13

CHAPTER 2

LITERATURE REVIEW

2.1 GENERAL

When structures (buildings/bridges) are subjected to strong

earthquake ground motions, they exhibit inelastic behaviour, which cannot be

assessed using an elastic analysis. A nonlinear analysis to evaluate the seismic

performance of structures considering the post-elastic behaviour is needed, to

predict the vulnerability of the structures, and their survivability under severe

earthquake ground motions. Various simplified nonlinear analysis methods to

estimate the maximum inelastic displacement demand of structures are

proposed in literature. The nonlinear static analysis or pushover analysis and

nonlinear dynamic time-history analysis have been carried out in this study.

2.2 NONLINEAR STATIC ANALYSIS

The traditional approach employs the linear static analysis, in

which an elastic analysis is used to determine the lateral seismic forces, which

are then reduced to inelastic design force levels by the response modification

factor. Though the approach had several shortcomings, it was accepted

for its simplicity, and the lack of alternative practical approaches

(Imbsen et al 1996).

Displacement based methods are the most effective and attractive

methods of analysis for predicting the demand imposed by earthquakes on the

structures, as they emphasise on the graphical evaluation of seismic

14

performance. The advent of a performance based design has brought the

nonlinear static analysis to the forefront. Nonlinear static methods have

appeared in national resource documents, such as the report on “Seismic

evaluation and retrofit of concrete buildings” (ATC-40, 1996) and the

FEMA-273 NEHRP (National Earthquake Hazard Reduction Program)

guidelines for the seismic rehabilitation of buildings (FEMA 1997).

Krawinkler and Seneviratna (1998) conducted a detailed study that discusses

the pros and cons of the pushover analysis. The pushover analysis has been

seen in number of references (Kim and D’More 1999). This procedure shows

the behaviour of the structures after they begin to crack and yield in response

to realistic earthquake motions. According to Chandler and Mendis (2000),

the focus on displacement-based methods in finding the inelastic

displacement of a structure, helps in predicting the ultimate failure and

damage potential. This shift of attention, from the traditional elastic force

based methods of seismic analysis towards displacement-based methods,

addresses the uncertainties in the current codal practices in terms of the

seismic capacity of a structure and its seismic demand (Ahmed Ghobarah

2001). The methods of nonlinear static analysis, are the capacity spectrum

method, CSM (ATC-40 1996), the displacement coefficient method, DCM

(FEMA 273 1997), the secant method (COLA 1995), the N2 method

(Fajfar 1999, 2000) and the modal pushover analysis (Chopra and Goel 2001).

2.2.1 Capacity Spectrum Method (CSM)

The capacity spectrum method, a performance-based seismic

analysis technique was originally developed by Freeman et al (1975). It is a

graphical procedure for estimating the structure load-deformation

characteristics, and for predicting earthquake damage and structure

survivability. This method was recommended by the ATC-40 (1996) as a

displacement-based design and assessment tool for any structure. The

15

conceptual development of the CSM was explained in detail in the ATC-40

document in procedures A, B and C, which are presented in sections 8.2.2.1.1

to 8.2.2.1.3. The procedure compares the capacity of a structure (in the form

of a pushover curve) with the demand imposed by the earthquake on the

structure (in the form of response spectra). Freeman (2004) reviewed the

development of the CSM which has the added advantage of visualising the

relationship between the demand and capacity.

The capacity of a structure and the earthquake demand imposed

upon it are dependent. One source of mutual dependence is evident from the

capacity curve itself. The structure yields as the demand increases. The

stiffness of the structure decreases and its period lengthens. The conversion of

the capacity curve into an ADRS (Acceleration Displacement Response

Spectrum) format makes it easy to visualise. Since the seismic accelerations

depend on the period, the demand also changes as the structure yields. The

capacity curve converted to the ADRS format, is named the capacity spectrum

which is shown in Figure 2.1.

