evaluation of seismic capacity of the flyovers in …

EVALUATION OF SEISMIC CAPACITY OF THE FLYOVERS IN

BANGLADESH

MD. ABDUL KADER

DEPARTMENT OF CIVIL ENGINEERING

DHAKA UNIVERSITY OF ENGINEERING AND TECHNOLOGY, GAZIPUR

August, 2010

i

EVALUATION OF SEISMIC CAPACITY OF THE FLYOVERS IN

BANGLADESH

A Thesis

by

MD. ABDUL KADER

submitted to the Department of Civil Engineering,

Dhaka University of Engineering and Technology, Gazipur,

in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE IN CIVIL ENGINEERING

August, 2010

ii

The thesis titled Evaluation of Seismic Capacity of the Flyovers in Bangladesh submitted by Md. Abdul Kader, Student No. 052101 (P), Session 2005-2006 has been accepted as satisfactory in partial fulfillment of the requirements for the degree of Master of Science in Civil Engineering on August 23, 2010.

BOARD OF EXAMINERS

1. Dr. Md. Ganesh Chandra Saha

Professor& Head Department of Civil Engineering Dhaka University of Engineering and Technology (DUET), Gazipur

Chairman

2. Md. Nuruzzaman

Professor Department of Civil Engineering Dhaka University of Engineering and Technology (DUET), Gazipur

Member

3. Dr. Mohammad Abdur Rashid

Professor Department of Civil Engineering Dhaka University of Engineering and Technology (DUET), Gazipur

Member

4. Dr. Md. Mozammel Hoque

Associate Professor Department of Civil Engineering Dhaka University of Engineering and Technology (DUET), Gazipur

Member (Supervisor)

5. Dr. Syed Ishtiaq Ahmad

Associate Professor Department of Civil Engineering Bangladesh University of Engineering and Technology (BUET), Dhaka

Member (External)

iii

CANDIDATE’S DECLARATION

It is hereby declared that this thesis or any part of it has not been submitted elsewhere for the award of any degree or diploma.

Candidate’s Signature

(Md. Abdul Kader)

iv

DEDICATED TO

MY MOTHER

v

ACKNOWLEDGMENT

It is a great pleasure of the author to acknowledge respect and gratitude to his

supervisor Dr. Md. Mozammel Hoque, Associate Professor, Department of Civil

Engineering, DUET, Gazipur for his kind cooperation, constant encouragement,

valuable advice and guidance throughout the research work. He is also greatly

acknowledged for his constructive suggestions. Without his whole-hearted

supervision, this work would not have been possible.

The author wishes to express gratitude to Professor Dr. Ganesh Chandra Saha, Head,

Department of Civil Engineering, DUET, Gazipur, for encouragement and for

providing fund and facilities to conduct this research.

The author would like to thank Professor Md. Nuruzzaman, and Professor Dr.

Mohammad Abdur Rashid, Department of Civil Engineering, DUET, Gazipur, and

Dr. Syed Ishtiaq Ahmad, Associate Professor, Department of Civil Engineering,

Bangladesh University of Engineering and Technology (BUET), Dhaka, who

provided him with information, comments, corrections and criticisms pertaining to the

preparation of this thesis. He would also like to thank Professor Dr. Md. Showkat

Osman, Mr. Md. Abdus Salam and Mr. Md. Rezaul Karim, Faculty of Civil

Engineering, and Mr. Firoz Ahmed Pathowary, M.Sc. Student, DUET, Gazipur for

their assistance during this research work. He would also like to thank all other

faculty members of DUET for their valuable encouragement to perform this research.

The author would like to thank Md. Enamul Hoque, Local Government Engineering

Department (LGED), Dhaka, Md. Shabbir Hassan Khan and Md. Nazmul Islam, Road

and Highway Department (RHD), Dhaka to give their suggestions, information and

providing design data and drawing for this research work.

Finally, the author would like to thank his mother, brothers and family members for

their constant support and encouragement. Without their support, this work would

never have been completed.

August, 2010 Author

vi

ABSTRACT

The objectives of the study are to evaluate the seismic capacity of the sub-structural

members and the systems. In the study, among four flyovers built in Bangladesh,

Mohakhali and Khilgaon flyovers are used as model flyovers. The capacities are

evaluated analytically both in terms of strength and deformation. Nonlinear static

analyses: sectional analyses and pushover analyses are carried out along longitudinal

and transverse directions for achieving the objectives. The moment-curvature

relationships are obtained from sectional analyses, while load-displacement

relationships are obtained from the pushover analyses. The yield and ultimate bending

moment capacity and associated deformations of different members are obtained from

the moment-curvature relationship. The expected failure mode of the piers is obtained

from shear and bending capacity of the piers. The uncertainty of material strength on

the capacity has been addressed using Latin Hypercube sampling technique. Finally, a

capacity hierarchy factor for piers and pile foundations are estimated.

The range of normalized lateral strength of the piers of Mohakhali flyover as found

from the analytical investigation is 0.52W to 1.18W in transverse direction and 0.3W

to 0.86W in longitudinal direction. On the contrary, that for Khilgaon flyover is

0.17W to 0.39W in both the directions. It has been found from the failure mode

analyses of the piers of Mohakhali flyover that all the piers except three are expected

to fail in bending mode, while almost all of the piers of Khilgaon flyover are expected

fail in shear mode. It implies that adequate warnings will not be found before collapse

of the Khilgaon flyover under a large magnitude earthquake. The range of normalized

lateral strength of the substructure of Mohakhali flyover is 0.52W to 0.76W; 0.3W to

0.71W in transverse and longitudinal direction, respectively, whereas that for

Khilgaon flyover is 0.17W to 0.39W. For Mohakhali flyover, fifteen piers out of

eighteen in transverse direction and five out of eighteen in longitudinal direction

possess lateral strength larger than that of the respective pile foundations, while

sixteen out of thirty six piers’ strengths are found larger than those of the respective

pile foundation for Khilgaon flyover. The capacity hierarchy factors found for many

cases in both the flyovers less than one indicates that the possible damages under a

large magnitude of earthquake are like to occur in the pile foundations which will be

very difficult to go for inspection and necessary repair.

vii

CONTENTS Page No.

TITLE PAGE iCERTIFICATION PAGE iiDECLARATION iiDEDICATION ivACKNOWLEDGEMENTS vABSTRACT viCONTENTS viiLIST OF FIGURES xLIST OF TABLES xvChapter 1: INTRODUCTION 1 1.1

1.2 1.3 1.4 1.5

Background Objective of the Study Methodology 1.3.1 Moment-Curvature Relationship 1.3.2 Load-Displacement Relationship 1.3.3 Failure Mode 1.3.4 Capacity Hierarchy Scope and Limitation of the Study Contents of the Study

134455566

Chapter 2: MODEL FLYOVERS 8

2.1 2.2 2.3 2.4 2.5

Model Flyovers Mohakhali Flyover Khilgaon Flyover Materials Properties 2.4.1 Constitutive Model of Concrete 2.4.2 Constitutive Model for Reinforcing Steel 2.4.3 Soil Property Conclusion

88

101414161718

Chapter 3: SECTIONAL ANALYSIS 20 3.1

3.2 3.3 3.4

Introduction Sectional Analysis Methods 3.3.1 Sectional Analysis Result and Discussions

2020212124

viii

Page No.

3.5

3.4.1 Moment-Curvature Relationship of the Pier Sections 3.4.2 Moment-Curvature Relationship of the Pile Sections 3.4.3 Moment-Curvature Relationship of the Pile Caps Sections 3.4.4 Characteristic Moment and Curvature of the Piers Conclusion

2426283233

Chapter 4: PUSHOVER ANALYSIS 34 4.1

4.2 4.3 4.4 4.5 4.6 4.7

Background Pushover Analysis 4.2.1 Analytical Procedure used for Piers 4.2.2 Analytical Procedure used for Piles Procedure for sub-structural system 4.3.1 Analytical model of substructure Analytical Model of Sub-Structural Members and System 4.4.1 Pier with Bottom End Fixed 4.4.2 Pile Foundation 4.4.3 Substructure of Flyover Parameters Estimation for Analytical Model 4.5.1 Weight of Superstructure 4.5.2 Material Properties 4.5.3 Yield Moment of Pile Cap and Pile Body 4.5.4 Soil Spring 4.5.5 Adequacy of Thickness for Rigidity of pile cap Results and Discussions 4.6.1 Load-Displacement Relationship of the Piers 4.6.2 Load-Displacement Relationship of the Pile Foundations 4.6.3 Load-Displacement Relationship of Substructure Conclusion

343435363737414141424343434343474848545762

Chapter 5: LATERAL STRENGTH AND DUCTILITY 64 5.1

5.2 5.3 5.4 5.5

Introduction Evaluation of Lateral Strength of Piers 5.2.1 Shear Capacity of piers 5.2.2 Flexural Capacity of piers Ductility of piers Failure Mode of piers Analytical Results

64646467686970

ix

Page No.

5.6 5.7

5.5.1 Bending Strength of the Piers 5.5.2 Shear Strength of the Piers 5.5.3 Lateral Strength of Pile Foundations 5.5.4 Lateral Strength of Substructures 5.5.5 Failure Mode of Piers 5.5.6 Ductility of piers 5.5.7 Probability of shear Failure Hierarchy Factor Conclusions

707274757779818284

Chapter 6: THE EFFECT OF VARIABILITY OF MATERIALS STRENGTH

86

6.1 6.2 6.3 6.4 6.5 6.6 6.7

Introduction Statistical Parameters of Material Properties Latin Hypercube Sampling Methodology Statistical Tests 6.5.1 Chi-Square Test 6.5.2 Kolmogorov-Smirnov (K-S) Test Results and Discussions 6.6.1 Moment Curvature Relationship 6.6.2 Load Displacement Relationship 6.6.3 Statistical Distribution Conclusions

8686888989899090909295

106

Chapter 7: CONCLUSION AND RECOMMENDATIONS FOR FURTHER STUDY

107

7.1 7.2 7.3

Introduction Conclusions Recommendations for further study

107107108

REFERENCES 109

SYMBOLS AND NOTATIONS 113

x

LIST OF FIGURES

Fig. No.

Title of Figure PageNo.

2.1 Piers layout of Mohakhali flyover 8

2.2 Elevation and cross-section of a typical pier of Mohakhali flyover 9

2.3 Piers layout of Khilgaon flyover 11

2.4 Elevation and cross-section of a typical pier of Khilgaon flyover 12

2.5 Stress-strain relationship for concrete 16

2.6 Stress-strain relationship for steel 16

2.7 Bore log of sub soil used for Mohakhali flyover 17

2.8 Bore log of sub soil used for Khilgaon flyover 18

3.1 Overview of numerical sectional analysis of a RC section 21

3.2 Relation between bending moment and curvature 24

3.3 Moment-curvature relationship of the Mohakhali flyover pier bottomsection in transverse direction

25

3.4 Moment-curvature relationship of the Mohakhali flyover pier bottom section in longitudinal direction

25

3.5 Moment-curvature relationship of pier bottom section of Khilgaon flyover

26

3.6 Cross-section of Mohakhali flyover pile 26

3.7 Moment-curvature relationship of the Mohakhali flyover pile section 27

3.8 Moment-curvature relationship of the Khilgaon flyover pile sections 28

3.9 Moment-curvature relationship of the Mohakhali flyover pile cap section in longitudinal direction

28

3.10 Moment-curvature relationship of the Mohakhali flyover pile cap sections in transverse direction

28

3.11 Moment-curvature relationship of the Khilgaon flyover pile cap sections in transverse direction

30

3.12 Moment-curvature relationship of the Khilgaon flyover pile cap sections in longitudinal direction

31

xi

Fig. No.


4.1 Simple pushover analysis 34

4.2 Numerical evaluation of flexural and shear component of displacement 36

4.3 Schematic diagram of the fiber model of pier cross-section 38

4.4 Element geometry of fiber element 38

4.5 Analytical model of pier 39

4.6(a) Element geometry of inelastic element 39

4.6(b) Elastic element stiffness 40

4.7(a) Pile with surrounding soil 40

4.7(b) Pile with spring soil model 41

4.8 Analytical model of a flyover pier 41

4.9 Analytical model of a flyover pile foundation 42

4.10 Analytical model of a flyover substructure 42

4.11 Pile resistance characteristics 44

4.12 Isolated pile cap 48

4.13 Load displacement relationship of the piers at top of Mohakhali flyoverin transverse direction

50

4.14 Load displacement relationship of the piers at top of Mohakhali flyoverin longitudinal direction

51

4.15 Load displacement relationship of the piers at top of Khilgaon flyover 53

4.16 Load displacement relationship of the pile foundation at center of pilecap of Mohakhali flyover in transverse direction

54

4.17 Load displacement relationship of the pile foundation at center of pile cap of Mohakhali flyover in longitudinal direction

54

4.18 Load displacement relationship of the pile foundation at center of pile cap of Khilgaon flyover in transverse direction

56

4.19 Load displacement relationship of the system at top of pier of Mohakhali flyover in transverse direction

58

4.20 Load displacement relationship of the system at top of pier ofMohakhali flyover in longitudinal direction

59

xii

Fig. No.


4.21 Load displacement relationship of the system at top of pier of Khilgaon flyover

61

5.1 Determination of effective height of a section 67

5.2 Numerical evaluation of bending capacity of pier 67

5.3 Evaluation of Failure Mode, Lateral Strength and Ductility Capacityfor a RC member

69

5.4 Lateral strength of Mohakhali flyover for piers under bending intransverse direction

70

5.5 Normalized Lateral strength of Mohakhali flyover for piers underbending in transverse direction

70

5.6 Lateral strength of Mohakhali flyover for piers under bending inlongitudinal direction

70

5.7 Normalized Lateral of Mohakhali flyover for piers under bending inlongitudinal direction

70

5.8 Lateral strength Khilgaon flyover piers under bending 71

5.9 Normalized lateral strength Khilgaon flyover piers under bending 71

5.10 Lateral strength of Mohakhali flyover for piers under shear intransverse direction

72

5.11 Normalized lateral strength of Mohakhali flyover for piers under shearin transverse direction

72

5.12 Lateral strength of Mohakhali flyover for piers under shear in longitudinal direction

72

5.13 Normalized Lateral strength of Mohakhali flyover for piers under shearin longitudinal direction

73

5.14 Shear strength Khilgaon flyover piers under shear 73

5.15 Normalized lateral strength Khilgaon flyover piers under shear 73

5.16 Lateral strength of pile foundation of Mohakhali flyover 74

5.17 Normalized lateral strength of pile foundation of Mohakhali flyover 74

5.18 Lateral strength Khilgaon flyover pile foundation 74

5.19 Normalized lateral strength of Khilgaon flyover pile foundation 75

xiii

Fig. No.


5.20 Lateral strength of Mohakhali flyover substructure in transversedirection

75

5.21 Normalized Lateral strength of Mohakhali flyover substructure intransverse direction

75

5.22 Lateral strength of Mohakhali flyover substructure in longitudinal direction

76

5.23 Normalized Lateral strength of Mohakhali flyover substructure inlongitudinal direction

76

5.24 Lateral strength of Khilgaon flyover substructure 76

5.25 Normalized lateral strength of Khilgaon flyover for substructure 76

5.26 Curvature ductility of Mohakhali flyover in transverse direction 79

5.27 Curvature ductility of Mohakhali flyover in longitudinal direction 79

5.28 Displacement ductility of Mohakhali flyover piers in transversedirection

79

5.29 Displacement ductility of Mohakhali flyover piers in longitudinaldirection

80

5.30 Hierarchy factor of the piers of Mohakhali flyover in transversedirection

83

5.31 Hierarchy factor of the piers of Mohakhali flyover in longitudinaldirection

84

5.31 Hierarchy factor of the piers of Khilgaon flyover 84

6.1 Relationship between mean value and characteristic value 87

6.2 Moment-curvature relationship of Mohakhali flyover piers intransverse direction

91

6.3 Moment-curvature relationship of Mohakhali flyover piers in longitudinal direction

91

6.4 Moment-curvature relationship of Khilgaon flyover piers 91

6.5 Load-displacement relationship of Mohakhali flyover piers intransverse direction

92

6.6 Load-displacement relationship of Mohakhali flyover piers in 93

xiv

Fig. No.


longitudinal direction

6.7 Load-displacement relationship of Khilgaon flyover piers 94

6.8 Load-displacement relationship of Mohakhali flyover pile inlongitudinal direction

95

6.9 Load-displacement relationship of Mohakhali flyover pile in transverse direction

95

6.10 Statistical distribution test of Mohakhali flyover piers in longitudinaldirection

96

6.11 Statistical distribution test of Mohakhali flyover piers in transversedirection

96

6.12 Statistical distribution test of Mohakhali flyover piers in longitudinal direction

97

6.13 Statistical distribution test of Mohakhali flyover piers in transversedirection

100

6.14 Statistical distribution test of Khilgaon flyover piers 105

xv

LIST OF TABLES

Table No.

Title of Table Page No.

2.1 Sizes and reinforcements of different members of Mohakhali flyover 10

2.2 Sizes and reinforcements of different members Khilgaon flyover 13

2.3 Design strengths of the materials are used 14

3.1 Moment and curvature of Mohakhali flyover piers 32

3.2 Moment and curvature of Khilgaon flyover piers 32

4.1 Strength characteristic displacement of pier of Mohakhali flyover 61

4.2 Strength characteristic displacement of pier of Khilgaon flyover 62

5.1 Average shear stress of concrete 69

5.2 Modification factors for effective height (d) of a pier section. 69

5.3 Modification factor in relation to axial tensile reinforcement ratio ptC 69

5.4 Failure mode of Mohakhali flyover pier in the transverse direction 82

5.5 Failure mode of Mohakhali flyover pier in the longitudinal direction 82

5.6 Failure mode of Khilgaon flyover piers 83

5.7 Curvature ductility of Mohakhali flyover piers 85

5.8 Displacement ductility of Mohakhali flyover piers 86

5.9 Probability of shear failure of Mohakhali flyover 86

5.10 Probability of shear failure of Khilgaon flyover 87

Chapter 1

INTRODUCTION

1.1 BACKGROUND

Flyover is a lifeline structure that plays a vital role in surface mode of transportation

in a densely populated city where the traffic congestion is very high. In case of natural

disaster i.e., earthquake, cyclone, flood, and storm surge, flyovers over the cities play

a very important role for evacuation, and offer emergency routes for rescue, first aid,

medical services, firefighting, and transporting relief goods to the refugees as well.

However, the flyovers are one of the most vulnerable structures in a highway system.

Thus, the vulnerability, reliability or safety of a highway system largely depends on

the safety of the flyovers.

Flyovers are designed for different loads: a) traffic load; b) wind load; and c) seismic

load. Design is to be done with a view to make the flyovers safe and cost effective,

and the designed flyovers are expected to be able to congregate the requirements

under predicted loads. The design variables for flyovers are not deterministic; rather

they are uncertain in nature. The uncertainty of seismic load is very large in nature.

Reinforced Concrete (RC) flyovers are commonly available in the world. Severe

damages and collapses of structures were observed in the past earthquakes (Hwang et

al., 2000). Most of the collapses or damages of the highway bridges have recently

occurred, in the seismic prone area like Japan, the United States of America, Turky,

Iran and South Asia, are due to seismic loading. For instance, more than 3000 bridge

structures suffered damages in the past earthquakes only in Japan since the 1923

Kanto earthquake (Kawashima et al, 1997). The Hyogo-ken Nanbu Earthquake of 17

January 1995 caused destructive damages to many highway bridges. Numerous

studies (Choi et al., 2004; Kim and Shinozuka, 2004; Kim and Feng, 2003; Karim and

Yamazaki, 2001; Shinozuka et al., 2000, Dymiotis et al. 1999) have been carried out

to asses the vulnerability of bridges structures. However, all earthquakes that occur

are not of same characteristics. Some earthquakes occur very frequently, some occur

in moderate interval, while some are very rare. Some are small in intensity; some are

moderate, while some of them are very large.

2

Bangladesh is an earthquake prone country; there is a high probability of occurrence

of major earthquakes due to existence of active faults (Ansary, 2001; Bolt 2001; Ali

and Choudhury, 1994; 1992). Earthquakes are of stochastic nature that means small to

large magnitude earthquake may occur an anytime.

The earthquake resistant structural design philosophy currently employed all over the

world calls for ensuring different performance levels depending on the characteristics

of earthquakes (Tanabe, 1999). In fact, the design specifications (JSCE, 2000;

Eurocode 8, 1998; CalTrans, 1999) define two levels of earthquakes based on the

probability of occurrence of earthquakes. The seismic performance of flyover, that is

bridge structures, should ensure that there should be no damage due to a moderate

earthquake which may occur several times during the service life of structure. In

contrast, the structures should not collapse but be repairable under a severe

earthquake which may not be encountered during the service life. In order to ensure

the expected performances the flyovers should possess adequate strength and

ductility. More elaborately, for ensuring no damage under a moderate earthquake the

flyovers should possesses sufficient strength so that the responses lie within the elastic

limit, while for no collapse condition the structures should be adequately ductile so

that inelastic energy dissipation can be achieved without affecting the integrity or

stability of the structural system. Therefore, it is necessary to evaluate the strength

and ductility of flyovers to assess the seismic performance, seismic vulnerability of

the flyovers.

The ductility of flyovers primarily depends on the failure mode in which the structure

is expected to fail. Two different modes in which a structure may fail due to

earthquake are shear mode and flexural mode. In shear mode, inadequate ductility will

be observed and hence collapse will occur without giving sufficient warning. In

contrast, flyovers are expected to behave in a ductile manner in the case flexural

failure. In order to minimize losses, losses in terms of life and property, due to

earthquake, it is expected sufficient time for warning even if the structure collapses. It

is found from history (Hashimoto et al., 2005; Karim and Yamazaki, 2001) that

numerous bridge structures failed in shear mode. The expected failure mode depends

on the lateral strengths for different mode. In the case of the flyovers in Bangladesh,

the lateral strength in different modes and possible failure mode has not yet been

3

investigated. Further, the variability of the design parameters for instance material

strengths are inevitable in nature. Due to the variability, the strengths and ductility are

supposed to vary. The effect of variability of the design parameters on the strength

and ductility has not yet also been studied.

