i PUSHOVER ANALYSIS OF BRIDGE BENT – OVERVIEW AND CASE STUDY Credit Seminar Report by Palak Jaykumar Shukla (Roll Number:P10ST516) Under the Supervision of Prof. C. D. Modhera DEPARTMENT OF APPLIED MECHANICS SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY SURAT-395007 GUJARAT (INDIA)
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i
PUSHOVER ANALYSIS OF BRIDGE BENT –
OVERVIEW AND CASE STUDY
Credit Seminar Report
by
Palak Jaykumar Shukla
(Roll Number:P10ST516)
Under the Supervision of
Prof. C. D. Modhera
DEPARTMENT OF APPLIED MECHANICS
SARDAR VALLABHBHAI NATIONAL INSTITUTE OF
TECHNOLOGY
SURAT-395007
GUJARAT (INDIA)
ii
Acceptance Certificate
APPLIED MECHANICS DEPARTMENT
SARDAR VALLABHBHAI NATIONAL INSTITUTE OF TECHNOLOGY
The credit seminar report titled “Pushover analysis of bridge bent- overview and case
study” submitted by Palak Jaykumar Shukla (Roll No. P10ST516) may be accepted for
evaluation.
Supervisor:
Prof. C. D. Modhera
Date:7-10-11
Place: SVNIT, Surat.
iii
Acknowledgement
I express my indebtness to Prof. C. D. Modhera who first motivated me in choosing this
interesting topic and then devoted his time and helped me at every stage during the
preparation of this credit seminar report. I thank him for all his precious time that he has spent
with me during the course of preparation of this report and the critical suggestions that he
made for its improvement. I also thank Prof. M .K. Desai for his help and encouragement
during preparation of this report.
Date : 7-10-11
Palak Jaykumar Shukla
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Abstract
The literatures are available on the seismic evaluation procedures of multi-storeyed
buildings using nonlinear static (pushover) analysis. There is no much effort available in
literature for seismic evaluation of existing bridges although bridge is very important structure
in any country. There are presently no comprehensive guidelines to assist the practicing
structural engineer to evaluate existing bridges and suggest design and retrofit schemes. In
order to address this problem, the aims of present study was to carry out seismic evaluation
case study for Rail/road Bridge bent using nonlinear static (pushover) analysis.
The first chapter focuses on the introduction of the subject, where as second chapter
deals with the literature review about pushover procedures, its applications, limitations and
some alternative methods.
The third chapter includes statement of problem for case study. The modelling of
bridge bent is done with software SAP2000 to perform pushover analysis using SAP2000.
In forth chapter the conclusions are drawn from results of analysis are mentioned and
also indicate future scope of study.
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CONTENTS
Acceptance Certificate ii
Acknowledgement iii
Abstract iv
Contents v
List of figures vi
List of tables vi
Chapter 1 Introduction 1
1.1 GENERAL 1
Chapter 2 Literature Review 3
2.1 Definition of the Non-Linear Static Procedure (Pushover Analysis) - FEMA 273 3
2.2 Performing the Non – Linear Static Procedure (Pushover Analysis) 4 2.3 Use of Pushover Results 10
Figure 2-5 Bilinear Relation of Base Shear vs. Roof Displacement Plot ................................. 9 Figure 2-6 Schematic representation of Capacity Spectrum Method (ATC 40) ....................... 9
Figure 3-1 Geometry of problem selected............................................................................. 14 Figure 4-1 Basic SAP2000 model without pushover data ..................................................... 15
Figure 4-2 Frame Hinge Property ......................................................................................... 16 Figure 4-3 Assign pushover hinge ..................................................................................... 16
Figure 4-4 Pushover load case data ...................................................................................... 17 Figure 5-1 Pushover Curve................................................................................................... 18
2.2 Performing the Non – Linear Static Procedure (Pushover Analysis)
The steps in performing the Non – Linear Static Procedure or Pushover Analysis are:
1) Determine the gravity loading and the vertical distribution of the lateral loads.
2) Determine the desired Building Performance Level.
3) Calculate the Seismic Hazard.