Initial structural period, Ti

Performance point

Spectral displacement, Sd

Reduced seismic demandCapacity spectrum

Equivalent period, Teq

Initial seismic demand ( 5% damping)(effective damping)

Increasing damping

Figure 2.1 Capacity-demand spectra

16

Another source of mutual dependence between the capacity anddemand is effective damping ( eff). As the structure yields in response to aseismic demand, it dissipates energy with hysteretic damping. Structures thathave large stable hysteresis loops during cyclic yielding dissipate more thanthose with pinched loops which are caused by the degradation of the strengthand stiffness. Since the energy that is dissipated need not be stored in thestructure, the damping has the effect of diminishing the displacement demand.The CSM characterises the seismic demand initially, using a 5% dampedelastic response spectrum (Figure 2.1). This spectrum is plotted in ADRSformat showing the spectral acceleration as a function of the spectraldisplacement. This format allows the demand spectrum to be "overlaid" onthe capacity spectrum for the structure. The intersection of the demand andcapacity spectrum, if located in the linear range of the capacity, would definethe actual displacement of the structure; however this is not normally the caseas most of the structures exhibit some inelastic behaviour.

2.2.1.1 Performance point or performance displacement

A bilinear representation of the capacity spectrum is needed toestimate the effective damping and appropriate reduction of the spectraldemand. The spectral acceleration points (api,dpi) on the capacity spectrumshown in Figure (2.2) are assumed to be the trial performance point todevelop the reduced demand response spectrum. Using the spectralacceleration and displacement from this point, the reduction factors that havethe effect of pulling the demand spectrum down from the 5% elasticspectrum, to account for the hysteretic energy dissipation associated with thespecific point, are calculated. If the reduced demand spectrum intersects thecapacity spectrum at or near the initial assumed point, then it is the solutionfor the unique performance point where the capacity equals the demand.Thus, in the capacity spectrum, the point at which both the curves interactwith the same effective damping, is the performance point which defines thedemand imposed on the structure.

17

The position of the performance point must satisfy two

relationships: a) the point must lie on the capacity spectrum curve, and b) the

point must lie on a spectral demand curve, reduced from the elastic, 5%

damped design spectrum, which represents the nonlinear demand at the same

structural displacement. Spectral reduction factors are given in terms of

effective damping ( eff) which is calculated based on the shape of the capacity

curve, the estimated displacement demand, and the resulting hysteresis loop.

2.2.1.2 Effective damping ( eff)

When a structure is subjected to an earthquake ground motion, it

would be driven into the inelastic range. The damping that occurs during this

period can be viewed as a combination of viscous damping that is inherent in

the structure and hysteretic damping. Hysteretic damping is related to the area

inside the loops that are formed when the earthquake force (base shear) is

plotted against the structure displacement. Hysteretic damping can be

represented as equivalent viscous damping using equations that are available

in the literature (Chopra 1995; ATC-40 1996).

The equivalent viscous damping, eq , associated with a maximum

displacement of dpi as shown in Figure 2.2, was estimated from the following

Equation 2.1.

05.0oeq (2.1)

where,

o - hysteretic damping represented as equivalent viscous

damping.

18

The viscous damping inherent in the structure is 5% (assumed to be

constant). According to Chopra (1995), the term o can be calculated using

Equation 2.2.

Capacity spectrum

Bilinear representationof capacity spectrum

Kinitial

Keffective

ay

api

ESO

dpi

ED

Spectral displacement, Sddy

Figure 2.2 Damping for spectral reduction

SO

Do E

E41

(2.2)

where,

ED = energy dissipated by the structure in a single cycle of

motion = the area enclosed by a single hysteresis loop.

19

Eso = maximum strain energy associated with that cycle of

motion = the area of the hatched triangle.

The physical significance of the terms ED and Eso in Equation 2.2 is

illustrated in Figure 2.2.

Referring to Figure 2.2

piypiyD adda4E (2.3)

/2daE pipiSO (2.4)

Thus,pipi

piypiyo da

adda63.7 (2.5)

Equivalent viscous damping, 5da

adda63.75

pipi

piypiy0eq (2.6)

The idealized hysteresis loop shown in Figure 2.2 is a reasonable

approximation for a ductile-detailed structure subjected to relatively short

duration ground shaking, i.e., cycles not enough to significantly degrade

elements, and with equivalent viscous damping less than approximately 30%.

