear structural system in which the girder is supported elastically at points along its length by inclined cable stays. A wide variety of geometric configurations have been utilized in cable-stayed bridge construction, depending on the site conditions and utility. The concept of a cable-stayed bridge is rather simple. The bridge carries mainly vertical loads acting on the girder with the stay cables providing intermediate supports for the girder so that it can span large distances. The basic structural form of a cable-stayed bridge is a series of overlapping triangles comprising the pylon or the tower, the cables and the girder. All these members are under predominantly axial forces, with the cables under tension and both the pylon and the girder under compression.

Modern cable-stayed bridges present a three-dimensional system consisting of stiffening girders, transverse and longitudinal bracings, orthotropic-type deck and supporting parts such as towers in compression and inclined cables in tension. The important characteristic of such a threedimensional structure is the full participation of the transverse construction in the work of the main longitudinal structure. This means a considerable increase in the moment of inertia of the construction, which permits a reduction in the depth of the girders, and economy in steel.

3

1.3.1 Arrangement of Stay Cables

According to the various longitudinal arrangements, cable-stayed bridges can be divided into three basic systems – radial, harp and fan pattern (Fig. 1.2). However, except in very long span structures, cable configuration does not have a major effect on the behavior of the bridge.

(a) Radial System

(b) Harp System

(c) Fan System Fig. 1.2 Cable Arrangement Systems

1.3.2 Position of Cables in Space

With respect to the various positions in space, which may be adopted for the planes in which the cable stays are disposed, there are two basic arrangements: two-plane systems and single-plane systems (Fig. 1.3)

4

(a) Two Vertical Plane

(b) Two Inclined Plane

(c) Single Plane System

Fig. 1.3 Space Positions of Cables

1.3.3 Types of Tower

The various possible types of tower or pylon may be of the form of: (a) trapezoidal portal frames; (b,c) twin towers; (d) single towers; (e) A towers; and (f,g) side towers (Fig. 1.4)

Fig. 1.4 Different Shapes of Pylon

With single towers or twin towers with no cross-member, the tower is stable in the lateral direction as long as the level of the cable anchorages is situated above the level of the base of the tower. In the event of the lateral displacement of the top of the tower due to wind forces, the length of the cables is increased and the resulting increase in tension provides a restoring force. Longitudinal moment of the tower is restricted by the restraining effect of the cables fixed at the saddles or tower anchorages.

1.3.4 Deck

Most cable-stayed bridges have orthotropic decks, which differ from one another only as far as the cross-sections of the longitudinal ribs, and the spacing of the cross-girders is concerned. The orthotropic deck performs as the top chord of the main girders or trusses. It may be considered as 5

one of the main structural elements, which lead to the successful development of modern cablestayed bridges. 1.3.5 Main Girders and Trusses

The three basic types of main girders or trusses presently being used for cable-stayed bridges are steel girders, trusses and reinforced or pre-stressed concrete girders (Fig. 1.5).

(a) Steel Girder

(b) Truss Girder

(c) Pre-stressed Concrete Girder

Fig. 1.5 Types of Main Girders

1.4 Notable Cable Stayed Bridges

A list of the world’s prominent cable-stayed bridges constructed in various countries and their salient features (in decreasing order of their main span length) are compiled in Table 1.1.

6

Rank

Photograph

Longest span

Year Pylons

306 m

1,088 m (3,570 ft)

2008

2

293 m

1,018 m (3,340 ft)

2009

2

890 m (2,920 ft)

1999

2

856 m (2,808 ft)

1995

2

230.5 m

800 m (2,625 ft)

2009

2

People's Republic of China

270 m

730 m (2,395 ft)

2009

2


270 m

628 m (2,060 ft)

2001

2

Name

Location

Country

[1]

Sutong Bridge

Suzhou, Nantong


[2]

Stonecutters Bridge

Rambler Channel

Hong Kong

[3]

Tatara Bridge

Seto Inland Sea

Japan

[4]

Pont de Normandie

Le Ha vre

France

[5]

Incheon Bridge

Incheon

South Ko Korea rea

[6]

Shanghai Yangtze River Bridge

Shanghai

[7]

Second Nanjing Yangtze Bridge

Nanjing, Jiangsu

7

Height of pylon

220 m

214.77 m

[8]

Jintang Bridge

Zhoushan Archipelago


202.5 m

620 m (2,034 ft)

2009

2

[9]

Yangpu Bridge

Shanghai


223 m

602 m (1,975 ft)

1993

2

[10]

Bandra-Worli Sea Link

Mumbai

India

126 m

600 m (1,969 ft)

2009

2

[11]

Taoyaomen Bridge

Zhoushan


151 m

580 m (1,903 ft)

2003

2

[12]

Rio-Antirio Bridge

Rio

Greece

163 m

560 m (1,837 ft) (3 spans)

2004

4

[13]

Stromsund Bridge

Inderøy

Norway

153.4 m

530 m (1,739 ft)

1991

2

[14]

Kanchanaphisek Bridge

Bangkok

Thailand

187.6 m

500 m (1,640 ft)

2007

2

[15]

Oresund Bridge

Copenhagen, Sweden

DenmarkSweden

204 m

490 m (1,608 ft)

1999

2

8

2. LITERATURE REVIEW 2.1 Concept of Cable Stayed Bridges

The basic idea of a cable-stayed bridge is the utilization of high strength cables to provide intermediate supports for the bridge girder so that the girder can span a much longer distance. This introduces high compressive stresses in both the bridge girder and the towers. Technically this is an excellent design concept and aesthetically, this has a very soothing effect on the landscape because of its extreme slender appearance. “A cable-stayed bridge is a statically indeterminate structure with a large degree of redundancy. The girder is like a continuous beam elastically supported at the points of cable attachments and supported on rollers at the towers. If the non-linearity due to factors such as large deflection, catenary action of cables, and beamcolumn interaction of the girder and tower elements are neglected, the structure can be assumed to be linearly elastic” (Agarwal, 1997). Cable-stayed bridges have been found to be economical in the range from 750 ft to 1500 ft, where normal girder bridges become too heavy and suspension bridges too short to be competitive. With increase in span of cable-stayed bridges, the overall structure turns light-weight and slender, with more sensitivity to lateral loads.

