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) Dr. G. R. Vesmawala Shobhit Bhatnagar (U06CE043) Prof. A. J. Shah C. Uma (U06CE056)

DEPARTMENT OF APPLIED MECHANICS S. V. NATIONAL INSTITUTE OF 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 bridge’s structural response under different load conditions.

In the present study, an attempt has been made to analyze the seismic response of cable-

stayed 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, Surat. 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 Page no. 1. Introduction 1 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

1 1 3 4 4 5 5 6 6

2. Literature review 9 2.1. Concept of Cable Stayed Bridges

2.2. Static Analysis of Cable Stayed 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 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

5. Results and Discussion 28 5.1 Introduction 28 5.2 Non-Linear Static Analysis 28 5.2.1 Time Period vs Mode Number Graphs 29 5.2.2 Frequency vs Mode Number Graphs 30 5.2.3 Pylon Top Deflection vs Pylon Height 31 5.2.4 Deck Deflection vs Pylon Height 32 5.2.5 Longitudinal Reaction Forces at Pylon Bottom 33 5.2.6 Lateral Reaction Forces at Pylon Bottom 34 5.2.7 Longitudinal Moment at Pylon Bottom 35 5.3 Non Linear Dynamic (Time History) Analysis 36 5.3.1 Maximum Deck Deflection 36 5.3.2 Axial Forces in Cables 39 5.3.3 Shell Stresses at Deck 41 6.

Conclusions

45

7.

References

46

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 of the bridge 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 Pylon 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 Pylon Models

5.6 Lateral Reaction at Pylon Bottom for A and H-shaped Pylon Models

5.7 Longitudinal Moment at Pylon Bottom for A and H-shaped Pylon Models

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

Histories

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

Histories

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

Histories

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

Histories

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



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



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

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

2

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

3

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 three-

dimensional 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.

4

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)

5

(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

6

one of the main structural elements, which lead to the successful development of modern cable-

stayed 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.

7

Rank Photograph Name Location Country

Height of pylon

Longest span Year Pylons

[1]

Sutong Bridge

Suzhou, Nantong

People's Republic of

China

306 m

1,088 m (3,570 ft) 2008 2

[2]

Stonecutters Bridge

Rambler Channel

Hong Kong

293 m 1,018 m (3,340 ft) 2009 2

[3]

Tatara Bridge

Seto Inland Sea Japan 220 m 890 m

(2,920 ft) 1999 2

[4]

Pont de Normandie

Le Havre France

214.77 m

856 m (2,808 ft) 1995 2

[5]

Incheon Bridge

Incheon South Korea 230.5 m 800 m (2,625 ft) 2009 2

[6]

Shanghai Yangtze

River Bridge

Shanghai


China

270 m

730 m (2,395 ft) 2009 2

[7]

Second Nanjing

Yangtze Bridge

Nanjing, Jiangsu


China

270 m 628 m (2,060 ft) 2001 2

8

[8]

Jintang Bridge

Zhoushan Archipelago

People's Republic of China

202.5 m 620 m (2,034 ft) 2009 2

[9]

Yangpu Bridge

Shanghai

People's Republic of China

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


China

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

Denmark- Sweden 204 m 490 m

(1,608 ft) 1999 2

9

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 beam-

column 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

10

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

11

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 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

12

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

13

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.

14

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.

Section A (m2) I (m4) E (kN/m2)

Girder (steel) 0.319 1.131 1.655E+08

Towers (steel) 0.312 0.623 1.655E+08

Cables

Cable No. A (m2) E (kN/m2)

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

15

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)

16

Fig. 3.3 Layout of 3D Model

Table 3.2 Sectional dimensions of the 3D model

Section A (m2) I (m4) E (kN/m2)

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. A (m2) E (kN/m2)

15, 54, 70, 64,15’,54’,

70’, 64’ 0.042 1.655E+08

45, 55, 71, 65 45’, 55’,

71’, 65’ 0.016 1.655E+08

46, 56, 72, 66 46’, 56’,

72’, 66’ 0.016 1.655E+08

15

15’ 54, 54’

70

70’

64, 64’

45

45’

55

71 65, 65’

55’ 71’

46 56, 56’

72, 72’ 66, 66’

46’

17

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

18

Table 3.3 First 10 mode shapes of the 2D mode

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

19

Table 3.4 First 10 mode shapes of the 3D model

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

20

Mode Time Period (sec)

Time Period

(sec) % Variation of

2D with 2D

(Nazmy)

% Variation of

2D with 3D

Model Number 2D model 3D model 2D (Nazmy)

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.