For conditions other than these, the idealized hysteresis loops are imperfect

(reduced in area or pinched). To enable the simulation of imperfect hysteresis

loops, the concept of effective viscous damping using a damping modification

factor, , is used. The factor, is a measure of the extent to which the actual

building hysteresis is well represented by the parallelogram (Figure 2.2) either

initially, or after degradation. The factor depends on the structural behaviour

of the structure, which in turn depends on the quality of the seismic resisting

system and the duration of ground shaking. The ATC-40 document simulates

three categories of structural behaviour, Types A, B and C. The structural

20

behaviour Type A represents stable, reasonably full hysteresis loops, and is

assigned a value of 1.0, Type B is assigned a value of32 and represents

a moderate reduction of area. Type C represents poor hysteretic behaviour

with a substantial reduction of the loop area and is assigned a value of31 .

Effective viscous damping , eff is given in Equation 2.7.

5da

adda63.75

pipi

piypiy0eff (2.7)

When the structure is expected to undergo only a moderate damage

before collapse, a 5% damped response spectrum is recommended. But a

response spectrum for a damping level between 10% and 20% is expected for

structures which are able to accommodate significant inelastic response to

account for hysteretic damping (Mahaney et al 1993). The capacity spectrum

analysis method assumes, that the elastic response spectra can be used

together with the inelastic capacity curve of a structure to determine the

seismic response (Yu et al 1999; Nathalie et al 2010).

2.2.2 Displacement Coefficient Method (DCM)

Newmark and Hall (1982) and Miranda (2000) proposed

procedures, in which the displacement modification factors are applied to the

maximum deformation of an equivalent elastic single-degree-of-freedom

(SDOF) system, to estimate the maximum inelastic displacement demand of

the multi-degree-of-freedom (MDOF) system. In the FEMA-273 document,

the DCM is used to characterize the displacement demand. This method

primarily estimates the elastic displacement of an equivalent SDOF system

assuming initial linear properties and damping for the ground motion

21

excitation under consideration. In this method, the demand is represented by

reducing the elastic demand spectra by the correction factors 0 1 2 3C ,C ,C ,C

to the inelastic demand spectra (constant-ductility demand spectrum) which

are more accurate than the elastic spectra, with equivalent viscous damping

(Fajfar 1999). The steps followed in the displacement coefficient method are

as follows:

A bilinear representation of the capacity curve is constructed as

shown in Figure 2.3.

The post-elastic stiffness, Ks as shown in Figure 2.3 is found.

The effective elastic stiffness, Ke by drawing a secant line

passing through a point on the capacity curve corresponding to

the base shear 0.6Vb is found.

The effective fundamental period (Teq) is found from

Equation 2.8.

eKiK

iTeqT (2.8)

Where, Ti - elastic fundamental period and Ki – initial stiffness.

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Ki

Ke

Ks

ty

Time period, sTeq

Sa

(b) Elastic Response Spectrum

Capacity curve

Bilinear representationof capacity curveInitial structural period

Lateral displacement,

Response spectrum

(a) Capacity curve

Figure 2.3 Schematic representation of the displacement coefficient

method

An effective fundamental period (Teq) is generated from the elastic

fundamental period (Ti) by a graphical procedure. This equivalent period

represents the linear stiffness of the equivalent SDOF system. The peak

elastic spectral displacement corresponding to this period, is calculated

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directly from the response spectrum representing the seismic ground motion

under consideration (Figure 2.3), as given in Equation 2.9.

a2

2eq

d S4T

S (2.9)

The expected maximum target displacement ( t) under the selected

seismic ground motion is given in Equation 2.10.

d3210t SCCCC (2.10)

Substituting Sd from Equation 2.9, Equation 2.10 can be rewritten

as Equation 2.11.

a2

2eq

3210t S4T

CCCC (2.11)

where,

C0 - modification factor to relate the spectral displacement

and building roof displacement.

C1 - modification factor to relate the expected maximum

inelastic displacements to the displacements calculated

for the linear elastic response.

C1 0.1 for 0eq TT (2.12)

RTT1R1.0

eq

0

for 0eq TT (2.13)

C1 need not exceed 2.0 for 0.1Teq second

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T0 - a 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.