4) Compute the maximum expected displacement or Target Displacement, δt.
Each of these steps are described in the sections following.
1) Determine the Vertical Distribution of the Lateral Loads
In addition to the gravity loads, the first thing that can be determined is the vertical
distribution of the lateral loads. The gravity loads to be used in the Pushover Analysis are
dDcalculated by equation [1], while the vertical distribution of lateral loads is given by
the FEMA 273 Cvx loading profile reproduced as equation [2].
[1]
Where, QG is equal to the total gravity force, QD is equal to the total dead load effect, QL
is equal to the effective live load effect, defined as 25% of the unreduced live load, and
QS is equal to 70% of the full design snow load except where the design snow load is less
than thirty pounds per square foot in which case it is equal to 0.0.
[2]
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The Cvx coefficient represents the lateral load multiplication factor to be applied at floor level
x, wx represents the fraction of the total structural weight allocated to floor level x, hx is the
height of floor level x above the base, and the summation in the denominator is the sum of
these values over the total number of floors in the structure, n. The parameter k varies with
the structural fundamental period, T. k is 1.0 for T less than or equal to 0.5 seconds and 2.0
for T greater than or equal to 2.5 seconds. For shorter, stiffer structures, the fundamental
period will be small and the variation of the lateral loading over the height of the building will
approach the linear distribution for a k value equal to 1.0. For taller, more flexible structures,
the fundamental period will be greater and the variation of the lateral loading over the height
of the structure will approach the non – linear distribution for k equal to 2.0. The implication
of this is that for stiffer structures the higher mode response of the structure will be less
significant and the lateral loading can enforce purely first mode response. As the structure
becomes more flexible however, the higher mode effects become much more important and
the k value attempts to account for this by adjusting the lateral load distribution
2) Building Performance Level Determination
The next thing that may be determined is the Building Performance Level. The Building
Performance Level is the desired condition of the building after the design earthquake decided
upon by the owner, architect, and structural engineer, and is a combination of the Structural
Performance Level and the Non–Structural Performance Level. The Structural Performance
Level is defined as the post – event conditions of the structural building components. This is
divided into three levels and two ranges. The levels are, S – 1: Immediate Occupancy, S – 3:
Life Safety, and S – 5: Collapse Prevention, are shown in Table 2-1.
Table 2-1 Determination of Building Performance Level
Structural Level Non Structural Level Operational Immediate
Occupancy Life Safety Hazards
Reduced Damage Not Limited
N-A N-B N-C N-D N-E Immediate Occupancy S-1 1-A 1-B Range Between S-1 & S-3 S-2 Life Safety S-3 3-C Range Between S-3 & S-5 S-4 Collapse Prevention S-5 5-E
The owner, architect, and structural engineer can now decide what Building Performance
Level they want their building to achieve after a range of ground shakings which are expected
to occur at a given design location (Table 2-2). The values K and P shown in bold in Table 2-
2 correspond to the performance one achieves when designing by the Uniform Building Code
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(UBC). This corresponds to Life Safety after a 10% probability of exceedance in 50 year
event and Collapse Prevention after a 2% probability of exceedance in 50 year event,
respectively.
Table 2-2 Building Performance Level for Given Seismic Event
Seismic Event Building Performance Level 1-A 1-B 3-C 5-E
50% /50 years A B C D 20% /50 years E F G H 10% /50 years I J K L 2% /50 years M N O P
3) Calculation of the Seismic Hazard
An important parameter that must be determined for the Pushover Analysis is the Seismic
Hazard of a given location. The Seismic Hazard is a function of:
1) The Building Performance Level
2) The Mapped Acceleration Parameters (found from contour maps included with FEMA
273)
3) The Site Class Coefficients (which account for soil type)
4) The effective structural damping
5) The Fundamental Structural Period
the General Response Spectrum can be formulated for the design event being considered. The
General Response Spectrum is shown qualitatively in Figure 2-4.
