evaluation of seismic capacity of the flyovers in …
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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.
Title of Figure PageNo.
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.
Title of Figure PageNo.
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.
Title of Figure PageNo.
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.
Title of Figure PageNo.
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
Curvature (rad/m)x10-3
Mom
ent (
kN-m
)
P11, P12
0
40000
80000
120000
160000
0 3 6 9 12 15
Curvature (rad/m)x10-3
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
Curvature (rad/m)x10-3
Mom
ent (
kN-m
)
P09, P10, P13
0
20000
40000
60000
80000
100000
0 3 6 9 12 15
Curvature (rad/m)x10-3
Mom
ent (
kN-m
)
P11, P12
0
20000
40000
60000
80000
100000
0 3 6 9 12 15
Curvature (rad/m)x10-3
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
Curvature (rad/m)x10-3
Mom
ent (
kN-m
)
PML06-PML08, PML13
0
4000
8000
12000
16000
0 5 10 15 20
Curvature (rad/m)x10-3
Mom
ent (
kN-m
)
PML11, PML14
0
4000
8000
12000
16000
0 5 10 15 20
Curvature (rad/m)x10-3
Mom
ent (
kN-m
)
PML12, PML15, PML16
0
4000
8000
12000
16000
0 5 10 15 20
Curvature (rad/m)x10-3
Mom
ent (
kN-m
)
PR02-PR12, PS02-PS10, PM02-PM07
0
4000
8000
12000
16000
0 5 10 15 20
Curvature (rad/m)x10-3
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
Dia= 900mm10-25mmφ rod
0
200
400
600
800
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
Dia= 900mm10-20mmφ rod
0
200
400
600
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
Dia= 610mm10-25mmφ rod
0
200
400
600
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
28
Dia= 610mm10-16mmφ rod
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
)
PS03 & PS04SECTION 1200x6760
STEEL 54-T250
2000
4000
6000
8000
10000
12000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PS05 & PS09SECTION 1200x6760
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
)
PS 07SECTION 1200x6760
STEEL 80-T250
3000
6000
9000
12000
15000
18000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PS 08SECTION 1200x6760
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
)
PR03SECTION 1300x5180
STEEL 35-T250
1500
3000
4500
6000
7500
9000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PR11SECTION 1300x8110
STEEL 58-T250
3000
6000
9000
12000
15000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PR12SECTION 1300x8250
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
PS02 & PS10SECTION 1200x5300
STEEL 44-T250
2000
4000
6000
8000
10000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PS03 & PS04SECTION 1200x5800
STEEL 43-T250
2000
4000
6000
8000
10000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PS05 & PS09SECTION 1200x4850
STEEL 42-T250
2000
4000
6000
8000
10000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PS 06SECTION 1200x6000
STEEL 40-T250
2000
4000
6000
8000
10000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PS 07SECTION 1200x6760
STEEL 42-T250
2000
4000
6000
8000
10000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PS 08SECTION 1200x6760
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
)
PR02SECTION 1300x4060
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
)
PR 11SECTION 1300x5872
STEEL 51-T250
3000
6000
9000
12000
15000
0 4 8 12 16 20Curvature x10-3 (rad/m)
Mom
ent (
kN-m
)
PR 12SECTION 1300x5900
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)
Ultimate moment (kN-m)
Yield curvature (rad/km)
Ultimate curvature (rad/km)
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)
Ultimate moment (kN-m)
Yield curvature (rad/km)
Ultimate curvature (rad/km)
Pier ID
Yield moment(kN-m)
Ultimate moment (kN-m)
Yield curvature (rad/km)
Ultimate curvature (rad/km)
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
Elastic perfectly plastichorizontal spring
Inelastic beam element
Rigid pile cap
Elastic perfectly plastichorizontal spring
Inelastic beam element
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
4.6 RESULTS AND DISCUSSIONS
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
0 20 40 60 80 100Displacement (mm)
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
0 30 60 90 120 150Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P06-P10, P13 & P140
2000
4000
6000
8000
10000
12000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P11 & P120
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P06-P10 & P13, P140
2000
4000
6000
8000
10000
12000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P11, P120
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (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
)
PM 020
300
600
900
1200
1500
1800
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PM 030
400
800
1200
1600
2000
2400
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PM 040
300
600
900
1200
1500
1800
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PM 050
300
600
900
1200
1500
1800
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PM 060
300
600
900
1200
1500
1800