Another aspect of seismic performance as specified in design specifications is

reparability. In order to ensure the reparability, the damage in flyover structures due

to a major earthquake should be limited with respect to its position and extent. Since it

is difficult to detect and repair the damages in foundations, earthquake resistance

design specifications recommend that the primary inelastic behavior due to damage

should preferably be located in pier. This type of seismic design concept is called

“capacity design”, where the inelastic behavior should be limited to the predetermined

regions that can be easily inspected and repaired (JRA, 2002; CalTrans, 1999;

AASHTO, 1998). With a view to prepare a pre-earthquake plan and mitigating the

loss due to a future earthquake it is necessary to know the capacities of individual

members. However, the lateral load carrying capacity of different structural members

and capacity hierarchy has not yet been studied in the case of the flyovers in

Bangladesh.

1.2 OBJECTIVES OF THE STUDY

Behavior of flyovers under the seismic load has been a major point of interest for

engineers over a long period of time. On the basis of the background stated in the

previous section, the principal objectives of the present study are:

i. To evaluate yield moment, ultimate moment, yield curvature and ultimate

curvature of pier, pile body by carrying out nonlinear sectional analyses.

ii. To obtain the lateral load-deformation characteristics of piers, pile

foundation and the substructures by carrying out pushover analyses.

iii. To obtain the lateral strengths of the flyovers taking different possible

failure mode into considerations.

iv. To determine the failure modes of the piers under seismic loads to verify

whether adequate warnings before failure will be obtained or not.

4

v. To obtain the capacity hierarchy factors of the Mohakhali and Khilgaon

flyover to judge the extent and position of damage under a major

earthquake.

vi. To evaluate the effect of variability of material strengths on the strength

deformation characteristics of the flyovers.

1.3 METHODOLOGY

First of all, lateral strengths of the individual members of the elevated bridges are

evaluated. Nonlinear sectional analyses are carried out at first step to obtain the

moment-curvature relationship for different members, and pushover analyses are

conducted for individual members and substructure system in the subsequent step for

getting the load-displacement relationships. Bending strengths of the piers are

obtained from the result of sectional analysis, while the shear strengths of the piers are

calculated using code defined equations. Ductility of the members and system are

obtained using curvature and displacement results of the respective members. Failure

modes of the piers are evaluated from the results of bending and shear strengths of the

piers. The capacity hierarchies of the members are calculated using the results of

lateral strengths. The procedures used for obtaining moment-curvature and load-

displacement relationships are briefly described in following sections.

1.3.1 Moment-Curvature Relationship

The moment-curvature loop is usually obtained by calculating the moment and

curvature corresponding to a range of strains in the extreme fiber of the member. First

of all, the section is discretized into several fibers depending of the position steel and

confinement of the concrete. For an assumed strain in the extreme fiber, the neutral

axis depth is adjusted until the stresses in the concrete and steel, determined from the

strain profile and the stress-strain curves for the materials (Hoshikuma et al., 1997)

into account result in internal forces that balance the external forces acting on the

section. Then the moment and curvature corresponding to that strain profile are

calculated. Moment-curvature relationship is obtained. Drain 2DX (Prokash et al.,

1992) and Response-2000 (Bentz and Collins, 2000) will be used in the study. The

moment-curvature relationships are obtained for pier sections, sections of pile and pile

caps.

5

1.3.2 Load-Displacement Relationship

To obtain the lateral load-displacement characteristics for pier, pile foundation,

system of the fly-over in flexural mode, pushover analyses are carried out using Drain

2DX (Prokash et al., 1992). The analytical models in finite Element method used for

different cases are described in the following way:

According to the recommendation of seismic design specifications (Caltrans 1999;

Transit New Zealand, 1994; Eurocode 8, 1998) a design vibration unit instead of the

flyover is used in the study. The unit consists of a part of superstructure, pier, pile cap,

pile foundation and the surrounding soil. The effect of superstructure is modeled with

an inertia force acting at the pier head. Two different regions used for the pier are:

plastic region, elastic region. The range of plastic region is obtained from JRA (2002).

The plastic region is discretized into several slices. According to design specifications

(JRA, 2002) the number of slices is kept around fifty. The pile body is discretized into

several beam elements, and soils surrounding the pile body are modeled with spring

elements.

1.3.3 Failure Mode

To minimize the loss and ensure the reparability of the flyovers, adequate ductility is

expected in the pier. The piers in the fly-over may fail in different modes: a) flexural

failure: b) shear failure: c) shifting type from flexural damage to shear failure. Failure

due to shear occurs instantaneously that is brittle failure possesses a very little

ductility. That means collapses of the piers occur without giving warning, and hence,

such failures cause devastating effects in all respects. Failure modes of the piers are

determined using the lateral strength in shear and flexure following the method of

Specification of Highway Bridge (SHB) (JRA, 2002).

1.3.4 Capacity Hierarchy

The lateral strength of pier, pile foundation, and pile caps are evaluated and

compared, and finally the capacity hierarchies the both the flyovers will be obtained.

Finally, the capacity of the design vibration unit will obtained using the capacity of

individual members and the analytical model.

6

1.4 SCOPE AND LIMITATION OF THE STUDY

Megacity Dhaka is a highly populated one all over the world. To reduce the traffic

congestion, flyovers or under passes are being used. Some flyover namely, Mohakhali

Flyover, Khilgaon Flyover, Tongi Ahsanullah Master Flyover, Bijoy Sharoni Flyover

have been constructed and many others are in the queue to be constructed. At the time

inception, two flyovers were available, and hence, the study concentrates on the

Mohakhali and Khilgaon Flyovers only. Since the study deals only with the capacity,

hence the static analyses; sectional analysis and pushover analysis are carried out. In

the analyses, two dimensional analytical models are used considering the criticality of

loading and simplicity of analysis. To achieve objective of the research, the strength

and ductility of members and parts of flyovers are obtained from the analytical

investigations. It is worthy to mention that to study the insitu strengths of materials

are beyond the scope of our study. Hence analytical works are carried out on the basis

of belief that the design strengths are achieved at site and the empirical correlations of

design specifications with SPT and other engineering properties of materials holds

good for the underlying soil of the flyovers.

1.5 CONTENTS OF THE STUDY

The major focus of this research is to determine strength and ductility and also to

evaluate the failure mode of the flyovers. In order to maintain a systematic way and

clarity in the presentation of the study, the content of the study is summarized as

follows:

Chapter II deals with the model flyovers Mohakhali and Khilgaon. Dimension,

longitudinal reinforcement, transverse reinforcement of the piers, piles, and pile cap

are shown in this chapter. The materials strengths and its constitutive models are

described. Computerized analytical model of the flyovers members are describe in

this chapter.

In Chapter III, the sectional analysis results of the different members of the flyover

for instance pier, pile body, pile cap are represented and discussed.

Chapter IV is to describe the pushover analysis procedure of the flyover and related

results are presented and subsequently explained. From pushover analysis results, the

7

yield displacement, ultimate displacement, yield load, ultimate load of the flyover

members are found.

Chapter V summarizes the strength and ductility of the flyovers. The lateral strengths

of the flyover are found from the sectional analysis and pushover analysis results. The

yield, ultimate and allowable ductility are determined from yield, ultimate curvature

and displacement of the flyover. The failure mode is also determined.

Chapter VI is to describe the effect of variability of materials strength of the flyovers.

The moment-curvature and load displacement relationships are obtained for

variability of materials strength by using the sectional analysis and pushover analysis.

The mean, standard deviation, coefficient of variation, characteristic strength of

flyovers are determined from the moment-curvature and load-displacement

relationship.

Chapter VII deals with the conclusion of the research and recommendations for the

further study.

Chapter 2

MODEL FLYOVERS

2.1 MODEL FLYOVERS

With a view to mitigate traffic congestion in Dhaka city of Bangladesh, flyovers

namely, the Mohakhali flyover, Khilgaon flyover, Bijoy Sharoni flyover and Tongi

Ahsanulla Master flyover have already been constructed and many others are in the

queue to be constructed. Mohakhali and Khilgaon flyovers are used as model

flyovers in the study.

2.2 MOHAKHALI FLYOVER

Mohakhali Flyover is the first of its kind in Bangladesh was opened to traffic on

November 2004. The construction was started in December 2001.

Fig.2.1: Piers layout of Mohakhali flyover

The flyover is expected ease the traffic congestion at Mohakhali rail-crossing in mega

city Dhaka. It has two ramps, the length of ramp on the north of the flyover towards

the Airport Road is 147 m and another one length on the west, in front of Shaheen

College, of the flyover towards the Jahangir gate is 177 m. The flyover has a total

9

length of 1.12 Km with a total 19 (Nineteen) spans of pre-stressed segmental box

girder profile, which means that there are eighteen piers in the flyover. The span

length of four lane flyover varies from 26.0m to 63.0m, which is 17.9 m wide. The

piers are numbered P1 to P18. The piers, P5, P11, P16, are fixed with the decks and

others piers are connected with the decks by the Shock Transmission Unit (STU)

shown in Fig. 2.1. The piers are single column hammerhead type and cross-sections

are tapered in three different dimensions. The elevation and cross-section of a typical

pier is shown in Fig. 2.2.

Fig. 2.2: Elevation and cross-section of a typical pier of Mohakhali flyover

Section A-A

Length

Pier

Hei

ght (

Hp)

Wid

th

A A

Pile cap

1 2 3 ……….…n

1

2

3

…. m

ST

SL

L

Pile

Pile

Len

gth

(Lp)

Pile cap

B

h d

Covering 175mm

Side (longitudinal) view Side (transverse) view

10

The pier heights above the pile cap vary from 3.524 m to 10.62 m. The longitudinal

and transverse reinforcement ratios vary from 0.87% to 1.263% and 0.59% to 0.85%,

respectively. Pile foundation has been used in the flyover. Three different pile

arrangements have been used with three different pile caps shown in Fig. 2.2. The

details of piers, pile foundation, and pile cap have been presented in Table 2.1.

Table 2.1: Sizes and reinforcements of different members of Mohakhali flyover Pier Pile cap Pile

Pier ID Hp

(m) LxB (m)

Lρ (%)

sρ (%)

LxB (m)

h (m)

Lp (m)

Dp (mm)

SL (mm)

ST (mm) m n

P01 3.524 2.00 x 3.50 1.263 0.85 11.25 x 7.50 2.00 19.80 750 1900 1900 4 6 P02 4.852 2.00 x 3.50 1.263 0.85 11.25 x 7.50 2.00 19.50 750 1900 1900 4 6 P03 6.156 2.00 x 3.50 1.263 0.85 11.25 x 7.50 2.00 19.50 750 1900 1900 4 6

P04 7.310 2.00 x 3.50 1.263 0.85 11.25 x 7.50 2.00 19.50 750 1900 1900 4 6

P05 8.297 2.00 x 3.50 1.263 0.85 11.25 x 7.50 2.00 19.50 750 1900 1900 4 6 P06 9.118 2.00 x 3.50 1.263 0.85 11.25 x 9.40 2.25 19.55 750 1900 1900 5 6 P07 9.772 2.00 x 3.50 1.263 0.85 11.25 x 9.40 2.25 19.55 750 1900 1900 5 6 P08 10.260 2.00 x 3.50 1.263 0.85 11.25 x 9.40 2.25 19.75 750 1900 1900 5 6 P09 10.255 2.00 x 4.75 1.142 0.76 11.25 x 9.40 2.25 19.55 750 1900 1900 5 6 P10 9.517 2.00 x 4.75 1.142 0.76 11.25 x 9.40 2.25 19.55 750 1900 1900 5 6 P11 8.725 3.00 x 4.75 0.870 0.59 13.20 x 11.3 2.50 20.`00 750 1900 1900 6 7 P12 7.925 3.00 x 4.75 0.870 0.59 13.20 x 11.3 2.50 20.10 750 1900 1900 6 7 P13 9.426 2.00 x 4.75 1.142 0.76 11.25 x 9.40 2.25 20.05 750 1900 1900 5 6 P14 9.401 2.00 x 3.50 1.263 0.85 11.25 x 9.40 2.25 19.75 750 1900 1900 5 6 P15 8.843 2.00 x 3.50 1.263 0.85 11.25 x 9.40 2.25 19.05 750 1900 1900 5 6 P16 7.814 2.00 x 3.50 1.263 0.85 11.25 x 7.50 2.00 20.00 750 1900 1900 4 6 P17 5.685 2.00 x 3.50 1.263 0.85 11.25 x 7.50 2.00 21.80 750 1900 1900 4 6 P18 5.356 2.00 x 3.50 1.263 0.85 11.25 x 7.50 2.00 21.80 750 1900 1900 4 6

Notes: Hp , Lp: height of pier and length of pile respectively; LxB: sectional dimension of respective members; d: depth of pile cap; Lρ , sρ : longitudinal and transverse steel ratio percentages; SL, ST : pile spacing in longitudinal and transverse directions respectively; D: Diameter of pile body.

2.3. KHILGAON FLYOVER

To ease the nagging traffic congestion in the city center, the country's biggest fly-over

was constructed at the busy road-rail intersection near Khilgaon, connecting

Rajarbagh in the south, Malibagh in the west and Sayedabad in the east shown in Fig.

2.3. According to the Local Goverment Engneering Department (LGED), people of

the eastern region of Dhaka had to lose three and a half hours everyday, as the rail

crossing would close around 72 times a day to allow passage of trains. Those people

are now able to move without much delay.

11

Construction of this 1.9 Km long and 14 meters wide flyover, having 543 piles, began

in 2001 at a cost of Tk 81.75 crore, including expenses for land acquisition and

compensation to the affected people. The flyover has a 780-metre main bridge and

three ramps. The length of the flyover towards Sayedabad is 303 metres, Malibagh

190 metres and Rajarbagh 285 metres. The ramp towards Sayedabad is 220 metres,

Malibagh 202 metres and Rajarbagh 222 metres. The LGED built and opened to

traffic in March 2005.

Fig.2.3: Piers layout of Khilgaon flyover

The whole structure is of concrete girder with slab. The range of span lengths is 16.0

meters to 28 meters. The piers of flyover are of circular shape having of two different

diameters: 1.5 meters and 2.0 meters, and they are of with hammerhead type. The

elevation and cross-section of a typical pier is shown in Fig. 2.4. The piers heights

above the pile cap vary from 6.35 meters to 11.72 meters. The height, sectional

dimensions, longitudinal reinforcement, transverse reinforcement of the piers, pile

cap, and piles are presented in tabular form in Table 2.2.

Rajarbagh Arm

Sayedabad Arm

Malibagh Arm

Malibagh Loop

12

Fig. 2.4: Elevation and cross-section of a typical pier of Khilgaon flyover

Pier bent

Pier

Pier

Hei

ght (

Hp)

B

Pile cap

1 ………..……….…n

1

2…

….…

. m

ST

SL

L

Pile

Len

gth

(Lp)

Pile cap

Pile

D

h

Covering 175mm

d

Side(longitudinal) view Side(transverse) view

Pier cross-section

13

Table-2.2: Sizes and reinforcements of different members of Khilgaon flyover Pier Pile cap Pile

Pier ID Hp (m)

D (m)

Lρ (%)

stρ (%)

sbρ(%)

LxB (m)

h (m)

Lp (m)

Dp (mm)

SL (mm)

ST (mm) m n

PML03 6.844 1.50 1.61 0.11 0.30 3.27 x 3.27 0.85 32.0 610 2200 2200 2 2

PML04 7.319 1.50 1.61 0.11 0.30 3.27 x 3.27 0.85 32.0 610 2200 2200 2 2

PML05 8.658 1.50 1.61 0.11 0.30 - 32.0 610 - - - - PML06 9.650 1.50 2.64 0.11 0.30 4.07 x 4.07 1.05 32.0 610 2800 2800 2 2 PML07 10.220 1.50 2.64 0.11 0.30 4.07 x 4.07 1.05 32.0 610 3000 3000 2 2 PML08 10.844 1.50 2.64 0.11 0.30 3.87 x 4.73 1.35 32.0 610 1830 2800 3 2 PML11 11.276 1.50 2.82 0.11 0.30 5.47 x 3.12 1.20 32.0 610 4400 2020 2 2 PML12 11.081 1.50 2.82 0.11 0.30 4.72 x3.07 1.20 32.0 610 3650 2000 2 2 PML13 10.649 1.50 2.64 0.11 0.30 7.57 x 4.73 1.20 32.0 610 3250 1830 5 PML14 9.731 1.50 2.82 0.11 0.30 5.17 x 3.07 1.20 32.0 610 4100 2000 2 2 PML15 8.732 1.50 2.28 0.11 0.30 5.27 x 3.07 1.20 32.0 610 4200 2000 2 2 PML16 7.888 1.50 2.28 0.11 0.30 - - 32.0 610 - - - - PM02 7.286 2.00 0.91 0.085 0.23 3.61 x 6.76 30.0 900 2700 2250 2 3 PM03 7.194 2.00 0.91 0.085 0.22 4.36 x 6.11 1.30 30.0 900 2250 3000 2 3 PM04 7.076 2.00 0.91 0.085 0.22 3.61 x 6.76 1.30 30.0 900 2700 2250 2 3 PM05 6.975 2.00 0.91 0.085 0.22 3.61 x 6.76 1.30 30.0 900 2700 2250 2 3 PM06 6.893 2.00 0.91 0.085 0.22 3.61 x 6.76 1.30 30.0 900 2700 2250 2 3 PM07 6.351 2.00 0.91 0.085 0.22 3.61 x 6.76 1.30 30.0 900 2700 2250 2 3 PR02 7.338 2.00 0.91 0.085 0.22 4.06 x 6.76 1.30 30.0 900 2700 2700 2 3 PR03 7.208 2.00 0.91 0.085 0.22 5.18 x 5.18 1.30 30.0 900 3820 3820 2 2 PR04 7.234 2.00 0.91 0.085 0.22 3.61 x 6.76 1.30 30.0 900 2725 2250 2 3 PR05 7.301 2.00 0.91 0.085 0.22 5.95 x 8.11 1.30 20.85 900 2250 4590 2 4 PR06 7.329 2.00 0.91 0.085 0.22 5.11 x 8.11 1.30 20.85 900 2250 3750 2 4 PR07 7.119 2.00 0.91 0.085 0.22 - 1.30 20.85 900 - - 10 PR08 7.148 2.00 0.91 0.085 0.22 5.11 x 8.11 1.30 20.85 900 2250 3750 2 4 PR09 6.916 2.00 0.91 0.085 0.22 4.92 x 8.11 1.30 20.85 900 2250 3560 2 4 PR10 6.455 2.00 0.91 0.085 0.22 5.12 x 8.11 1.30 20.85 900 2250 3760 2 4 PR11 6.386 2.00 0.91 0.085 0.22 5.87 x 8.11 1.30 20.85 900 2250 4512 2 4 PR12 5.966 2.00 0.91 0.085 0.22 5.90 x8.25 1.30 20.85 900 2296 4540 2 4 PS02 7.416 2.00 0.91 0.085 0.22 5.30 x 6.76 1.30 30.0 900 2700 3940 2 3 PS03 7.408 2.00 0.91 0.085 0.22 5.80 x 6.76 1.20 30.0 900 2700 4440 2 3 PS04 7.424 2.00 0.91 0.085 0.22 5.80 x 6.76 1.20 30.0 900 2700 4440 2 3 PS05 7.389 2.00 0.91 0.085 0.22 4.85 x 6.76 1.20 30.0 900 2700 3490 2 3 PS06 7.409 2.00 0.91 0.085 0.22 6.00 x 6.76 1.20 30.0 900 2700 4640 2 3 PS07 7.362 2.00 0.91 0.085 0.22 6.90 x 6.76 1.20 30.0 900 2700 5540 2 3 PS08 7.292 2.00 0.91 0.085 0.22 6.40 x 6.76 1.20 30.0 900 2700 5040 2 3 PS09 7.319 2.00 0.91 0.085 0.22 4.85 x 6.76 1.20 30.0 900 2700 3940 2 3 PS10 7.325 2.00 0.91 0.085 0.22 5.30 x 6.76 1.20 30.0 900 2700 6180 2 3

14

Notes: Hp, Lp: Height of pier and length of pile, respectively; D: diameter of pier; LxB: length and width of pile cap; d: depth of pile cap; Lρ , sbρ , stρ : longitudinal, transverse at bottom of pier and transverse at top of pier steel ratio percentages; Dp: Diameter of pile body.

2.4 MATERIALS PROPERTIES

The strength-deformation characteristics of the flyovers depend largely on the

material strengths and deformation characteristic. Material strengths are obtained

from the design data collected form Roads and Highways Department and Local

Government Engineering Department during 2006-2009. It is believed that specified

design strengths are achieved in the materials used in the construction. The design

strengths of the piers, piles and pile caps are given in the Table 2.3.