2.2 Static Analysis of Cable Stayed Bridges

Extensive research has been done in the static analysis of cable stayed bridges, for their most suitable forms with changes to different parameters. A-shaped pylon has been shown to provide increased stiffness and stability to the overall structure. Although harp cable arrangement has been investigated to be statically less stable and uneconomical; its use in present day can be attributed to its pleasant aesthetical appearance. Fan cable arrangement has been found to be most economical and stable with lesser axial compressive force in the deck. The total weight of steel in stay cables is considerably less than the steel required in other members. The structural response changes with change in stay spacing. With increase in number of cables, maximum tension in cable decreases, but it affects the buckling behavior of the bridge. The fundamental critical load of the bridge is also affected by the number of cables (Wang, 1999). If cable spacing

9

is reduced by increasing the number of cables, then the live load moment in deck increases. However if cable spacing is increased, dead load moment increases with no significant effect on live load moments (George, 1999). For cable-stayed bridges with concrete decks, the most economical solution having minimum longitudinal moment is always the one having maximum number of cables. On the other hand, if a light steel or composite deck is chosen then for minimum longitudinal moments, more number of stays is not the best design solution. For static and dynamic stability, box girder has been found to be the most advantageous. Because of its slenderness, cable-stayed girder bridges with open plate girder cross section are very sensitive to winds, especially in erection stages when the main span simulates a cantilever and some cables are still ineffective.

2.3 Dynamic Behaviour of Cable Stayed Bridges

Although cable-stayed bridges are stable in static analysis, due to its slenderness and light-weight structure with increasing span, dynamic analysis is very important which too determines the feasibility of the structure. In general, there are three types of dynamic problems – aerodynamic stability, physiological effects and safety against earthquakes.

Aerodynamic behaviour of cable-stayed bridge determines, to a great extent, the safety of the bridge. There are three aerodynamic phenomenon that are responsible for dynamic response of bridge road deck – vortex shedding excitation torsional instability and buffeting by wind turbulence. In cable-stayed bridges, vibrations due to low wind and traffic cause inconvenience without damaging the structure and these are called physiological effects. Third and one of the important dynamic phenomenon is earthquakes. Safety against all the above mentioned dynamic effects determines the total feasibility of the project.

2.4. Non-Linear Analysis of Cable Stayed Bridges

Although the behavior of structural material used in cable-stayed bridges is linearly elastic, the overall load displacement relationship for the structure is non-linear under normal design loads. This non-linear behavior is a result of the non-linear axial force deformation relationships for the inclined cables due to the sag caused by their own dead weight; the non-linear axial and bending 10

force deformation relationship for the bending members which occurs due to the interaction of large bending and axial deformations in the members; and the large displacements which occur in the structure under normal design loads. All these effects are due to changes in geometry of the structure as it deforms (Fleming, 1980).

After applying both linear and nonlinear procedures for a wide variety of bridge geometries, it has been found (Fleming, 1980) that linear dynamic response, using the stiffness of the structure at the dead load deformed state, considering the nonlinear behavior of the structure during the application of static load and nonlinear dynamic response, considering the non-linear behavior of the structure during the application of the dead load, demonstrate almost the same dynamic behavior throughout the time loading. Damping can have have significant effect upon the response of the structure and should be considered, at least initially during the analysis.

2.5. Seismic Analysis of Cable Stayed Bridges

Ghaffar (1991) has given general guidelines for seismic analysis and design of cable stayed bridges. He has given different procedures to estimate earthquake loads considering both simplified and elaborate dynamic analysis.

From studies it has been found that, the input ground motion, whether it is uniform or non uniform should satisfy the following criteria: 1. Three or more sets of appropriate ground motion time histories should be used; they should contain at least 20 seconds of strong ground shaking or have a strong shaking duration of 6 times the fundamental period of the bridge, whichever is greater. 2. The ordinates of the input ground spectra should not be less than 90 percent of the design spectrum over the range of the first five periods of vibration of the bridge in direction being considered.

Because of hybrid structural system and the flexible, extended in plane configuration as well as three dimensionality of cable stayed bridges, earthquake excitations especially to non uniform motions, may introduce special features into the bridge response due to complicated interaction

11

between the three dimensional input motion and the whole structure. Three dimensionality and modal coupling cannot be captured in any two dimensional dynamic analysis. Therefore for proper representation and more accuracy, response of dynamic loading has to be obtained. T get the maximum response of the required quantities for dynamic loading minimum six modes has to be taken into consideration for displacement results, whereas practically 10 modes are required to calculate bending moments correctly. Generally both symmetric and anti symmetric modes contribute to the overall response of the structure, but contribution from symmetric modes are more than those of anti symmetric modes. Angle of incidence of ground motion has considerable effect on the response and depends upon both the nature of correlation function and the ratio between the three components of ground motion. The response of the bride is also influenced by tower deck inertia ratio, (Allam, Datta, 1998). With increase in tower-deck inertia ratio, both displacement and bending moment responses decrease in outer span, whereas there is no significant change in the inner span.

2.6 Need of Study

From the thorough review of literature, it has been found that although considerable amount of work has been done on seismic performance of cable-stayed bridges, still few areas have not been paid adequate attention. These are: 

Main research works have been concentrated on 2-pylon with equal side spans, although a large number of cable-stayed bridges are with two span single pylon.



Characterization of bridges under static and dynamic loads with variation in span length, pylon height and pylon shape.



Structural response of bridges subjected to various types of seismic loading.

2.7 Scope of Work

In this study, the following works have been conducted: 

Verification of the standard software - SAP2000 for analysis of cable stayed bridges 12



Comparison of 2D and 3D model for static analysis using SAP2000



Study of 3D computer model using SAP2000 for A and H shaped pylon for spans of 100 m, 200 m, 300 m, 400 m and 500m.



Comparison of static analysis results for different configuration of bridges - their mode shapes, time period, frequency, pylon top deflection, maximum deck deflection; and longitudinal reaction, lateral reaction and longitudinal moment at pylon bottom.



Comparison of time history analysis results for different configuration of bridges under the effects of 3 earthquakes response spectrum (Bhuj, El Centro and Utarkashi) - their stress diagrams, axial forces in stay cables and deck deflection at maximum peak ground acceleration of the above mentioned earthquakes.