R - ratio of the inelastic strength demand to calculate the

yield strength coefficient as given in Equation 2.14.

0y

a

C1

WVgS

R (2.14)

where,

Sa - the response spectrum acceleration determined at the

effective fundamental period of the structure.

Vy - Yield strength calculated using the capacity curve, where

the capacity curve is characterised by a bilinear relation.

W - Total dead load and anticipated live load

C2 - modification factor to represent the effect of the hysteresis

shape on the maximum displacement response.

C3 - modification factor to represent the increased

displacements due to second order effects.

dS - Spectral displacement.

Chopra and Goel (1999) developed the capacity-demand-diagram

method, using the constant-ductility demand spectrum, instead of the elastic

design spectrum in the ATC-40. According to Chopra and Goel, the ATC-40

(1996) procedure significantly underestimates the deformation of inelastic

systems for a wide range of Tn and ductility (µ) values, compared to the value

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determined from the inelastic design spectrum, using three different Ry- µ- Tn

equations (Tn - natural period, Ry - yield strength reduction factor,

µ- ductility), all of which provided similar results.

2.2.3 Secant Method

The secant method of design is derived from the “Substitute

Structure” procedure, similar to a methodology developed by Sozen and

others (Shibata and Sozen 1976). The principal advantages of this method are

that, it accounts for three dimensional effects, including torsion and

multi-loading, and that it accounts for higher mode effects. The main

disadvantage is that it is more time consuming than any other nonlinear static

method.

2.2.4 N2 Method

The N2 method (N stands for nonlinear analysis and 2 for two

mathematical models) was developed at the University of Ljubljana

(Fajfar 2000). The basic idea came from model developed by Saiidi and

Sozen (1981). It combines the pushover analysis of a MDOF model with the

response spectrum analysis of an equivalent SDOF system. Following

Bertero’s (Bertero 1995) idea, this method is formulated in the

acceleration-displacement format, which enables the visual representation of

the capacity spectrum method developed by Freeman (1975), with the sound

basis of the inelastic demand spectra. The N2 method, in its new format, is in

fact a variant of the capacity spectrum method based on inelastic spectra. The

inelastic demand spectra are determined from a typical smooth elastic design

spectrum. The reduction factors, which relate the inelastic spectra to the basic

elastic spectrum, are consistent with the elastic spectrum. The lateral load

pattern in the pushover analysis is related to the assumed displacement shape.

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This feature leads to a transparent transformation from a MDOF system to an

equivalent SDOF system.

2.3 MODAL PUSHOVER ANALYSIS (MPA)

The pushover analysis has been widely used for analysing the

seismic behaviour of any structures. But this method is limited by the

assumption that the response of the structure is controlled by its fundamental

mode. Paret et al (1996) and Sasaki et al (1998) suggested a multi-modal

pushover procedure in which the higher mode effects were considered. The

procedure comprises several pushover analyses under forcing vectors

representing the various modes deemed to be excited in the dynamic response.

The modal pushover analysis (MPA), an extension of the pushover

analysis, proposed by Chopra and Goel (2001) had been developed to include

the higher-“mode” contributions to seismic demands. According to the MPA,

(Chopra and Goel 2001; Chopra and Goel 2002; Moghadam and Tso 2002;

Jan et al 2003; Kappos et al 2005) the pushover analysis is performed for each

mode independently, wherein invariant seismic load patterns are defined

according to the elastic modal forces. As higher modes may reveal the failure

mechanism that is not detected by the first mode, a better understanding of the

structural performance considering the effect of higher modes becomes

mandatory (Chintanapakdee and Chopra 2003). The Modal pushover curves

are then plotted and can be converted to SDOF capacity diagrams, using

modal conversion parameters based on the same shapes. Then the response

quantities are separately estimated for each individual mode, and then

superimposed using an appropriate modal combination rule. The effectiveness

and the resulting estimates of the demand for elastic and inelastic buildings

demonstrated by the MPA procedure, were much better than those derived

27

from the Federal Emergency Management Agency (FEMA) force

distributions over a wide range of responses (Goel and Chopra 2004).