Table 2.3: Design strengths of the materials are used. Mohakhali flyover Khilgaon flyover

Concrete Reinforcing Steel Concrete Reinforcing Steel Part of

substructure cf ′

(MPa) cE

(MPa) ysf

(MPa) sE

(MPa) cf ′

(MPa) cE

(MPa) ysf

(MPa) sE

(MPa) Pier 32 26700 415 2x105 25 23700 415 2x105

Pile 25 23700 415 2x106 25 23700 415 2x105

Pile cap 25 23700 415 2x106 25 23700 415 2x105

2.4.1 Constitutive Model of Concrete

Sectional properties of the piers is related to the characteristics of the materials i.e.,

stress-strain relationship and strength of materials. For particular material strengths of

reinforcing steel and concrete, the moment-curvature relationship of a specific section

may vary for different constitutive relations. For this reason, a reasonably accurate

prediction model for stress-strain relationship of the materials has been a great

challenge over the years. In the early days, the stress-stress relationships for

unconfined concrete (Wang et al., 1978; Ahmad and Shah, 1982) had been used. With

the advancement of experimental facilities, along with experimental investigation, the

effect of confinement is now available in literatures (Mander et al.,1988a, 1988b,

Hoshikuma et al., 1997). One such model, which has been used extensively in recent

years, was developed by Hoshikuma et al. (1997). The descending branch of the

material law as well as the increase of strength and corresponding strain because of a

confining reinforcement is taken into consideration which is shown in Fig. 2.5. The

authors provided some insight into the behavior of tied columns under axial and

15

flexural loading. The model stress-strain curve consist the three parts i.e., an

ascending branch, falling branch and sustaining branch. The graphical presentation of

the Hoshikuma et al. (1997) model is given as below:

( )

( ) ( )⎪⎩

⎪⎨

⎧

≤−−

≤≤⎭⎬⎫

⎩⎨⎧

⎟⎟⎠

⎞⎜⎜⎝

⎛−

=

0 11

cuccccccdescc

ccccc

ccc

c

Ef

nE

f

εεεεε

εεεε

ε

p

(2.1)

where n = coefficient, desE = deterioration rate, and are given as

ccccc

ccc

fEE

n−

=εε

(2.2)

yhs

codes f

fE

ρ

2

2.11= (2.3)

The confinement effectiveness for circular sections may be represented as

yhscocc fff αρ8.3+= (2.4)

co

yhscc f

fρβε 033.0002.0 += (2.5)

where

cE : Modulus of elasticity of concrete (MPa);

cf : Compressive strength of concrete (MPa);

ccf : Compressive strength of concrete in confined condition (MPa);

cof : Compressive strength of concrete in unconfined condition (MPa);

yhf : Yield strength of hoop reinforcement (MPa);

cε : Strain of concrete;

ccε : Strain of concrete at peak stress of concrete;

cuε : Ultimate strain of concrete;

sρ : Volumetric ratio of hoop reinforcement.

βα , = modification factors depending on shape of cross section;

in which α and β = modification factors depending on confined sectional shape: for

circular α = 1.0 and β = 1.0; for square α = 0.2 and β = 0.4.

16

Fig.2.5: Stress-strain relationship for concrete.

2.4.2 Constitutive Model for Reinforcing Steel

The elastic perfectly plastic model for reinforcing steel is used in the study. The yield

strength is taken as the design yield strength used in the design. The modulus of

elasticity of reinforcing steel considered in the study is 52 10× MPa. The ultimate

strain used is 0.01 mm/mm. A stress-strain model used for reinforcing steel in the

study has been shown in Fig. 2.6.

Fig.2.6.Stress-strain relationship for steel

εcu

Stre

ss (M

Pa)

Strain εcc

0.8fcc

fcc

Strain (mm/mm)

Stre

ss (M

Pa)

415 MPa

0.01

17

2.4.3 Soil Property

Fig 2.7 and Fig 2.8 show the bore log of the under lying soil for Mohakhali and

Khilgaon flyover. Due to unavailability of detailed engineering property of the

underlying soil standard penetration test (SPT) based correlation are used in the study

which are based on experimental results and adopted by design specifications.

Fig.2.7: Bore log of sub-soil used for Mohakhali flyover

18

Fig.2.8: Bore log of sub-soil used for Khilgaon flyover

2.5 CONCLUSION

The analytical investigations carried out in the study are based on the material

strengths stated in the chapter. In order to take the material nonlinearity into

consideration, the constitutive models of the materials are described in details which

are used in the investigation. An elastic perfectly plastic model for reinforcing steel

and detail models for concrete considering the effect of confinements are used. In the

case of soil, the soil profile of the underlying soil based on standard penetration tests

19

(SPT) values are used in the study. Established empirical correlations from SPT have

been adopted for developing constitutive relations of soil springs for the development

of analytical models that will be discussed in the subsequent chapter.

Chapter 3

SECTIONAL ANALYSIS

3.1 INTRODUCTION

One of the aims of the study is to evaluate the lateral strength and ductility of different

members and parts of the flyovers. The lateral strength of a particular member

depends on the failure modes, which in turn depends on the lateral strengths in shear

and bending. Hence, it is necessary to obtain the bending strength of the members to

evaluate the lateral strengths. One of the ways to obtain lateral strength of any

structural member under bending is to carry out sectional analysis taking the material

nonlinearity into considerations. Apart from the strength, the deformation patterns of

the members in terms of curvature and the characteristic deformation can also be

obtained from the sectional analysis results.

3.2 SECTIONAL ANALYSIS

The sectional analyses are carried out to obtain the moment curvature relationships for

a reinforced concrete (RC) section. The moment-curvature relationship is usually

obtained by calculating the moment and curvature corresponding to a range of strains

in the extreme fiber of the member (Priestly, 1987; Memari et Al, 2005). For a given

strain in the extreme fiber, the neutral axis depth is adjusted until the stresses in the

concrete and reinforcing steel, determined from the strain profile and stress-strain

curve for the materials results in balanced internal forces which is shown in Fig. 3.1.

The sectional analyses for the pier, pile, pile cap sections are carried out for obtaining

the relationships between moment and curvature from which the yield moment

capacity and the ultimate moment capacity of the sections are obtained. Seismic

capacity of the substructure of the flyovers in bending are evaluated from ultimate

yield moments.

21

Fig. 3.1: Overview of numerical sectional analysis of a RC section

3.3 METHOD

Sectional analyses are carried out on the basis of fiber model of a Reinforced

Concrete (RC) section. RC sections are discretized into several fibers taking the

reinforcing steel, confined and unconfined concrete into considerations. Based on the

code (JRA, 2002) recommended value, the number of fibers taken in the study is

around fifty. In the fiber model, the materials nonlinearity, that is, nonlinear stress-

strain characteristic of concrete and reinforcing steel has been taken into account. The

constitutive model that is stress-strain relationships of confined and unconfined

concrete and reinforcing steel used in the sectional analysis have been described in

Chapter II. Characteristic strengths of the materials have also been detailed in of the

chapter earlier.

3.3.1 Sectional Analysis Procedure

To obtain the moment-curvature relationship of a RC section, the section is divided

into N slices. The steps for obtaining the moment-curvature relationships are as

follows:

i. The material properties and the constitutive relations of concrete and reinforcing

steel are selected first.

ii. The bending moment and curvature at cracking of concrete, and bending tension

strength of concrete are computed by using the following equations:

⎟⎟⎠

⎞⎜⎜⎝

⎛+=

i

ibtic A

NfSM (3.1)

ic

cc IE

M=φ (3.2)

3/25.0 ckbt ff = (3.3)

C1 C2

J

Section Stress Strain M = JT, φ = ε/y

T T=C1+C2

ε

y

Strain Strain Concrete Steel

Stre

ss

Stre

ss

22

where

Si : Section modulus of flyover pier cross-section with axial reinforcement in

the i-th section from the acting position of inertial force of the

superstructure also taken into consideration (mm3)

btf : Tensile strength of concrete in bending (N/mm2)

Ni : Axial force due to the weights of superstructure and substructure, acting

on the i-the section from the acting position of inertial force of

superstructure (N).

Ai : Section area for bridge pier in the i-th section from the acting position of

inertial force of superstructure, with axial reinforcement also taken into

consideration (mm2) Ec : Modulus of elasticity of concrete (N/mm2)

Ii : Moment of inertia of areas of flyover in the i-th section from the acting

position of inertial force of superstructure, taking the axial reinforcement

also taken into consideration (mm4)

ckf : Design compressive strength of concrete (N/mm2)

iii. The section of each element is divided into N divisions in the direction in which

inertial force acts, and on the assumption that fiber strain is in proportion to the

distance from the neutral axis obtained by assuming that the plane is preserved

and the stresses corresponding to the fiber strain are fixed within the respective

infinitesimal elements, a neutral axis to satisfy the equilibrium condition of

equation. (3.4) is obtained by trial calculation. The number of divisions in each

section is kept around 50.

sj

n

jsjcj

n

jcji AfAfN ∆+∆= ∑∑

== 11 (3.4)

where

sjcj σσ , : Stresses in concrete and reinforcing steel of the j-th infinitesimal

element (N/mm2)

sjcj AA ∆∆ , : Sectional areas of concrete and reinforcing steel in the j-th

infinitesimal element (N/mm2)

After obtaining the position of the neutral axis, bending moment and curvature are

obtained respectively by equation (3.5) and equation (3.6)

23

sj

n

jjsjcjj

n

jcji AxfAxfM ∆+∆= ∑∑

== 11

(3.5)

ocoi x/εφ = (3.6)

Mi : Bending moment acting on the i-th section from the acting position of

inertial force of superstructure (N-mm)

iφ : Curvature of the i-th section from the acting position of inertial force of

superstructure (rad/mm)

jx : Distance from concrete or reinforcing bar in the j-th infinitesimal element to

the centroid position (mm)

coε : Compressed edge strain of concrete (mm/mm)

ox : Distance from the compressed edge of concrete to the neutral axis (m)

The bending moment and curvature formed when the strain occurred in the axial

tensile reinforcement placed on the outermost edge of the section reaches yield

strain syε are obtained and are taken as initial yield moment yoM and initial yield

curvature yoφ .Besides, the bending moment and curvature formed when the strain

of concrete in the position of the axial compressive reinforcement on the

outermost edge reaches ultimate strain cuε are obtained and are taken as ultimate

moment uM and ultimate curvature uφ .

iv. The initial yield displacement 0yδ is calculated by equation (3.7) based on the

curvature distribution obtained when the initial yield horizontal strength Py0 is

acted on the height of the super structural inertia force.

2/)( 111

0 iii

m

iiiy yyyydy ∆+== −−

=∫ ∑ φφφδ (3.7)

v. Yield curvature yφ in the skeleton curve is calculated by equation (3.8)

uy yo

yo

MM

φ φ⎛ ⎞

= ⎜ ⎟⎜ ⎟⎝ ⎠

(3.8)

24

Fig.3.2: Relation between Bending Moment and Curvature

In the current study a finite element based program RESPONSE 2000 (Bentz and

Collins, 2000) has been used to analyze the sections.

3.4 RESULTS AND DISCUSSIONS

Sectional analysis of pier sections, pile section and the pile cap are carried out to obtain

the moment-curvature relationships of different members. From the moment-curvature

relationships, the respective yield moments, ultimate moments and respective curvatures

are obtained. Finally, the bending strength and ductility in terms of curvature are

obtained. Discussions on the moment-curvature relationship of the different members are

made in the subsequent subsections.

3.4.1 Moment-Curvature Relationships of the Pier Sections

Mohakhali flyover

As mentioned in Chapter II, three different cross-sections with different geometry,

longitudinal and transverse reinforcements are used in the Mohakhali flyover. The

pier sections are discretized into fibers in such a way that reinforcing steel, confined

and unconfined concrete is represented and the total number of fibers is around fifty.

Sectional analyses are carried out using the materials properties and constitutive

relations discussed in the chapter II. The moment curvature relations obtained by

sectional analyses on the basis of fiber models are presented in Fig. 3.3 and Fig.3.4.

My0

Mu

φy0 φy

Ben

ding

mom

ent

Curvature

25

P01-P08, P14-P18

0

30000

60000

90000

120000

150000

0 3 6 9 12 15

Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

P09, P10, P13

0

30000

60000

90000

120000

150000

0 3 6 9 12 15


Mom

ent (

kN-m

)

P11, P12

0

40000

80000

120000

160000

0 3 6 9 12 15


Mom

ent (

kN-m

)

Fig 3.3: Moment-curvature relationship of the Mohakhali flyover pier bottom section in transverse direction.

P01-P08, P14-P18

0

20000

40000

60000

80000

100000

0 3 6 9 12 15


Mom

ent (

kN-m

)

P09, P10, P13

0

20000

40000

60000

80000

100000

0 3 6 9 12 15


Mom

ent (

kN-m

)

P11, P12

0

20000

40000

60000

80000

100000

0 3 6 9 12 15


Mom

ent (

kN-m

)

Fig 3.4: Moment-curvature relationship of the Mohakhali flyover pier bottom section in longitudinal direction.

It is seen from the figures that the moment increases rapidly with increasing

curvatures initially, while the rate of increase becomes insignificant after an interval.

The reason for changing the relation is that reinforcing steel in the extreme tensile

layer reaches yield strength. The moment in the stage is termed as yield moment.

Moments are observed to increase further with curvature beyond the yield moment

due to the fact that the reinforcement in layers other than in extreme layers is yet to

reach yield strength. Further, a minor change in the slope is observed in the initial

linear regime. It is due to developing tension cracks in the cover concrete and hence

reduction of effective cross-sectional area occurs. It is also seen from the figures that

for a particular direction of analysis, the characteristic values of the moments and

curvatures are different for different piers due to change in geometry, quantity and

arrangement of longitudinal reinforcements and quantity of longitudinal

reinforcements. It is also observed form the figures that the ultimate moment of

transverse direction pier is higher than the ultimate moment in the longitudinal

direction.

Khilgaon flyover

Circular cross-sections with two different diameters and longitudinal reinforcements

(0.91% to 2.82%) and transverse reinforcements (0.22% to 0.30%) are used in

26

Khilgaon flyover, which has been mentioned in the previous chapter. Accordingly,

sectional analyses are conducted on the basis of fiber model described earlier.

PML03-PML05

0

4000

8000

12000

16000

0 5 10 15 20


Mom

ent (

kN-m

)

PML06-PML08, PML13

0

4000

8000

12000

16000

0 5 10 15 20


Mom

ent (

kN-m

)

PML11, PML14

0

4000

8000

12000

16000

0 5 10 15 20


Mom

ent (

kN-m

)

PML12, PML15, PML16

0

4000

8000

12000

16000

0 5 10 15 20


Mom

ent (

kN-m

)

PR02-PR12, PS02-PS10, PM02-PM07

0

4000

8000

12000

16000

0 5 10 15 20


Mom

ent (

kN-m

)

Fig 3.5: Moment-curvature relationship of pier bottom section of Khilgaon flyover.

The moment-curvature relationships of the piers of the Khilgaon flyover are shown in

the Fig.3.5. The trend of the moment-curvature relationships is similar to those of the

piers of Mohakhali flyover. The difference in moment strengths i.e., the yield moment

and ultimate moment are observed from the figures. This is due to difference in

diameter and amount and arrangement of longitudinal and transverse reinforcements.

3.4.2 Moment-Curvature Relationships of the Pile Sections

Mohakhali flyover

Pile lengths, and numbers have been mentioned in Table 2.1. Cross-section of pile

that has been used in Mohakhali flyover has been shown in Fig. 3.6.

Fig. 3.6: Cross-section of Mohakhali flyover pile

750 mm

18-φ20 mm bar

Clear cover 75 mm

27

Fig.3.7 is to present the moment-curvature relationships of the pile sections of the

Mohakhali flyover.

Dia= 750mm18-20mmφ rod

0

200

400

600

800

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

Fig. 3.7: Moment-curvature relationship of the Mohakhali flyover pile section

It is seen from the figure that the moment is found to increases rapidly with increasing

curvatures. At this stage, the moment is increased without significant increase in

curvature. After the initial stage, a remarkable change in the rate of change of moment

with respect to curvature is observed due to developing tension cracks in the cover

concrete and hence reduction of effective cross-sectional area. Even after the regime

of moderate slope, another transition can be seen from the figure which is due to

yielding of longitudinal reinforcement at the extreme fibers. After yielding of all the

reinforcements, no increase in the moment occurs that can be seen from the flat part

of the figure.

Khilgaon flyover

Two different cross-sections with different steel contents are used in the Khilgaon

flyover. Accordingly, sectional analyses of four cross-sections are carried out, and the

results of the sectional analyses are presented in terms of moment-curvature in Fig.

3.8


0

200

400

600

800

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


0

200

400

600

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


0

200

400

600

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

28


0

100

200

300

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

Fig. 3.8: Moment-curvature relationship of the Khilgaon flyover pile sections

The trend of the moment-curvature relationships is similar to the Mohakhali flyover

piles. Differences in yield and ultimate moments are observed from the figures. The

reasons for the differences are: difference in diameter, different quantity and

arrangements of longitudinal reinforcements, and varied confining reinforcements.

3.4.3 Moment-Curvature Relationships of the Pile Cap Sections

Mohakhali flyover

Three different thicknesses and reinforcements are used in the pile caps of Mohakhali

flyover. Accordingly, sectional analyses of the three pile cap sections along

longitudinal and transverse directions are carried out, and the results in the form of

moment curvature relations are presented in the form of moment-curvature relations

in Fig. 3.9 and Fig. 3.10.

P01-P05 & P15-P16SECTION 2000x7500

STEEL 74-T400

20000

40000

60000

80000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

P06-P10 & P13-P14SECTION 2250x9400

STEEL 93-T400

20000

40000

60000

80000

100000

120000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

P11, P12SECTION 2500x11300

STEEL 112-T400

30000

60000

90000

120000

150000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

Fig. 3.9: Moment-curvature relationship of the Mohakhali flyover pile cap section in longitudinal direction

P01-P05 & P15-P18SECTION 2000x11250

STEEL 112-T400

20000

40000

60000

80000

100000

120000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

P06-P10 & P13-P14SECTION 2250x11250

STEEL 112-T400

20000

40000

60000

80000

100000

120000

140000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

P11, P12SECTION 2500x13200

STEEL 131-T400

30000

60000

90000

120000

150000

180000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

Fig.3.10: Moment-curvature relationship of the Mohakhali flyover pile cap sections in transverse direction

29

It is seen from the figures that the moment increases maintaining a very steep slope

initially, while the rate of increase decreases after an interval. The reason for changing

the relation is the development of tension cracks in the cover concrete and hence

reduction of effective cross-sectional area. After a certain interval, reinforcing steel in

the extreme tensile layer reaches yield strength, and the rate of increasing moment

with curvature becomes very small as compared to others. The moment in the stage is

termed as yield moment. Moments are observed to increase further with curvature

beyond the yield moment due to the fact that the reinforcement in layers other than in

extreme layers is yet to reach yield strength. It is to be noted that yield moment

reaches at a curvature which is much less than that of pier and pile body. This may be

due to larger clear cover to reinforcements.

It is also seen from the figures that for a particular direction of analysis, the

characteristic values of the moments and curvatures are different for different piers

due to change in geometry, quantity and arrangement of longitudinal reinforcements

and quantity of longitudinal reinforcements. It is also observed form the figures that

the ultimate moment of transverse direction pier is higher than the ultimate moment in

the longitudinal direction.

Khilgaon flyover

Sectional analyses along the longitudinal and transverse directions of the pile caps of

thirteen different pile caps differing in reinforcement, thickness and sizes are

conducted, and the results are presented in the form of moment-curvature relationship

in Fig.3.11 and Fig.3.12. A trend observed for the pile caps of Khilgaon flyover can

be seen from Fig.3.11 and Fig.3.12, and the reason can be explained in the similar

way that utilized for the Mohakhali flyover.

PS02 & PS10SECTION 1200x6760

STEEL 44-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 54-T250

2000

4000

6000

8000

10000

12000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 39-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

30

PS 06SECTION 1200x6760

STEEL 59-T250

3000

6000

9000

12000

15000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 80-T250

3000

6000

9000

12000

15000

18000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 68-T250

3000

6000

9000

12000

15000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

PM02, PM05 & PM07SECTION 1300x6760

STEEL 29-T250

2000

4000

6000

8000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

PM03SECTION 1300x6110

STEEL 24-T250

2000

4000

6000

8000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

PM04 & PM06SECTION 1300x6760

STEEL 29-T250

1500

3000

4500

6000

7500

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

PR02SECTION 1300x6760

STEEL 24-T250

1500

3000

4500

6000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 35-T250

1500

3000

4500

6000

7500

9000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 58-T250

3000

6000

9000

12000

15000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 55-T250

3000

6000

9000

12000

15000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

Fig.3.11: Moment-curvature relationship of the Khilgaon flyover pile cap sections in transverse direction


STEEL 44-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 43-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 42-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 40-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 42-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 42-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

31

PM02, PM05 & PM07SECTION 1300x3610

STEEL 40-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

PM03SECTION 1300x4360

STEEL 42-T250

2000

4000

6000

8000

10000

12000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

PM04 & PM06SECTION 1300x3610

STEEL 43-T250

2000

4000

6000

8000

10000

12000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 40-T250

2000

4000

6000

8000

10000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

PR 03SECTION 1300x5180

STEEL 35-T250

1500

3000

4500

6000

7500

9000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 51-T250

3000

6000

9000

12000

15000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)


STEEL 51-T250

3000

6000

9000

12000

15000

0 4 8 12 16 20Curvature x10-3 (rad/m)

Mom

ent (

kN-m

)

Fig.3.12: Moment-curvature relationship of the Khilgaon flyover pile cap sections in longitudinal direction.

One can see in some of the figures that moment suddenly drops at curvature. This

might be due to extremely low value of reinforcing steel ratio. The reinforcing steel

ratios used in those cases are less than ρmin specified in design codes.

32

3.4.4 Characteristic Moments and Curvatures of the Piers

Mohakhali flyover

Table 3.1: Moment and curvature of Mohakhali flyover piers. Transverse direction Longitudinal direction

Pier ID

Yield moment (kN-m)

Ultimate moment (kN-m)

Yield curvature (rad/km)

Ultimate curvature (rad/km)

Yield moment(kN-m)




P01 58863 74581 1.24 3.35 35818 42687 2.00 5.56 P02 60863 77736 1.24 3.35 37118 44608 2.00 5.56 P03 61561 78014 1.24 3.35 37552 44767 2.00 5.56 P04 61111 78234 1.24 3.35 37241 44469 2.00 5.56 P05 63853 78393 1.24 3.35 37762 45027 2.00 5.56 P06 61267 78625 1.24 3.35 37354 45135 2.00 5.56 P07 64139 78756 1.24 3.35 37900 45211 2.00 5.56 P08 59084 75977 1.24 3.35 36518 43546 2.00 5.56 P09 98245 123875 1.00 2.38 45180 52125 1.90 5.56 P10 98513 127779 1.00 2.38 45711 53853 1.90 5.56 P11 126296 158082 0.96 2.20 85032 99828 1.50 3.54 P12 123630 143947 0.96 2.20 84502 99362 1.50 3.54 P13 98067 123778 1.00 2.38 45056 51985 1.90 5.56 P14 59185 75797 1.24 3.35 36103 43437 2.00 5.56 P15 61881 78567 1.24 3.35 38079 45106 2.00 5.56 P16 60084 78298 1.24 3.35 37398 44967 2.00 5.56 P17 61804 77904 1.24 3.35 37660 44713 2.00 5.56 P18 58693 74948 1.24 3.35 35828 42916 2.00 5.56

Table 3.2: Moment and curvature of Khilgaon flyover piers.

