G D ( )DM EDP G ( )EDP IM I - Department of Civil Engineering, METUcourses.ce.metu.edu.tr/ce7013/wp-content/uploads/sites/35/2015/05… · Tesfamariam—Handbook of seismic risk analysis

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Tesfamariam—Handbook of seismic risk analysis and management of civil infrastructure systems

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© Woodhead Publishing Limited, 2013

536

20Using a performance-based earthquake

engineering (PBEE) approach to estimate structural performance targets for bridges

Z . G Ü L E R C E , Middle East Technical University, Turkey

DOI: 10.1533/9780857098986.4.?

Abstract: Performance-based seismic evaluation aims to provide information on expected seismic performance of structures in measurable and meaningful terms for decision makers (performance objectives). The Pacific Earthquake Engineering Research (PEER) Center utilized the performance-based earthquake engineering (PBEE) framework by linking the earthquake scenarios, design ground motions, structural demand, and performance variables. PBEE procedure may be simplified by decoupling the evaluation of the design ground motion levels (probabilistic seismic hazard assessment, PSHA) and that of the structural demand due to design ground motions (probabilistic seismic demand assessment, PSDA). The objective of this study is to present the vector-valued implementation of PEER PBEE approach. Key points of PSDA methodology such as selection of engineering demand parameters along with the intensity measures, effects of ground motion record scaling, and construction of probabilistic seismic demand models, are discussed. Integration of vector-valued intensity measures into these models is explained in details. Results of a recent study investigating the effect of vertical ground motions on ordinary highway bridges was used to demonstrate the application of PEER PBEE framework for a particular class of structures.

Key words: Performance-based earthquake engineering (PBEE), probabilistic seismic demand models, engineering demand parameters, probabilistic seismic hazard assessment (PSHA), vertical ground motions.

20.1 Introduction

According to modern building codes (FEMA-273, 1997; FEMA-350, 2000), structures should be designed to ensure specific performance objectives under design ground motion levels. These performance objectives can be expressed in terms of annual frequency of exceeding a given level of struc-tural response (or structural demand parameter). Estimating the annual frequency of exceeding any structural demand parameter involves several steps, such as: computing the design ground motions according to the accepted hazard levels in the code, evaluating the structural response due

Tesfamariam_2687_c20_main.indd 536 1/31/2013 6:57:40 PM


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Using a performance-based earthquake engineering (PBEE) 537


to the design ground motions, and comparing the calculated structural response parameters with the performance objectives. The Pacific Earth-quake Engineering Research (PEER) Center utilized the performance-based earthquake engineering (PBEE) framework that covers the essential elements of this problem and encouraged the research efforts on each step towards the evaluation of structural performance objectives (Cornell and Krawinkler, 2000; Stewart et al., 2002). The PBEE framework components and the links among the earthquake scenarios, design ground motions, structural response, and performance variables are explained in detail in the following section.

Studies in the past decade (e.g. Luco, 2002; Baker and Cornell, 2003; Tothong and Cornell, 2006) demonstrated that the PBEE procedure may be simplified by decoupling the evaluation of the design ground motion levels (probabilistic seismic hazard assessment) and that of the structural response due to design ground motions (i.e. probabilistic seismic demand assessment, PSDA). The details of the PSDA methodology, especially dis-cussions on controversial issues, such as selecting proper ground motions and intensity measures, are presented in Section 20.3. Recent advances in the field call for the use of vector-valued intensity measures and vector-valued probabilistic seismic hazard assessment tools; therefore Section 20.4 is devoted to the vector-valued probabilistic seismic hazard/demand assess-ment procedure. An application of the PBEE framework for the evaluation of near-fault vertical ground motion effects on the ordinary highway bridges is provided in Section 20.5 to further explain the theoretical background given in the previous sections. Finally, the vital parts of the PBEE frame-work that requires further research and elaboration are discussed briefly in the last section.

20.2 Performance-basedseismicevaluationframework(PEERapproach)

Performance-based seismic evaluation is a process that results in a realistic understanding of seismic risk due to future earthquakes. It aims to provide information on expected seismic performance of structures in measurable and meaningful terms for decision makers. To meet this objective, the PEER Center developed a PBEE framework that integrates a series of distinct and logically related parts of the problem. According to Stewart et al. (2002), PBEE includes the evaluation of the distributions of four fun-damental variables conditioned on the previous variable in the chain: (1) evaluating the probability of exceeding ground motion intensity measures (IMs) at a site given earthquake scenarios, (2) distribution of engineering demand parameters (EDPs) given a particular set of IMs,


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538 Handbook of seismic risk analysis


(3) distribution of damage measures (DMs; e.g. physical condition of a damaged element) given EDPs, and (4) evaluation of the probability of exceeding decision variables (DVs; e.g. human or collateral loss, post-earth-quake repair time, and other parameters of interest to an owner) given appropriate DMs. These four steps in the performance-based design meth-odology are linked through the theorem of total probability. The relations between key variables in the PEER-PBEE framework are illustrated in Fig. 20.1 and the PBEE integral is presented in Equation 20.1:

ν νDV G DV DM G DM EDP G EDP IM IM( ) = ( ) × ( ) × ( ) × ( )∫∫∫ d d [20.1]

where ν(DV) is the annual rate of exceeding the decision variable; G(DV|DM) is the probability of exceeding the decision variable given the damage measure; dG(DM|EDP) is the probability of exceeding the damage measure given the engineering demand parameter; dG(EDP|IM) is the probability of exceeding the engineering demand parameter given the intensity measure; and ν(IM) is the annual rate of exceeding the ground motion intensity measure (Cornell and Krawinkler, 2000; Krawinkler, 2002). Integrals in Equation 20.1 represent the variability in each key item, conditioned on the previous variable in the chain. In order to simplify the problem, the distributions of the variables are assumed to be indepen-dent of each other. For example, DM is a function of EDP only and knowl-edge of IM provides no additional information (Baker and Cornell, 2003). This assumption permits the user to treat each part of the integral separately.