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PM 070
300
600
900
1200
1500
1800
2100
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PR 020
400
800
1200
1600
2000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PR 030
400
800
1200
1600
2000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PR 040
300
600
900
1200
1500
1800
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
56
PR 050
500
1000
1500
2000
2500
3000
3500
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)PR 06
0
500
1000
1500
2000
2500
3000
3500
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PR 080
500
1000
1500
2000
2500
3000
3500
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PR 090
500
1000
1500
2000
2500
3000
3500
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PR 100
500
1000
1500
2000
2500
3000
3500
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PR 110
5001000150020002500300035004000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PR 120
5001000150020002500300035004000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 020
500
1000
1500
2000
2500
3000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 030
500
1000
1500
2000
2500
3000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)PS 04
0
500
1000
1500
2000
2500
3000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 050
500
1000
1500
2000
2500
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 060
500
1000
1500
2000
2500
3000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 070
500
1000
1500
2000
2500
3000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 080
500
1000
1500
2000
2500
3000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 090
500
1000
1500
2000
2500
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 100
500
1000
1500
2000
2500
3000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PS 110
500
1000
1500
2000
2500
3000
0 25 50 75 100 125Displacement (mm)
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
0 20 40 60 80 100Displacement (mm)
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
0 20 40 60 80 100Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (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
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)
Ultimate displacement
(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
P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18
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
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.5.12: 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
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
(V/W
)
Shear (JRA) CapacityShear (AASHTO) Capacity
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
)
Shear (JRA) CapacityShear (AASHTO) Capacity
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
)
Shear (JRA) CapacityShear (AASHTO) Capacity
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
P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18Pier ID
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
P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18Pier ID
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
P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18Pier ID
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
Transverse direction Longitudinal direction Pier ID
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
Transverse direction Longitudinal direction Pier ID
Ultimate displacement
ductility
*Allowable displacement
ductility
Allowable displacement
ductility
Ultimate displacement
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
P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18Pier ID
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
P01 P02 P03 P04 P05 P06 P07 P08 P09 P10 P11 P12 P13 P14 P15 P16 P17 P18Pier ID
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 RESULTS AND DISCUSSIONS
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
Curvature (rad/m)x10-3
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
Mohakhali flyover piers
P010
4000
8000
12000
16000
20000
24000
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P040
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P050
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P060
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P070
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
PIer-080
2000400060008000
100001200014000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P090
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P100
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (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 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P130
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
Latr
eral
Loa
d (k
N)
P170
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P180
3000
6000
9000
12000
15000
18000
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)P10
0
3000
6000
9000
12000
15000
0 25 50 75 100 125Displacement (mm)
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.