Pier ID Yield

moment (kN-m)




Pier ID

Yield moment(kN-m)




PML03 6473 8113 2.50 10.50 PR09 10845 13435 1.86 7.10 PML04 6646 8358 2.52 10.50 PR10 10786 13387 1.86 7.10 PML05 6687 8413 2.52 10.50 PR11 10845 13460 1.86 7.20 PML06 9255 12103 2.77 12.71 PR12 10303 11914 1.86 6.42 PML07 9234 12067 2.72 12.72 PS02 11628 13564 1.86 7.10 PML08 10340 12776 2.77 13.98 PS03 11628 13564 1.86 7.10 PML11 9833 12635 2.85 13.10 PS04 11628 13564 1.86 7.10 PML12 8420 10690 2.85 11.92 PS05 11628 13513 1.86 7.20 PML13 9228 12062 2.77 12.72 PS06 11602 13460 1.86 7.20 PML14 9880 12731 2.85 13.10 PS07 11602 13460 1.86 7.20 PML15 8434 10705 2.60 11.92 PS08 11602 13460 1.86 7.20 PML16 8434 10705 2.52 10.50 PS09 11602 13460 1.86 7.20 PR02 10963 13662 1.86 7.20 PS10 11602 13460 1.86 7.20 PR03 10789 13387 1.86 7.10 PM02 11628 13564 1.86 7.10 PR04 10789 13387 1.86 7.10 PM03 11628 13564 1.86 7.10 PR05 10789 13387 1.86 7.10 PM04 11628 13564 1.86 7.10 PR06 10789 13387 1.86 7.10 PM05 11484 13250 1.86 7.10 PR07 10789 13387 1.86 7.10 PM06 11484 13250 1.86 7.10 PR08 10845 13435 1.86 7.10 PM07 11484 13250 1.86 7.10

33

3.5 CONCLUSION

Sectional analysis of the sections of piers, pile body, pile caps have been conducted

on the basis of fiber model of the respective sections. The moment-curvature are

presented and discussed, and finally yield and ultimate moment and associated

curvatures are obtained from the moment-curvature relationship and listed in tabular

form. The moments and curvature ill be utilized for evaluating lateral strength and

ductility.

Chapter 4

PUSHOVER ANALYSIS

4.1 BACKGROUND

With a view to achieve the goal of evaluating ductility of the flyovers in terms of

displacement, it is necessary to obtain the load-displacement relationships. One of the

ways to get load-displacement relationship is to carry out pushover analysis. The

detail methodology adopted for pushover analysis for different members and system

is described in the following section.

4.2 PUSHOVER ANALYSIS

A pushover analysis is a nonlinear static analysis wherein monotonically increasing

lateral loads are applied to the structure till a target displacement is achieved or the

structure is unable to resist further loads. The load-displacement relationship is

usually obtained by applying the load and calculates the corresponding displacement

at a particular point(s) of structure of the structure shown in Fig. 4.1. The pushover

analyses have been conducted for the pier, pile foundation, and whole substructure to

obtain the relationships between load and displacement and finally the ductility. The

yield load capacity and the ultimate load capacity of the respective structures are

obtained from the load-displacement relationship. In this study, pushover analyses are

carried out to evaluate the seismic capacity and the ductility of the flyovers.

Fig. 4.1: Simple pushover analysis

Pi

i∆

W W

h

35

4.2.1 Analytical Procedure Used for Piers

In this case, the pier base is assumed as fixed. The pier is divided into N numbers

slices along its height to obtain the load-displacement relationship at the top of the

flyovers piers. Fifty slices are recommended in design specification (JRA, 2002). The

load displacement relationships at the top of the pier are obtained using the moment-

curvature and shear stress-strain relations. Fig. 4.2 shows the numerical evaluation of

the flexural and shear components of displacement. The steps for obtaining the force-

displacement relationships are as follows:

i. The pier is divided into N slices along its height

ii. The moment-curvature relations for each cross-section are obtained through

sectional analysis

iii. The horizontal load P is applied at the top of the pier

iv. The bending moment diagrams of the pier for the applied load P is drawn

v. The curvature from bending moment and moment-curvature diagram is

obtained

vi. The displacement δ at the top of the pier is estimated using the following

equation:

∑=

××=N

iii ddy

1φδ (4.1)

vii. In a similar way, several forces P are applied and the corresponding

displacements are obtained. Finally, using these values, the load–displacement

relationship at pier top is obtained.

36

Fig. 4.2: Numerical evaluation of flexural and shear component of displacement

4.2.2 Analytical Procedure Used for Pile

To obtain the load-displacement relationship at the center of pile cap, an analytical

model is developed in two-dimensions. In the analytical model, first of all, a single

pile is modeled taking the surrounding soil into considerations. A pile body is divided

into N numbers of segments along its length. The divided pile segments are modeled

using elasto-plastic beam elements with strain hardening. The surrounding soils are

modeled with soil springs, and the parameters of soil spring will be discussed in the

subsequent sub-section. The analytical model is developed in DRAIN-2DX. The load

displacement relationships at the center of the pile are obtained using the moment-

curvature and shear stress-strain relations. The steps for obtaining the load-

displacement relationships are as follows:

i. The pile is divided into N numbers of segments along its height

ii. The moment-curvature relations for each cross-section are obtained through

sectional analysis

iii. The horizontal load P is applied at the top of the pier

iv. The bending moment diagrams of the piles is drawn for the applied load P

v. The curvature is obtained from bending moment and moment-curvature

diagram

Section i

1 2

N-1 N

Mi Φi Vi γi

Applied load on column

Bending moment diagram

Curvature diagram

Shear force diagram

Shear strain diagram

di

P

W

37

vi. The displacement δ at the center of the pile cap is estimated using the

following equation:

∑=

××=N

iii ddy

1φδ (4.2)

vii. In a similar way, several loads P are applied and the corresponding

displacements are obtained. Finally, using these values, the load–

displacement relationship at center of pile cap is obtained.

4.3 PROCEDURE FOR SUB-STRUCTURAL SYSTEM

A highway substructure comprises of a Shock Transmission Unit (STU) or rubber

bearing, a pier, group of piles, surrounding soil, and a pile cap used for transmitting

and distributing the load from the pier. With a view to carry out pushover analysis of

flyover substructure, an analytical model of the substructure including pile foundation

system is to develop using finite elements.

4.3.1 Analytical Model of Substructure

An analytical model, used in this study, capable of expressing the all the structural

and material properties of the flyovers. The nonlinearities of the members are

incorporated into a two-dimensional nonlinear analytical model of the flyovers that

are developed using DRAIN-2DX (Prokash et al, 1992). The flyovers have two part;

superstructure and substructure. The effect of superstructure has been modeled by its

weight at pier top and the substructure consisting of a reinforced concrete STU or

rubber bearing, RC pier, RC pile cap, and cast-in-place RC piles, and surrounding

soil of the piles are modeled using different elements defined in DRAIN (Prokash et

al, 1992) which will be described in the subsequent sections.

Shock Transmission Unit (STU)

STU used for connecting superstructure mass to substructure is modeled with a link

element. The strength and stiffness of the link element is obtained from the sectional

and material properties of STU.

38

Pier

The piers are modeled using the DRAIN-2DX fiber beam-column element. In

modeling the fiber beam-column element, the pier is divided longitudinally into two

zones: deformable and rigid zone. The deformable zone is specified in the pier part

within the pile cap and pier head, while the rigid zone is specified within the pile cap

and pier head. In order to obtain the nature of the deformable zone, the cross-section

of the pier is discretized into a number of fibers. The fibers represent three zones: a)

reinforcing steel; b) confined concrete; and c) unconfined concrete. The stress-strain

relationships of concrete and reinforcing steel described in Chapter II are used and

assigned to each fiber. A schematic fiber model of the column is shown in Fig. 4.3.

Fig. 4.3: Schematic diagram of the fiber model of pier cross-section

The element geometry is shown in Fig.4.4. The element consists of deformable part,

elastic part, plastic hinge zone and optional rigid end zones.

Fig.4.4: Element geometry of fiber element

39

Fig.4.5: Analytical model of pier

Pile Cap and Pile Body

The pile cap is modeled with a simple inelastic beam element with moment-curvature

relationships. The moment-curvature relations obtained from sectional analyses are

given to the element as input.

The pile body is modeled with beam element with concentrated plastic hinge at the

ends. Each of the pile body is discretized into a number of elements where each

element is of length around 1.5 m. The element geometry is shown in Fig.4.6 (a). The

element consists essentially of an elastic beam, two rigid-plastic hinges at the ends of

this beam.

Fig.4.6(a): Element geometry of inelastic element

Nonlinear fiber beam element

Rigid portion of pier

Plastic hinge zone

Deformable portion of pier

Pile cap

Pile

Elastic zone

40

Fig.4.6 (b): Elastic element stiffness

Soil

The soil surrounding the pile body and the pile cap is modeled with nonlinear springs

with elastic perfectly plastic load-displacement characteristics in both axial and

transverse direction of the pile body. The piles and pile foundation with surrounding

soil model as a spring is shown in Fig.4.7 (a) and Fig.4.7 (b). The stiffness and the

ultimate load are estimated by using the method of Specification of Highway Bridges

in Japan (SHB) (JRA, 1996) that is described in section 4.5.4. The load displacement

characteristics of the springs of the axial resistance and the transverse resistance of

pile body are obtained from SHB (JRA, 2002) and shown in Fig. 4.11. In DRAIN-

2DX, the spring element connects two nodes which are identical coordinates, i.e. a

zero length element.

Fig.4.7 (a): Pile with surrounding soil

Surrounding Soil

Surrounding Soil

Pile body

Pile cap

41

Fig.4.7 (b) Pile with spring soil model

4.4 ANALYTICAL MODEL OF SUB-STRUCTURAL MEMBERS AND

SYSTEM

4.4.1 Pier with Bottom End Fixed

In this study, different types of analytical model have been used. An analytical model

of a pier is shown in the following Fig. 4.8. The pier is considered as fixed at bottom.

Fig. 4.8: Analytical model of a flyover pier

4.4.2 Pile Foundation

An analytical model of pile foundation is shown in the Fig.4.9. The pile foundation is

modeled considering the pier and pile cap are rigid.

Pile cap

Elastic perfectly plastichorizontal spring

Pile body

Elastic perfectly plasticvertical spring

Plastic hinge length (Lp) Pl

astic

hin

ge

leng

th =

4Lp

Superstructure weight

Pier

42

Fig. 4.9: Analytical model of a flyover pile foundation

4.4.3 Substructure of Flyover

An analytical model of a substructure of flyover is shown in the Fig.4.10.

Fig. 4.10: Analytical model of a flyover substructure

Nonlinear fiber beam element

Weight from the superstructure

Inelastic beam element



Rigid pile cap



Rigid pier

Superstructure weight

Elastic perfectly plasticvertical spring

43

4.5 PARAMETERS ESTIMATION FOR ANALYTICAL MODEL

4.5.1 Weight o Superstructure

The superstructure weight has been estimated by calculating the volume of the

different component and multiplied by the unit weight of materials. The unit weight of

reinforced concrete is 150 pcf and the unit weight of wearing coat materials 120 pcf is

used.

4.5.2 Material Properties

The materials strength of concrete and steel are considered to the design strengths of

respective material in the respective flyover. The design strengths of concrete and

steel have been discussed in Chapter II. The constitutive model of concrete and steel

have also described in Chapter II.

4.5.3 Yield Moment of Pile Cap and Pile Body

Yield moment is obtained from the sectional analysis of the pile and pile cap using the

section of respective member with material properties. The sectional analysis is

carried out in Chapter III. As started earlier, the yield moment of a particular section

is that moment which produces yield strain in the outer most reinforcing steel. Elastic

perfectly plastic model has been used for pile cap and pile body.

4.5.4 Soil Spring

The soil surrounding the pile body and the pile cap is modeled by using springs with

strength and stiffness in axial. The axial springs are elastic perfectly plastic with an

initial gradient being the axial spring constant vK and with an ultimate capacity PNU

against push-in and an ultimate capacity PTU against pull-out of the spring shown in

Fig. 4.11 (a). The transverse springs are also elastic perfectly plastic with an initial

gradient being the coefficient of horizontal ground reaction HEk and with an ultimate

unit horizontal ground reaction HUp shown in Fig. 4.11 (b).

44

Fig.4.11: Pile resistance characteristics

Axial spring

The axial spring constant vK of a pile is designed as axial capacity of piles which

generates a unit displacement at the pile head. The axial spring constant vK have been

calculated by the following equation

LEAaK PP

v = (4.3)

where

vK : Axial spring constant of a pile (N/mm)

PA : Cross-sectional area of the pile (mm2)

PE : Modulus of elasticity of the pile concrete (MPa)

L : Pile length (mm)

a : Cast-in-place piles constant and it has been calculated using the following

equation

15.0)/(031.0 += DLa (4.4)

D : Diameter of the pile body (mm)

The ultimate axial bearing capacity NUP against push-in and the ultimate bearing

capacity TUP against pull-out are calculated from the following equations

),min( PUUNU RRP = (4.5)

Ultimate axial bearing capacity against push-in

(a) Axial pile resistance characteristics

Ultimate transverse bearing capacity against push-in/pull-out

PHU

Transverse displacement (mm)

tan-1kHE

Coe

ffic

ient

of t

rans

vers

e gr

ound

reac

tion

P H (M

Pa)

(b) Transverse pile resistance characteristics

Ultimate axial bearing capacity against pull-out

P(N)

PNU

Displacement (mm) of pile head in axial direction

PTU

tan-1KVE

Pile head reaction

45

),min( PUUTU PWPP += (4.6)

iidU fLUAqR ∑+= (4.7)

iiU fLUP ∑= (4.8)

sycckPU AfAfR += 85.0 (4.9)

syPU AfP = (4.10)

Where

NUP : Ultimate axial capacity in Newton’s (N) against push-in

TUP : Ultimate axial capacity against pull-out (N)

UR : Ultimate bearing capacity of the pile against push-in considering the soil

parameters (N)

UP : Ultimate bearing capacity of the pile against pull-out considering the soil

parameters (N)

W : Effective weight of the pile (N)

PUR : Ultimate bearing capacity of pile against push-in considering the pile body

(N)

PUP : Ultimate bearing capacity of pile against pull-out considering the pile body

(N)

ckf : Design standard strength of concrete (MPa)

cA : Cross-sectional area of concrete (mm2)

yf : Yield strength of steel (MPa)

sA : Cross-sectional area of steel (mm2)

A : Cross-sectional area of pile tip (mm2)

dq : Ultimate bearing capacity per unit area to be borne by a pile tip (N)

U : Circumferential length of the pile (mm)

iL : Thickness of a layer for which skin friction force is taken into account (mm)

if : Maximum skin friction force per unit area of a layer taking the skin friction

force into account (MPa)

46

Transverse spring

The coefficient of horizontal ground reaction HEk has been calculated by the following

equation

HkkHE kk αη= (4.11)

4/3

0 300

−

⎟⎠⎞

⎜⎝⎛= H

HHBkk (4.12)

00 3001 EkH α= (4.13)

NE 8.20 = (4.14)

Where

HEk : Coefficient of horizontal ground reaction (N/mm3)

Hk : Coefficient of horizontal ground reaction (N/mm2)

0Hk : Coefficient of horizontal sub-grade reaction (N/mm3)

0E : Modulus of deformation (MPa) of a soil layer

α : A coefficient is assumed 2

kα : Correction factor of horizontal ground reaction around a single pile.

kη : Correction factor of horizontal ground reaction with the group of piles effect

taken into account. Assumed kη =2/3

N : Slandered Penetration Number (SPT) value

HB : Equivalent loading width of a foundation (mm), for pile foundation HB is

calculated as follows

β/DBH = (4.15)

β : Characteristics value of foundation (mm-1) is calculated by the following

equation

EIDkH

4=β (4.16)

EI : Rigidity of the foundation (N-mm2)

D : Loading width of a foundation perpendicular to a load working direction

The upper limit of unit horizontal ground reaction HUP has been calculated by the

following equation

47

UPPHU pP αη= (4.17)

φφ

sin1sin1

−+

=Up (4.18)

HUP : Upper limit of unit horizontal ground reaction (N/mm2)

Up : Passive soil pressure (N/mm2)

pα : Correction factor of upper limit of unit horizontal ground reaction around a

single pile.

pη : Correction factor of upper limit of unit horizontal ground reaction with the

group of piles effect taken into account.

φ : Angle of friction depends on SPT value, determine by the following

equation

1518 += Nφ (4.19)

N : Slandered Penetration Number (SPT) value

4.5.5 Adequacy of thickness for Rigidity of Pile Cap

The pile cap is treated as a rigid body considering the influence of rigidity of the pile

cap on the subgrade reaction and pile reaction. If the pile cap satisfies equation (4.20),

it may be deemed a rigid body. Even in cases such as a staggered arrangement, where

the numbers of piles in the rows are different, and the judgment as to whether or not it

is a rigid body may be made by substituting the numbers of piles into n and m in

equation (4.21) if the piles are arranged uniformly.

0.1≤βλ (4.20)

Where 43

3Eh

k=β (mm-1)

⎩⎨⎧

=foundation pile a of case n the.........i

foundation spread a of case n the.........i

p

V

kk

k

vk : Coefficient of vertical subgrade reaction (MPa)

pk : Coefficient of equivalent subgrade reaction (MPa)

BDmnKk Vp = (4.21)

48

VK : Axial spring constant of one pile (N/mm)

D : Pile cap width (mm)

B : Pile cap length (mm)

n : numbers of pile columns

m : numbers of pile rows

E : Modulus of elasticity of concrete of pile cap (MPa)

h : pile cap thickness (mm)

λ : Equivalent protrusion length of pile cap (mm), determined according to pile

cap type as follow:

),max( bl=λ (4.22)

where 2/Dl = if 2/Dl ≥

2/Bb = if 2/Bb ≥

Fig. 4.12: Isolated pile cap


Pushover analysis of the pier, pile and substructure are carried out in the study to

obtain the force-displacement relationships of the respective members. From the

results of the pushover analysis the horizontal capacity at yielding and crushing of the

extreme fiber concrete under compression are obtained and termed as yielding load

and ultimate horizontal load.

4.6.1 Load-Displacement Relationship of the Piers

Pushover analyses of the piers are carried out considering the pier fixed at bottom.

D

B

l

b

49

Mohakhali flyover

Three different cross-sections with different dimensions, longitudinal and transverse

reinforcement are used in the Mohakhali flyover discussed in the Chapter II. The fiber

model is made taking the material and geometric nonlinearity into considerations for

pushover analysis. The materials properties are also described in Chapter II. For

geometric nonlinearity is considered taking the P-∆ effect into considerations. The

pushover analyses are carried out of the piers in both directions of the piers:

longitudinal direction, transverse direction.

P010

4000

8000

12000

16000

20000

24000

0 20 40 60 80 100Displacement (mm)

Late

ral L

oad

(kN

)

P020

3000

6000

9000

12000

15000

18000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

P030

3000

6000

9000

12000

15000

18000

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

P040

3000

6000

9000

12000

15000

18000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

P050

3000

6000

9000

12000

15000

18000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

P060

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P070

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P080

3000

6000

9000

12000

15000

18000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

P090

3000

6000

9000

12000

15000

18000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

P100

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P110

3000

6000

9000

12000

15000

18000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

P120

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225Displacement (mm)

Late

ral L

oad

(kN

)

P130

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P140

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P150

3000

6000

9000

12000

15000

18000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

50

P160

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225Displacement (mm)

Late

ral L

oad

(kN

)P17

0

3000

6000

9000

12000

15000

18000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

P180

3000

6000

9000

12000

15000

18000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

Fig.4.13: Load displacement relationship of the piers at top of Mohakhali flyover in transverse direction

P010

2000

40006000

800010000

1200014000


Late

ral L

oad

(kN

)

P020

2000

40006000

800010000

1200014000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

P030

2000

40006000

800010000

1200014000

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

P0402000

40006000

800010000

1200014000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

P0502000

40006000

800010000

1200014000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

P060

2000

400060008000

10000

1200014000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P070

2000

40006000

800010000

1200014000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P080

2000400060008000

100001200014000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

P090

2000

40006000

800010000

1200014000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

P1002000400060008000

100001200014000

0 45 90 135 180 225 270

Displacement (mm)

Load

(kN

)

P1102000400060008000

100001200014000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

P120

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225Displacement (mm)

Late

ral L

oad

(kN

)

P130

200040006000

8000100001200014000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P140

200040006000

8000100001200014000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P150

200040006000

8000100001200014000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

51

P160

2000

400060008000

10000

1200014000

0 45 90 135 180 225Displacement (mm)

Late

ral L

oad

(kN

)P17

02000

400060008000

10000

1200014000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

P180

2000

400060008000

10000

1200014000


Late

ral L

oad

(kN

)

Fig.4.14: Load displacement relationship of the piers at top of Mohakhali flyover in longitudinal direction

The load-displacement relationships of the piers are graphically presented in Fig.4.13

and Fig.4.14. A common trend found in the figures. The trend is, the load increases

linearly with displacement up to a certain limit. The reason for that relationship is the

elastic properties of concrete and reinforcing steel. After yielding the outer most fiber

steel, the slope of load-displacement is mild with respect to within that limit because

of the inner steel yet not yield. After that the slope is found almost zero. It is seen

from the figures, that ultimate lateral loads are higher in transverse direction with

respect to the longitudinal direction of the flyover because of the sectional rigidity in

transverse direction is high.