Figure 2-4 General Response Spectrum
The General Response Spectrum is a function of the many site and design event specific
parameters which are related by a complicated system of equations. However, once it has
been developed, since it is a function only of site location parameters and the design event
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under consideration, it becomes a very useful tool as it describes the maximum acceleration a
structure, with a given fundamental period, must endure during the design event.
4) Calculation of the Target Displacement
There are two approaches to calculate target displacement:
a) Displacement Coefficient Method (DCM) of FEMA 356
The Target Displacement, i.e. the maximum displacement the structure is expected to undergo
during the design event, can now be obtained. The target displacement is calculated from the
following equation:
[3]
Where the value C0 is a modification factor that relates spectral displacement and
likely building roof displacement. Values for C0 are tabulated in FEMA 273 as a function of
the total number of stories of the structure.C1 is a modification factor which relates expected
maximum inelastic displacements to displacements calculated for linear elastic response.
Values for C1 are obtained from:
[4]
[5]
Te is the effective fundamental period of the structure and is defined as given in equation
[11]. To is the characteristic period of the response spectrum, defined as the period associated
with the transition from the constant acceleration segment of the spectrum to the constant
velocity segment of the spectrum and is calculated as
[6]
Where Bs and Bl are Damping Coefficients given in FEMA 273
SXS is the final design short period spectral response acceleration parameter, and
SX1is the final design spectral response acceleration parameter at a one second period, can be
determined from:
[7]
[8]
Where Fa ,Fv,Ss and Sl is tabulated in FEMA 273
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R is the ratio of elastic strength demand to calculated yield strength coefficient. Values for R
are obtained from:
[9]
Sa is the Response Spectrum Acceleration, in g’s, (where g must be in consistent units,
usually in/s2 ) at the effective fundamental period and damping ratio of the building in the
direction under consideration . Vy is the yield strength calculated using the results of the
Pushover Analysis, where the non – linear force – displacement curve of the building is
characterized by a bilinear relation as shown in figure 2-5. W is the total dead load and
anticipated live load, as calculated by equation 1. C0 is as defined above .
C2 is a modification factor that represents the effect of hysteresis shape on the
maximum displacement response of the structure. Values for C2 are tabulated in FEMA 273
and are a function of Building Performance Level, framing type, and the fundamental period
of the structure.
C3 is a modification factor to represent increased displacements due to dynamic P – ∆
effects. For buildings with positive post – yield stiffness, C3 shall be set equal to 1.0. For
buildings with negative post – yield stiffness, C3 shall be calculated from:
[10]
Values for R and Te are obtained from equations [9] and [11] respectively, and α is the ratio
of post – yield stiffness to effective elastic stiffness, where the non – linear force –
displacement relation is characterized by a bilinear relation as shown in figure 2-5.
The effective fundamental period of the structure in the direction under consideration, Te
may be calculated from:
[11]
Where T is the elastic fundamental period of the structure (in seconds), in the direction under
consideration, calculated by elastic dynamic analysis. Ki is the elastic lateral stiffness of the
building in the direction under consideration and is found from the initial stiffness of the non
– linear base shear vs. roof displacement curve as shown in Figure 2-5. Ke is the effective
lateral stiffness of the building in the direction under consideration and is defined as the slope
of the line which connects the point of intersection of the post – yield stiffness line with the
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horizontal line at the yield base shear value to zero, while intersecting the original base shear
vs. roof displacement curve at 60% of the yield base shear value. Ki and Ke are shown in
Figure 2-5.
b) Capacity Spectrum Method (ATC 40)
The basic assumption in Capacity Spectrum Method is also the same as the previous one. That
is, the maximum inelastic deformation of a nonlinear SDOF system can be approximated
from the maximum deformation of a linear elastic SDOF system with an equivalent period
and damping. This procedure uses the estimates of ductility to calculate effective period and
damping. This procedure uses the pushover curve in an acceleration-displacement response
spectrum (ADRS) format. This can be obtained through simple conversion using the dynamic
properties of the system. The pushover curve in an ADRS format is termed a ‘capacity
spectrum’ for the structure. The seismic ground motion is represented by a response spectrum
in the same ADRS format and it is termed as demand spectrum (Figure 2-6).