Khilgaon flyover piers
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
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P06-P10 & P13 0
4000
8000
12000
16000
20000
24000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P11 & P120
4000
8000
12000
16000
20000
24000
0 25 50 75 100 125Displacement (mm)
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
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P06-P10, & P130
4000
8000
12000
16000
20000
24000
0 25 50 75 100 125Displacement (mm)
Late
ral L
oad
(kN
)
P11 & P120
4000
8000
12000
16000
20000
24000
0 25 50 75 100 125Displacement (mm)
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
Mohakhali flyover piers
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
52894 55178 57463 59747 62031 64316 66600Ultimate Moment (kN-m)
n i or e
i
Observed frequencyTheoretical frequency COV µ σ Mck
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)
Observed CDFTheoretical CDF
P11 and P12
0
1
2
3
4
5
6
7
92498 96533 100568 104603 108638 112673 116708Ultimate Moment (kN-m)
n i or e
i
Observed frequencyTheoretical frequency COV µ σ Mck
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
90000 99000 108000 117000Ultimate Moment (kN-m)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
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
79960 82660 85360 88060 90760 93460 96160Ultimate Moment (kN-m)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Mck
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
77000 82000 87000 92000Ultimate Moment (kN-m)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
P09, P10, P13
0
1
2
3
4
5
6
7
129620 134120 138620 143120 147620 152120 156620Ultimate Moment (kN-m)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Mck
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
125000 135000 145000 155000Ultimate Moment (kN-m)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
P11, P12
0
1
2
3
4
5
6
7
149310 155540 161770 168000 174230 180460 186690Ultimate Moment (kN-m)
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)
Observed CDFTheoretical CDF
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)
Observed CDFTheoretical CDF
97
P10
01
2345
67
5367 5631 5894 6157 6421 6684 6948Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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)
Observed CDFTheoretical CDF
P11
01
2345
67
10136 10647 11157 11667 12178 12688 13198Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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
9500 10500 11500 12500 13500Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
Fig.6.12: Statistical distribution test of Mohakhali flyover piers in longitudinal direction
P01
01
2345
67
18259 19121 19984 20846 21709 22571 23434Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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
16000 18000 20000 22000 24000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
P02
01
2345
67
13243 13836 14429 15022 15615 16208 16801Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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
12000 13000 14000 15000 16000 17000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
P03
0
1
2
3
4
5
6
10433 10905 11378 11850 12323 12795 13268Ultimate Lateral Load (kN)
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)
observed CDFTheoretical CDF
P04
01
2345
67
8770 9160 9550 9940 10330 10720 11110Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
8200 8700 9200 9700 10200 10700 11200Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
98
P05
01
2345
67
7729 8082 8435 8788 9140 9493 9846Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
7200 7900 8600 9300 10000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
P06
01
2345
67
7036 7364 7692 8019 8347 8675 9003Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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
6500 7000 7500 8000 8500 9000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
P07
01
2345
67
6570 6874 7178 7483 7787 8091 8395Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
6000 6500 7000 7500 8000 8500Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
P08
01
2345
67
6269 6553 6837 7120 7404 7687 7971Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
6000 6500 7000 7500 8000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
P09
0
1
2
3
4
5
6
10872 11366 11859 12352 12846 13339 13832Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
10000 11000 12000 13000 14000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
P10
01
2345
67
11775 12333 12892 13450 14009 14567 15126Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
11000 12000 13000 14000 15000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
99
P11
01
2345
67
12743 13298 13853 14408 14963 15518 16073Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
12000 13000 14000 15000 16000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
observed CDFTheoretical CDF
P12
01
2345
67