It is also found from the figures, that ultimate lateral loads are different for the

different piers. The reason for the different loads is the difference in heights, cross-

sectional dimensions, and amount and arrangement of longitudinal reinforcement. In

addition, the strength and stiffness of the piers in transverse direction of the piers are

found larger than those for the longitudinal direction of the flyover. The reason can be

explained in the same as done for the case of explaining the difference in strength for

individual piers. It is also seen from the figures that the lateral strength of pier among

transverse direction large than those the longitudinal direction. In addition, the

stiffness of the piers in transverse direction is also higher in the transverse direction as

compared to those in the longitudinal direction. The reason for it is due to alignment

of the pier cross-section and longitudinal reinforcement,

Khilgaon flyover

Two different circular cross-sections with four different longitudinal reinforcements

and two transverse reinforcements are used in the Khilgaon flyover which has been

mentioned in details in Chapter II.

52

PLM 030

300

600

900

1200

1500

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)PLM 04

0

300

600

900

1200

1500

0 40 80 120 160 200Displacement (mm)

Late

ral L

oad

(kN

)

PLM 050

300

600

900

1200

1500

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

PLM 060

300

600

900

1200

1500

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

PLM 070

300

600

900

1200

1500

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

PLM 080

300

600

900

1200

1500

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

PLM 110

300

600

900

1200

1500

0 55 110 165 220 275 330Displacement (mm)

Late

ral L

oad

(kN

)

PLM 120

300

600

900

1200

1500

0 55 110 165 220 275 330Displacement (mm)

Late

ral L

oad

(kN

)

PLM 130

300

600

900

1200

1500

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)PLM 14

0

300

600

900

1200

1500

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

PLM 150

300

600

900

1200

1500

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

PLM 160

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM 020

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM 030

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM 040

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM 050

400

800

1200

1600

2000

0 40 80 120 160 200Displacement (mm)

Late

ral L

oad

(kN

)

PM 060

400

800

1200

1600

2000

2400

0 40 80 120 160 200Displacement (mm)

Late

ral L

oad

(kN

)

PM 070

400

800

1200

1600

2000

2400

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PR-020

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR-030

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR-040

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

53

PR 050

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)PR 06

0

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR 07

0

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR 080

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR 090

400

800

1200

1600

2000

0 40 80 120 160 200Displacement (mm)

Late

ral L

oad

(kN

)

PR 100

400

800

1200

1600

2000

2400

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PR 110

400

800

1200

1600

2000

2400

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PR 120

400800

12001600200024002800

0 25 50 75 100 125 150 175Displacement (mm)

Late

ral L

oad

(kN

)

PS 020

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)PS 03

0

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS 040

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS 050

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS 060

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS 070

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS 080

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS 090

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS 100

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS 110

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

Fig. 4.15: Load displacement relationship of the piers at top of Khilgaon flyover

The load-displacement relationships of the piers are shown in Fig.4.15. A common

trend is found in the figures. It is found from the figures that load increases linearly

with displacement up to a certain limit. The reason for that relationship is the elastic

properties of concrete and reinforcing steel. The rate of changing force with

displacement decreases with displacement after that limit and a smooth transition was

54

observed in this case. The reason for smooth transition is due to circular arrangement

of reinforcement. Due to the arrangement, progressive yielding of different layers’ is

reinforcing steel. In the case of rectangular section, a large number of reinforcing steel

is positioned in the extreme layers, and hence a sudden transition is observed. It is

also found from the figures, that ultimate lateral forces are different for the different

piers. The reason for the different forces is the different height of the piers.

4.6.2 Load-Displacement Relationship of the Pile Foundations

Mohakhali flyover

The lengths, sections, and number of piles are described in Table 2.1. In Fig.4.16 and

Fig.4.17 are to present the load-displacement relationships of the pile foundations of

Mohakhali flyover. In this case, the pier and pile cap are made rigid, and the load is

applied at the top of the pier. The displacement is recorded at the center of pile cap.

P01-P05 & P15-P180

2000

4000

6000

8000

10000


Late

ral L

oad

(kN

)

P06-P10, P13 & P140

2000

4000

6000

8000

10000

12000


Late

ral L

oad

(kN

)

P11 & P120

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

Fig.4.16: Load displacement relationship of the pile foundation at center of pile cap of Mohakhali flyover in transverse direction

P01-P05 & P15-P180

2000

4000

6000

8000

10000


Late

ral L

oad

(kN

)

P06-P10 & P13, P140

2000

4000

6000

8000

10000

12000


Late

ral L

oad

(kN

)

P11, P120

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

Fig.4.17: Load displacement relationship of the pile foundation at center of pile cap of Mohakhali flyover in longitudinal direction.

It is found from the figures that load increases linearly with displacement up to a

certain limit. The reason for that relationship is the elastic properties of individual

piles and soil springs. The rate of changing force with displacement decreases with

displacement after that limit and a transition was observed. The reason for smooth

transition is due to progressive yielding piles arranged in different layers.

55

Khilgaon flyover

Pile length, sections, and numbers are described in Table 2.2. In Fig.4.18 is to present

the load-displacement relationships of the pile of the Khilgaon flyover.

PML 030

200

400

600

800

1000

1200

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PML 040

200

400

600

800

1000

1200

0 40 80 120 160 200Displacement (mm)

Late

ral L

oad

(kN

)

PML 060

200

400

600

800

1000

1200

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

PML 070

200

400

600

800

1000

1200

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

PML 080

200

400

600

800

1000

1200

1400


Late

ral L

oad

(kN

)

PML 110

200

400

600

800

1000

1200

0 55 110 165 220 275 330Displacement (mm)

Late

ral L

oad

(kN

)

PML 120

200

400

600

800

1000

1200

0 55 110 165 220 275 330Displacement (mm)

Late

ral L

oad

(kN

)

PML 140

200

400

600

800

1000

1200

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

PML 150

200

400

600

800

1000

1200

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

PM 020

300

600

900

1200

1500

1800


Late

ral L

oad

(kN

)

PM 030

400

800

1200

1600

2000

2400


Late

ral L

oad

(kN

)

PM 040

300

600

900

1200

1500

1800


Late

ral L

oad

(kN

)

PM 050

300

600

900

1200

1500

1800


Late

ral L

oad

(kN

)

PM 060

300

600

900

1200

1500

1800


Late

ral L

oad

(kN

)

PM 070

300

600

900

1200

1500

1800

2100


Late

ral L

oad

(kN

)

PR 020

400

800

1200

1600

2000


Late

ral L

oad

(kN

)

PR 030

400

800

1200

1600

2000


Late

ral L

oad

(kN

)

PR 040

300

600

900

1200

1500

1800


Late

ral L

oad

(kN

)

56

PR 050

500

1000

1500

2000

2500

3000

3500


Late

ral L

oad

(kN

)PR 06

0

500

1000

1500

2000

2500

3000

3500


Late

ral L

oad

(kN

)

PR 080

500

1000

1500

2000

2500

3000

3500


Late

ral L

oad

(kN

)

PR 090

500

1000

1500

2000

2500

3000

3500


Late

ral L

oad

(kN

)

PR 100

500

1000

1500

2000

2500

3000

3500


Late

ral L

oad

(kN

)

PR 110

5001000150020002500300035004000


Late

ral L

oad

(kN

)

PR 120

5001000150020002500300035004000


Late

ral L

oad

(kN

)

PS 020

500

1000

1500

2000

2500

3000


Late

ral L

oad

(kN

)

PS 030

500

1000

1500

2000

2500

3000


Late

ral L

oad

(kN

)PS 04

0

500

1000

1500

2000

2500

3000


Late

ral L

oad

(kN

)

PS 050

500

1000

1500

2000

2500


Late

ral L

oad

(kN

)

PS 060

500

1000

1500

2000

2500

3000


Late

ral L

oad

(kN

)

PS 070

500

1000

1500

2000

2500

3000


Late

ral L

oad

(kN

)

PS 080

500

1000

1500

2000

2500

3000


Late

ral L

oad

(kN

)

PS 090

500

1000

1500

2000

2500


Late

ral L

oad

(kN

)

PS 100

500

1000

1500

2000

2500

3000


Late

ral L

oad

(kN

)

PS 110

500

1000

1500

2000

2500

3000


Late

ral L

oad

(kN

)

Fig.4.18: Load displacement relationship of the pile foundation at center of pile cap of Khilgaon flyover.

A similar to that observed in the case Mohakhali flyover can be observed from the

figures, and the reasons can also be explained in a way similar to that of the

57

Mohakhali flyover. Additional features that observed in the case of Khilgaon flyover

is that the transition from elastic to plastic are abrupt. The reason is that, the piles are

arranged in less number of rows, and yielding of either the pile body or soil springs

occur within short intervals.

4.6.3 Load-Displacement Relationships of Substructure

The seismic capacity of the flyover comes from the capacity of the substructure.

Hence, the pushover analyses of the substructures are carried out taking the strength-

deformation characteristics of all the members. It is to mention that in a flyover,

different members of flyovers for instance, STU or rubber bearings, pier, pile cap, and

pile foundations are arranged in a series, and hence failure of any one will cause the

failure of the flyover. To obtain the load displacement relationship of the substructure,

pushover analyses of the substructures are carried out.

Mohakhali flyover

P010

2000

4000

6000

8000

10000


Late

ral L

oad

(kN

)

P020

2000

4000

6000

8000

10000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

P030

2000

4000

6000

8000

10000

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

P040

2000

4000

6000

8000

10000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

P050

2000

4000

6000

8000

10000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

P060

2000

4000

6000

8000

10000

12000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P070

2000

4000

6000

8000

10000

12000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P080

2000

4000

6000

8000

10000

12000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

P090

2000

4000

6000

8000

10000

12000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

P100

2000

4000

6000

8000

10000

12000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P110

3000

6000

9000

12000

15000

18000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

P120

4000

8000

12000

16000

0 45 90 135 180 225Displacement (mm)

Late

ral L

oad

(kN

)

58

P130

3000

6000

9000

12000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)P14

0

2000

4000

6000

8000

10000

12000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P150

2000

4000

6000

8000

10000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

P160

2000

4000

6000

8000

10000

0 45 90 135 180 225Displacement (mm)

Late

ral L

oad

(kN

)

P170

2000

4000

6000

8000

10000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

P180

2000

4000

6000

8000

10000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

Fig.4.19: Load displacement relationship of the substructure top of Mohakhali flyover in transverse direction

P010

2000

4000

6000

8000

10000


Late

ral L

oad

(kN

)

P020

2000

4000

6000

8000

10000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

P030

2000

4000

6000

8000

10000

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

P040

2000

4000

6000

8000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

P050

1500

3000

4500

6000

7500


Late

ral L

oad

(kN

)

P060

1500

3000

4500

6000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P070

1000

2000

3000

4000

5000

6000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P080

1500

3000

4500

6000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

P090

1500

3000

4500

6000

7500


Late

ral L

oad

(kN

)

P100

1500

3000

4500

6000

7500

9000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P110

3000

6000

9000

12000

15000


Late

ral L

oad

(kN

)

P120

3000

6000

9000

12000

15000

18000

0 45 90 135 180 225Displacement (mm)

Late

ral L

oad

(kN

)

P130

1500

3000

4500

6000

7500

9000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P140

1500

3000

4500

6000

7500

9000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

P150

1500

3000

4500

6000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

59

P160

1500

3000

4500

6000

7500

0 45 90 135 180 225Displacement (mm)

Late

ral L

oad

(kN

)

P170

2000

4000

6000

8000

10000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

P180

2000

4000

6000

8000

10000

0 25 50 75 100 125 150Displacement (mm)

Late

ral L

oad

(kN

)

Fig.4.20: Load displacement relationship of the substructure top of Mohakhali flyover in longitudinal direction.

Fig. 4.19 and Fig. 4.20 show the load-displacement relationship at the top of the

substructures of Mohakhali flyover. It is seen from the figures that load increases with

the increase in displacement upto a certain limit. After that limit, the displacement

increases without increase in load. This is due to yielding of any of the members

arranged in series. It can also be seen that the strength and stiffness of the

substructures reduces remarkable as compared to that of the piers. This might be due

to the fact that, the strength and stiffness of the substructures depend on the stiffness

of the weakest members in the series, and yielding of the substructure initiates with

the yield of any of the members.

Khilgaon flyover

PML 030200

400600800

10001200

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PML 040

200

400

600

800

1000

0 40 80 120 160 200Displacement (mm)

Late

ral L

oad

(kN

)

PML 060

200

400

600

800

1000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

PML 070

200

400

600

800

1000

1200

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

PML 080

200400600800

100012001400

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

PML 110

200

400

600

800

1000

1200

0 55 110 165 220 275 330Displacement (mm)

Late

ral L

oad

(kN

)

PML 120

200

400

600

800

1000

1200

0 55 110 165 220 275 330Displacement (mm)

Late

ral L

oad

(kN

)

PML 140

200

400

600

800

1000

1200

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

PML 150

200

400

600

800

1000

1200

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

60

PM020

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)PM 03

0

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM040

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM050

500

1000

1500

2000

0 40 80 120 160 200Displacement (mm)

Late

ral L

oad

(kN

)

PM060

500

1000

1500

2000

0 40 80 120 160 200Displacement (mm)

Late

ral L

oad

(kN

)

PM070

500

1000

1500

2000

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PR020

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR030

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR040

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR050

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR060

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR080

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR090

500

1000

1500

2000

2500

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR100

500

1000

1500

2000

2500

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR110

500

1000

1500

2000

2500

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS020

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS030

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS040

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS050

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS060

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS070

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

61

PS080

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)PS09

0

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PS100

500

1000

1500

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

Fig.4.21: Load displacement relationship of the substructure top of Khilgaon

flyover

Fig. 4.21 presents the load-displacement relationship at the top of the substructures of

Khilgaon flyover. It is seen from the figures that load increases with the increase in

displacement upto a certain limit. After that limit, the displacement increases without

increase in load. This is due to yielding of any of the members arranged in series. It

can also be seen that the strength and stiffness of the substructures reduces remarkable

as compared to that of the piers. This might be due to the fact that, the strength and

stiffness of the substructures depend on the stiffness of the weakest members in the

series, and yielding of the substructure initiates with the yield of any of the members.

Table 4.1: Strength characteristic displacement of pier of Mohakhali flyover

Transverse direction Longitudinal direction Pier ID

Yield load (*W)

Ultimate load (*W)

Yield displacement

(mm)

Ultimate displacement

(mm)

Yield load (*W)

Ultimate load (*W)

Yield displacemen

t (mm)


(mm)

P01 1.40 1.70 4.96 7.46 0.80 1.09 9.44 15.32

P02 0.82 1.09 10.22 16.16 0.45 0.68 17.18 29.43

P03 0.61 0.88 16.14 26.77 0.33 0.54 27.58 47.71

P04 0.49 0.75 23.90 39.75 0.27 0.46 41.48 65.72

P05 0.41 0.67 31.58 52.69 0.23 0.40 52.16 79.92

P06 0.36 0.60 37.94 64.00 0.21 0.36 61.85 92.53 P07 0.33 0.56 42.47 72.84 0.18 0.34 71.37 104.38

P08 0.36 0.63 50.03 83.82 0.21 0.38 81.28 116.02

P09 0.63 1.00 32.41 53.00 0.28 0.48 70.93 106.63

P10 0.60 0.92 28.34 45.69 0.27 0.45 60.29 93.29

P11 0.48 0.74 26.76 39.50 0.32 0.62 29.09 52.67 P12 0.54 0.81 21.52 31.67 0.40 0.68 24.79 43.88

P13 0.62 1.01 33.48 50.46 0.30 0.51 63.78 96.45

P14 0.38 0.70 40.75 68.63 0.22 0.41 67.12 98.81

P15 0.34 0.62 35.33 59.67 0.22 0.37 59.53 89.23

P16 0.43 0.70 28.52 46.96 0.25 0.43 46.80 72.84

P17 0.66 0.95 15.22 24.01 0.37 0.58 24.40 41.80

P18 0.80 1.15 12.63 20.23 0.47 0.74 21.85 37.11

62

Table 4.2: Strength characteristic displacement of pier of Khilgaon flyover

Pier ID Yield load (*W)

Ultimate load (*W)

Yield displace-

ment (mm)

Ultimate displace-

ment (mm)

Pier ID

Yield load (*W)

Ultimate load (*W)

Yield displace-

ment (mm)

Ultimate displace-

ment (mm)

PML03 0.41 0.55 52.50 91.31 PR09 0.26 0.37 39.10 72.72

PML04 0.28 0.40 60.60 102.16 PR10 0.33 0.43 36.50 67.70

PML05 0.23 0.31 63.50 113.07 PR11 0.32 0.43 34.80 66.23

PML06 0.30 0.43 87.40 156.55 PR12 0.67 0.90 31.90 56.67

PML07 0.31 0.41 96.00 169.84 PS02 0.24 0.33 41.30 77.54

PML08 0.18 0.24 106.00 194.02 PS03 0.24 0.34 41.40 77.60

PML11 0.30 0.40 114.00 201.26 PS04 0.24 0.33 41.60 77.88

PML12 0.27 0.36 105.00 177.83 PS05 0.25 0.34 41.20 77.99

PML13 0.29 0.40 102.00 178.67 PS06 0.25 0.35 41.40 78.29

PML14 0.33 0.45 89.90 161.82 PS07 0.26 0.35 41.10 77.74

PML15 0.33 0.46 69.80 128.22 PS08 0.26 0.35 41.00 77.27

PML16 0.39 0.52 59.70 104.67 PS09 0.26 0.35 41.10 77.51

PR02 0.24 0.33 41.30 77.81 PS10 0.26 0.36 40.60 77.05

PR03 0.27 0.37 40.50 75.65 PM02 0.25 0.34 38.00 73.56

PR04 0.27 0.37 40.60 75.89 PM03 0.25 0.35 39.50 74.58

PR05 0.26 0.36 40.90 76.54 PM04 0.27 0.36 40.30 74.76

PR06 0.27 0.36 41.00 76.78 PM05 0.29 0.40 38.40 72.33

PR07 0.27 0.38 39.00 73.68 PM06 0.30 0.41 38.00 71.50

PR08 0.26 0.37 39.10 73.94 PM07 0.30 0.41 38.00 68.66

4.7 CONCLUSION

Pushover analysis of piers, pile foundations and whole substructures are carried out to

obtain lateral load displacement relationships of the model flyovers. To do pushover

analysis, analytical models are developed using Drain-2DX. Element description and

parameter estimation procedures are described in details in the chapter. Load-

displacement relationships are obtained for piers fixed at bottoms, pile foundations,

and the whole substructures are obtained and presented in graphical forms. The yield

63

load and yield displacement, and the ultimate load and ultimate displacements are

obtained from the analyses and presented in tabular forms.

Chapter 5

LATERAL STRENGTH AND DUCTILITY

5.1 INTRODUCTION

Lateral loads are induced in the structural members and/or system under seismic

events. To withstand under an earthquake of moderate to large magnitude earthquake,

the structure should possesses adequate strength. In addition, to minimize the

consequences, the structural members or system should be ductile enough so that

adequate warning before collapse of the structure can be observed. Hence, both the

strength and ductility are expected for achieving earthquake resistant design.

Reinforced concrete members fail mainly in two modes: a) flexural failure; b) shear

failure under lateral load. Lateral strength of a structure member depends on expected

failure mode. Lateral strength of a pier can be estimated by obtaining shear strength

and flexural strength. Failure due to shear occurs instantaneously without giving

sufficient warning, and such failures cause devastating effects in all respects. Hence,

the large ductility of a structural members or systems is highly expected. Ductility is a

mechanical property used to describe the extent to which materials can be deformed

plastically without fracture. In shear mode inadequate ductility will be observed and

hence collapse will occur without sufficient warning. In contrast, members are

expected to behave in a ductile manner in the case flexural failure. In order to

minimize losses due to earthquake, it is expected sufficient time for warning even if

the structure collapses. It is found from history (Hashimoto et al., 2005; Karim and

Yamazaki, 2001) that numerous bridge structures failed in shear mode. The shear

strength, flexural strengths and hence the lateral strength and ductility are evaluated in

this chapter.

5.2 EVALUATION OF LATERAL STRENGTH OF PIERS

5.2.1 Shear Capacity of Piers

The lateral strengths of the flyovers in shear are estimated from the SHB (JRA, 2002)

and the AASHTO standard specifications (2007) adapted equations. The JRA

65

standard specification, shear strength in a member is resisted by concrete and shear

reinforcements. The shear strengths of the flyovers members are calculated by the

JRA code adapted equations 5.1 to 5.3

sc VVV += (5.1)

bdcccV cptecc τ= (5.2)

sfA

V syws

)cos(sin θθ += (5.3)

where

V : Shear strength (N)

cV : Shear strength resisted by concrete (N)

sV : Shear strength borne by hoop ties (N).

cτ : Average shear stress that are borne by concrete (N/mm2). Values in Table 5.1

shall be used.

cc : Modification factor on the effects of alternating cyclic loading .Cc is taken as

0.6 for type 1 earthquake ground motion and 0.8 for type 2.

ec : Modification factor in relation to the effective height (d) of a section obtained

from Table 5.2

ptc : Modification factor in relation to the axial tensile reinforcement ratio tρ .

Values are obtained from Table 5.3

b : Width of the section perpendicular to the direction in calculating shear

strength (mm).

d : Effective height of the section parallel to the direction in calculating shear

strength (mm).

tρ : Axial tensile reinforcement ratio.

Aw : Sectional area of hoop type arranged with and interval of α and angle (mm2)

syf : Yield strength of hoop ties (N/mm2)

θ : Angle formed between hoop ties and the vertical axis (degree) s : Spacing of hoop ties (mm)

66

Table 5.1: Average shear stress of concrete Design compressive strength of concrete (MPa) 21 24 27 30 40

Average shear stress of concrete (MPa) 0.33 0.35 0.36 0.37 0.41

Table 5.2: Modification factors for effective height (d) of a pier section. Effective height(mm) Below 1000 3000 5000 Above1000

eC 1.0 0.7 0.6 0.5

Table 5.3: Modification factor in relation to axial tensile reinforcement ratio ptC

Tensile reinforcement ratio (%) 0.2 0.3 0.5 Above 1.0

ptC

0.9 1.0 1.2 1.5

The shear strengths for the failure mode shifting type from flexural damage to shear

failure are obtained by using the modification factor equals 1.0 by using the method

of JRA (2002).