PSDM

Ground motionintensity

Damagemeasure

Earthquakescenarios

Engineeringdemand parameter

FragilityPSHA

Definition ofseismic sources

• Definition of suitableground motionintensity measures

•

Nonlinear structuralanalysis

•

Performanceassessment

•• Seismic sourcecharacterization

• Selection of properground motionpredictionequations

20.1 PEER performance-based earthquake engineering framework (PSDM = probabilistic seismic demand model, PSHA = probabilistic seismic hazard analysis).



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20.2.1 Probabilistic seismic hazard analysis (PSHA) and ground motion intensity measures

The first part of the problem is the estimation of the annual rate of ex-ceeding IM. In other words, seismic hazard analysis represents a link be-tween earthquake scenarios and ground motion intensity in Fig. 20.1. Traditionally, probabilistic seismic hazard analysis (PSHA) due to a point source is carried out by evaluating the following equation (Cornell, 1968; McGuire, 2004):

ν IM z N f M f M R P IM z M R M RM R

RM

>( ) = ⋅ ( ) ( ) >( ) × ×∫∫min , , d d [20.2]

where R is the distance from the source to site; M is the earthquake mag-nitude; Nmin is the annual rate of earthquakes with magnitude greater than or equal to the minimum magnitude; fM(M) and fR(M, R) are the probability density functions for the magnitude and distance, and P(IM>z|M, R) is the probability of observing an IM greater than z for a given earthquake mag-nitude and distance. The IM is the quantification of the characteristics of a ground motion that are important to the structural response. PSHA requires the definition of seismic sources close to the specific site and characteriza-tion of these seismic sources by appropriate probability density functions and recurrence models. After defining a suite of earthquake scenarios, the range of ground motions for each earthquake scenario is estimated and the annual rate of each combination of earthquake scenario and ground motion is computed. The probability that the IM will exceed z, P(IM>z|M, R), is obtained from the ground motion prediction equation (GMPE) and includes an implicit integration over the ground motion variability. The probability that the IM will exceed z is given by:

P IM z M R f P IM z M R>( ) = ( ) × >( ) ×∫, , ,εε

ε ε εd [20.3]

where the epsilon (ε) is the number of standard deviations above or below the median, fε(ε) is the probability density function for the epsilon (given by the standard normal distribution) and P(IM>z|M, R, ε) is either 0 or 1. In this formulation, P(IM>z|M, R, ε) identifies earthquake scenarios and ground motion combinations that lead to IM greater than z. The hazard (Equation 20.2) can then be written as:

ν

ε ε εεε

IM z

N f M f M R f P IM z M R M RM R

RM

>( )= ⋅ ( ) ( ) ( ) >( ) × × ×∫∫∫min , , , d d d

[20.4]


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While this form is more complicated than Equation 20.1, it has the advan-tage of clearly showing that the hazard integral accounts for aleatory vari-ability in three main parameters of scenario earthquake: magnitude, distance, and epsilon (Gülerce and Abrahamson, 2010). Using Equation 20.4, the annual rate of exceeding any ground motion intensity measure may be calculated using the seismic source models and GMPEs suitable for the region.

20.2.2 Engineering demand parameter

The earthquake hazard is associated with direct consequences (damage to property or loss of function) and indirect consequences (such as loss of productivity or jobs) of earthquakes (McGuire, 2004). In order to predict the economic loss resulting from earthquake ground motions in a building or any other structure, it is necessary to predict the response of the structure when subjected to ground motions of different intensity levels (Baker and Cornell, 2003). To evaluate the seismic performance of structures, uncer-tainties in the ground motions and nonlinear structural responses need to be considered. The most direct way to make probabilistic estimates of earthquake damage is to express it as a function of M and R, as:

EDP N f M f M R P EDP y M R M Rm r

MR

( ) = ⋅ ( ) × ( ) × >( ) × ×∫∫min , , d d [20.5]

where P(EDP > y|M, R) is the probability of observing an EDP greater than y for a given earthquake magnitude and distance. PSHA procedures can be used to directly estimate earthquake damage using this form of the hazard equation. The concept is similar to the conventional PSHA, but the GMPE is replaced with a structure-specific response prediction equation. To develop a structure-specific response prediction model, the structure should be analyzed for a large number of ground motions. The results of nonlinear dynamic analysis can be used to model the distribution of struc-tural demand with magnitude and distance by regression analysis. The implicit assumptions in this method are (Baker and Cornell, 2003): (1) the functional form of the regression equation, (2) the lack of dependence of EDP on the source characteristics not contained in the vector of indepen-dent variables (e.g., rupture duration), and (3) the lack of dependence of EDP on the geometry of the fault relative to the site. The need for hundreds of dynamic analysis to obtain a reliable estimate for the structure- and response-specific prediction model and the complications involved in the modeling process called for a simplified procedure that treats the ground motion hazard and structural response independently. Cornell and co- workers (Baker and Cornell, 2003; Tothong and Cornell, 2006) proposed



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the probabilistic seismic demand model (PSDM) approach where the results of nonlinear dynamic analyses for a specific structure are used to evaluate behavior of important EDPs in terms of the IM levels. The main idea of these studies was to develop PSDMs for particular structures and to provide the annual frequency of exceeding a given structural engineering demand measure y conditioned on IM, as:

ν νEDP y IM f EDP IM IMEDP

IM

>( ) = ( ) × ( )∫ d [20.6]

where fEDP(EDP | IM) is the probabilistic seismic demand model for a particular EDP and IM. PSDMs represent the connection between IMs and EDPs as shown in Fig. 20.1 and provide information about the probability of exceeding the pre-determined critical levels of EDPs for a particular class of structures. These models may be used as risk-based design tools, since they present the variability in the structural demand parameters for specified ground motion intensities. Moreover, when coupled with PSHA, PSDMs can be used to compute structural demand hazard curves (Mackie and Stojadinovic, 2003). PSDMs may be incorporated into the hazard inte-gral to directly estimate the annual probability of exceeding a certain EDP (Gülerce and Abrahamson, 2010). For an EDP that depends on only a scalar ground motion IM, the hazard integral for the EDP can be written as:

νε

ε

εεEDP y

N f M f M R f

P EDP y EDP IM M R

M R

RM>( ) =⋅ ( ) ( ) ( )

> ( )[ ]

∫∫∫min ,

ˆ , , ,, σ εln d d dEDP M R( )

[20.7]

where ED̂P(IM(M, R, ε)) is the median EDP and σlnEDP is the standard deviation of ln(EDP) for a given IM (EDP is modeled as lognormal variate). This approach combines the site-specific ground motion hazard with the structural responses of interest from nonlinear dynamic analyses of the given structure. The final result of the hazard integral given in Equation 20.7 is a structural demand hazard curve representing the annual probabil-ity of exceeding a specified value of EDP.