21

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 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

22

4.2 Material Properties

4.2.1 Concrete

� Grade: M 45

� Modulus of elasticity: 33541020 kN/m2

� Poisson’s ratio: 0.2

� Coefficient of thermal expansion: 1.170E-05 /0C

� Shear modulus: 12900392 kN/m2

4.2.2 Steel

� Modulus of elasticity: 1.999E+08




4.2.3 Steel Tendon (for cable)

� Modulus of elasticity: 1.580E+08




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).

23

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

24

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 non-

prismatic 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







4.5.3 Pylon Top







4.6 Cable Section

� Material: Steel tendon (for cable)

� Diameter: 0.2 m

25

4.7 Support Conditions:

� Deck supports: Hinge restraints at one end and roller restraints 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 displacement 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 spectrum (for peak value of equivalent static force and base

shear);

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

� Pseudo-displacement spectrum (for peak deformation).

26

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: 26th

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

4.9.2 The El Centro Earthquake

� Location: Southern California

� Year: 18th May, 1940



� Excitation type: Medium


� Step size: 0.02



Peak Ground Acceleration vs Time (Bhuj)

Peak Ground Acceleration vs Time (El Centro)

27

4.9.3 The Uttarkashi Earthquake

� Location: Tehri Region, Himalaya

� Year: 20th Oct, 1991



� Excitation type: Short


� Step size: 0.02



Peak Ground Acceleration vs Time (Uttarkashi)

28

5. RESULTS AND DISCUSSION

5.1 Introduction

To understand the static non-linear behaviour 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. Non-

linearity 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 non-

linear 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 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.

29

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

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 period.

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10

Tim

e Per

iod

(sec

)

Mode NumberPylon Height = Span/2

H-shaped PylonTime Period vs Mode Number

100m

200m

300m

400m

500m

01234567

1 2 3 4 5 6 7 8 9 10

Tim

e Pe

riod

(sec

)


100m

200m

300m

400m

500m 01234567

1 2 3 4 5 6 7 8 9 10

Tim

e Pe

riod

(sec

)


100m

200m

300m

400m

500m

01234567

1 2 3 4 5 6 7 8 9 10

Tim

e Per

iod

(sec

)


100m

200m

300m

400m

500m 01234567

1 2 3 4 5 6 7 8 9 10

Tim

e Per

iod

(sec

)


100m

200m

300m

400m

500m

0

2

4

6

8

1 2 3 4 5 6 7 8 9 10

Tim

e Pe

riod

(sec

)


A-shaped PylonTime Period vs Mode Number

100m

200m

300m

400m

500m

30

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

A-shaped Pylon

Frequency vs Mode Number

H-shaped Pylon

Frequency vs Mode Number

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.

00.5

11.5

22.5

3

1 2 3 4 5 6 7 8 9 10

Freq

uenc

y (c

yc/se

c)


100m

200m

300m

400m

500m 00.5

11.5

22.5

3

1 2 3 4 5 6 7 8 9 10

Freq

uenc

y (c

yc/se

c)


100m

200m

300m

400m

500m

00.5

11.5

22.5

33.5

4

1 2 3 4 5 6 7 8 9 10

Frea

quen

cy (c

yc/s

ec)


100

200

300

400

500 00.5

11.5

22.5

3

1 2 3 4 5 6 7 8 9 10

Freq

uenc

y (c

yc/s

ec)


100m

200m

300m

400m

500m

00.5

11.5

22.5

33.5

44.5

1 2 3 4 5 6 7 8 9 10

Freq

uenc

y (c

yc/s

ec)


100m

200m

300m

400m

500m 00.5

11.5

22.5

33.5

4

1 2 3 4 5 6 7 8 9 10

Freq

uenc

y (c

yc/se

c)


100m

200m

300m

400m

500m

31

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

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 H-

pylons. Thus, it can be accurately inferred that A-shaped pylons have higher stiffness that its

counterpart H-shaped pylons.