13

3. VALIDATION STUDY AND COMPARISON OF 2D AND 3D ANALYSIS 3.1 Introduction

In order to test and validate the analysis results obtained in this work, two dimensional and three dimensional models of cable-stayed bridges have been considered in this chapter. Sectional dimensions of the bridge elements and other parameters have been taken from the PhD work of Nazmy and Sadek (1987). After analyzing the above models in SAP2000, the mode shapes and their corresponding time periods have been compared with the results given in the above mentioned work. Furthermore, time periods of 2D and 3D models have also been compared. 3.2 The 2-Dimensional Model

This model is very much the same as that used by Nazmy. The configuration of the towers and cables lie in a single plane at the centre of the deck. The cables have a harp-type configuration. The bridge has a centre span of 335.28 m (1100 feet) and two side spans of 137.16 m (450 feet) each. The pylon height above deck level is 60.96 m (200 feet) and below deck level is 15.24 m (50 feet). The towers are assumed to be fixed to the piers and rigidly connected to the deck girder at the deck level. The deck girder is simply supported at the end abutments. Figure 3.1 shows the general configuration of this cable-stayed bridge model. Table 3.1 shows the member properties of this bridge model. 2

4

2

Section

A (m )

I (m )

E (kN/m )

Girder (steel)

0.319

1.131

1.655E+08

Towers (steel)

0.312

0.623

1.655E+08

Cables 2

2

Cable No.

A (m )

E (kN/m )

15, 54, 70, 64

0.042

1.655E+08

45, 55, 71, 65

0.016

1.655E+08

46, 56, 72, 66

0.016

1.655E+08

Table 3.1 Sectional dimensions of the 2D model

14

Fig. 3.1 Layout of 2D Model

3.3 The 3-Dimensional Model

The 3D model has similar sections as that of the 2D bridge model. The pylon is A-shaped (as shown in Figure 3.2) and the cable arrangement is harp type. The bridge has the same centre span of 335.28 m (1100 feet) and two side spans of 137.16 m (450 feet) each. The pylon height above deck level is 60.96 m (200 feet) and below deck level is 15.24 m (50 feet). A horizontal beam has been provided in the pylon at the deck level. The towers are assumed to be fixed to the piers and rigidly connected to the deck girder at the deck level. The deck girder is simply supported at the end abutments. Figure 3.3 shows the general configuration of this cable-stayed bridge model. Table 3.2 shows the member properties of this bridge model. Figure 3.5 shows the static deformation of the model under dead load. This deformed shape is based on the non-linear analysis approach with P-Delta geometric non-linearity parameters.

Fig. 3.2 Pylon Layout (All dimensions in m)

15

70

65, 65’

71

15

70’ 55

45

55’

66, 66’

72, 72’

71’ 54, 54’

15’ 45’

46’

46

56, 56’

Fig. 3.3 Layout of 3D Model

Section

A (m )

2

I (m )

4

E (kN/m )

2

Girder (steel)

0.319

1.131

1.655E+08

Towers (steel)

0.312

0.623

1.655E+08

Horizontal Beam (steel)

0.139

0.170

1.655E+08

Cables

Cable No. 15, 54, 70, 64,15’,54’, 70’, 64’ 45, 55, 71, 65 45’, 55’, 71’, 65’ 46, 56, 72, 66 46’, 56’, 72’, 66’

A (m )

2

E (kN/m )

2

0.042

1.655E+08

0.016

1.655E+08

0.016

1.655E+08

Table 3.2 Sectional dimensions of the 3D model

16

64, 64’

3.4 Analysis Results

The mode shapes obtained from the 2D bridge model are shown in table 3.3 and those of the 3D are shown in table 3.4. Time periods of the corresponding mode shapes are also shown in the respective tables. A comparison of the time periods obtained from the analysis results of the 2D model and those obtained by Nazmy (1987) have been shown in table 3.5. Figure 3.4 shows the static deformation of the 2D model under dead load and Figure 3.5 shows the static deformation of the 3D model under same dead load. These deformed shapes are based on the non-linear analysis approach with P-Delta geometric non-linearity parameters.

Fig. 3.4 Deflection of 2D Model

Fig.3.5 Deflection of 3D Model

17

Mode Number

Mode Shape (2D Model)

Time Period (sec)

1

3.0126

2

2.1246

3

1.4692

4

1.3331

5

1.0265

6

0.7997

7

0.5569

8

0.5288

9

0.5121

10

0.4919

Table 3.3 First 10 mode shapes of the 2D mode 18

Mode Number

Mode Shape (3D Model)

Time Period (sec)

1

2.8842

2

2.0286

3

1.4055

4

1.2676

5

0.9729

6

0.7642

7

0.5281

8

0.5298

9

0.4867

10

0.4692 Table 3.4 First 10 mode shapes of the 3D model

19

Time Period (sec)

Mode

Time Period

% Variation of

% Variation of

(sec)

2D with 2D

2D with 3D

Number

2D model

3D model

2D (Nazmy)

(Nazmy)

Model

1

3.0126

2.8842

3.0795

2.172

4.261

2

2.1246

2.0286

2.1851

2.769

4.517

3

1.4692

1.4055

1.5071

2.514

4.332

4

1.3331

1.2676

1.3633

2.217

4.916

5

1.0265

0.9729

1.0562

2.811

5.212

6

0.7997

0.7642

0.8258

3.163

4.443

7

0.5569

0.5281

0.5695

2.215

5.171

8

0.5528

0.5251

0.5694

2.919

5.011

9

0.5121

0.4687

0.5292

3.221

4.927

10

0.4919

0.4658

0.5054

2.667

5.305

Table 3.5 Comparison of time periods of the 2D and 3D models

3.5 Validation:

The 2D model when compared with the 2D model of Nazmy, the % variation in time period comes out to be within the tolerable limits of 2.172 to 3.22. Also, the maximum % variation of the 2D and 3D models’ time period comes out to be 5.305 with minimum variation at 4.261 which is well acceptable for the purpose of validating our analysis results.