2.4 PREVIOUS STUDIES ON THE ANALYSIS OF BRIDGES

Priestly et al (1996) reviewed the bridge damages caused by

earthquakes, and identified basic design deficiencies which were the direct

consequences of the elastic design philosophy. The design deficiencies

identified were i) an underestimation of seismic displacements due to the

usage of gross section member stiffness ii) wrong moment shape patterns

under combined gravity and low seismic force levels iii) mislocation of the

point of contra flexure and iv) negligence of the concepts of ductility and

capacity design in the elastic design process.

Regarding the global geometric modeling of the bridge

components, the geometric discretisation effort increases significantly from

the lumped parameter models (LPM) to the structural component models

(SCM) and on to the finite-element models (FEM). The LPM, SCM and FEM

are shown in Figure 2.4.

In the LPM, the mass, stiffness and damping are conveniently

lumped or concentrated at discrete locations. The elements are idealized to

represent the prototype bridge behaviour. In the SCM, the idealized structural

system is connected to resemble the geometry of the bridge prototype. The

superstructure is represented by a single line of multiple three-dimensional

frame elements (spine-type configuration) which passes through the centroid

of the superstructure, and it remains elastic for lateral loadings.

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X

Y

Z

Lumped Parameter Modeling (LPM) Structural Component Modeling (SCM)

Finite Element Modeling (FEM)

Figure 2.4 Geometric discretisation

In the FEM, the actual geometry domain of the bridge is discretized with a

large number of small elements.

Several analytical tools are available for assessing the seismic

vulnerability of the existing bridges, each of which incorporates different

assumptions and varies in the complexity of application. The following are

the literature reviews on the application of various analytic methods for

assessing the seismic behaviour of bridge structures.

Paret et al (1996) used approximate inelastic procedures for

estimating the critical mode of two 17-storey steel frame buildings, damaged

by the 1994 Northridge Earthquake. For the two frames, the pushover analysis

based only on the first mode load pattern was inadequate to identify the actual

29

damage. It was concluded that the multi-modal-pushover analysis (MMP) can

be useful for structures with a significant higher-order modal response to

identify the failure mechanisms. The multi-modal pushover analysis was used

for estimating the Modal Criticality Index (MCI). The MCI is a value which is

used to identify the critical vibration mode which causes the failure of the

structure. Mathematically, it is the ratio of the spectral acceleration (Sa) value

for the demand and the spectral acceleration (Sa) value for the capacity. The

capacity-demand spectra of the strong column building (Paret et al 1996) for

calculating the MCI, are shown in Figure 2.5.

Figure 2.5 Capacity-demand spectra of the strong column building

(Paret et al 1996)

The capacity curves of the building structure were generated

individually for mode#1, mode#2 and mode#3. The demand curve used was a

5% damped response spectrum, developed from the earthquake ground

motions recorded near the site. The structure’s capacity for each mode was

compared with the earthquake demand, using the capacity spectrum method.

Sasaki et al (1998) evaluated two steel frame buildings (17-storey

and 12-storey) using the multi-modal-pushover (MMP) procedure and

30

capacity spectrum method (CSM). The load patterns used were based on the

elastic mode shapes of buildings. It was found that, the MMP results match

the actual damage closer than the pushover procedures, as pushover

procedures does not account for the higher-order modal response.

Yu et al (1999) evaluated the seismic performance and survivability

of two bridges, namely the Moses Lake bridge and Mercer Slough bridge in

Washington, using the elastic analysis, inelastic pushover analysis, capacity

spectrum method and nonlinear time-history analysis. The results of the

analyses were used to evaluate the advantages, limitations, and ease of

application of each approach, for the seismic analysis. The force and

displacement demands of the Moses Lake bridge and Mercer Slough bridge

for three different seismic ground motions from historical earthquake records,

were determined. The earthquake records used were the 1940 El Centro

Earthquake record, the 1949 Olympia Earthquake record and the 1995 Kobe

Earthquake record. Under imposed ground motions the survivability of the

bridge structures was checked. Regarding the survivability of the structure the

pushover analysis and the nonlinear dynamic time-history analysis provided

the same conclusion.