14031 14648 15266 15883 16500 17117 17734Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
13000 14000 15000 16000 17000 18000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
P13
0
1
2
3
4
5
6
10904 11399 11894 12389 12884 13379 13874Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
10000 11000 12000 13000 14000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
P14
01
2345
67
6812 7119 7426 7733 8041 8348 8655Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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
6500 6900 7300 7700 8100 8500 8900Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
P15
01
2345
67
7242 7576 7910 8243 8577 8911 9245Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
6800 7200 7600 8000 8400 8800 9200Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
P16
01
2345
67
8207 8579 8951 9323 9694 10066 10438Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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)
Observed CDFTheoretical CDF
100
P17
01
2345
67
11320 11856 12392 12927 13463 13999 14535Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
10500 11500 12500 13500 14500Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
P18
0
1
2
3
4
5
6
11996 12536 13077 13618 14158 14699 15240Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical f requency COV µ σ Pck
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
11200 12200 13200 14200 15200Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
Fig.6.13: Statistical distribution test of Mohakhali flyover piers in transverse direction
Khilgaon flyover piers
PML03, PML04, PML05
0
12
3
4
56
7
8
7016 7418 7820 8222 8624 9026 9428Ultimate Moment (kN-m)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Mck
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
6900 7400 7900 8400 8900 9400Ultimate Moment (kN-m)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML06,PML07, PML08, PML13
0
12
3
4
56
7
8
11268 11882 12497 13111 13725 14339 14953Ultimate Moment (kN-m)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Mck
7.16% 12632 905 11143
λ ξ
9.44 0.072
PML06,PML07, PML08, PML13
0.0
0.2
0.4
0.6
0.8
1.0
1.2
10500 11500 12500 13500 14500Ultimate Moment (kN-m)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML11, PML14
0
1
2
3
4
5
6
7
11764 12415 13067 13719 14371 15023 15675Ultimate Moment (kN-m)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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
11400 12400 13400 14400 15400Ultimate Moment (kN-m)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML12, PML15, PML16
0
12
3
4
56
7
8
7016 7418 7820 8222 8624 9026 9428Ultimate Moment (kN-m)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Mck
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
6900 7400 7900 8400 8900 9400Ultimate Moment (kN-m)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
101
Sayedabad and Rajarbagh pier
0
12
3
4
56
7
8
7016 7418 7820 8222 8624 9026 9428Ultimate Moment (kN-m)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Mck
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
6900 7400 7900 8400 8900 9400Ultimate Moment (kN-m)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PR03
0
1
2
3
4
5
6
1210 1270 1331 1392 1453 1514 1575Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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
1100 1200 1300 1400 1500 1600Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PR04
01
2345
67
1205 1266 1326 1387 1447 1508 1568Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.56% 1370 104 1199λ ξ
7.22 0.075
PR040.0
0.2
0.4
0.6
0.8
1.0
1.2
1100 1200 1300 1400 1500 1600Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PR05
01
2345
67
1194 1254 1314 1374 1434 1494 1554Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.53% 1357 102 1189λ ξ
7.21 0.075
PR050.0
0.2
0.4
0.6
0.8
1.0
1.2
1100 1200 1300 1400 1500 1600Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PR06
01
2345
67
1189 1249 1309 1368 1428 1488 1547Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.55% 1352 102 1184λ ξ
7.21 0.075
PR060.0
0.2
0.4
0.6
0.8
1.0
1.2
1100 1200 1300 1400 1500 1600Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PR07
01
2345
67
1225 1287 1348 1410 1472 1534 1595Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.60% 1394 106 1219λ ξ
7.24 0.076
PR070.0
0.2
0.4
0.6
0.8
1.0
1.2
1150 1250 1350 1450 1550 1650Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
102
PR08
01
2345
67
1220 1281 1343 1404 1466 1527 1588Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.55% 1387 105 1215λ ξ
7.23 0.075
PR080.0
0.2
0.4
0.6
0.8
1.0
1.2
1150 1250 1350 1450 1550 1650Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PR09
01
2345
67
1262 1325 1389 1453 1517 1581 1645Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.54% 1432 108 1254λ ξ
7.26 0.075
PR090.0
0.2
0.4
0.6
0.8
1.0
1.2
1200 1300 1400 1500 1600 1700Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PR10
01
2345
67
1358 1432 1506 1580 1654 1728 1802Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