20

4. MODELING OF CABLE STAYED BRIDGES 4.1 Introduction

In this study, the effect of span length, pylon height and pylon shape on the behaviour of cable-stayed bridges have been investigated. The study was carried out for two-inclined two-inclined plane system and two-vertical plane system bridges i.e. for both A-shaped pylon and H-shaped pylon. Span lengths of 100 m, 200 m, 300 m, 400 m and 500 m with pylon heights of span/2, span/3 and span/4 have been considered. The deck is designed as concrete section with steel truss as girder section. The models have been analysed for dead load (static) as well as dynamic loads under the effect of load time histories of Bhuj, El Centro and Uttarkashi earthquakes.

Fig. 4.1 Schematic diagram representing various sections of the bridge model

21

4.2 Material Properties 4.2.1 Concrete 

Grade: M 45



Modulus of elasticity: 33541020 kN/m



Poisson’s ratio: 0.2



Coefficient of thermal expansion: 1.170E-05 / C



Shear modulus: 12900392 kN/m

2

0

2

4.2.2 Steel 

Modulus of elasticity: 1.999E+08 1. 999E+08










0

2

4.2.3 Steel Tendon (for cable) 

Modulus of elasticity: 1.580E+08 1. 580E+08










0

2

4.3 Modelling of Deck 

Element: Shell – Thin



Grade of concrete: M 45



Thickness:

Membrane: 0.3 m Bending: 0.3 m

4.4 Modelling of Girder

The steel truss girder comprises of horizontal beams, edge beams, internal brace and external brace (Fig. 4.1).

22

4.4.1 Horizontal Beam 

Material: M 45 concrete



Section: Rectangular



Outside depth: 6.7 m



Outside width: 5.448 m



Flange thickness: 2.232 m



Web thickness: 1.82 m

4.4.2 Edge Beam 

Material: Steel



Section: I



Outside height: 1.0 m



Top flange width: 0.5 m



Top flange thickness: 0.025 m






Bottom flange width: 0.5 m



Bottom flange thickness: 0.025 m

4.4.3 External Brace •

Material: Steel

•

Section: Circular

•

Outside diameter: 0.3 m

•

Wall thickness: 3.0E-03

4.4.4 Internal Brace •

Material: Steel

•

Section: Circular

•

Outside diameter: 0.1524 m

•

Wall thickness: 6.0E-03

23

4.5 Modelling of Pylon

The sub-structure pylon is a non-prismatic section which ranges from the pylon bottom section to the pylon interior section whereas the super-structure pylon is the same nonprismatic section ranging from the pylon interior section to the pylon top section. 4.5.1 Pylon Bottom 




Section: Box/Tube













4.5.2 Pylon Interior 




Section: Box/Tube













4.5.3 Pylon Top 




Section: Box/Tube













4.6 Cable Section 

Material: Steel tendon (for cable)



Diameter: 0.2 m

24

4.7 Support Conditions: 

Deck supports: Hinge restraints at one end and roller r estraints estraints on the other end



Pylon Base: Fixed restraints



Link between deck and horizontal beam: Dampers of stiffness 3000 kN-m and damping coefficient of 5000.

Fig. 4.2 The 3D CSB model in SAP 2000

4.8 Time History Analysis

Time history analysis can be defined as the study of the behavior of a structure as a response to acceleration, velocity or d isplacement isplacement of the structure during a given period of vibration. It is basically the study of the seismic response of a structure and the analysis can be linear as well as non-linear. The response of the structure can be plotted by three graphs: 

Pseudo-acceleration Pseudo-acceleration spectrum (for peak value of o f equivalent static force and base shear);



Pseudo-velocity Pseudo-velocity spectrum (for peak value of equivalent strain energy stored); and



Pseudo-displacement Pseudo-displacement spectrum (for peak deformation). 25

A response spectrum is simply a plot of the peak or steady-state response (displacement, velocity or acceleration) of oscillator(s) of varying natural frequency that are forced into motion by the same base vibration or shock. 4.9 Time History Functions Used:

Three load time histories with different characteristics have been used in the currently analysis, namely – past earthquakes at Bhuj, El Centro and Uttarkashi. 4.9.1 The Bhuj Earthquake 

Location: Kachchh Peninsula



Year: 26 Jan, 2001



Magnitude: 7.6 (on Richter scale)



Duration: 109.97 sec



Excitation type: Long



Number of steps: 26706



Step size: 0.0005



Time history type: Modal



Occurrence of maximum acceleration: 46.622 sec

th

Peak Ground Acceleration vs Time (Bhuj)

4.9.2 The El Centro Earthquake 

Location: Southern California



Year: 18th May, 1940






Duration: 31.1 sec



Excitation type: Medium






Step size: 0.02







Peak Ground Acceleration vs Time (El Centro)

26

4.9.3 The Uttarkashi Earthquake 

Location: Tehri Region, Himalaya



Year: 20 Oct, 1991






Duration: 6.22 sec



Excitation type: Short






Step size: 0.02







th

Peak Ground Acceleration vs Time T ime (Uttarkashi)

27

5. RESULTS AND DISCUSSION 5.1 Introduction

To understand the static non-linear behaviour b ehaviour of cable-stayed bridges, the bridge models have been subjected to dead load and modal load combinations and their responses studied. The parameters studied are - time period and natural frequency of different mode shapes; maximum pylon top deflection and maximum deck deflection; longitudinal and lateral reaction at pylon bottom and longitudinal moment at pylon bottom for spans of 100 m to 500 m with varying pylon height and pylon shape. Similarly, dynamic behaviour has been investigated by applying load time histories of Bhuj (2006), Uttarkashi (1991) and Elcentro (1940) to the above models. The parameters observed in this case are - maximum deck deflection, shell stresses at deck and maximum axial forces in cables.

5.2 Non-Linear Static Analysis

To be static, is to be simply constant with time. Thus, a static load is any load whose magnitude, direction, and/or position does not vary with time. Similarly, the structural response to a static load, i.e. the resulting stresses and deflections are also static. Nonlinearity in a structure refers to the changing of the stiffness co-efficient with change in load conditions. Even though the material properties of the cable-stayed bridge behave in a linearly elastic manner, the overall load-displacement relationship for the structure is nonlinear under normal design static loads. Although the loads do not vary with time but different loads induce different stiffness in the structure, which leads to a more complex analysis - the non-linear static stat ic analysis.

The time periods, natural frequencies, deformed configurations and bending moments of the cable-stayed bridge models obtained by the non-linear static analysis are shown below. The above mentioned parameters are seen for H-shaped and A-shaped pylons. The spans considered are 100 m, 200 m, 300 m, 400 m and 500 m with varying pylon heights of span/2, span/3 and span/4.