The AASHTO Standard Specifications (2007) adapted the following equations based

on 45 degree truss model in determination of the nominal shear strength of

reinforcement concrete columns.

scn VVV += (5.4)

bdf

AfPV c

gcc 6

10 ′⎟⎟⎠

⎞⎜⎜⎝

⎛

′= (5.5)

sdfA

V syws = (5.6)

In the case of AASHTO, the effect of many parameters has not been accounted for as

done in JRA equations.

Determination of effective height (d) for circular sections

The determination of effective height d of a circular section of a member is shown in

Fig. 5.1. For rectangular section, the effective height is a distance from the

compression edge to the position of the center of gravity of the tensile reinforcement

having neglected the lateral reinforcement. For the circular section, it is substituted

with an equivalent square section having an equivalent area as the circular section and

67

then the distance between the compression edge and the center of gravity of the

tensile reinforcement in the squire section is taken as the effective height.

Fig. 5.1: Determination of effective height of a section

5.2.2 Flexural Capacity of Piers

Flexural strengths of the piers are obtained by using ultimate moment capacities of the

pier. In this case, pier bases are assumed to be fixed, that means the flexibility of the

foundations are ignored.

Ultimate moment capacities are found from the Sectional analysis results, and the

lateral strengths in flexure are found from Equation (5.7) considering the pier as a

single degree of freedom system shown in Fig.5.2.

P

uu H

MP = (5.7)

where

uP =ultimate lateral strength in bending, uM =ultimate moment capacity of the pier

section obtained from Sectional analysis, PH = height of the pier.

Fig. 5.2: Numerical evaluation of bending capacity of pier

Pu

Hp

Mu

d

Effective width

68

5.3 DUCTILITY OF PIERS

Ductility is a mechanical property used to describe the extent to which materials can

be deformed plastically without fracture. Larger ductility is expected for all the

structures. Ductility largely depends on the expected mode of failure: shear failure

and bending failure. Ductility is of two types: curvature ductility and displacement

ductility. Displacement ductility can be related to curvature ductility. In the study,

both curvature ductility and displacement ductility are estimated in terms of ultimate

and allowable ductility. Ultimate curvature/displacement ductility is defined as the

ratio of ultimate curvature/displacement to yield curvature/displacement and the

allowable ductility is obtained (JRA, 2002; 1998) by equations 5.8, 5.9.

y

yuac αφ

φφµ

−+= 1 (5.8)

y

yuad αδ

δδµ

−+= 1 (5.9)

Where

acµ : Allowable curvature ductility of the reinforced concrete section, uφ : Ultimate

curvature of the reinforced concrete section, yφ : Yield curvature of the reinforced

concrete section, adµ : Allowable displacement ductility of a concrete member, uδ :

Ultimate displacement of the reinforced concrete member, yδ : Yield displacement of

the reinforced concrete member and α : Safety factor, 3.0 is used in the study.

Ultimate displacement of a member is calculated by the following equation

)2/()( ppyuyu LhL −××−+= φφδδ (5.10)

where

pL : Plastic hinge length calculated by the following equation

DhLp 1.02.0 −= (5.11)

In which DLD p 5.01.0 ≤≤

D : Sectional depth (mm) (D is diameter of a circular section, or the height of a

rectangular section in the analytical direction)

69

5.4 FAILURE MODE OF PIERS

Failure mode of the reinforced concrete column can be evaluated by the following

equation

⎪⎭

⎪⎬

⎫

<≤<

≤

failureShear : yielding flexuralafter failureShear :

failure bendingor Flexural :

0

0

us

su

u

PVVPV

VP (5.12)

Where

uP : Lateral strength of a reinforced concrete column, as specified in above (N)

V : Shear strength of a reinforced concrete column, as specified in above (N)

0sV : Shear strength of a reinforced concrete column calculated by assuming that

the modification factor on the effects of repeated alternative loads is equal to

1.0.

A flow diagram is shown to evaluate the lateral strength and ductility of the flyover.

Fig. 5.3: Evaluation of Failure Mode, Lateral Strength and Ductility Capacity

for a RC Member.

70

5.5 ANALYTICAL RESULTS

5.5.1 Bending Strengths of the Piers

Mohakhali Flyover

0300060009000

1200015000180002100024000

P01 P03 P05 P07 P09 P11 P13 P15 P17

Pier ID

Late

ral S

tren

gth

(kN

)

0.0

0.4

0.7

1.1

1.4

1.8

P01 P03 P05 P07 P09 P11 P13 P15 P17

Pier ID

Late

ral S

tren

gth

(Pu/W

)

Fig.5.4: Lateral strength of Mohakhali flyover for piers under bending in transverse direction

Fig.5.5: Normalized Lateral strength of Mohakhali flyover for piers under bending in transverse direction

0

3000

6000

9000

12000

15000

P01 P03 P05 P07 P09 P11 P13 P15 P17Pier ID

Late

ral S

tren

gth

(kN

)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

P01 P03 P05 P07 P09 P11 P13 P15 P17

Pier ID

Late

ral S

tren

gth

(Pu/W

)

Fig.5.6: Lateral strength of Mohakhali flyover for piers under bending in longitudinal direction

Fig.5.7: Normalized Lateral of Mohakhali flyover for piers under bending in longitudinal direction

The variations of lateral strengths in bending for the piers Mohakhali flyover in both

transverse and longitudinal directions can be seen from Fig. 5.4 to Fig. 5.7. In Fig. 5.4

and Fig. 5.6 the lateral strengths in KN are presented. One can easily see the

differences in the magnitude of the lateral strengths. To quantify the variations, the

lateral strengths are normalized by the respective weights from the superstructures

and piers themselves. The variations of normalized lateral strengths can be seen from

Fig. 5.5 and Fig. 5.7. It is seen from the figures that the strength varies from 0.52W to

1.77W along the transverse direction, while that for the longitudinal direction is

0.30W to 1.01W.

71

Khilgaon flyover

0

500

1000

1500

2000

2500

PML03

PML05

PML07

PML11

PML13

PML15

PR02PR04

PR06PR08

PR10PR12

PS03PS05

PS07PS09

PS11PM03

PM05PM07

Pier ID

Late

ral S

tren

gth

(kN

)

Fig.5.8: Lateral strength Khilgaon flyover piers under bending

0.00

0.15

0.30

0.45

0.60

0.75

0.90

PML03

PML05

PML07

PML11

PML14

PML16

PR03PR05

PR08PR10

PR12PS03

PS05PS07

PS09PS11

PM03PM05

PM07

Pier ID

Late

ral S

tren

gth

(Pu/W

)

Fig.5.9: Normalized lateral strength Khilgaon flyover piers under bending

The variations of lateral strengths in bending for the piers Khilgaon flyover can be

seen from Fig. 5.8 and Fig. 5.9. Fig. 5.8 shows the lateral strengths in KN, while the

normalized lateral strengths are presented in Fig. 5.9. As can be seen from the figure,

the normalized lateral strengths of the piers vary from 0.24W to 0.74W.

72

5.5.2 Shear Strength of the Piers

Mohakhali Flyover

0

4000

8000

12000

16000

20000

24000

28000

P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18

Pier ID

Late

ral S

tren

gth

(kN

)Shear (JRA) CapacityShear (AASHTO) Capacity

Fig.5.10: Lateral strength of Mohakhali flyover for piers under shear in transverse direction

0.0

0.4

0.8

1.2

1.6

2.0


Pier ID

Late

ral S

trem

gth

(V/W

)

Shear (JRA) CapacityShear (AASHTO) Capacity

Fig.5.11: Normalized lateral strength of Mohakhali flyover for piers under shear in transverse direction

0

4000

8000

12000

16000

20000

24000


Pier ID

Late

ral S

tren

gth

(kN

)


Fig.5.12: Lateral strength of Mohakhali flyover for piers under shear in

longitudinal direction

73

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4


Pier ID

Late

ral S

tren

gth

(V/W

)


Fig.5.13: Normalized Lateral strength of Mohakhali flyover for piers under shear

in longitudinal direction

Khilgaon flyover

0

400

800

1200

1600

2000

2400

PML03

PML05

PML07

PML11

PML13

PML15

PR02PR04

PR06PR08

PR10PR12

PS03PS05

PS07PS09

PM02PM04

PM06

Pier ID

Late

ral S

tren

gth

(kN

)


Fig.5.14: Shear strength Khilgaon flyover piers under shear

0.00

0.10

0.20

0.30

0.40

0.50

0.60

PML03

PML05

PML07

PML11

PML13

PML15

PR02PR04

PR06PR08

PR10PR12

PS03PS05

PS07PS09

PM02PM04

PM06

Pier ID

Late

ral S

tren

gth

(V/W

)


Fig.5.15: Normalized lateral strength Khilgaon flyover piers under shear

Fig. 5.10 to Fig. 5.15 shows the variation of shear strengths of different piers of

Mohkhali and Khilgaon flyover. Different shear capacities of same piers are observed

using SHB (JRA, 2002) and AASHTO (2007) equations. It can easily be found that

74

the SHB equations are more conservative but more rigorous. From now and on, the

discussions will be made with respect to SHB capacities. From Fig. 5.11, it is seen

that the normalized shear strength varies from 0.89W to 1.36W in the transverse

direction of the piers, while that for the longitudinal directions are observed in Fig.

5.13 to vary from 0.66W to 0.87W. However, the shear strength of the piers of

Khilgaon flyover ranges from 0.17W to 0.39W.

5.5.3 Lateral Strength of Pile Foundation

Mohakhali Flyover

0

3000

6000

9000

12000

15000

P01 P03 P05 P07 P09 P11 P13 P15 P17

Pier ID

Late

ral S

tren

gth

(kN

)

0.0

0.2

0.4

0.6

0.8

1.0

P01 P03 P05 P07 P09 P11 P13 P15 P17

Pier ID

Late

ral S

tren

gth

(Pu/W

)

Fig.5.16: Lateral strength of pile foundation of Mohakhali flyover

Fig.5.17: Normalized lateral strength of pile foundation of Mohakhali flyover

Khilgaon flyover

0

700

1400

2100

2800

3500

4200

PML03

PML05

PML07

PML11

PML14

PML16

PR03PR05

PR08PR10

PR12PS03

PS05PS07

PS09PS11

PM03PM05

PM07

Pier ID

Late

ral S

tren

gth

(kN

)

Fig.5.18: Lateral strength Khilgaon flyover pile foundation

75

0.00

0.30

0.60

0.90

1.20

1.50

PML03

PML05

PML07

PML11

PML14

PML16

PR03PR05

PR08PR10

PR12PS03

PS05PS07

PS09PS11

PM03PM05

PM07

Pier ID

Late

ral s

tren

gth

(Pu/W

)

Fig.5.19: Normalized lateral strength of Khilgaon flyover pile foundation

Lateral strengths of pile foundation of the substructures of both the flyovers are

obtained from pushover analyses on the basis of analytical models and methods

described earlier. The obtained lateral strengths are normalized by the weight of the

super-structures and piers. The normalized strengths of the different substructures,

named according the name of the piers, are presented in Fig. 5.17 and Fig. 5.19. The

capacity of the pile foundation of Mohakhali flyover varies from 0.54W to 0.87W

which are observed from Fig. 5.17. The variation of the normalized lateral strengths

of pile foundation can be seen from Fig. 5.19. It is seen from the figure that the

strengths vary from 0.23W to 1.36W.

5.5.4 Lateral Strength of Substructure

Mohakhali Flyover

0

3000

6000

9000

12000

15000

P01 P03 P05 P07 P09 P11 P13 P15 P17

Pier ID

Late

ral S

tren

gth

(kN

)

0.00

0.20

0.40

0.60

0.80

1.00

1.20

P01 P03 P05 P07 P09 P11 P13 P15 P17

Pier ID

Late

ral S

tren

gth

(Pu/W

)

Fig.5.20: Lateral strength of Mohakhali flyover substructure in transverse direction

Fig.5.21: Normalized Lateral strength of Mohakhali flyover substructure in transverse direction

76

0

3000

6000

9000

12000

15000

P01 P03 P05 P07 P09 P11 P13 P15 P17Pier ID

Late

ral S

tren

gth

(kN

)

0.0

0.2

0.4

0.6

0.8

P01 P03 P05 P07 P09 P11 P13 P15 P17

Pier ID

Late

ral S

tren

gth

(Pu/W

)

Fig.5.22: Lateral strength of Mohakhali flyover substructure in longitudinal direction

Fig.5.23: Normalized Lateral strength of Mohakhali flyover substructure in longitudinal direction

Khilgaon flyover

0

300

600

900

1200

PML03

PML05

PML07

PML11

PML14

PML16

PR03PR05

PR08PR10

PR12PS03

PS05PS07

PS09PS11

PM03PM05

PM07

Pier ID

Late

ral S

tren

gth

(kN

)

Fig.5.24: Lateral strength of Khilgaon flyover substructure

0.00

0.10

0.20

0.30

0.40

PML03

PML05

PML07

PML11

PML14

PML16

PR03PR05

PR08PR10

PR12PS03

PS05PS07

PS09PS11

PM03PM05

PM07

Pier ID

Late

ral s

tren

gth

(Pu/W

)

Fig.5.25: Normalized lateral strength of Khilgaon flyover for substructure

Analytical models of the substructures are developed in DRAIN-2DX and pushover

analyses are carried out accordingly. The results have been presented in Chapter IV.

The lateral strengths both in magnitude and normalized forms are presented for both

Khilgaon and Mohakhali flyover in the above figures. It is seen from Fig. The

77

normalized lateral strength for the substructure along transverse and longitudinal

directions can be seen from Fig. 5.21 and Fig. 5.23. One can easily see the variation

of normalized lateral strengths of the substructures of Khilgaon flyover from Fig.

5.31. It is seen from the figure that the minimum strength is 0.17W.

5.5.5 Failure Mode of Piers

Table 5.4: Failure mode of Mohakhali flyover pier in the transverse direction Pier No. Pu (kN) Vc (kN) Vs

(kN) V

(kN) Vso

(kN) Failure Mode

P01 21164 1609 12364 13972 15045 Shear failure P02 16022 1609 12364 13972 15045 Shear failure P03 12673 1609 12364 13972 15045 Flexural Failure P04 10702 1609 12364 13972 15045 Flexural Failure P05 9448 1609 12364 13972 15045 Flexural Failure P06 8623 1609 12364 13972 15045 Flexural Failure P07 8059 1609 12364 13972 15045 Flexural Failure P08 7405 1609 12364 13972 15045 Flexural Failure P09 12079 1995 16885 18880 20210 Flexural Failure P10 13426 1995 16885 18880 20210 Flexural Failure P11 18118 2992 16885 19877 21872 Flexural Failure P12 18164 2992 16885 19877 21872 Flexural Failure P13 13132 1995 16885 18880 20210 Flexural Failure P14 8063 1609 12364 13972 15045 Flexural Failure P15 8885 1609 12364 13972 15045 Flexural Failure P16 10020 1609 12364 13972 15045 Flexural Failure P17 13703 1609 12364 13972 15045 Flexural Failure P18 13993 1609 12364 13972 15045 Shifting failure

Table 5.5: Failure mode of Mohakhali flyover pier in the longitudinal direction Pier No. Pu (kN) Vc (kN) Vs

(kN) Vs

(kN) Vso

(kN) Failure Mode

P01 12113 2006 8326 10331 11668 Shear failure P02 9194 2006 8326 10331 11668 Flexural Failure P03 7272 2006 8326 10331 11668 Flexural Failure P04 6143 2006 8326 10331 11668 Flexural Failure P05 5427 2006 8326 10331 11668 Flexural Failure P06 4950 2006 8326 10331 11668 Flexural Failure P07 4627 2006 8326 10331 11668 Flexural Failure P08 4244 2006 8326 10331 11668 Flexural Failure P09 5083 2722 8326 11048 12862 Flexural Failure P10 5659 2722 8326 11048 12862 Flexural Failure P11 11442 3421 12666 16087 18368 Flexural Failure P12 12538 3421 12666 16087 18368 Flexural Failure P13 5515 2722 8326 11048 12862 Flexural Failure P14 4620 2006 8326 10331 11668 Flexural Failure P15 5101 2006 8326 10331 11668 Flexural Failure P16 5755 2006 8326 10331 11668 Flexural Failure P17 7865 2006 8326 10331 11668 Flexural Failure P18 8013 2006 8326 10331 11668 Flexural Failure

78

Table 5.6: Failure mode of Khilgaon flyover pier

Pier No. Pu (kN) Vc (kN) Vs (kN) Vs (kN) Vso (kN) Failure Mode

PML03 1194 484 284 768 1091 Shear Failure PML04 1130 484 284 768 1091 Shear Failure PML05 962 484 284 768 1091 Shifting failure PML06 1250 525 284 809 1159 Shear Failure PML07 1204 525 284 809 1159 Shear Failure PML08 1189 525 284 809 1159 Shear Failure PML11 1143 525 284 809 1159 Shifting failure PML12 963 525 284 809 1159 Shifting failure PML13 1126 525 284 809 1159 Shifting failure PML14 1301 525 284 809 1159 Shear Failure PML15 1218 525 284 809 1159 Shear Failure PML16 1348 525 284 809 1159 Shear Failure PR03 1972 673 383 1056 1505 Shear Failure PR04 1965 673 383 1056 1505 Shear Failure PR05 1947 673 383 1056 1505 Shear Failure PR06 1935 673 383 1056 1505 Shear Failure PR07 1992 673 383 1056 1505 Shear Failure PR08 1996 673 383 1056 1505 Shear Failure PR09 2063 673 383 1056 1505 Shear Failure PR10 2202 673 383 1056 1505 Shear Failure PR11 2239 673 383 1056 1505 Shear Failure PR12 2119 673 383 1056 1505 Shear Failure PS02 1644 673 383 1056 1505 Shear Failure PS03 1869 673 383 1056 1505 Shear Failure PS04 1865 673 383 1056 1505 Shear Failure PS05 1874 673 383 1056 1505 Shear Failure PS06 1856 673 383 1056 1505 Shear Failure PS07 1867 673 383 1056 1505 Shear Failure PS08 1885 673 383 1056 1505 Shear Failure PS09 1878 673 383 1056 1505 Shear Failure PS10 1877 673 383 1056 1505 Shear Failure PM02 1900 673 383 1056 1505 Shear Failure PM03 1925 673 383 1056 1505 Shear Failure PM04 1957 673 383 1056 1505 Shear Failure PM05 1985 673 383 1056 1505 Shear Failure PM06 2009 673 383 1056 1505 Shear Failure PM07 2180 673 383 1056 1505 Shear Failure

79

5.5.6 Ductility of Piers

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18Pier ID

Cur

vatu

re D

uctil

ityUltimate curvature ductilityAllow able curvature ductility (Far f ield earthquake)Allow able curvature ductility (Near f ield earthquake)

Fig.5.26: Curvature ductility of Mohakhali flyover in transverse direction

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0


Cur

vatu

re D

uctil

ity

Ultimate curvature ductilityAllow able curvature ductility (Far f ield earthquake)Allow able curvature ductility (Near field earthquake)

Fig.5.27: Curvature ductility of Mohakhali flyover in longitudinal direction

0.0

0.5

1.0

1.5

2.0

2.5


Dis

plac

emen

t Duc

tility

Ultimate displacement ductilityAllow able displacement ductility (Far f ield earthquake)Allow able displacement ductility (Near f ield earthquake)

Fig.5.28: Displacement ductility of Mohakhali flyover piers in transverse direction

80

0.0

0.5

1.0

1.5

2.0

2.5


Dis

plac

emen

t Duc

tility

Ultimate displacement ductilityAllow able displacement ductility (Far f ield earthquake)Allow able displacement ductility (Near f ield earthquake)

Fig.5.29: Displacement ductility of Mohakhali flyover piers in longitudinal direction

Table 5.7: Curvature ductility of Mohakhali flyover piers


Ultimate curvature ductility

*Allowable curvature ductility

Allowable curvature ductility

Ultimate curvature ductility

*Allowable curvature ductility

Allowable curvature ductility

P02 1.0 1.0 1.0 2.78 2.19 1.59 P03 2.70 2.13 1.57 2.78 2.19 1.59 P04 2.70 2.13 1.57 2.78 2.19 1.59 P05 2.70 2.13 1.57 2.78 2.19 1.59 P06 2.70 2.13 1.57 2.78 2.19 1.59 P07 2.70 2.13 1.57 2.78 2.19 1.59 P08 2.70 2.13 1.57 2.78 2.19 1.59 P09 2.38 1.92 1.46 2.93 2.28 1.64 P10 2.38 1.92 1.46 2.93 2.28 1.64 P11 2.29 1.86 1.43 2.36 1.91 1.45 P12 2.29 1.86 1.43 2.36 1.91 1.45 P13 2.38 1.92 1.46 2.93 2.28 1.64 P14 2.70 2.13 1.57 2.78 2.19 1.59 P15 2.70 2.13 1.57 2.78 2.19 1.59 P16 2.70 2.13 1.57 2.78 2.19 1.59 P17 2.70 2.13 1.57 2.78 2.19 1.59 P18 1.0 1.0 1.0 2.78 2.19 1.59

Note: * allowable ductility has been calculated considering far field earthquake

81

Table 5.8: Displacement ductility of Mohakhali flyover piers



ductility

*Allowable displacement

ductility

Allowable displacement

ductility


ductility

*Allowable displacement

ductility

Allowable displacement

ductility P02 1.0 1.0 1.0 1.71 1.48 1.24 P03 1.66 1.44 1.22 1.73 1.49 1.24 P04 1.66 1.44 1.22 1.58 1.39 1.19 P05 1.67 1.45 1.22 1.53 1.35 1.18 P06 1.69 1.46 1.23 1.50 1.33 1.17 P07 1.71 1.48 1.24 1.46 1.31 1.15 P08 1.68 1.45 1.23 1.43 1.28 1.14 P09 1.64 1.42 1.21 1.50 1.34 1.17 P10 1.61 1.41 1.20 1.55 1.36 1.18 P11 1.48 1.32 1.16 1.81 1.54 1.27 P12 1.47 1.31 1.16 1.77 1.51 1.26 P13 1.51 1.34 1.17 1.51 1.34 1.17 P14 1.68 1.46 1.23 1.47 1.31 1.16 P15 1.69 1.46 1.23 1.50 1.33 1.17 P16 1.65 1.43 1.22 1.56 1.37 1.19 P17 1.58 1.38 1.19 1.71 1.48 1.24 P18 1.0 1.0 1.0 1.70 1.47 1.23

Note: * allowable ductility has been calculated considering far field earthquake

5.5.7 Probability of shear Failure

Table 5.9: Probability of shear failure of Mohakhali flyover

Pier ID. Flexural Failure

Shifting failure

Shear Failure Pier ID. Flexural

Failure Shifting failure

Shear Failure

P01 0 8 17 P10 25 0 0 P02 22 3 0 P11 25 0 0 P03 25 0 0 P12 25 0 0 P04 25 0 0 P13 25 0 0 P05 25 0 0 P14 25 0 0 P06 25 0 0 P15 25 0 0 P07 25 0 0 P16 25 0 0 P08 25 0 0 P17 25 0 0 P09 25 0 0 P18 17 8 0

82

Table 5.10: Probability of shear failure of Khilgaon flyover

Pier ID. Flexural Failure

Shifting failure

Shear Failure Pier ID. Flexural

Failure Shifting failure

Shear Failure

PML03 0 23 2 PR10 0 0 25 PML04 0 25 0 PR11 0 0 25 PML05 9 16 0 PR12 0 0 25 PML06 0 5 20 PS02 0 0 25 PML07 0 11 14 PS03 0 0 25 PML08 0 21 4 PS04 0 0 25 PML11 0 20 5 PS05 0 0 25 PML12 0 19 6 PS06 0 0 25 PML13 0 2 23 PS07 0 0 25 PML14 0 25 0 PS08 0 0 25 PML15 0 7 18 PS09 0 0 25 PML16 0 2 23 PS10 0 0 25 PR03 0 0 25 PM02 0 0 25 PR04 0 0 25 PM03 0 0 25 PR05 0 0 25 PM04 0 0 25 PR06 0 0 25 PM05 0 0 25 PR07 0 0 25 PM06 0 0 25 PR08 0 0 25 PM07 0 0 25 PR09 0 0 25

Table 5.9 describes the total number of failure occurs in different piers of Mohakhali

and Khilgaon flyover. It can be seen from the Table that only a single pier of

Mohakhali flyover is expected to fail in shear mode and its probability is 68%, and the

probability of shifting type failure that is from bending failure to shear failure may

occur in two columns including the first one. It is to mention that the absolute shear

capacity of those two piers are very high, and hence the real probability of shear or

shifting type failure is almost negligible, since such a shear force demand may never

be happened.