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 )
Observed CDFTheoretical CDF
PR11
01
2345
67
1368 1438 1508 1578 1648 1718 1787Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.56% 1552 117 1359λ ξ
7.34 0.075
PR110.0
0.2
0.4
0.6
0.8
1.0
1.2
1300 1400 1500 1600 1700 1800Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PR12
01
2345
67
1465 1541 1617 1692 1768 1843 1919Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.51% 1662 125 1457λ ξ
7.41 0.075
PR120.0
0.2
0.4
0.6
0.8
1.0
1.2
1350 1450 1550 1650 1750 1850 1950Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PM02
01
2345
67
1200 1260 1319 1378 1438 1497 1556Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.50% 1361 102 1193λ ξ
7.41 0.075
PM020.0
0.2
0.4
0.6
0.8
1.0
1.2
1100 1200 1300 1400 1500 1600Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
103
PM03
01
2345
67
1212 1273 1334 1395 1456 1517 1578Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.55% 1378 104 1207λ ξ
7.23 0.075
PM030.0
0.2
0.4
0.6
0.8
1.0
1.2
1150 1250 1350 1450 1550Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PM04
01
2345
67
1233 1295 1357 1419 1481 1543 1605Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.55% 1401 106 1227λ ξ
7.24 0.075
PM040.0
0.2
0.4
0.6
0.8
1.0
1.2
1150 1250 1350 1450 1550 1650Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PM05
01
2345
67
1251 1314 1377 1440 1503 1567 1630Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.48% 1423 106 1248λ ξ
7.26 0.075
PM050.0
0.2
0.4
0.6
0.8
1.0
1.2
1150 1250 1350 1450 1550 1650Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PM06
012345678
1266 1335 1404 1473 1542 1611 1680Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.53% 1438 108 1260λ ξ
7.27 0.075
PM060.0
0.2
0.4
0.6
0.8
1.0
1.2
1200 1300 1400 1500 1600 1700Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PM07
0
1
2
3
4
5
6
1370 1436 1501 1567 1632 1697 1763Ultimate Lateral Load (kN)
n i o
r ei
Observed frequencyTheoretical frequency COV µ σ Pck
7.55% 1561 118 1367λ ξ
7.35 0.075
PM070.0
0.2
0.4
0.6
0.8
1.0
1.2
1300 1400 1500 1600 1700 1800Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML03
0
1
2
3
4
5
6
7
1275 1340 1404 1469 1533 1598 1662Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency COV µ σ Pck
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
1200 1300 1400 1500 1600 1700Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
104
PML04
0
1
2
3
4
5
6
7
1191 1251 1310 1370 1430 1490 1549Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency COV µ σ Pck
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
1100 1200 1300 1400 1500 1600Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML05
0
1
2
3
4
5
6
7
1004 1053 1102 1151 1199 1248 1297Ultimate Lateral Load (kN)
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
950 1050 1150 1250 1350Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML06
0
1
2
3
4
5
6
7
900 947 995 1043 1090 1138 1186Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
950 1050 1150 1250 1350Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML07
0
1
2
3
4
5
6
847 887 927 967 1007 1047 1087Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
820 920 1020 1120Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML08
01
23
45
67
8
815 861 907 953 999 1046 1092Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
760 860 960 1060Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML11
0
1
2
3
4
5
6
1201 1258 1315 1373 1430 1487 1544Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
1160 1260 1360 1460 1560Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
105
PML12
0
1
2
3
4
5
6
7
780 815 851 887 923 958 994Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
750 800 850 900 950 1000Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML13
0
1
2
3
4
5
6
7
822 864 906 948 990 1032 1074Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
780 880 980 1080Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML14
0
1
2
3
4
5
6
7
1395 1463 1531 1598 1666 1734 1802Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
1300 1400 1500 1600 1700 1800Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML15
0
1
2
3
4
5
6
7
995 1043 1092 1140 1189 1237 1286Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
950 1050 1150 1250 1350Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
PML16
0
1
2
3
4
5
6
1104 1158 1213 1268 1322 1377 1432Ultimate Lateral Load (kN)
n i or e
i
Observed frequencyTheoretical frequency
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
1050 1150 1250 1350 1450Ultimate Lateral load (kN)
F E(e
m) o
r Sn(
e m)
Observed CDFTheoretical CDF
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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113
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.
114
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).
115
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.
116
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