28

5.2.1 Time Period vs Mode Number Graphs for A an d H Pylon Models

H-shaped Pylon Time Period vs Mode Number

A-shaped Pylon Time Period vs Mode Number 8 ) c e s 6 ( d o i 4 r e P e 2 m i T 0

8 ) c e s 6 ( d o 4 i r e P 2 e m i T 0

100m 200m 300m 400m 1 2 3 4 5 6 7 8 9 10

500m

100m 200m 300m 400m 1 2 3 4 5 6 7 8 9 10

Mode Number

Mode Number

Pylon Height = Span/2


7 ) c 6 e s ( d 5 o 4 i r e P 3 e 2 m i T 1 0

7 ) c 6 e s ( d 5 o 4 i r e 3 P e 2 m i T 1 0

100m 200m 300m 400m 500m 1 2 3 4 5 6 7 8 9 10

100m 200m 300m 400m 1 2 3 4 5 6 7 8 9 10

Mode Number

Mode Number



7 ) c 6 e s ( 5 d o 4 i r e 3 P e 2 m i T 1 0

7 ) c 6 e s ( 5 d o 4 i r e 3 P e 2 m i T 1 0

100m 200m 300m 400m 500m 1 2 3 4 5 6 7 8 9 10

500m

500m

100m 200m 300m 400m 500m 1 2 3 4 5 6 7 8 9 10

Mode Number

Mode Number



Fig. 5.1 Time Period vs Mode Number Graphs for A and H-shaped Pylon Models From the above graphs we can clearly deduce that time period of vibration of cable-stayed bridges (under static load) increases with increase in span and decreases with decrease in height of pylon, irrespective of the mode shapes. Moreover, pylon shape does not seem to have any significant effect on the time p eriod. 29

5.2.2 Frequency vs Mode Number Graphs for A and H Pylon Models

A-shaped Pylon

H-shaped Pylon

Frequency vs Mode Number

Frequency vs Mode Number

3 ) c e s 2.5 / c 2 y c ( y 1.5 c n 1 e u 0.5 q e r 0 F

100m 200m 300m 400m 500m

3 ) c e s 2.5 / c 2 y c ( 1.5 y c n 1 e u0.5 q e r 0 F

1 2 3 4 5 6 7 8 9 10

100 200 300 400 500

4.5 4 3.5 3 2.5 2 1.5 1 0.5 0

300m 400m 500m 1 2 3 4 5 6 7 8 9 10

3 ) c e s 2.5 / c y 2 c ( y 1.5 c n e 1 u q0.5 e r F 0

100m 200m 300m 400m 500m 1 2 3 4 5 6 7 8 9 10

1 2 3 4 5 6 7 8 9 10 Mode Number Pylon Height = Span/3

) c e s / c y c ( y c n e u q e r F

200m

Mode Number Pylon Pylon Height = Span/2

Mode Number Pylon Height = Span/2

4 ) c 3.5 e s / c 3 y c 2.5 ( y c 2 n 1.5 e u 1 q a e r 0.5 F 0

100m

Mode Number Pylon Height = Span/3

100m 200m 300m 400m 500m 1 2 3 4 5 6 7 8 9 10

4 ) c 3.5 e s 3 / c y2.5 c ( y 2 c n1.5 e u 1 q0.5 e r 0 F

100m 200m 300m 400m 500m 1 2 3 4 5 6 7 8 9 10



Fig. 5.2 Frequency vs Mode Number Graphs for A and H-shaped Pylon Models The above graphs clearly depict that the frequency of vibration of cable-stayed bridges under dead load is higher for lesser span. Although pylon shape does not have any considerable effect on the frequency, pylon height does have – with decrease in pylon height, frequency is seen to increase. increase.

30

5.2.3 Pylon Top Deflection vs Pylon Height for A and H Pylon Models

Pylon Top Deflection vs Pylon Height for A-shaped pylon

0.001

) m0.0008 ( n o 0.0006 i t c e l f e 0.0004 D

100m 200m 300m 400m

0.0002

500m

0 L/4

L/3

L/2

Pylon Height (where L = Span) Pylon Top Deflection vs Pylon Height for H-shaped pylon

0.0035 0.003 ) m0.0025 ( n 0.002 o i t c e 0.0015 l f e D 0.001 0.0005

100m 200m 300m 400m 500m

0 L/4

L/3

L/2

Pylon Height (where L = Span)

Fig. 5.3 Pylon Top Deflections for A and H-shaped Pylon Models The above graphs show that the A-shaped pylon deflects less at its top than its corresponding H-shaped pylon irrespective of span length. Also, it can be observed that for pylon heights of span/2, the pylon top deflection is less in case of A-pylons and much higher in case of Hpylons. Thus, it can be accurately inferred that A-shaped pylons have higher stiffness that its counterpart H-shaped p ylons. ylons.

31

5.2.4 Deck Deflection vs Pylon Height for A and H Pylon Models

Maximum Deck Deflection vs Pylon Height for A-shaped pylon

1.6 1.4 ) 1.2 m ( n 1 o i 0.8 t c e l 0.6 f e D 0.4 0.2 0

100m 200m 300m 400m 500m

L/4

L/3 Pylon Height (where L = Span)

L/2

Maximum Deck Deflection vs Pylon Height for H-shaped pylon

1.5 1.3 ) 1.1 m ( n 0.9 o i 0.7 t c e l 0.5 f e D 0.3 0.1 -0.1

100m 200m 300m 400m 500m L/4

L/3

L/2

Pylon Height (where L = Span)

Fig. 5.4 Maximum Deck Deflections for A and H-shaped Pylon Models From the above graphs, it can be noticed that the maximum deflection of deck increases with increase in the span which is quite obvious otherwise also. Moreover, pylon shape and pylon height seem to have not much effect on the maximum deck deflection although variations are observed in deflection values of the models with pylon height of span/2. In addition, the 500 m model with pylon of span/2 height seems to have much lower deck deflection values irrespective of the shape of pylon p ylon,, when compared to the other models.