The number of different modes of failures in different piers for statically 25 different

but nominally identical piers can be seen from Table 5.10. It is seen from the Table

that almost all the piers are expected to fail in shear and the probability of such failure

is almost one.

5.6 HIERARCHY FACTOR

Earthquake resistant design methodology all over the world calls for ensuring

reparability after a large magnitude earthquake. In order to ensure reparability, the

damage in the flyovers under a major earthquake should be limited with respect to its

position and extent. Since, it is difficult to detect and repair the damages in

83

foundations, earthquake resistant design specifications recommend that the primary

inelastic behavior should preferably be located in piers. This type of seismic design is

called “capacity design” where the inelastic behavior should be limited to

predetermined regions that can easily be inspected and repair. The capacity design

approach is adopted in many earthquake resistant design specifications (AASHTO,

2007; JRA, 2002; CalTrans, 2001). To ensure such reparability, it is expected that the

lateral strength of pier should be less than that of the foundation. To verify whether,

the hierarchy of capacity is maintained or not, a factor named as “hierarchy factor” is

introduced in this study and which defined as the ratio of the lateral strength of a pile

foundation to that of the respective pier.

It is rational that the hierarchy factor is more than 1, and it is 1.1 in SHB (JRA, 2002),

and 1.15 in AASHTO (2007).

The results of the hierarchy factor for different piers are presented in Fig.5.30 to

Fig.5.31 for Mohakhali. It can be seen from the figures that fifteen piers out of

eighteen in transverse direction and five piers out of eighteen in longitudinal direction

of Mohakhali possess the hierarchy factor less than one. It indicates that the damages

in the substructures are expected to occur in the pile foundations which are

unexpected due to complexity in inspection and necessary repair.

0.000

0.250

0.500

0.750

1.000

1.250

1.500


Hei

rarc

hy F

acto

r

Fig.5.30: Hierarchy factor of the piers of Mohakhali flyover in transverse direction

84

0.500

0.750

1.000

1.250

1.500

1.750

2.000

2.250


Hei

rarc

hy F

acto

r

Fig.5.31: Hierarchy factor of the piers of Mohakhali flyover in longitudinal

direction

The results of the hierarchy factor for different piers are presented in Fig.5.32 for

Khilgaon flyover. The variation of hierarchy factor for the substructures of Khilgaon

flyover can be seen from Fig.5.32. The range of the factor lies within 0.62 to 1.72.

Sixteen out of thirty six substructures damages are expected to occur in the pile

foundations.

0.50

0.75

1.00

1.25

1.50

1.75

2.00

PML03

PML05

PML07

PML11

PML14

PML16

PR03PR05

PR08PR10

PR12PS03

PS05PS07

PS09PM02

PM04PM06

Pier ID

Hei

rarc

hy F

acto

r

Fig.5.32: Hierarchy factor of the piers of Khilgaon flyover

5.7 CONCLUSIONS

Lateral strengths and Ductility for instance pier, pile foundation and sub structural

system are evaluated on the nonlinear static analysis results of sectional analysis and

pushover analysis results are utilized for pile foundation and sub structural system.

The following conclusions can be drawn from the analytical investigations.

85

Almost all of piers of Khilgaon flyovers are expected to fail in shear mode

under a major earthquake, while most of the piers of Mohakhali flyover will

fail in flexural mode.

The normalized lateral strength of the pier Mohakhali flyover lies within

0.52W to 1.18W in transverse direction and 0.30W to 0.86W in longitudinal

direction, while that the Khilgaon flyover lies within 0.17 W to 0.39W.

The normalized lateral strengths of pile foundation of Mohakhali flyover

ranges from 0.54W to 0.87W in both directions. In contrast, that for the

Khilgaon flyover is 0.23W to 1.36W.

The lateral strength of the substructure ranges from 0.30W to 0.76W for

Mohakhali flyover and 0.17W to 0.39W for Khilgaon flyover.

It has been found from the investigation that a large number of pile

foundation possess lateral strengths less than that of the respective piers of

Mohakhali flyover.

The ultimate curvature ductility of piers lies within 2.29 to 2.93 in Mohakhali

flyover and 3.45 to 5.05 for Khilgaon flyover.

The hierarchy factor of Mohakhali flyover lies within 0.42 to 1.27 in

transverse direction and 0.65 to 2.14 in longitudinal direction, while that for

the Khilgaon flyover is 0.62 to 1.72. However, the values for many

substructures are less than one.

Chapter 6

THE EFFECT OF VARIABILITY OF MATERIALS

STRENGTH

6.1 INTRODUCTION

The uncertainties of design and load parameters are inevitable in nature. Due to the

inherent uncertainty of design variable, the capacity or strength varies. This chapter

describes the method of evaluating the effect of the variability of the load and design

variables on the seismic capacity of the bridge substructure of the flyovers. The

statistical parameter of the design and load variables is selected at the first step to

evaluate the effect of variability of design variables; sampling technique is used in the

subsequent steps which will be discussed in this chapter.

6.2 STATISTICAL PARAMETERS OF MATERIAL PROPERTIES

The variability of the design parameters related to resistance are used in this study.

The variability concerning section dimensions such as the height and width of a

section, the depth of concrete cover and the amount of reinforcement are ignored due

to the less significant effects (Frangopol et al, 1996). The variability of the

fundamental random variables belonging to three basic materials: concrete, soil, and

reinforcing steel are used. For concrete, compressive strength and modulus of

elasticity are considered as the fundamental random variables. The fundamental

random variables related to reinforcing steel are yield strength and modulus of

elasticity. SPT N-value is used as the fundamental random variables for soil.

For the compressive strength of concrete, normal probability distribution has been

found best suitable by many investigators (Mirza, 1996; Mirza et al, 1979). In this

study, the construction quality is assumed as well controlled and the Coefficient of

Variation (COV) is selected as 11% (Mirza et al, 1979). The mean strength of

concrete is often related to its characteristic strengths, and that relationship is shown

in Fig. 6.1. Hence, the relationship between the mean and characteristic strengths can

be written as

87

)1( cncmck Vkff −= (6.1)

where

fcm and fck : mean and characteristic values of compressive strength of concrete; Vc :

COV for concrete strength; kn : a factor depending on the type of statistical

distribution.

Fig.6.1: Relationship between mean value and characteristic value

For the normal distribution and 5% level of significance nk equals 1.645 and the

Equation (6.2) turns into

)645.11( ccmck Vff −= (6.2)

Different statistical distribution for the yield strength of reinforcing steel has been

proposed by different researchers: Low and Hao (2001) (normal); Galambos and

Ravindra (1978) (lognormal), and Mirza and McGregor (1979) (beta distribution).

However, the normal distribution is more appropriate for yield strength of

reinforcement at 95% confidence level (Arafah, 1997). Hence, the normal distribution

for yield strength of reinforcing steel is used in this study. Galambos and Ravindra

(1978) recommended a COV of yield strength of reinforcing steel equal to 8-12%.

Considering progress of manufacturer’s control over quality with time, a lower value

of COV i.e., 8% is selected for this study.

From the characteristic value of yield strength the mean value fym is evaluated

considering that characteristic value at 5% fractal. The relationship is defined by

)645.11( r

ykym V

ff

−= (6.3)

88

Kulhway (1992), Phoon (1999), and Rackwitz (2000) summarized the nature of

distributions and COV ranges of soil properties for different types of deposits. They

reported that COV of SPT N-values lies within the range of 15-45%. It is worthy to

note that the average of SPT N-values along the depth, instead of N- values at points

for a certain significant depth, controls the side friction, end bearing resistance of a

pile and the spring constants of the ground. Honjo et al (2000) and Vanmarcke (1977)

recommended reducing the variance SPT N-values averaged over depth. Kulhway

(1992) and Phoon (1999) proposed a COV of SPT N-value 30% for sandy layer.

Hence, COV of SPT N-value is assigned to 30% in this study. Normal distribution is

assumed for the averaged SPT N-values, because it is more likely to follow the

normal distribution following the central limit theorem.

6.3 LATIN HYPERCUBE SAMPLING

Among different methods of evaluating the effect of material variability on the

structural capacity and response, the sampling technique is adopted in this study for

simplicity and accuracy. Latin Hypercube Sampling (LHS) one of the most advanced

sampling techniques. LHS was first proposed by McKay et al. (1979) and has been

further developed for different purpose by several researchers (Iman and Conover,

1982; Olsson and Sandberg, 2002, Owen, 1994; stein, 1987; Ziha, 1995). To facilitate

the presentation of the LHS importance sampling, the original and most simple form

of the sampling plan for general Monte Carlo simulation purpose in presented below.

The desired accuracy of the estimated distribution function determines the number of

realizations required. Let N define the required number of realizations and K the

number of random variables. The sampling space is then K-dimension. An

KN × matrix P, in which each of the K columns is a random permutation of 1, 2,

3,……..,N, and an KN × matrix R of independence random numbers from the

uniform (0,1) distribution are established. These matrices form the basic sampling

plan, represented by the matrix S as

)(1 RPN

S −= (6.4)

Each element of S, ijs , is then mapped according to its target marginal distribution as

)(ˆ 1ijxij sFx

j

−= (6.5)

89

where 1−jxF represents the inverse of the target cumulative distribution function for

variable j. A vector ]ˆ...............ˆ ˆ[ˆ 21 ikiiij xxxx = now contains input data for one

deterministic computation.

6.4 METHODOLOGY

Samples are generated by LHS using the statistical parameters of the design and load

variable. For a particular combination material and load parameters, the capacity of

pier, pile body, pile foundation and the substructure have been evaluated using

nonlinear static analysis which has been discussed in the earlier chapter. Moment-

curvature relationships of RC section of piers, piles, footing have been evaluated and

presented in the subsequent sections. Finally the statistical parameters of the capacity

indicators for instance the yield moment capacity, ultimate moment capacity, shear

strength, bending strength and lateral strengths are presented in the results section.

6.5 STATISTICAL TESTS

In order to obtain the type of statistical distribution those verify the goodness of fit,

statistical test is conducted on the results of nonlinear static analysis of statically

different and nominally identical flyover. Two different tests namely Kolmogorov-

Smirnov (K-S) tests and Chi-Square tests are conducted. (Halder and Mahadavan,

2000).

6.5.1 Chi-Square Test

In the χ2 goodness –of- fit test, the range of the n observed data is divided into m

intervals, and the number of times (ni) the random variable is observed in the i th

interval is counted (i = 1 to m ). Observed frequencies n1, n2, ….. nm of m intervals of

the random variable are then compared with the corresponding theoretical frequencies

e1, e1, …… em of an assumed distribution. It can be shown (Hoel, 1962) that the quantity

( )∑=

−m

i i

ii

een

1

2

(6.6)

Approaches the χ2 distribution with f= m-1-k degrees of freedom as the total sample

points n tends to α. Here, m is the number of intervals and k is the number of

distribution parameters estimated from the data. The number of degrees of freedom f

90

is a parameter of the χ 2 distribution. A significance level α is selected. Significance

level levels between 1% and 10% are common. A significance level of 5% implies

that for 5 out of a total of 100 different samples, the assumed theoretical distribution

cannot be an acceptable model.

6.5.2 Kolmogorov-Smirnov (K-S) Test

The K-S test compares the observed cumulative frequency and the CDF of an

assumed theoretical distribution. The first step is to arrange the data in increasing

order. Then the maximum difference between the two cumulative distribution

functions of the ordered data can be estimated as

( ) ( )[ ]iniXn xSxFD −= max (6.7)

Where ( )iX xF is the theoretical CDF of the assumed distribution at the i th

observation of the ordered sample ix , and ( )in xS is the corresponding stepwise CDF

of the observed order samples. According to the K-S test, if the maximum difference

nD is less than or equal to the tabulated value αnD , the assumed distribution is

acceptable at the significance level α.

The advantage of the K-S test over the χ 2 test is that it is not necessary to divide the

data into intervals, thus the error or judgment associated with the number of size of

the interval is avoided.


6.6.1 Moment Curvature Relationship

Mohakhali flyover piers

Generating 25 combinations of statistically different but normally identical samples,

the nonlinear static analyses are carried out. One of the nonlinear analyses carried out

for the pier sections is sectional analysis based on fiber model. The results for a

particular pier section are presented in Fig. 7.2 to Fig. 7.4 by superposition.

91

P01-P08 & P14-P180

30000

60000

90000

120000

150000

180000

0 5 10 15 20 25Curvature (rad/m)X10-3

Mom

ent (

kN-m

)P09, P10, & P13

0

30000

60000

90000

120000

150000

180000

0 5 10 15 20 25Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

P11 & P120

40000

80000

120000

160000

200000

0 5 10 15 20 25Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

Fig.6.2: Moment-curvature relationship of Mohakhali flyover piers in transverse direction

P01-P08 & P14-P180

20000

40000

60000

80000

100000

120000

0 5 10 15 20 25

Curvature(rad/m)x10-3

Mom

ent (

kN-m

)

P09, P10, & P13 0

20000

40000

60000

80000

100000

120000

0 5 10 15 20 25Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

P11 &P120

20000

40000

60000

80000

100000

120000

0 5 10 15 20 25


Mom

ent (

kN-m

)

Fig.6.3: Moment-curvature relationship of Mohakhali flyover piers in longitudinal direction

Khilgaon flyover piers

PML03, PML04, PML050

4000

8000

12000

16000

20000

0 5 10 15 20 25Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

PML06,PML07, PML08, PML130

4000

8000

12000

16000

20000

0 5 10 15 20 25Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

PML11, PML140

4000

8000

12000

16000

20000

0 5 10 15 20 25Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

PML12, PML15, PML160

4000

8000

12000

16000

20000

0 5 10 15 20 25Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

Sayedabad and Rajarbagh pier0

4000

8000

12000

16000

20000

0 5 10 15 20 25Curvature (rad/m)x10-3

Mom

ent (

kN-m

)

Fig.6.4: Moment-curvature relationship of Khilgaon flyover piers

92

6.6.2 Load Displacement Relationship


P010

4000

8000

12000

16000

20000

24000


Late

ral L

oad

(kN

)

P020

3000

6000

9000

12000

15000

18000

0 25 50 75 100 125

Displacement (mm)

Late

ral L

oad

(kN

)

P030

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P040

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P050

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P060

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P070

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

PIer-080

2000400060008000

100001200014000


Late

ral L

oad

(kN

)

P090

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P100

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P110

3000

6000

9000

12000

15000

18000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

P120

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P130

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P140

3000

6000

9000

12000

15000

18000

0 25 50 75 100 125

Displacement (mm)

Late

ral L

oad

(kN

)

P150

3000

6000

9000

12000

15000

18000

0 25 50 75 100 125

Displacement (mm)

Late

ral L

oad

(kN

)

P160

3000

6000

9000

12000

15000

18000


Latr

eral

Loa

d (k

N)

P170

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

P180

3000

6000

9000

12000

15000

18000


Late

ral L

oad

(kN

)

Fig.6.5: Load-displacement relationship of Mohakhali flyover piers in transverse direction

93

P080

3000

6000

9000

12000

15000


Late

ral L

oad

(kN

)P10

0

3000

6000

9000

12000

15000


Late

ral L

oad

(kN

)

P110

3000

6000

9000

12000

15000

0 25 50 75 100 125

Displacement (mm)

Late

ral L

oad

(kN

)

Fig.6.6: Load-displacement relationship of Mohakhali flyover piers in longitudinal direction

In longitudinal and transverse direction can be observed for piers of Mohakhali and

Khilgaon flyover, the reason for the narrow range in the linear part is due to elastic

behavior of material in the regime. The inelastic behavior is much sensitive to the

variation of material property and P-∆ effect and hence, a wide band for all the piers

of both Mohakhali and Khilgaon flyover can be observed.


PM020

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM030

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM040

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM050

300

600

900

1200

1500

1800


Late

ral L

oad

(kN

)

PM060

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PM070300600900

1200150018002100

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PR030

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR040

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR050

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR060

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR070

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

PR080

300

600

900

1200

1500

1800

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

94

PR090

300

600

900

1200

1500

1800

0 50 100 150 200

Displacement (mm)

Late

ral L

oad

(kN

)PR10

0

300

600

900

1200

1500

1800

0 30 60 90 120 150 180

Displacement (mm)

Late

ral L

oad

(kN

)

PR110

300

600

900

1200

1500

1800

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PR120

400

800

1200

1600

2000

0 25 50 75 100 125 150 175Displacement (mm)

Late

ral L

oad

(kN

)

PML-030

400

800

1200

1600

2000

0 30 60 90 120 150 180Displacement (mm)

Late

ral L

oad

(kN

)

PML-040

400

800

1200

1600

2000

0 40 80 120 160 200

Displacement (mm)

Late

ral L

oad

(kN

)

PML-050

400

800

1200

1600

2000

0 50 100 150 200 250Displacement (mm)

Late

ral L

oad

(kN

)

PML-060

400

800

1200

1600

2000

0 45 90 135 180 225 270Displacement (mm)

Late

ral L

oad

(kN

)

PML-070

400

800

1200

1600

2000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

PML-080

400

800

1200

1600

2000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

PML-110

400

800

1200

1600

2000

0 55 110 165 220 275 330Displacement (mm)

Late

ral L

oad

(kN

)

PML-120

400

800

1200

1600

2000

0 55 110 165 220 275 330Displacement (mm)

Late

ral L

oad

(kN

)

PML-130

400

800

1200

1600

2000

0 50 100 150 200 250 300Displacement (mm)

Late

ral L

oad

(kN

)

PML-140

400

800

1200

1600

2000

0 50 100 150 200 250Dsplacement (mm)

Late

ral L

oad

(kN

)

PML-150

400

800

1200

1600

2000

0 35 70 105 140 175 210Displacement (mm)

Late

ral L

oad

(kN

)

Fig. 6.7: Load-displacement relationship of Khilgaon flyover piers

Similar trends in the results of pushover analysis are observed as observed in the case

of sectional analysis results and can be explained as similar way.