20.2.3 Damage measure

The last step of the PEER-PBEE approach is to evaluate the distribution of DMs with respect to the EDP levels. In current practice and building codes, damage measures are typically not continuous, rather a set of discrete damage states (FEMA-273, 1997; Miranda and Aslani, 2003). To assess the extent of incurred damage, fragility functions are assigned for discrete damage states, which provide the probability of exceeding a damage state for a given EDP level (represented by the third chain in Fig. 20.1). With


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structural capacity information (fragility curves), the PSDA results can be used to compute the annual frequency of exceeding a specified damage state. Several methodologies for generating seismic fragility curves have been developed over the years, including empirical, elastic-spectral, non-linear static, and non-linear dynamic approaches (Nielson and DesRoches, 2007). Development of these component-specific and region-specific fragil-ity functions is a topic of current and future research.

20.3 Probabilisticseismicdemandanalysis(PSDA)

PSDA (Luco, 2002) involves a series of steps including: the selection of ground motion dataset for nonlinear analyses; modeling a class of structures by varying important structural parameters considered in design; selection of proper EDPs and IMs; and formulating the relationship between EDP and IM (i.e. PSDMs). Each one of these steps has been examined by various researchers in the past decade and substantial progress has been made on vital parts of the process, such as nonlinear dynamic analysis of structural models and selection of suitable IMs and EDPs. Some controversial issues like selection and scaling of ground motions for nonlinear dynamic analysis and functional shape of PSDMs are topics of ongoing discussions and require further research efforts.

The selection of the time histories in nonlinear dynamic analysis is criti-cal, since the spectral shape of the ground motion has a substantial effect on the structural demand due to nonlinear response of the soil and struc-ture. It is common practice to select the ground motion time histories based on seismological properties, such as magnitude and distance to the fault. Similar site conditions, style of faulting, and directivity effects may also be considered in the selection process. Additionally, the scale factor required to scale the time series to the design ground motion level may be consid-ered. In general, scale factors closer to unity are preferred and many ground motion experts recommend a limit on the amount of scaling applied (Watson-Lamprey and Abrahamson, 2006). The issues on ground motion selection and scaling for PSDA are closely related with IM selection process. Selection of the proper IM is critical in terms of reducing the uncertainty in prediction of the structural response, decreasing the number of required nonlinear dynamic analyses, and decreasing the effect of ground motion selection and scaling on the PSDM. An IM containing information on the ground motion characteristics (e.g. spectral shape) and the structure (e.g. fundamental period) is preferable, since it will reflect the main characteris-tics of seismological and structural parameters on the structural response effectively.

Many researchers focused on the selection process and proposed quan-titative measures for choosing the most appropriate IM for the structure



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among the candidate IMs. According to Luco and Cornell, (2007), the desired properties of an IM include efficiency, sufficiency, scaling robustness, and feasibility. An efficient IM is defined as one that results in a relatively small variability in the structural response given the level of IM; therefore, the variance of the PSDM is the quantitative measure of the efficiency. An IM is defined as efficient if the variance of the PSDM is small and using an efficient IM will reduce the number of nonlinear dynamic analyses. A suf-ficient IM is defined as one that brings negligible conditional dependency to the structural response from the previous chain of Equation 20.1. In other words, if an IM is sufficient, the EDP will be independent of the earthquake parameters (earthquake magnitude, distance, etc.). Sufficiency of an IM is desirable, because it reduces the complexity in the PSDA cal-culation as well as the record selection procedure (Luco and Cornell, 2007). If an insufficient IM is used and the selected records do not represent the hazard at the site, the performance-based seismic evaluation will be biased (Tothong and Cornell, 2006).

Scaling robustness is used to define the possibility of introducing bias to the structural demand by scaling the time histories used in nonlinear dynamic analyses. If an IM is robust, no statistically strong relationship exists between the structural responses and the scale factors used for the scaling of records. Finally, feasibility of the ground motion measure should be considered for selecting an appropriate IM. Since the annual rate of exceeding the IM is determined by PSHA, the IM should be computable in terms of scenario earthquake parameters. Hazard maps and hazard curves are available or easily computable for some IMs, but other IMs require more effort, or even structure-specific information for their deter-mination. An IM with higher efficiency may be less desirable on the basis of feasibility (Giovenale et al., 2004). Peak ground acceleration (PGA) and spectral acceleration (Sa(T1)) at the fundamental period (T1) of the struc-ture are the most commonly adopted IMs in the structural engineering applications (Shome and Cornell, 1999; Nielson and DesRoches, 2007; Padgett et al., 2008).

Following the selection of ground motions and IMs, sample models of the particular class of structures are generated by varying key design parame-ters (e.g. model dimensions, damping ratio, material strengths, etc.) within the allowable range of code requirements. The selected and scaled time histories are employed in nonlinear dynamic analyses of these sample models, and important response parameters are monitored throughout the analysis. Typically, peak values of responses are recorded and plotted versus the peak value of the IM for that ground motion for each analysis (Mackie and Stojadinovic, 2003). The EDPs are selected among the peak values of monitored structural response parameters; however the results of previous analysis or experiments on that class of structures may be used as a basis


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for selection. These parameters may be related to the global (e.g. drift ratio), intermediate (e.g. moments, deformations at the bearings, and column cur-vature ductility), or local (e.g. stresses) elements of the structure.