32

5.2.5 Longitudinal Reaction Forces at Pylon Bottom for A and H Pylon Models

Longitudinal Reaction at Pylon Bottom (A-Pylon) 100000 90000

L/4 P1

) N 80000 k ( e c r 70000 o F

L/4 P2 L/3 P1 L/3 P2 L/2 P1

60000

L/2 P2 50000 100m

200m

300m

400m

500m

Span (m)

Longitudinal Reaction at Pylon Bottom (H-Pylon) 18000 15000

L/4 P1

) N 12000 k ( e c r 9000 o F 6000

L/4 P2 L/3 P1 L/3 P2 L/2 P1

3000

L/2 P2

0 100m

200m

300m

400m

500m

Span (m)

Fig. 5.5 Longitudinal Reaction at Pylon Bottom for A and H-shaped Pylon Models These graphs indicate that differing heights of pylon do not have any significant effect on the magnitude of forces (in longitudinal direction) at the pylon base although pylon shape does have an immense effect. The forces fo rces in case of A-shaped A-s haped pylons are considerably higher when compared to the same forces in H-shaped pylons. In addition, the magnitude of these forces increases with increasing span in case of A pylons and vice-versa for H pylons. Also, it can be seen that for both legs of the pylon structure i.e. P1 and P2, the force magnitudes are one and the same. 33

5.2.6 Lateral Reaction Forces at Pylon Bottom for A and H Pylon Models

Lateral Reaction at Pylon Bottom Bot tom (A-Pylon) 6000 5000

L/4 P1

) N 4000 k ( 3000 e c r o F 2000

L/4 P2

1000

L/2 P2

L/3 P1 L/3 P2 L/2 P1

0 100m

200m

300m

400m

500m

Span (m) Lateral Reaction at Pylon Bottom Bott om (H-Pylon) 7000 6000

L/4 P1

) 5000 N k ( 4000 e c r 3000 o F

L/4 P2 L/3 P1 L/3 P2 L/2 P1

2000

L/2 P2

1000 0 100m

200m

300m

400m

500m

Span (m)

Fig. 5.6 Lateral Reaction at P ylon ylon Botto m for A and H-shaped Pylon Models From the above graphs, it can be seen that the magnitude of forces (in lateral direction) at the pylon base decreases as span length increases although there is no substantial effect of the pylon shape on the magnitude. In addition, the force magnitudes increase with decreasing pylon height. Furthermore, the magnitudes are different at the two legs of the p ylon (i.e. at P1 and P2) unlike in the case o f longitudinal forces seen previously. previously.

34

5.2.7 Longitudinal Moment at Pylon Bottom for A and H Pylon Models

Longitudinal Moment at Pylon Bottom (A-Pylon) 700000

) m N k ( t n e m o M

600000

L/4 P1

500000

L/4 P2

400000

L/3 P1

300000

L/3 P2

200000

L/2 P1

100000

L/2 P2

0 100m

200m

300m

400m

500m

Span (m) Longitudianl Moment at Pylon Bottom (H-Pylon) 350000 300000

) m N k ( t n e m o M

L/4 P1

250000

L/4 P2

200000

L/3 P1

150000

L/3 P2 L/2 P1

100000

L/2 P2 50000 0 100m

200m

300m

400m

500m

Span (m)

Fig. 5.7 Longitudinal Moment at P ylon Bottom for A and H-shaped Pylon Models The increase in longitudinal moment is almost linear with increasing span of the bridge in case of H-shaped pylon but models with A-shaped pylon show an erratic increase in longitudinal moment. It can also be observed that neither the height of pylon nor the pylon legs have much influence on the longitudinal moment values in case of H pylons. On the other hand, moment values for A-shaped pylons vary with both the pylon height as well as the pylon leg of the structure. In general, longitudinal moment is seen to increase with decreasing pylon height. 35

5.3 Non Linear Dynamic (Time History) Analysis

Dynamic may be defined simply as time-varying; thus a dynamic load is any load whose magnitude, direction, and/or position varies this time. Similarly, the structural response to a dynamic load, i.e. the resulting stresses and deflections, is also time-varying, or dynamic. Static-loading condition may be looked upon as a special form of dynamic loading. Dynamic analysis is basically divided into two approaches for evaluating structural response to dynamic loads: deterministic and non-deterministic. non-deterministic. A structural-dynamic problem differs from its static-loading counterpart in its time- varying nature of the dynamic problem. Because both loading and response vary with time, it is evident therefore that a dynamic problem does not have a single solution, as a static problem does; instead a succession of solutions corresponding to all times of interest must be established. Thus a dynamic analysis is clearly more complex and time consuming than a static analysis. Although if the the motions are so so slow that the the inertial forces are negligibly small, the analysis of response for any desired instant of time may be made by static structural analysis procedures even though the loa d and response response may ma y be time varying.

The models were subjected to different load time histories and their maximum deck deflections, cable forces and deck stresses have been plotted for the bridges with A and Hshaped pylons. The load time time histories used were were of Bhuj, El Centro and Uttarkashi. Uttarkashi. The pylon heights used are the same as in the static analysis.

5.3.1 Maximum Deck Deflection for A and H Pylon Models

The response has been plotted with zero end at the hinge supports and the 500 m end at the roller supports of the girder . From the following graphs, it can be b e ascertained that irrespective of pylon shape, the deck deflection is minimum at the mid-span for all the three earthquakes. This can be explained by the presence of dampers connecting the bridge deck and the horizontal beam of the pylon structure. The maximum values of deck deflection are seen at the middle section of the side spans but deflection values are slightly different for A and H pylons depending on the load time history applied. However, the maximum and minimum deck deflection is observed to be under Uttarkashi earthquake and Bhuj earthquake respectively, in both A-shaped and H-shaped pylon models.