95

Mohakhali flyover piles

P01-P05 & P14-P18 0

4000

8000

12000

16000

20000

24000


Late

ral L

oad

(kN

)

P06-P10 & P13 0

4000

8000

12000

16000

20000

24000


Late

ral L

oad

(kN

)

P11 & P120

4000

8000

12000

16000

20000

24000


Late

ral L

oad

(kN

)

Fig. 6.8: Load-displacement relationship of Mohakhali flyover pile in longitudinal direction

P01-P05 & P14-P180

4000

8000

12000

16000

20000

24000


Late

ral L

oad

(kN

)

P06-P10, & P130

4000

8000

12000

16000

20000

24000


Late

ral L

oad

(kN

)

P11 & P120

4000

8000

12000

16000

20000

24000


Late

ral L

oad

(kN

)

Fig. 6.9: Load-displacement relationship of Mohakhali flyover pile in transverse direction

6.6.3 Statistical Distribution

The statistical natures of the capacity are checked for type of distribution for which

the results fit at the first step, and the statistical parameters of the distribution type are

evaluated in the subsequent steps. For verifying the distribution type statistical test for

goodness of fit are conducted. In the tests for goodness of fit, it is assumed that either

normal or lognormal distribution fit well. For that reason, Kolmogorv-Smirnov (K-S)

and Chi-Square tests (Halder and Mahadevan, 2000) are carried out and relevant

results are presented


P01-08 and P14-P18

0

1

2

3

4

5

6

7

44755 46534 48313 50093 51872 53652 55431Ultimate Moment (kN-m)

n i or e

i

Observed frequencyTheoretical frequency COV µ σ Mck

5.81% 49639 2884 44894

λ ξ

10.81 0.058

P01-P08 and P14-P180.0

0.2

0.4

0.6

0.8

1.0

1.2

44000 46000 48000 50000 52000 54000Ultimate Moment (kN-m)

F E(e

m) o

r Sn(

e m)

Observed CDFTheoretical CDF

96

P09, P10 and P14

0

1

2

3

4

5

6

7


n i or e

i


6.31% 59019 3728 52886

λ ξ

10.98 0.063

P09, P10 and P130.0

0.2

0.4

0.6

0.8

1.0

1.2

52000 57000 62000 67000Ultimate Moment (kN-m)

F E(e

m) o

r Sn(

e m)


P11 and P12

0

1

2

3

4

5

6

7


n i or e

i


6.43% 102723 6607 91854

λ ξ

11.54 0.064

P11 and P120.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


Fig.6.10: Statistical distribution test of Mohakhali flyover piers in longitudinal direction

P01-08 and P14-P18

0

1

2

3

4

5

6


n i o

r ei


5.57% 86867 4846 78894

λ ξ

11.37 0.056

P01-P08 and P14-P180.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P09, P10, P13

0

1

2

3

4

5

6

7


n i o

r ei


7.70% 4006 309 3499

λ ξ

11.86 0.058

P09, P10, P130.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P11, P12

0

1

2

3

4

5

6

7


n i o

r ei

Observed frequencyTheoretical f requency

COV µ σ Mck

7.49% 5988 449 5250

λ ξ

12.00 0.059

P11, P120.0

0.2

0.4

0.6

0.8

1.0

1.2

145000 155000 165000 175000 185000Ultimate Moment (kN-m)

F E(e

m) o

r Sn(

e m)


Fig.6.11: Statistical distribution test of Mohakhali flyover piers in transverse direction

P08

01

2345

67

3528 3709 3891 4072 4253 4434 4615Ultimate Lateral Load (kN)

n i o

r ei

Observed frequencyTheoretical frequency COV µ σ Pck

7.70% 4006 309 3499

λ ξ

8.29 0.077

P080.0

0.2

0.4

0.6

0.8

1.0

1.2

3300 3600 3900 4200 4500 4800Ultimate Lateral load (kN)

F E(e

m) o

r Sn(

e m)


97

P10

01

2345

67


n i o

r ei


7.49% 5988 449 5250

λ ξ

8.69 0.075

P100.0

0.2

0.4

0.6

0.8

1.0

1.2

5000 5500 6000 6500 7000Ultimate Lateral load (kN)

F E(e

m) o

r Sn(

e m)


P11

01

2345

67


n i o

r ei


7.69% 11363 874 9925

λ ξ

9.33 0.077

P110.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


Fig.6.12: Statistical distribution test of Mohakhali flyover piers in longitudinal direction

P01

01

2345

67


n i o

r ei


7.28% 20300 1477 17870

λ ξ

9.92 0.073

P010.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)

observed CDFTheoretical CDF

P02

01

2345

67


n i o

r ei


7.27% 14745 1072 12982

λ ξ

9.60 0.073

P020.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P03

0

1

2

3

4

5

6


n i o

r ei

Observed frequencyTheoretical f requency COV µ σ Pck

7.27% 11622 845 10232

λ ξ

9.35 0.073

P030.0

0.2

0.4

0.6

0.8

1.0

1.2

9800 10400 11000 11600 12200 12800 13400Ultimate Lateral load (kN)

F E(e

m) o

r Sn(

e m)


P04

01

2345

67


n i o

r ei


7.27% 9788 712 8617

λ ξ

9.18 0.073

P040.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


98

P05

01

2345

67


n i o

r ei


7.28% 8624 627 7591

λ ξ

9.06 0.073

P050.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P06

01

2345

67


n i o

r ei


7.28% 7847 571 6908

λ ξ

8.96 0.073

P060.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P07

01

2345

67


n i o

r ei


7.27% 7322 533 6446

λ ξ

8.89 0.073

P070.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P08

01

2345

67


n i o

r ei


7.27% 6974 507 6139

λ ξ

8.85 0.073

P080.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P09

0

1

2

3

4

5

6


n i o

r ei


7.38% 12140 896 10666

λ ξ

9.40 0.074

P090.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P10

01

2345

67


n i o

r ei


7.38% 13115 968 11523

λ ξ

9.48 0.074

P100.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


99

P11

01

2345

67


n i o

r ei


7.11% 14178 1007 12520

λ ξ

9.56 0.071

P110.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P12

01

2345

67


n i o

r ei


7.12% 15615 1112 13786

λ ξ

9.65 0.071

P120.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P13

0

1

2

3

4

5

6


n i o

r ei


7.38% 12175 899 10697

λ ξ

9.40 0.074

P130.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P14

01

2345

67


n i o

r ei


7.27% 7611 554 6700

λ ξ

8.93 0.073

P140.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P15

01

2345

67


n i o

r ei


7.27% 8091 588 7123

λ ξ

9.00 0.073

P150.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P16

01

2345

67


n i o

r ei


7.27% 9156 666 8061

λ ξ

9.12 0.073

P160.0

0.2

0.4

0.6

0.8

1.0

1.2

7500 8500 9500 10500Ultimate Lateral load (kN)

F E(e

m) o

r Sn(

e m)


100

P17

01

2345

67


n i o

r ei


7.27% 12585 915 11079

λ ξ

9.44 0.073

P170.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


P18

0

1

2

3

4

5

6


n i o

r ei


7.27% 13358 972 11760

λ ξ

9.50 0.073

P180.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


Fig.6.13: Statistical distribution test of Mohakhali flyover piers in transverse direction


PML03, PML04, PML05

0

12

3

4

56

7

8


n i o

r ei


7.26% 8151 591 7177

λ ξ

9.00 0.072

PML03, PML04, PML050.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)



0

12

3

4

56

7

8


n i o

r ei


7.16% 12632 905 11143

λ ξ

9.44 0.072


0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML11, PML14

0

1

2

3

4

5

6

7


n i o

r ei


7.16% 13403 959 11823

λ ξ

9.50 0.072

PML11, PML140.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML12, PML15, PML16

0

12

3

4

56

7

8


n i o

r ei


7.24% 11068 801 9749

λ ξ

9.31 0.072

PML12, PML15, PML160.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


101

Sayedabad and Rajarbagh pier

0

12

3

4

56

7

8


n i o

r ei


7.39% 11532 852 10128

λ ξ

9.35 0.074

Sayedabad and Rajarbagh pier

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PR03

0

1

2

3

4

5

6


n i o

r ei


7.99% 1386 111 1204

λ ξ

7.23 0.080

PR030.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PR04

01

2345

67


n i o

r ei


7.56% 1370 104 1199λ ξ

7.22 0.075

PR040.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PR05

01

2345

67


n i o

r ei


7.53% 1357 102 1189λ ξ

7.21 0.075

PR050.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PR06

01

2345

67


n i o

r ei


7.55% 1352 102 1184λ ξ

7.21 0.075

PR060.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PR07

01

2345

67


n i o

r ei


7.60% 1394 106 1219λ ξ

7.24 0.076

PR070.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


102

PR08

01

2345

67


n i o

r ei


7.55% 1387 105 1215λ ξ

7.23 0.075

PR080.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PR09

01

2345

67


n i o

r ei


7.54% 1432 108 1254λ ξ

7.26 0.075

PR090.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PR10

01

2345

67


n i o

r ei


8.04% 1528 123 1326λ ξ

7.33 0.080

PR100.0

0.2

0.4

0.6

0.8

1.0

1.2

1250 1350 1450 1550 1650 1750 1850U lt imat e Lat eral load ( kN )


PR11

01

2345

67


n i o

r ei


7.56% 1552 117 1359λ ξ

7.34 0.075

PR110.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PR12

01

2345

67


n i o

r ei


7.51% 1662 125 1457λ ξ

7.41 0.075

PR120.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PM02

01

2345

67


n i o

r ei


7.50% 1361 102 1193λ ξ

7.41 0.075

PM020.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


103

PM03

01

2345

67


n i o

r ei


7.55% 1378 104 1207λ ξ

7.23 0.075

PM030.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PM04

01

2345

67


n i o

r ei


7.55% 1401 106 1227λ ξ

7.24 0.075

PM040.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PM05

01

2345

67


n i o

r ei


7.48% 1423 106 1248λ ξ

7.26 0.075

PM050.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PM06

012345678


n i o

r ei


7.53% 1438 108 1260λ ξ

7.27 0.075

PM060.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PM07

0

1

2

3

4

5

6


n i o

r ei


7.55% 1561 118 1367λ ξ

7.35 0.075

PM070.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML03

0

1

2

3

4

5

6

7


n i or e

i


7.56% 1449 109 1268

λ ξ

7.27 0.075

PML030.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


104

PML04

0

1

2

3

4

5

6

7


n i or e

i


7.53% 1355 102 1186

λ ξ

7.21 0.075

PML040.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML05

0

1

2

3

4

5

6

7


n i or e

i

Observed frequencyTheoretical frequency

COV µ σ Pck

7.54 1145 86.4 1003

λ ξ

7.04 0.075 PML05

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML06

0

1

2

3

4

5

6

7


n i or e

i


COV µ σ Pck

7.53% 1023 77.0 896

λ ξ

6.92 0.075 PML06

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML07

0

1

2

3

4

5

6


n i or e

i


COV µ σ Pck

7.53% 970 73 850

λ ξ

6.87 0.075 PML07

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML08

01

23

45

67

8


n i or e

i


COV µ σ Pck

7.53% 914 68 801

λ ξ

6.81 0.075 PML08

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML11

0

1

2

3

4

5

6


n i or e

i


COV µ σ Pck

7.36% 1362 100 1196

λ ξ

7.21 0.074

PML110.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


105

PML12

0

1

2

3

4

5

6

7


n i or e

i


COV µ σ Pck

7.50% 895 67 784

λ ξ

6.79 0.075 PML12

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML13

0

1

2

3

4

5

6

7


n i or e

i


COV µ σ Pck

7.51% 931 70 816

λ ξ

6.83 0.075 PML13

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML14

0

1

2

3

4

5

6

7


n i or e

i


COV µ σ Pck

7.36% 1578 116 1386

λ ξ

7.36 0.073 PML14

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML15

0

1

2

3

4

5

6

7


n i or e

i


COV µ σ Pck

7.53% 1135 85 994

λ ξ

7.03 0.075 PML15

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


PML16

0

1

2

3

4

5

6


n i or e

i


COV µ σ Pck

7.54% 1257 95 1100

λ ξ

7.13 0.075 PML16

0.0

0.2

0.4

0.6

0.8

1.0

1.2


F E(e

m) o

r Sn(

e m)


Fig. 6.14: Statistical distribution test of Khilgaon flyover piers

It is found from the statistical test results that the seismic capacity of the piers and pile

foundations and substructure fit for both normal and lognormal distribution up to 95%

level of significance. However, normal distribution fits better that of the lognormal

type. The reason may be due to consideration of the material parameters as normally

distributed the variability measured in terms of Coefficient of Variation (COV). COV

for the ultimate moment capacity lies within 5% to 8%.

106

6.7 CONCLUSIONS

With a view to obtain the effect of variability of material parameters analytically, the

variability of the material parameters related to resistance for instance, strength,

modulus of elasticity has been taken into considerations. To achieve the goals, 25

nominally identical but statistically different flyovers are generated for each using

Latin Hypercube sampling technique. Nonlinear static analyses are carried out for all

the sample flyovers generated. On the basis of the results obtained on the generated

flyovers, the following conclusions are drawn:

The effect variability on the elastic range is not prominent, while that for the inelastic

part is significant.

The statistical tests of goodness of fit shows that the response of the statistically

different bridges can be modeled using both Normal and Log-Normal distributions for

high degree of confidence for instance, it is valid upto 95% confidence interval.

However, Normal distribution fits better for further higher levels of significances for

instance 98% or more.

The COV of the responses that is, lateral strength of the statistically different bridges

as obtained from the investigation ranges from 5 to 8%

Chapter 7

CONCLUSIONS AND RECOMMENDATIONS FOR

FURTHER STUDY

7.1 INTRODUCTION

The main objectives the study was to evaluate the seismic capacity of the flyovers in

Bangladesh. To reach the destination, Mohakhali and Khilgaon flyover have been

used as model flyovers. The nonlinear static analyses are carried out deterministically

at first step, and probabilistically in the subsequent step. In the nonlinear analyses

different analytical models are used to achieve the objectives of the research. The

major conclusions that are derived from the study can be summarized in the next

section.

7.2 CONCLUSIONS

The major conclusions derived in the study are as follows:

i. The minimum lateral strength of the substructures of Mohakhali flyover is

0.30W, while that for the Khilgaon flyover is 0.17W.

ii. The ranges of the lateral strengths of piers, pile foundations, substructures of

Mohakhali flyover are 0.30W to 1.18W, 0.54W to 0.87W, 0.30W to 0.76W

respectively, while the ranges of Khilgaon flyover are 0.17W to 0.39, 0.23W

to 1.36W, 0.17W to 0.39W respectively.

iii. The lateral strengths of the substructures, piers and pile foundations of

Khilgaon flyover are found significantly smaller than those of Mohakhali

flyover.

iv. All the piers of Khilgaon flyover are expected to fail in shear mode which

will exhibit brittle collapse that implies inadequate warning before collapse

of the flyover will be obtained under a major earthquake. In contrast, only

one pier out of eighteen piers along the longitudinal direction and three piers

along the transverse direction of Mohakhali flyover are supposed fail in

shear mode, and the rest of the piers are expected to fail in flexural mode.

108

v. The curvature ductility of the piers of Mohakhali flyover ranges from 2.29 to

2.93, while that for the Khilgaon flyover lies within a range from 3.45 to

5.05.

vi. The displacement ductility of the piers of Mohakhali flyover ranges from

1.47 to 1.81, while that for the Khilgaon flyover is 1.0.

vii. For Mohakhali flyover, fifteen piers out of eighteen in transverse direction

and five out of eighteen in longitudinal direction possess lateral strength

larger than that of the respective pile foundations, while sixteen out of thirty

six piers’ strengths are found larger than those of the respective pile

foundation for Khilgaon flyover. It indicates that the damages in the

substructures are expected to occur in the pile foundations which are

unexpected due to complexity in inspection and necessary repair.

7.3 RECOMMENDATIONS FOR FURTHER STUDY

The following recommendations are made for further study.

i. The study may be extended other flyovers that have already been

constructed and those are being constructed.

i. Three dimensional analytical models of the flyovers may be made for

obtaining the lateral strength and ductility of the flyovers.

ii. Experimental verifications of the analytical results could be done using

Shaking table and full scale physical models

iii. Nonlinear time history analyses may be conducted to obtain the response

data and compare with the capacity

iv. Seismic vulnerability analyses could be carried out.

109

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SYMBOLS AND NOTATIONS

A : Cross-sectional area of pile tip.

cA : Cross-sectional area of concrete.

Ai : Section area for bridge pier in the i-th section from the acting position of

inertial force of superstructure, with axial reinforcement also taken into

consideration.

PA : Cross-sectional area of the pile.

sA : Cross-sectional area of steel.

Aw : Sectional area of hoop type arranged with and interval of α and angle.

a : Cast-in-place piles constant.

B : Width of a respective member.

HB : Equivalent loading width of a foundation.

b : Width of the section perpendicular to the direction in calculating shear

strength.

COV : Coefficient of Variance.

cc : Modification factor on the effects of alternating cyclic loading.

ce : Modification factor in relation to the effective height (d) of a section (Table-

5.2)

cpt : Modification factor in relation to the axial tensile reinforcement ratio tρ .

(Table-5.3)

D : Diameter of respective member/ height of a rectangular section in the

analytical direction.

D : Loading width of a foundation perpendicular to a load working direction.

Dn : Difference between the two cumulative distribution functions of the ordered

data.

Dp : Diameter of pile body.

d : Effective height/depth of a section.

di : Distance from top of pier/pile.

E0 : Modulus of deformation of a soil layer.

Ec : Modulus of elasticity of concrete.

EI : Rigidity of the foundation.

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Ep : Modulus of elasticity of the pile concrete.

e : Theoretical frequency. 1−

jxF : Inverse of the target cumulative distribution functions for variable j.

FX(xi) : Theoretical CDF of the assumed distribution at the i th observation of the

ordered sample ix .

fbt : Tensile strength of concrete in bending.

fck : Design compressive strength of concrete.

f ’c : Design strength of concrete.

fc : Compressive strength of concrete.

fcc : Compressive strength of concrete in confined condition.

fcm : mean values of compressive strength of concrete.

fco : Compressive strength of concrete in unconfined condition.

fi : Maximum skin friction force per unit area of a layer.

fy : Yield strength of reinforcement.

fyh : Yield strength of hoop reinforcement.

fyk : Design yield strength of reinforcing steel.

fym : mean value of yield strength the reinforcing steel.

Hp : Height of a pier.

h : Total depth of pile cap.

Ii : Moment of inertia of areas of flyover in the i-th section from the acting

position of inertial force of superstructure, taking the axial reinforcement

also taken into consideration.

K : Number of random variables.

Kv : Axial spring constant of a pile.

kv : Coefficient of vertical subgrade reaction.

kp : Coefficient of equivalent subgrade reaction.

kH : Coefficient of horizontal ground reaction.

kHE : Coefficient of horizontal ground reaction.

kH0 : Coefficient of horizontal sub-grade reaction.

kn : a factor depending on the type of statistical distribution.

L : Length of respective member.

Li : Thickness of a layer for which skin friction force is taken into account.

Lp : Length of pile body (incase of description of flyover).

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Lp : Plastic hinge length (incase of determination plastic hinge zone)

Mi : Bending moment acting on the i-th section from the acting position of

inertial force of superstructure.

Mu : Ultimate moment.

My0 : Initial yield moment.

Mck : Characteristic moment.

m : Number of piles in a column (incase of pile arrangement).

m : Intervals (incase of statistical analysis).

N : Standard Penetration Test (SPT) value (incase of soil parameter).

N : Number of slice/segment (incase of pushover analysis).

N : Required number of realizations (incase of statistical analysis).

Ni : Axial force due to the weights of superstructure and substructure, acting on

the i-the section from the acting position of inertial force of superstructure.

n : Number of piles in a row (incase of pile arrangement).

n : Observed data (incase of statistical analysis)

P : Lateral load.

Pa : Allowable ultimate lateral load.

Pck : Characteristic lateral load.

PHU : Upper limit of unit horizontal ground reaction.

PNU : Ultimate axial capacity against push-in.

PPU : Ultimate bearing capacity of pile against pull-out considering the pile body

(N)

PTU : Ultimate axial capacity against pull-out

Pu : Ultimate lateral strength.

PU : Ultimate bearing capacity of the pile against pull-out considering the soil

parameters.

pU : Passive soil pressure.

Py0 : Initial yield horizontal strength.

qd : Ultimate bearing capacity per unit area to be borne by a pile tip (N)

RU : Ultimate bearing capacity of the pile against push-in considering the soil

parameters.

RPU : Ultimate bearing capacity of pile against push-in considering the pile body.

S : Matrices form of basic sampling plan.

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Si : Section modulus of flyover pier cross-section with axial reinforcement in the

i-th section from the acting position of inertial force of the superstructure

also taken into consideration.

SL : Spacing of pile in longitudinal direction.

( )in xS : Corresponding stepwise CDF of the observed order samples.

ST : Spacing of pile in transverse direction.

s : Spacing of hoop ties.

U : Circumferential length of the pile.

V : Shear strength of reinforced member.

Vc : Shear strength resisted by concrete (incase of evaluation of shear strength).

Vc : Coefficient of variance for concrete (incase of statistic analysis).

Vr : Coefficient of variance for reinforcing steel (incase of statistic analysis).

Vs : Shear strength borne by hoop ties.

Vs0 : Shear strength of a reinforced concrete column calculated by assuming that

the modification factor on the effects of repeated alternative loads is equal to

1.0.

W : Effective weight of the pile.

xj : Distance from concrete or reinforcing bar in the j-th infinitesimal element to

the centroid position.

xo : Distance from the compressed edge of concrete to the neutral axis.

ijx̂ : A vector contains input data for one deterministic computation.

α : Safety factor (incase of evaluation of ductility).

α : A coefficient (incase of horizontal ground reaction calculation of soil)

α : Level of significance(incase of statistic analysis).

βα , : Modification factors depending on shape of cross section (incase of

constitutive model of concrete).

kα : Correction factor of horizontal ground reaction around a single pile.

pα : Correction factor of upper limit of unit horizontal ground reaction around a

single pile.

β : Characteristics value of foundation (incase of horizontal ground reaction

around).

χ2 : Chi-Square Test.

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δ : Lateral displacement of reinforced member.

δu : Ultimate displacement of the reinforced concrete member.

δy : Yield displacement of the reinforced concrete member.

δyo : Initial yield displacement of reinforced member.

cε : Strain of concrete.

ccε : Strain of concrete at peak stress of concrete.

coε : Compressed edge strain of concrete.

cuε : Ultimate strain of concrete.

kη : Correction factor of horizontal ground reaction with the group of piles effect

taken into account.

pη : Correction factor of upper limit of unit horizontal ground reaction with the

group of piles effect taken into account.

φ : Angle of friction depends on SPT value.

iφ : Curvature of the i-th section from the acting position of inertial force of

superstructure.

uφ : Ultimate curvature of the reinforced concrete section.

yφ : Yield curvature of the reinforced concrete section.

yoφ : Initial yield curvature.

µ : Mean value.

acµ : Allowable curvature ductility of the reinforced concrete section.

adµ : Allowable displacement ductility of a concrete member.

σ : Standard Deviation.

λ : Log mean value (incase of statistical distribution).

λ : Equivalent protrusion length of pile cap (incase of rigidity check of pile

cap).

ξ : Log standard deviation.

θ : Angle formed between hoop ties and the vertical axis.

Lρ : Longitudinal steel ratio.

sρ : Volumetric ratio of hoop reinforcement or transverse steel ratio.

tρ : Axial tensile reinforcement ratio.

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sjcj σσ , : Stresses in concrete and reinforcing steel of the j-th infinitesimal element.

cτ : Average shear stress that are borne by concrete (Table 5.1).

sjcj AA ∆∆ , : Sectional areas of concrete and reinforcing steel in the j-th infinitesimal

evaluation of seismic capacity of the flyovers in …

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