During the SAC project performed on steel moment frame buildings (FEMA, 2000), PSDMs were defined as lognormally distributed (Cornell et al., 2002). Therefore, the relation between EDPs and IMs can be pre-sented with a linear dependence in log–log space. The recommended func-tional form of the PSDMs is:

ln lnEDP c IM c( ) = × ( ) + +1 2 σPSDM [20.8]

where c1 and c2 are the regression coefficients and σPSDM is the standard deviation of the model. After EDPs and IMs are selected, regression analy-ses are performed for each EDP–IM pair to estimate the regression coef-ficients c1 and c2. Better predicting models (more efficient models) are selected according to the dispersion measures for the fit (Mackie and Sto-jadinovic, 2003). Recent studies (Nielson and DesRoches, 2007; Padgett et al., 2008) found that the PGA is the most appropriate IM for highway bridges and the PSDMs coupling several EDPs with the horizontal PGA were presented.

20.4 Vector-valuedprobabilisticseismichazardassessment(PHSA)

In addition to the common IMs (PGA and Sa(T1)) more complex param-eters like vector-based IMs have been examined by past researchers (Shome and Cornell, 1999; Bazurro and Cornell, 2002). According to Baker and Cornell (2005), using Sa only as the IM is not adequate to characterize the ground motion intensity since there is a large variability in the structural response due to different ground motions with the same Sa at a particular period. The reason for that variability is the spectral shape; using Sa(T1) as IM does not take Sa values at other periods into account. Sa at other periods has an impact on inelastic structures (due to period lengthening) and multi-degree-of-freedom systems with multiple modal periods. Baker and Cornell (2005) proposed ε as a proxy for spectral shape and recommended the use of vector-based IM, IM(Sa,ε), coupling Sa(T1) at the fundamental period of the structure with ε. If an EDP depends on two different IMs (IM1 and IM2), the rate of exceeding a specific value of EDP can be computed using the conditional distribution of EDP given IM1 and IM2. Therefore Equation 20.6 becomes:

ν

ν

EDP y IM IM f EDP y IM IM

f IM IM IM

EDP

IM

IM

>( ) = >( )

× ( ) ( )

∫1 2 1 2

2

1 1 2 2

, ,

d

[20.9]



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where fEDP(EDP > y | IM1, IM2) is the PSDM for a particular EDP and two IMs, fIM1(IM1|IM2) is the probability density function for the first IM condi-tioned on the other and dν(IM2) is the annual rate of exceeding a specified value of IM2 (within some small increment). For EDPs depending on a vector of ground motion parameters, a vector-valued PSHA is required (Bazzurro and Cornell, 2002). The vector-valued seismic hazard integral for calculating the structural demand is (Gülerce and Abrahamson, 2010):

ν

ε ε εε εεε

EDP y

N f M f M R f fM R IM IM IMIM IM

IM

>( ) =

⋅ ( ) ( ) ( ) ( )∫min , 1 2

2

1 2 1

IIMRM

IM IM

EDP

P EDP y EDP IM M R IM M R

1

2

∫∫∫>( ( ) ( )[ ]

)ˆ

1 1 2

ln

, , , , , ,

d

ε εσ MM R IM IMd d d1 2ε ε

[20.10]

where εIM1 and εIM2 are the epsilons for IM1 and IM2, fεIM1 (εIM1) is the prob-ability density function for εIM1, and fεIM2 (εIM2|εIM1) is the probability density function for εIM2 conditioned on εIM2. The form in Equation 20.10 differs from the formulation given by Bazzurro and Cornell (2002) such that the double integral in Equation 20.10 is carried out over εIM1 and εIM2, rather than IM1 and IM2. The modified formulation clearly shows that the correla-tion of the variability of the intensity measures needs to be considered (Gülerce and Abrahamson, 2010). fεIM1(εIM1) is given by the standard normal distribution, whereas fεIM2(εIM2|εIM1) is conditioned on the value of εIM1 and depends on the correlation of εIM1 and εIM2. The covariance of εIM2 with respect to εIM1 is computed from the correlation of the normalized residuals from the GMPE regression analysis. The probability density function for εIM2 can be defined as a function of εIM1 as:

ε ρ ε σε εIM IMT T T IM IM2 2 1 1 2 1( ) = × =( ) ± [20.11]

where ρ is the correlation coefficient and σεIM2|εIM1 is the standard deviation

of the correlation. A vector-IM-based analysis facilitates the record selec-tion process, assuming that no dependence upon any other parameters is remained (Baker and Cornell, 2005). Using a vector-valued IM also decreases the variability in the PSDM and thus the number of required nonlinear analyses to develop the PSDM model. However, to perform vector-valued PSHA, the correlation of residuals of the ground motion model (or models) across the periods needs to be known. Only a few of the recent GMPEs provide these correlation coefficients to be used in vector-valued analysis (e.g., Abrahamson and Silva, 2008; Gülerce and Abraham-son, 2011). Another drawback of the vector-valued IMs is the inability of the standard PSHA software packages to run vector-valued calculations.


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20.5 Performance-basedseismicevaluationofordinaryhighwaybridges

During 2004–2008, a detailed analytical study on the seismic response of ordinary highway bridges was conducted at UC Davis, California. The primary objective of the study was to quantify the effects of near-fault verti-cal ground motions on the seismic response of single bent, two-span highway overpasses. Results from the study were published in a series of papers; Kunnath et al. (2008), Gülerce and Abrahamson (2010), and Gülerce et al. (2012), focusing mainly on the structural demand due to vertical ground motion component. The study followed the PEER-PBEE framework by developing PSDMs for the particular structures using scalar- and vector-valued IMs, and by performing many scalar- and vector-valued PSHA anal-yses to address the question of when the vertical component should be included in design. Results of that study will be used as an example to explain an application of the PBEE methodology. The flowchart of the major steps is provided in Fig. 20.2.

The initial step of the process was the selection of ground motion record-ings to be used in the nonlinear dynamic analyses of the bridge prototype. A subset of 56 near-fault recordings (rupture distance smaller than 50 km) having the strongest horizontal components (horizontal PGA is greater than or equal to 0.5 g) was selected from the PEER-NGA database (Chiou

Step 1: Selection ofground motion

recordings

Step 2: Modeling thebridge prototype andnonlinear dynamic

analyses

Step 3: Selection ofEDPs and IMs

Step 4: BuildingPDSMs

(scalar and vector-valued)

Step 5: Scalar andvector-valued PSHAincluding the PDSMs

Step 6: Structuraldemand hazard

curves

20.2 PBEE framework for evaluating the vertical ground motion effects on the structural response.



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et al., 2008). Since the primary objective of the study was to address the effect of vertical ground motions, more near-fault recordings with either large or small vertical components compared to the horizontal component were added to the dataset. The resulting set of 114 ground motions (228 horizontal and 114 vertical components) were used in the nonlinear dynamic and regression analyses (Step 1 in Fig. 20.2).