Floren and Jamshid (2001) reviewed the developments in

performance-based design for buildings, and investigated the effects of this

design approach, speci cally as it is applied to bridges. The paper unified the

performance levels and developed a performance matrix for bridges, based on

the performance matrix established for buildings in “Vision 2000”. Various

types of designs, such as those based on strength, deformation, nonlinear

behaviour, and energy, which can be used to meet the speci ed performance

levels in the seismic design of highway bridges, was discussed. The seismic

performance levels proposed by the authors for bridges are shown in

Table 2.1.

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Table 2.1 Seismic performance levels

Sl.No.

Performance levels Description

1. Immediate Occupancy(IO)

Minimal damage occurs. Minor inelasticresponse may occur. Damage is restrictedto narrow flexural cracking in concrete andpermanent deformations are not apparent.

2. Life Safety

(LS)

Some structural damage occurs. Concretecracking, reinforcement yield, and minorspalling of the cover concrete is evidentdue to the inelastic response. Limiteddamage such that the structure can beessentially restored to its pre-earthquakecondition.

3. Collapse prevention(CP)

Significant damage occurs. Concretecracking, reinforcement yield, and majorspalling may require closure for repair.Permanent offsets may occur. Partial orcomplete replacement may be required.

Abeysingye et al (2002) determined the inelastic response of the

Greveniotikos bridge during a design-level earthquake using the nonlinear

pushover analysis. A three dimensional finite element model of the bridge

was used. Parametric studies on the foundation stiffness, P effect and

plastic hinge properties were carried out to evaluate the effects of different

assumptions made in structural modeling and analysis. Different foundation

stiffness did not result in a significant variation in the expected inelastic

displacement. The P effect during the structural deterioration was

substantially negligible in the bridge. While various properties of plastic

hinges and pier cross section were used, the difference in the global response

32

was observed, but this difference was lesser than the result obtained by

varying the foundation stiffness.

Cosmin et al (2003) investigated the collapse behaviour of a three

span prestressed reinforced concrete bridge of 115 meters length built in the

northeastern part of Portugal, over the Alva River, using the pushover

analysis. The behaviour of the bridge structure at all stages of loading, from

the initial application of loads up to and beyond the collapse condition, was

studied. An insight into the pushover methodology described in the ATC-40

document (1996), FEMA-273 (1997) and EC8 (2000), was also presented.

Symans et al (2003) evaluated the effectiveness of various

commercially available computer programs namely, SAP2000, and

GT-STRUDL, for performing practical displacement-based seismic analysis

of highway bridges. A secondary objective was to identify the fundamental

differences between force-based and displacement-based methods of analysis,

particularly as they apply to highway bridges. The experience gained by

utilizing the computer software revealed that some programs are well suited

to displacement-based analysis, both from the point of view of being efficient

and providing insight into the behavior of plastic hinges.

Itani and Liao (2003) employed the nonlinear modal pushover

procedure in the analysis of a three-dimensional (3D) nonlinear finite-element

model of the Dry Wash bridge, and the results were used as the baseline in the

parametric studies. The superstructure of the bridge consisted of a bridge deck

and a support system of bents. Due to the large in-plane rigidity, the

superstructure was assumed as a rigid body for lateral loadings. Short spanned

bridges are very stiff in superstructure and can be modeled with spine beam

elements that represent effective stiffness characterisation. The

three-dimensional nonlinear finite-element model of the Dry Wash bridge

with spine element is shown in Figure 2.6.

33

Figure 2.6 3D model of Dry Wash Bridge (Itani and Liao 2003)

The effective moment of inertia was based on the cracked section,

and the effective flexure stiffness (Priestley et al 1996) used, is shown in

Equation 2.6.

geff 0.5II (2.6)

where, Ieff = Effective moment of inertia

Ig = Gross moment of inertia

The global responses investigated include structural displacement

and ductility. Retrofitting methods included steel jacketing of the columns,

foundation, and abutment retrofit. The corresponding parameters representing

the structural elements included linear foundation springs, nonlinear abutment

springs, and various column-jacketing plans. The results were analysed by

conducting parametric study to evaluate the effects of different retrofit

schemes on the bridge global behaviour.