36

Maximum Deflection of Deck (A-Pylon): Bhuj 2.00E-01 1.80E-01 1.60E-01 ) 1.40E-01 m 1.20E-01 ( n 1.00E-01 o 8.00E-02 i t c e 6.00E-02 l f e 4.00E-02 D 2.00E-02 0.00E+00

Span/2 Span/3 Span/4

0

100

200

300

400

500

Distance (m)

Maximum Deflection of Deck (A-Pylon): El Centro 7.00E-02 6.00E-02

) m ( n o i t c e l f e D

5.00E-02 4.00E-02


3.00E-02 2.00E-02 1.00E-02 0.00E+00 0

100

200

300

400

500

Distance (m)

Maximum Deflection of Deck (A-Pylon): Uttarkashi


9.00E-03 8.00E-03 7.00E-03 6.00E-03 5.00E-03 4.00E-03 3.00E-03 2.00E-03 1.00E-03


0.00E+00 0

100

200

300

400

500

Distance (m)

Fig. 5.8 Maximum Deck Deflection for A-Pylon Models under Different Load Time Histories

37

Maximum Deflection of Deck (H-Pylon): Bhuj


2.00E-01 1.80E-01 1.60E-01 1.40E-01 1.20E-01 1.00E-01 8.00E-02 6.00E-02 4.00E-02 2.00E-02 0.00E+00


0

100

200

300

400

500

Distance (m)

Maximum Deflection of Deck (H-Pylon): El Centro 6.00E-02


5.00E-02 4.00E-02 Span/2 Span/3 Span/4

3.00E-02 2.00E-02 1.00E-02 0.00E+00 0

100

200

300

400

500

Distance (m)

Maximum Deflection of Deck (H-Pylon): Uttarkashi 8.00E-03


7.00E-03 6.00E-03 5.00E-03 Span/2 Span/3 Span/4

4.00E-03 3.00E-03 2.00E-03 1.00E-03 0.00E+00 0

100

200

300

400

500

Distance (m)

Fig. 5.9 Maximum Deck Deflection for H-Pylon Models under Different Load Time Histories

38

5.3.2 Axial Forces in Cables for A and H Pylon Models

The models were also checked for axial forces present in the stay cables. The response has been plotted with the outermost cable at the roller support end of the girder numbered one and subsequently increasing as we move towards the hinged end. The total number of cables in one plane is 36 with equal spacing of 12.5 m throughout the length of the deck.

Axial Force in Cables (A-Pylon): Bhuj

) N2000 K ( e 1500 c r o F1000 l a i x 500 A

Span/2 plane1 Span/2 plane2 Span/3 plane1 Span/3 plane2 Span/4 plane1 Span/4 plane2

0 0

5

10

15

20

25

30

35

40

Cable Position (Nos.)

Axial Force in Cables (A-Pylon): El Centro

) N2500 k ( 2000 e c r 1500 o F l 1000 a i x 500 A


0 0

5

10

15

20

25

30

35

40


Axial Force in Cables (A-Pylon): Uttarkashi

) N 100 k ( 80 e c r 60 o F l 40 a i x A 20


0 0

5

10

15

20

25

30

35

40


Fig. 5.10 Axial Forces in Cables for A-Pylon Models under Different Load Time Histories Cables 18 and 19 which are the ones closest to the pylon and of the shortest length are having minimum axial force (tensile in nature). This is irrespective of t he shape of the pylon. Plane 1 39

and 2 which refers to the two different planes of the cable-stayed bridge model, too, does not seem to influence the magnitude of axial forces in the cables. It can a lso be seen that the axial forces in the cables are least under the Uttarkashi load time history and this applies to models with both A and H-shaped pylons. Whereas magnitude of forces under Uttarkashi load case are within 100 kN, the same forces in the other two cases ranges from 250 kN to 2000 kN. Although pylon height seems not to influence the cable axial forces in A-shaped pylon models, deviations can be observed ob served in the H-shaped ones.

Axial Force in Cables (H-Pylon): Bhuj ) 2.00E+03 N K ( 1.50E+03 e c r1.00E+03 o F l 5.00E+02 a i x A0.00E+00

Span/2 plane1 Span/2 plane2 Span/3 plane1 Span/3 plane2 Span/4 plane1 0

10

20

30

40

Span/4 plane2


Axial Force in Cables (H-Pylon): El Centro ) 2500 N k ( 2000 e c r 1500 o F 1000 l a i x 500 A

Span/2 plane1 Span/2 plane2 Span/3 plane1 Span/3 plane2 Span/4 plane1

0

Span/4 plane2 0

5

10

15

20

25

30

35

40


Axial Force in Cables (H-Pylon): Uttarkashi U ttarkashi ) N k ( e c r o F l a i x A

100 Span/2 plane1 Span/2 plane2 Span/3 plane1 Span/3 plane2 Span/4 plane1 Span/4 plane2

80 60 40 20 0 0

5

10

15

20

25

30

35

40


Fig. 5.11 Axial Forces in Cables for H-Pylon Models under Different Load Time Histories

40

5.3.3 Shell Stresses at Deck for A and H Pylon Models

The deck has been checked for both compressive and tensile stresses developing under the effect of all the three load time histories at peak ground acceleration (PGA). The occurrence of PGA for the Bhuj, El Centro and Uttarkashi earthquake are 46.622 sec, 11.472 sec and 1.481 sec respectively. The left end of the stress diagrams refers to the roller end of the bridge deck and the right end to the hinged support end.

From the graphs (Fig. 5.12 – Fig. 5.14) of shell stresses at deck, it can be observed that the maximum values are under load time histories of the Bhuj earthquake. The maximum stress values in case of A-p ylon models models are: Span/2 = 1.7 - 2.5 MPa Span/3 = 4.0 - 5.0 MPa Span/4 = 3.6 - 4.5 MPa A-shaped Pylons: Shell Stresses at Deck (A-Pylon): Bhuj

Shell Stresses at Deck (A-Pylon): El Centro

Shell Stresse Stressess at Deck Deck (A-P lon): Uttarkash Uttarkashii

Fig. 5.12 Deck Stresses for A-shaped pylon of Height = Span/2

41

Shell Stresses at Deck (A-Pylon): Bhuj


Shell Stresses at Deck (A-Pylon): Uttarkashi


Shell Stresses at Deck (A-Pylon): Bhuj


Shell Stresses at Deck (A-Pylon): Uttarkashi


42

H-shaped Pylons: Shell Stresses at Deck (H-Pylon): Bhuj

Shell Stresses at Deck (H-Pylon): El Centro

Shell Stresses at Deck (H-Pylon): Uttarkashi

Fig. 5.15 Deck Stresses for H-shaped pylon of Height = Span/2

Shell Stresses at Deck (H-Pylon): Bhuj



Fig. 5.16 Deck Stresses for H-shaped pylon of Height = Span/3 43

Shell Stresses at Deck (H-Pylon): Bhuj



Fig. 5.17 Deck Stresses for H-shaped pylon of Height = Span/4

From the graphs (Fig. 5.15 – Fig. 5.17) of shell stresses at deck, it can be observed that the maximum values are under load time histories of the Bhuj earthquake. The maximum stress values in case of H-p ylon models models are: Span/2 = Span/3 = Span/4 =