Kunnath et al. (2008) used a portion of the widening project of the Camino Del Norte Bridge located in California as a typical ordinary stan-dard highway bridge and representative of the class of structures. The prototype was a single bent bridge with two spans of 30.95 m and 30.52 m in length and many configurations were generated by changing the dimen-sion of the bridge model without violating the code specifications on allowed dimensional and balanced stiffness requirements. The nonlinear dynamic analyses were carried out in two stages (Gülerce et al., 2012): in Stage 1, only horizontal components of the motion were applied as a base case, while in Stage 2, both horizontal and vertical components were applied simultane-ously to monitor the increase of the response due to the inclusion of the vertical component (Step 2 in Fig. 20.2). Analysis results of the bridge con-figuration with horizontal first-mode fundamental periods of 0.27 and 0.46 seconds (in longitudinal and transverse directions, respectively) and vertical first-mode fundamental period of 0.12 seconds are presented in the follow-ing sections.

Results of the nonlinear dynamic analyses showed that some structural response parameters were amplified significantly, when the vertical accel-erations were incorporated, such as the axial force demand in the column and moment demands in the girder both at the mid-span and at the face of the bent cap (Kunnath et al., 2008). On the other hand, some commonly examined structural response parameters, such as lateral displacements of the deck, were not influenced significantly by the inclusion of vertical motions (Gülerce et al., 2012). The parameters that were affected signifi-cantly due to vertical accelerations were selected as the structural response measures to be used in the PBEE framework. Similar to the previous prac-tice (Mackie and Stojadinovic, 2003), peak values of the response history were chosen as EDPs (after normalizing with the dead load values to offer a rational basis for comparison of different bridge configurations). Type 1-EDPs were defined using positive peaks of the axial load in the column, moments at the mid-span and at the support as:

Type EDPDemand

DL1 1− =

( )max,Stage [20.12]

where (Demand)max, Stage1 is the positive extreme value of the demand versus time history at the base case (without the vertical accelerations) and DL is the corresponding dead load value as shown in Fig. 20.3(a).


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Type 2-EDPs were defined using the negative peaks of the moment demand time histories at the mid-span and at the support to account for the effects of negative bending due to vertical ground motions as:

Type EDPDemand DL

DL2 1− =

( ) −max,Stage [20.13]

where (Demand)min, Stage1 is the negative extreme of the demand versus time history at the base case (Gülerce et al., 2012; Fig. 20.3(b)). To define a pos-sible relation between the ground motion characteristics and the EDPs, characteristics of the ground motions which produced the largest adverse effects were studied systematically. As mentioned in the previous sections, Sa(T1) at the fundamental period of the structure along with PGA is the most commonly preferred IM due to its efficiency, sufficiency, and feasibility. Sa at the vertical, transverse and longitudinal periods of the bridge were selected as IMs for this study. The effects of ground motion selection and scaling, defined by the scaling robustness property of an IM, can be checked by evaluating the dependency of EDP on the scale factor. The ground motions used in this study were scaled to an example target spectrum (Abrahamson and Silva 1997 model prediction, M7, D5km) to examine this property using two different approaches; in the first approach, only one scale factor per set (a set is defined as two horizontal and vertical ground motion components) was used, while in the second approach, separate scale factors for different components of the ground motion record sets were used. The distribution of an example Type 2-EDP with respect to the scale factors is given in Fig. 20.4 for one-scale-factor case and in Fig. 20.5 for separate-scale-factor case, respectively.

–2

–1

0

1

2

3

4

–2

–1

0

1

2

3

4

0 10 20 30 40

Dem

and

/ d

ead

load

Time (s)0 10 20 30 40

Dem

and

/ d

ead

load

Time (s)

(a) (b)

Demandmax_Stage 1

Demandmin_Stage 1

Demandmax_Stage 2

Demandmin_Stage 2

20.3 Typical demand-time histories from (a) Stage 1 and (b) Stage 2 of nonlinear dynamic analyses.



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The scatter of the EDP in Fig. 20.4 indicates that there is large variability in the response for the records that have similar scale factors. Therefore, even if the scale factor for a record is near unity, that record may not give a good estimate of the average response. The variability of response is not sensitive to the scaling of the records, since the amount of scatter in the EDP does not change with the scale factor. By using separate scale factors

–3

–2

–1

0

1

0.1 1 10 100

ln E

DP

(T

ype

2)

Scale factor

20.4 Distribution of Type 2-EDP (in natural log units) vs. scale factor for the one-scale-factor case.

–3

–2

–1

0

1

0.1 1 10 100

ln E

DP

(T

ype

2)

Scale factor

20.5 Distribution of Type 2-EDP (in natural log units) vs. scale factor for the separate-scale-factor case.


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for different ground motion components, the scatter in EDP is reduced significantly (Fig. 20.5). Still, some of the records with large-scale factors produce response close to average, and some of the records with scale factors close to unity generate response values greater than or less than the average. That leads to the conclusion that it is possible to obtain unbiased results even for large-scale factors, if the records are selected appropriately and that factors other than the scale factor, such as proper selection of IMs, are critical. Figures 20.4 and 20.5 validate the scaling robustness property of the selected IMs and ground motion recordings in the UC Davis study.