34

Kappos et al (2005) analyzed the Krystallopigi bridge - a twelve

span structure of 638m total length that crosses a valley in northern Greece

using the inelastic standard pushover analysis, the modal pushover analysis

(MPA) as well as the nonlinear time-history analysis. In the MPA, pushover

analysis was carried out separately for each significant mode, and the

contributions from the individual modes to calculate the response quantities

(displacements, drifts etc.) were combined, using an appropriate combination

rule (SRSS or CQC). The MPA provides a significantly improved estimate

with respect to the maximum displacement pattern, reasonably matching the

results of the more refined nonlinear time-history analysis, even for increasing

levels of earthquake loading that trigger an increased contribution of the

higher modes.

McDaniel (2006) assessed the seismic vulnerability of typical

pre-1975 Washington State Department of Transportation (WSDOT)

prestressed concrete multi-column bent bridges. Three pre-1975 WSDOT

bridges were modeled as spine models with nonlinear column elements and

expansion joints. Soil-structure-interaction was considered in the study. The

spine model developed by the author is shown in Figure 2.7.

Figure 2.7 Spine model (McDaniel 2006)

35

The effects of non-traditional retrofit schemes were evaluated in

finding the global response of the bridges. The vulnerability of

non-monolithic bridge decks and shear-dominated bridge columns in the

pre-1975 WSDOT prestressed concrete multi-column bent bridges as well as

the importance of including soil-structure-interaction, were highlighted in the

research.

Lupoi et al (2007) studied the applicability of the MPA proposed by

Chopra et al (2001) for the assessment of a highway viaduct built in the

sixties, with a total length equal to 420m, having 11 spans each of 33m and a

continuous reinforced concrete deck pinned over the piers. Differences

between the nodal displacements estimated by the MPA, and those by the

nonlinear time-history analysis were found to be in the order of 15%,

independently of the intensity level of the ground motion.

Cardone et al (2007) used the adaptive pushover analysis, referred

to as the “adaptive capacity spectrum”, for two numbers of simply supported

span viaducts in an Italian motorway network. A series of fragility curves,

which describe the seismic vulnerability of the bridge under a probabilistic

perspective was reported as the result.

Muljati and Warnitchai (2007) evaluated the inelastic seismic

response of multi-span concrete bridges, using the modal pushover analysis

(MPA). The performance of the study bridge using the MPA in a nonlinear

range, showed a similar tendency with the MPA in a linear range. The MPA

results provided an acceptable accuracy besides simplicity.

Shatarat et al (2008) evaluated the difference in the global response

of the bridge with two nonlinear static analysis methods (capacity spectrum

and displacement coefficient method)). The effectiveness of various nonlinear

software packages (GT-STRUDL version25, SAP2000 nonlinear version 7.0,

36

ADINA 800-node version) was evaluated. Among the software’s, SAP2000

provided better results with the advantage of less complexity in modeling and

analysis. The capacity spectrum method was considered advantageous over

the displacement coefficient method, as it gives the graphical representation

of the behaviour of the structure.

Fu and AlAyed (2008) aimed at studying the applicability of a

nonlinear static procedure, by implementing the displacement coefficient

method (DCM) in bridges. The accuracy and reliability of the method was

checked using the nonlinear time-history analysis. A three span continuous

bridge was analyzed for two levels of seismic intensities (design level and

maximum considered earthquake). The nonlinear static analysis gave

conservative results when compared to the nonlinear time history analysis at

the design Level, while it provided more conservative results at the maximum

considered earthquake level.

ElGawady et al (2009) investigated the seismic performance of a

reinforced concrete bridge with prestressed hollow core piles, using the

nonlinear static and dynamic analyses. A three dimensional spine model of

the bridge was developed using SAP2000, including modeling of the bridge

bearings, expansion joints, and soil-structural interaction. Due to the higher

mode effects, the results obtained from the nonlinear static analysis were

found to be incomparable with those from the nonlinear dynamic analysis.