0

0.0002

0.0004

0.0006

0.0008

0.001

L/4 L/3 L/2

Def

lect

ion

(m)

Pylon Height (where L = Span)

Pylon Top Deflection vs Pylon Height for A-shaped pylon

100m200m300m400m500m

00.00050.001

0.00150.002

0.00250.003

0.0035

L/4 L/3 L/2

Def

lect

ion

(m)


Pylon Top Deflection vs Pylon Height for H-shaped pylon

100m200m300m400m500m

32

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

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, when compared to the other models.

00.20.40.60.8

11.21.41.6

L/4 L/3 L/2

Def

lect

ion

(m)


Maximum Deck Deflection vs Pylon Height for A-shaped pylon

100m200m300m400m500m

-0.10.10.30.50.70.91.11.31.5

L/4 L/3 L/2

Def

lect

ion

(m)


Maximum Deck Deflection vs Pylon Height for H-shaped pylon

100m200m300m400m500m

33

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

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 in case of A-shaped 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.

50000

60000

70000

80000

90000

100000

100m 200m 300m 400m 500m

Forc

e (k

N)

Span (m)

Longitudinal Reaction at Pylon Bottom (A-Pylon)

L/4 P1

L/4 P2

L/3 P1

L/3 P2

L/2 P1

L/2 P2

0

3000

6000

9000

12000

15000

18000

100m 200m 300m 400m 500m

Forc

e (k

N)

Span (m)

Longitudinal Reaction at Pylon Bottom (H-Pylon)

L/4 P1

L/4 P2

L/3 P1

L/3 P2

L/2 P1

L/2 P2

34

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

Fig. 5.6 Lateral Reaction at Pylon Bottom 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 pylon (i.e. at P1

and P2) unlike in the case of longitudinal forces seen previously.

0

1000

2000

3000

4000

5000

6000

100m 200m 300m 400m 500m

Forc

e (k

N)

Span (m)

Lateral Reaction at Pylon Bottom (A-Pylon)

L/4 P1

L/4 P2

L/3 P1

L/3 P2

L/2 P1

L/2 P2

0

1000

2000

3000

4000

5000

6000

7000

100m 200m 300m 400m 500m

Forc

e (k

N)

Span (m)

Lateral Reaction at Pylon Bottom (H-Pylon)

L/4 P1

L/4 P2

L/3 P1

L/3 P2

L/2 P1

L/2 P2

35

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

Fig. 5.7 Longitudinal Moment at Pylon 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.

0

100000

200000

300000

400000

500000

600000

700000

100m 200m 300m 400m 500m

Mom

ent (

kN-m

)

Span (m)

Longitudinal Moment at Pylon Bottom (A-Pylon)

L/4 P1

L/4 P2

L/3 P1

L/3 P2

L/2 P1

L/2 P2

0

50000

100000

150000

200000

250000

300000

350000

100m 200m 300m 400m 500m

Mom

ent (

kN-m

)

Span (m)

Longitudianl Moment at Pylon Bottom (H-Pylon)

L/4 P1

L/4 P2

L/3 P1

L/3 P2

L/2 P1

L/2 P2

36

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.

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 motions are so slow that 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 load and response may 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 H-

shaped pylons. The load time histories used were of Bhuj, El Centro and 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 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.

37

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

0.00E+002.00E-024.00E-026.00E-028.00E-021.00E-011.20E-011.40E-011.60E-011.80E-012.00E-01

0 100 200 300 400 500

Def

lect

ion

(m)

Distance (m)

Maximum Deflection of Deck (A-Pylon): Bhuj

Span/2Span/3Span/4

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

7.00E-02

0 100 200 300 400 500

Def

lect

ion

(m)

Distance (m)

Maximum Deflection of Deck (A-Pylon): El Centro

Span/2Span/3Span/4

0.00E+001.00E-032.00E-033.00E-034.00E-035.00E-036.00E-037.00E-038.00E-039.00E-03

0 100 200 300 400 500

Def

lect

ion

(m)

Distance (m)