20.5.1 PSDMs for scalar IMs

Different PSDMs for the horizontal ground motion effects on the seismic response of ordinary highway bridges are available in the literature (Mackie and Stojadinovic, 2003; Nielson and DesRoches, 2007; Padgett et al., 2008). The UC Davis study extends the previous studies in two important aspects: the analysis focused on the combined effects of vertical and horizontal ground motion components, and the functional form of the seismic demand models was not pre-determined as given in Equation 20.8 (Gülerce et al., 2012). Nonlinear dynamic analyses results were used in regression analysis to develop scalar-valued PSDMs given only horizontal IMs for the first stage and to develop vector-valued PSDMs given both horizontal and verti-cal IMs for the second stage (Step 4 in Fig. 20.2). From preliminary analyses, it was found that a quadratic dependence on the IMs was applicable to (e.g. Fig. 20.6) Type 1-EDPs, whereas a linear dependence was adequate for Type 2-EDPs, as given in Equations 20.14 and 20.15.

ln lnEDP Type c IM c1 1 22

1( )[ ] = × ( ) −[ ] ± σStage [20.14]

ln EDP Type c c IM2 3 4 1( )[ ] = + ( ) ± σStage [20.15]

where c1, c2, c3 and c4 are the model coefficients estimated using nonlinear regression, and σStage1 is the standard deviation of the scalar PSDMs at the first stage. Note that the functional form of the PSDMs for Type 2-EDPs is identical to the form recommended by Cornell and Krawinkler (2000). Figure 20.6 presents the actual data points (in terms of the natural loga-rithms of Type 1-EDPs) and two PSDMs that model the same structural response data using different IMs; Sa at the longitudinal period of the bridge and Sa at the transverse period of the bridge. The actual data points in Fig. 20.6 clearly show the quadratic dependence of Type 1-EDPs to the selected IMs. The scatter in the actual structural response data increases when the horizontal Sa at the transverse period of the bridge model was used. Gülerce et al. (2011) analyzed all bridge configurations and concluded that the



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horizontal Sa at the longitudinal period is a better predictor of the response in terms of efficiency. Figure 20.7 presents the actual data points and two PSDMs that model the same structural response data using different IMs; Sa at the longitudinal period of the bridge and Sa at the transverse period

0.01

0.1

1

0.1 1

ln E

DP

(T

ype

1)

IM (horizontal Sa )

Data wrt Sa (long)

Longitudinal model

Data wrt Sa (trans)

Transverse model

20.6 Distribution of Type 1-EDP (in natural log units) vs. the transverse and longitudinal IMs.

–4

–3

–2

–1

0

1

0.01 0.1 1

ln E

DP

(T

ype

2)

IM (horizontal Sa)

Data wrt Sa (long)

Longitudinal model

Data wrt Sa (trans)

Transverse model

20.7 Distribution of Type 2-EDP (in natural log units) vs. the transverse and longitudinal IMs.


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of the bridge for Type 2-EDPs. For this type of EDPs, no curvature of the actual data was observed and the linear dependence in log-log space was sufficient. Similar to Type 1-EDPs, Sa at the longitudinal period of the bridge model was found to be more efficient for this type of EDP. The variations in the bridge model, regression coefficients, and standard deviations of PSDMs for both longitudinal and transverse IMs can be found in Gülerce et al. (2011).

20.5.2 Scalar PHSA for structural demand

Using the Type 1- and Type 2-PSDMs developed at Stage 1 (no vertical accelerations), PSHA was conducted by directly incorporating the PSDMs into the PSHA integral. For Type 1- and Type 2-EDPs that depend on scalar IMs only, Equation 20.7 was modified as (Gülerce and Abrahamson, 2010):

ν

εεε

EDP Stage z

N f M f M R f

P EDP Stage z

M R

RM

_

,

( _

min

1 1>( ) =

⋅ ( ) ( ) ( )

>

∫∫∫EEDP Stage

S M R M RStage

ˆ _

, , , )

1

1a d d dε σ ε( )( )

[20.16]

where ν(EDP_Stage1 > z) represents the annual rate of exceeding the specified value of EDP without the effect of vertical accelerations. Figure

0.0001

0.001

0.01

0.1

1

1 10

An

nu

al r

ate

of

exce

eden

ce

Type 1-EDP

Axial loaddemand

Axial loadcapacity

Span momentdemand

Span momentcapacity

Supportmomentdemand

Supportmomentcapacity

20.8 Structural demand hazard curves at Stage 1for an example near fault site for Type 1-EDPs.



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20.8 shows the structural demand hazard curves for Type 1-EDPs (axial load demand and positive moment demands at the mid-span and at the support) for a near-fault site. Similarly, Fig. 20.9 presents the structural demand hazard curves for Type 2-EDPs (negative moment demands at the mid-span and at the support) for the same site. The broken lines in both figures indicate the capacity defined as the onset of inelastic deformation/yielding recalling that the girders in a highway bridge are designed to remain elastic under seismic action (Kunnath et al., 2008) for each EDP. Details of the analysis, fault system, and source characteristics used for the assessment were explained in Gülerce and Abrahamson (2010). Acc-ording to Figs 20.8 and 20.9, none of the Type 1-EDPs exceed the corre-sponding capacities at Stage 1. There is a chance of exceeding the capacity for Type 2-EDPs, but these probabilities are small (less than 0.0001 annual probabilities).

20.5.3 Vector-valued PSDMs and vector-valued PSHA

The increase in the EDPs (defined by Equations 20.12 and 20.13) when the vertical ground motions were included in the nonlinear dynamic analyses (Stage 2) can be described by vertical factor (VF) (Gülerce et al., 2012). Similar to the EDPs at Stage 1, two types of vertical factors were defined and normalized as shown in Equations 20.17 and 20.18. Type 1-VF repre-sents the amplification for Type 1-EDPs, while Type 2-VF is for Type 2-EDPs:

0.000001

0.00001

0.0001

0.001

0.01

0.1

1

0.1 1 10

An

nu

al r

ate

of

exce

eden

ce

Type 2-EDP

Spanmomentdemand

Spanmomentcapacity

Supportmomentdemand


20.9 Structural demand hazard curves at Stage 1 for an example near fault site for Type 2-EDPs.


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Type VF

Demand

DLType EDP

11

2

− =

( )

−

max,Stage

[20.17]

Type VF

Demand DL

DLType EDP

22

2

− =

( ) −

−

max,Stage

[20.18]

where (Demand)max,Stage2 and (Demand)min,Stage2 are the positive and negative extremes of the demand versus time history at Stage 2. Since both horizon-tal and vertical ground motion components have impact on the structural response at this stage, distribution of the vertical factors with respect to the vertical and horizontal Sa was checked to investigate the trends of the data as shown in Figs 20.10 and 20.11 for Type 1-VFs and Type 2-VFs, respectively.