Shatarat and Assaf (2009) determined the seismic vulnerability of a

multi-span-simply-supported prestressed bridge, in order to develop the

required retrofit measure. The seismic vulnerability of the bridge was

evaluated using two seismic evaluation methods, presented in the federal

highway administration (FHWA) seismic retrofitting manual for highway

bridges, namely, Method C and Method D2.

37

Figure 2.8 Spine model (Shatarat and Assaf 2009)

The idealized mathematical model of the bridge was created using SAP2000.

The superstructure is represented by a single line of multiple

three-dimensional frame elements (i.e., a spine-type configuration), which

passes through the centroid of the superstructure (Figure 2.8). Each of the

columns and the tie beams are represented by three-dimensional frame

elements, which pass through the geometric center of the section. The results

of the seismic analyses demonstrated that Method C and Method D2 vary

markedly in terms of the information they provide to the bridge designer,

regarding the vulnerability of the bridge columns.

Rahai et al (2010) evaluated the seismic performance of two

models of prestressed concrete bridges employing the capacity spectrum

method (CSM) and displacement coefficient method (DCM). The

displacement controlled pushover analysis was used to find the capacity of the

structure. The DCM (which is recommended for buildings) results were found

to be acceptable, and at the same time more conservative than the CSM

results.

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Paraskeva and Kappos (2009) suggested an improvement to the

MPA procedure, that the deformed shape of the structure responding

inelastically to the considered earthquake level is used in lieu of the elastic

mode shape. The procedure is then verified by applying it to the bridge of

100m long three-span Overpass Bridge. The bridges were assessed using the

response spectrum, the standard pushover analysis (SPA), the MPA and the

nonlinear response history analysis for spectrum-compatible motions, and

they concluded that the MPA provides a good estimate of the maximum

inelastic deck displacement for several earthquake intensities, while the SPA

could not well predict the inelastic deck displacements of bridges, wherever

the contribution of the first mode to the response of the bridge was relatively

low.

Moni and Alam (2010) considered several retrofitting provisions on

three column reinforced concrete bridge bent in Canada which was designed

before 1965 with inadequate seismic detailing. As the bridge bent designed

only for gravity load failed to meet the seismic standards, several retrofitting

techniques such as steel jacketing, CFRP jacketing and steel bracing were

considered to improve the seismic performance. The nonlinear pushover

analysis was conducted for the original and retrofitted frames. An artificial

ground motion record was used to evaluate the dynamic response of these

structures. The seismic demand/capacity ratio, drift ratio, ductility has been

estimated. The best retrofitting technique has been proposed for such

multi-column bridge bents designed only for gravity load.

Ryan and Richins (2011) conducted a study on a three-span,

pre-stressed concrete girder bridge that crosses the Legacy Highway in

Farmington, Utah. The existing legacy bridge, which was designed as a

standard bridge for a 2500-year return period earthquake, was evaluated as an

essential bridge for a 1000-year return period earthquake. Subsequently, the

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bridge was redesigned and evaluated as a seismically isolated bridge. The

non-linear finite element analysis program SAP2000 was used, to evaluate

both the demand and capacity of the bridge structure. A linear spine model of

the bridge was developed for the demand analysis to determine the demands

on the existing structure, while a nonlinear model of the individual bents was

developed for the pushover analysis and capacity determination. Inspection

and maintenance practices for seismically isolated bridges were discussed.

2.5 INFERENCE FROM THE LITERATURE

The accuracy of the analytical results depends highly on the

element chosen for modeling.

A simple spine model is sufficient to represent the super

structure behaviour of straight medium length bridges.

The displacement-controlled nonlinear static analysis (the

pushover analysis) can predict the structural behaviour closer to

the experimental results.

The capacity spectrum method is a tool to predict the seismic

response of any structure.

The nonlinear analysis can be performed using standard

software packages.

The contribution of higher modes has to be considered in the

analysis to evaluate the exact behaviour of the structure for

seismic loading. The modal pushover analysis, wherein the

number of modes which contribute to 90% mass participation,

has to be performed on the structure to evaluate the criticality of

the structure.

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The nonlinear time-history analysis and the pushover analysis

could provide the same conclusions regarding the survivability

of the bridge structure. But processing and evaluating the output

from a nonlinear time history analysis require considerable

effort.