2.1 - 2.9 MPa 3.7 - 5.2 MPa 3.3 - 4.5 MPa

44

6. CONCLUSION:

Looking to the increased popularity of cable-stayed bridges, it is obvious that there is a need for more comprehensive comprehensive investigations of analysis and design of these contemporary contemporar y bridges. In order to have a proper understanding of the seismic behaviour of these bridges, 3-D earthquake analysis has been performed considering a variety of time histories like short, medium and long duration having different PGA value and different earthquake magnitudes. Vertical excitation which is usually ignored in the seismic analysis of buildings, drastically affects the response response of cable-s c able-stayed tayed bridges, b ridges, and hence the first ten major contributory modes were studied so as to obtain the most fundamental movements. Three dimensional models have been used to realistically model the complex geometry and configuration of towers and also to represent the actual dynamic behaviour of the bridge for seismic analysis. The conclusions drawn from the present study are: 

Cable-stayed bridges are highly non-linear structures under the effect of their own dead weight, where the structure stiffness increases with increasing load.



As the span of the bridge increases, time period increases i.e. natural frequency decreases because of decrease in stiffness of the bridges.



A-shaped pylons have more stiffness under static loads as ascertained from their low deflection values when compared to H-shaped pylons.



Deflection of deck under static loads is influenced by pylon shape as well as pylon height as shown by results.



Under dead load conditions, longitudinal forces and longitudinal moments at the pylon base reach upper limits of 95 MN and 600 MN-m respectively in select cases.



Cables of different lengths with different natural frequencies experience varying axial forces depending upon their relative positions. po sitions.



Shell stresses developing at the deck due to varying load time histories are both tensile and compressive in nature, but are within acceptab le limits.

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7. R EFERENCE EFERENCES S

Books:

1. Clough, R., Penzien, J.,”Dynamics of Structures”, Second Edition, Mc Graw-Hill International Editions, Civil Engineering Series. 2. Troitsky, M.S., “Cable Stayed Bridges: Theory and Design”, Crosby Lockwood Staples, London, 1977. Journal Articles:

1. Adeli, H., Zhang, J., “Fully Nonlinear Analysis of Composite Girder Cable-Stayed Bridges”, Computers & Structures, Vol. 54. No.2. pp. 267-277, 1995. 2. Allam, S.M, Datta, T.K., “Seismic Behaviour of Cable-Stayed Bridges Under MultiComponent Component Random Ra ndom Ground Motion”, Civil Engineering Department, Indian Institute of Technology, Delhi, New Delhi, India.

3. Chen, S.S. , Bingnan, Bingnan, S. , Feng, Y.Q. ,” Nonlinear Nonlinear Transient Response Response of Stay Cable With Viscoelasticity Damper in Cable-Stayed Bridge”, Applied Mathematics and Mechanics, English Edition, Vol. 25, No 6, Jun 2004.

4. Ghaffar, A., Nazmy, A.S. “Three-Dimensional Nonlinear Static Analysis of CableStructures, Vol. 34, No. 2. pp. 257-271, 1990. Stayed Bridges” , Computers and Structures,

5. Freire, A.M.S., Lopes, A.V., Negrao, J.H.O.,” Geometrical Nonlinearities on the Static Analysis of Highly Flexible Steel Cable-Stayed Bridges”, Computers and Structures 84, pp. 2128–2140, 2006.

6. Fleming, J.F., “Nonlinear Static Analysis of Cable-Stayed Bridge Structures”, Computers and Structures, Vol. 11, pp. 621-635, 1979.

7. Kanok-Nukulchai, W., Hong, G., “Nonlinear Modelling of Cable-Stayed Bridges”, J. Construct. Steel Research, Vol.26, pp. 249-266, 1993.

8. Nazmy, A.S., “Nonlinear Earthquake response Analysis of Cable Stayed Bridges subjected to multiple support excitations”, University Microfilms International Journal, Princeton University, 1987.

9. Tuladhar, R., Dilger,W.H., ”Effect Of Support Conditions on Seismic Response of Cable-Stayed Bridges”, Canadian Journal of Civil Engineering, Vol.26 , pp. 631-645, 1999.

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10. Ganev, T., Yamazaki, F., Ishizaki, H., Kitazawa, M., “Response Analysis of the Higashikobe Bridge and Surrounding Soil In The 1995 Hyogoken-Nanbu Earthquake”, Earthquake Engineering and Structural Dynamics Vol.27 , pp. 557-576, 1998. 11. Ren, W.X., Peng, X.L., Lina, Y.Q.,” Experimental and Analytical Studies on Dynamic Characteristics of a Large Span Cable-Stayed Bridge”, Engineering Structures, Vol.27 , pp. 535-548, 2005.

12. Agarwal, T.P., “Cable Stayed Bridges- Parametric Study”, Journal of bridge engineering, pp. 61-67, May 1997. 13. Fleming, J., Egeseli E., “Dynamic Behaviour of Cable Stayed Bridge”, Earthquake Engineering and structural dynamics, Vol. 8, pp. 1-16, 1980.

14. Wang, P., Yang, C., “Parametric Studies on Cable Stayed Bridges”, Computers and Structures, Vol. 60 , pp. 243-260, 1996

15. George, H., “Influence of Deck Material on Cable Stayed Bridges to Live Loads”, Journal of bridge engineering, pp. 136-142, May 1999. Websites:

1. “List of Largest Cable Stayed Bridges” http://en.wikipedia.org/wiki/List_of_largest_cable-stayed_bri http://en.wikipedia.org/wiki/Lis t_of_largest_cable-stayed_bridges, dges, Accessed Sep. 18, 2009) 2. “Basic Concepts of Cable Stayed S tayed Bridges” (http://www.stbridge.com.cn/English/ , Oct. 7, 2009) 3. “Structural Details of Various Bridges” (http://en.structurae.de/structures/stype/index.cfm?ID=1, Sep. 19, 2009)

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Non-Linear Time History Analysis of Cable Stayed Bridges

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