Figures 20.10 and 20.11 reveal that both types of vertical factors increase as the vertical Sa increases. The structural response is also dependent on the horizontal Sa, but this dependence is weaker than the dependence on the vertical Sa. The PSDMs were constructed using different functional forms for Type 1- and Type 2-VFs by including two IMs instead of one; the hori-zontal Sa at the horizontal (either transverse or longitudinal) period of the bridge (SaH) and the vertical Sa at the vertical period of the bridge (SaV). The PSDMs for the Type 1-VFs and Type 2-VFs are given in Equations 20.19 and 20.20, respectively.

(a) (b)

–0.5

0

0.5

1.0

1.5

2.0

–3 –2 –1 0 1 2 3

ln (

Typ

e 1-

VF

)

ln (

Typ

e 1-

VF

)

ln (IM1=SaH)

–0.5

0

0.5

1.0

1.5

2.0

–3 –2 –1 0 1 2 3ln (IM2=SaV)

20.10 Distribution of Type 1-VF with respect to (a) horizontal spectral accelerations and (b) vertical spectral acceleration.



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ln

ln

.

ln ln

Type VF

ifS

Sc

if S

S

S

1

0

0 0 1

1

−( ) =

→ ≤ ( )

→ ≤

−

a

a

a

a

a

V

H

H

V

H

cc c c S VF1 2 3( )

× + × ( )[ ]± →

ln aH otherwiseσ

[20.19]

ln

ln

Type VF

ifSS

c

c cS

S

V

H

VF

2

0 4

5 6

_

otherwi

a

a

a

a

V

H

( ) =→ ≤

+ ×

± →σ sse

[20.20]

where c1 to c6 are the model coefficients determined by nonlinear regression (Gülerce et al., 2012). Since PSDMs depend on a vector of IMs, vector-valued PSHA was required at this stage. Gülerce and Abrahamson (2010) modified the vector-valued hazard integral given in Equation 20.10 for structural demand as:

ν

ε ε εε εεε

EDP Stage z

N f M f M R f fM R

RM

_

,min H V HH V

VH

2 >( ) =

⋅ ( ) ( ) ( ) ( )∫∫∫∫∫[ > ( )(

( )] )P EDP Stage z EDP S M R

S M R M RVF

_ , , ,

, , , d d d da H

a V H

H

V

2 ˆ εε σ ε εεV

[20.21]

(a) (b)

–1

0

1

2

3

4

–3 –2 –1 0 1 2 3

ln (

Typ

e 2-

VF

)

ln (

Typ

e 2-

VF

)

ln (IM1=SaH)

–1

0

1

2

3

4

–3 –2 –1 0 1 2 3

ln (IM2=SaV)

20.11 Distribution of Type 2-VF with respect to (a) horizontal spectral accelerations and (b) vertical spectral accelerations.


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where v(EDP_Stage2 > z) represents the annual rate of exceeding the specified value of EDP including the vertical accelerations, SaH is the hori-zontal Sa at the horizontal period of the structure, SaV is the vertical Sa at the vertical period of the structure, εH is the epsilon for horizontal ground motion model, εV is the epsilon for the vertical ground motion model, fεH(εH) is the probability density function for εH, and fεV(εV|εH) is the probability density function for εV conditioned on εH. The structural demand hazard curves for Type 1-EDPs and Type 2-EDPs for the same site at Stage 2 are presented in Figs 20.12 and 20.13, respectively.

Figure 20.12 shows that the probability of exceeding the axial load capac-ity at the column and the positive mid-span moment capacity increased significantly, but these EDPs still remained below the capacity. Annual probability of exceeding the support moment capacity (for both Type1- and Type 2-EDPs) was computed as 0.001 at this stage (Figs 20.12 and 20.13). Negative mid-span moment demand was identified as the most critical parameter under the influence of vertical accelerations and the annual probability of exceeding the negative mid-span moment capacity was sub-stantial (equal to 0.01) at Stage 2 (Fig. 20.13).

20.6 Futuretrends

Recent research results and ongoing discussions on the main components of the PBEE framework including the PSHA, development of PSDMs, use

0.0001

0.001

0.01

0.1

1

1 10

An

nu

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Type 1-EDP

Axial loaddemand

Axial loadcapacity

Span momentdemand

Span momentcapacity

Supportmomentdemand





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of scalar or vector-valued PSDMs to build structural demand hazard curves, and performance objectives were discussed in the previous sections. Even though numerous studies focusing on the vital parts of the problem in the last decade brought in major improvement, there is still significant need for research efforts, especially on the PSHA procedure and its applications. Standardized procedures for estimating the annual rate of exceeding the specific levels of ground motion IMs are crucial in the implementation of the PBEE methodology. PSHA and its main components (seismic source characterization models and ground motion prediction models) are rapidly evolving with the increasing number of special projects (e.g. nuclear power plants, bridges and high-rise structures). However, for many faults around the world, critical seismic source parameters, such as fault geometry, mean characteristic earthquake, slip rate, reoccurrence intervals, time since the most recent large earthquake, etc., are either unknown or poorly con-strained by existing seismological data. The current practice of using areal sources to overcome the obstacle of the unknown source characteristics introduces significant epistemic uncertainty to seismic hazard calculations in many regions; therefore, studies on developing improved seismic source models should be encouraged.

On the other hand, uncertainty introduced by ground motion prediction models is considerably higher than any other parameter models included in the hazard integral. New and improved GMPEs in terms of additional prediction parameters (depth of the source, basin effects, site-dependent

0.0001

0.001

0.01

0.1

1

0.1 1 10

An

nu

al r

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of

exce

eden

ce

Type 2-EDP

Span moment demand

Span moment capacity

Support moment demand

Support moment capacity



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standard deviations, etc.), statistical approach, and a well-constrained global database were developed for California (Abrahamson et al., 2008). In the near future, new models will be developed for other tectonic environments (e.g. subduction zones). The applicability of these new and improved global GMPEs in PSHA studies of other regions is an important issue and should be investigated by comparing the magnitude, distance, and site effects scaling of these global models by regional databases.