Maximum Deflection of Deck (A-Pylon): Uttarkashi

Span/2Span/3Span/4

38

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

0.00E+002.00E-024.00E-026.00E-028.00E-021.00E-011.20E-011.40E-011.60E-011.80E-012.00E-01

0 100 200 300 400 500

Def

lect

ion

(m)

Distance (m)

Maximum Deflection of Deck (H-Pylon): Bhuj

Span/2Span/3Span/4

0.00E+00

1.00E-02

2.00E-02

3.00E-02

4.00E-02

5.00E-02

6.00E-02

0 100 200 300 400 500

Def

lect

ion

(m)

Distance (m)

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

Span/2Span/3Span/4

0.00E+00

1.00E-03

2.00E-03

3.00E-03

4.00E-03

5.00E-03

6.00E-03

7.00E-03

8.00E-03

0 100 200 300 400 500

Def

lect

ion

(m)

Distance (m)

Maximum Deflection of Deck (H-Pylon): Uttarkashi

Span/2Span/3Span/4

39

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.

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 the shape of the pylon. Plane 1

0

500

1000

1500

2000

0 5 10 15 20 25 30 35 40

Axi

al F

orce

(KN

)

Cable Position (Nos.)

Axial Force in Cables (A-Pylon): Bhuj

Span/2 plane1

Span/2 plane2Span/3 plane1Span/3 plane2

Span/4 plane1Span/4 plane2

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35 40

Axi

al F

orce

(kN

)


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

Span/2 plane1Span/2 plane2Span/3 plane1Span/3 plane2Span/4 plane1Span/4 plane2

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40

Axi

al F

orce

(kN

)


Axial Force in Cables (A-Pylon): Uttarkashi

Span/2 plane1

Span/2 plane2

Span/3 plane1

Span/3 plane2

Span/4 plane1

Span/4 plane2

40

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 also 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 in the H-shaped ones.

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

0.00E+00

5.00E+02

1.00E+03

1.50E+03

2.00E+03

0 10 20 30 40

Axi

al F

orce

(KN

)


Axial Force in Cables (H-Pylon): Bhuj

Span/2 plane1

Span/2 plane2

Span/3 plane1

Span/3 plane2

Span/4 plane1

Span/4 plane2

0

500

1000

1500

2000

2500

0 5 10 15 20 25 30 35 40

Axi

al F

orce

(kN

)


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

Span/2 plane1

Span/2 plane2

Span/3 plane1

Span/3 plane2

Span/4 plane1

Span/4 plane2

0

20

40

60

80

100

0 5 10 15 20 25 30 35 40

Axi

al F

orce

(kN

)


Axial Force in Cables (H-Pylon): Uttarkashi

Span/2 plane1Span/2 plane2Span/3 plane1Span/3 plane2Span/4 plane1Span/4 plane2

41

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-pylon 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:

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

Shell Stresses at Deck (A-Pylon): Bhuj

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

Shell Stresses at Deck (A-Pylon): Uttarkashi

42









43

H-shaped Pylons:

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


Shell Stresses at Deck (H-Pylon): Bhuj

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

Shell Stresses at Deck (H-Pylon): Uttarkashi




44


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-pylon models are:

Span/2 = 2.1 - 2.9 MPa Span/3 = 3.7 - 5.2 MPa Span/4 = 3.3 - 4.5 MPa




45

6. CONCLUSION:

Looking to the increased popularity of cable-stayed bridges, it is obvious that there is a need

for more comprehensive investigations of analysis and design of these contemporary 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 of cable-stayed bridges, 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.

Shell stresses developing at the deck due to varying load time histories are both

tensile and compressive in nature, but are within acceptable limits.

46

7. REFERENCES

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 Multi-

Component Random Ground Motion”, Civil Engineering Department, Indian Institute

of Technology, Delhi, New Delhi, India.

3. Chen, S.S. , Bingnan, S. , Feng, Y.Q. ,” Nonlinear Transient 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 Cable-

Stayed Bridges”, Computers and Structures, Vol. 34, No. 2. pp. 257-271, 1990.

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.

47

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_bridges, Accessed Sep. 18,

2009)

2. “Basic Concepts of Cable Stayed 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)

Non-Linear Time History Analysis of Cable Stayed Bridges

Documents

3d csb model

mode numbertime

load time

normal design

maximum stress

el centroaxial

3 pylon height

equal side