Figure 2-5 Bilinear Relation of Base Shear vs. Roof Displacement Plot
Figure 2-6 Schematic representation of Capacity Spectrum Method (ATC 40)
The equivalent period (Teq) is computed from the initial period of vibration (Ti) of the
nonlinear system and displacement ductility ratio (μ). Similarly, the equivalent damping ratio
(βeq) is computed from initial damping ratio (ATC 40 suggests an initial elastic viscous
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damping ratio of 0.05 for reinforced concrete building) and the displacement ductility ratio
(μ).
2.3 Use of Pushover Results
Pushover analysis has been the preferred method for seismic performance evaluation of
structures by the major rehabilitation guidelines and codes because it is conceptually and
computationally simple. Pushover analysis allows tracing the sequence of yielding and failure
on member and structural level as well as the progress of overall capacity curve of the
structure. The expectation from pushover analysis is to estimate critical response parameters
imposed on structural system and its components as close as possible to those predicted by
nonlinear dynamic analysis. Pushover analysis provides information on many response
characteristics that cannot be obtained from an elastic static or elastic dynamic analysis.
These are;
a) estimates of interstory drifts and its distribution along the height
b) determination of force demands on brittle members, such as axial force demands on
columns, moment demands on beam-column connections
c) determination of deformation demands for ductile members
d) identification of location of weak points in the structure (or potential failure modes)
e) consequences of strength deterioration of individual members on the behaviour of
structural system
f) identification of strength discontinuities in plan or elevation that will lead to changes
in dynamic characteristics in the inelastic range
g) verification of the completeness and adequacy of load path
h) Pushover analysis also exposes design weaknesses that may remain hidden in an
elastic analysis. These are story mechanisms, excessive deformation demands,
strength irregularities and overloads on potentially brittle members.
2.4 Limitations of Pushover analysis
Many publications have demonstrated that traditional pushover analysis can be an extremely
useful tool, if used with caution and acute engineering judgment, but it also exhibits
significant shortcomings and limitations, which are summarised below:
a) One important assumption behind pushover analysis is that the response of a MDOF
structure is directly related to an equivalent SDOF system. Although in several cases the
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response is dominated by the fundamental mode, this cannot be generalised. Moreover,
the shape of the fundamental mode itself may vary significantly in nonlinear structures
depending on the level of inelasticity and the location of damages.
b) Target displacement estimated from pushover analysis may be inaccurate for structures
where higher mode effects are significant. The method, as prescribed in FEMA 356,
ignores the contribution of the higher modes to the total response.
c) It is difficult to model three-dimensional and torsional effects. Pushover analysis is very
well established and has been extensively used with 2-D models However, little work has
been carried out for problems that apply specifically to asymmetric 3-D systems, with
stiffness or mass irregularities. It is not clear how to derive the load distributions and how
to calculate the target displacement for the different frames of an asymmetric building.
Moreover, there is no consensus regarding the application of the lateral force in one or
both horizontal directions for such buildings.
d) The progressive stiffness degradation that occurs during the cyclic nonlinear earthquake
loading of the structure is not considered in the present procedure. This degradation leads
to changes in the periods and the modal characteristics of the structure that affect the
loading attracted during earthquake ground motion.
e) Only horizontal earthquake load is considered in the current procedure. The vertical
component of the earthquake loading is ignored; this can be of importance in some cases.
There is no clear idea on how to combine pushover analysis with actions at every
nonlinear step that account for the vertical ground motion.
f) Structural capacity and seismic demand are considered independent in the current method.
This is incorrect, as the inelastic structural response is load-path dependent and the
structural capacity is always associated with the seismic demand.
2.5 Alternate Pushover analysis procedure
Pushover Analysis procedure, as explained in FEMA 356, is primarily meant for regular
buildings with dominant fundamental mode participation. There are many alternative
approaches of pushover analysis reported in the literature to make it applicable for different
categories of irregular buildings.
These comprise (i) Modal pushover analysis (Chopra and Goel, 2001),
(ii) Modified modal pushover analysis (Chopra et. al., 2004),