Recent studies (Baker and Cornell, 2005; Gülerce and Abrahamson, 2010) suggested that use of scalar-IMs may not be adequate in character-izing the ground motion intensity and nonlinear structural response, thus there is a need to develop vector-valued IMs and PSDMs. Integrating the vector-valued PSDMs into the PSHA integral will require vector-valued hazard computations, therefore standard PSHA software packages should be modified to develop the capability of performing vector-valued calcula-tions. Also, the correlation of residuals of the ground motion model (or models) across the periods should be provided by the developers of future predictive models to be able to perform vector-valued probabilistic seismic hazard and demand assessments. Finally, controversial issues such as selec-tion and scaling of ground motions for nonlinear dynamic analysis and functional shape of PSDMs are topics of ongoing discussions and require further research efforts.

20.7 Acknowledgments

Funding for the UC Davis was study provided by the California Depart-ment of Transportation (Caltrans) under Contract No. 59A0434 and grate-fully acknowledged. The author is thankful to Dr Norman Abrahamson, Dr Sashi K. Kunnath and Dr Emrah Erduran for their contribution to the UC Davis study presented in this chapter.

20.8 ReferencesAbrahamson, N. A., and Silva, W. J., 1997. Empirical response spectral attenuation

relations for shallow crustal earthquakes, Seismological Research Letters 68, 94–127.

Abrahamson, N. A., and Silva, W. J., 2008. Summary of the Abrahamson and Silva NGA ground-motion relations, Earthquake Spectra 24, 67–98.

Abrahamson, N. A., Atkinson, G., Boore, D., Bozorgnia, Y., Campbell, K., Chiou, B., Idriss, I. M., Silva, W. J., and Youngs, R., 2008. Comparisons of the NGA ground-motion relations, Earthquake Spectra 24, 45–66.

Baker, J., and Cornell, C. A., 2003. Uncertainty specification and propagation for loss estimation using FOSM method, Pacific Earthquake Engineering Research Center, University of California, Berkeley, PEER 2003-07.



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Baker, J., and Cornell, C. A., 2005. A vector-valued ground motion intensity measure consisting of spectral acceleration and epsilon, Earthquake Engineering and Structural Dynamics 34, 1193–1217.

Bazzurro, P., and Cornell, C. A., 2002. Vector-valued probabilistic seismic hazard analysis (VPSHA), in Proceedings, 7th U.S. National Conference on Earthquake Engineering, Paper #61, Boston, Massachusetts.

Cornell, C. A., 1968. Engineering seismic risk analysis, Bulletin of Seismological Society of America 58, 1583–1606.

Cornell, C. A., and Krawinkler, H., 2000. Progress and challenges in seismic perfor-mance assessment, Peer Center News, 3(2).

Cornell, C. A., Jalayer, F., Hamburger, R. O., and Foutch, D. A., 2002. Probabilistic basis for 2000 SAC/FEMA steel moment frame guidelines, Journal of Structural Engineering 128(4), 526–533.

Chiou, B., Darragh, R., Gregor, N., and Silva, W., 2008. NGA project strong motion database, Earthquake Spectra 24, 23–44.

FEMA, 1997. NEHRP guidelines for the seismic rehabilitation of buildings, Report No. FEMA-273, Federal Emergency Management Agency, Washington, DC.

FEMA, 2000. Recommended seismic design criteria for new steel moment-frame buildings, Report No. FEMA-350, SAC Joint Venture, Federal Emergency Man-agement Agency, Sacramento, CA.

Giovenale, P., Cornell, C. A., and Esteva, L., 2004. Comparing the adequacy of alter-native ground motion intensity measures for the estimation of structural responses. Earthquake Engineering and Structural Dynamics 33, 951–979.

Gülerce, Z., and Abrahamson, N. A., 2010. Vector-valued probabilistic seismic hazard assessment for the effects of vertical ground motions on the seismic response of highway bridges, Earthquake Spectra 26, 999–1016.

Gülerce, Z., and Abrahamson, N. A., 2011. Site-specific design spectra for vertical ground motion, Earthquake Spectra 27, 1023–1047.

Gülerce, Z., Erduran, E., Kunnath, S., and Abrahamson, N. A., 2012. Seismic demand models for probabilistic risk analysis of near fault vertical ground motion effects on ordinary highway bridges, Earthquake Engineering and Structural Dynamics 41, 159–175.

Kunnath, S. K., Erduran, E., Chai, Y. H., and Yashinsky, M., 2008. Effect of near-fault vertical ground motions on seismic response of highway overcrossings, Journal of Bridge Engineering 13, 282–290.

Krawinkler, H., 2002. A general approach to seismic performance assessment. Pro-ceedings, International Conference on Advances and New Challenges in Earth-quake Engineering Research, ICANCEER 2002, Hong Kong, August 19–20, 2002.

Luco, N., 2002. Probabilistic seismic demand analysis, Smrf connection fractures, and near-source effects. Ph.D. Dissertation, Department of Civil and Environmental Engineering, Stanford University, Stanford, CA.

Luco, N., and Cornell, C. A., 2007. Structure-specific scalar intensity measures for near-source and ordinary earthquake ground motions, Earthquake Spectra, 23, 357–392.

Mackie, K., and Stojadinovic, B., 2003. Seismic demands for performance-based design of bridges, Pacific Earthquake Engineering Research Center, University of California, Berkeley, PEER 2003–16.

McGuire, R. K., 2004. Analysis of seismic hazard and risk, EERI Monograph, Earth-quake Engineering Research Center.


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Miranda, E., and Aslani, H., 2003. Probabilistic response assessment for building-specific loss estimation, Pacific Earthquake Engineering Research Center, Univer-sity of California, Berkeley, PEER 2003-03.

Nielson, B. G., and DesRoches, R., 2007. Seismic fragility methodology for highway bridges using a component level approach, Earthquake Engineering and Struc-tural Dynamics 36, 823–839.

Padgett, J. E., Nielson, B. G., and DesRoches, R., 2008. Selection of optimal intensity measures in probabilistic seismic demand models of highway bridge portfolios. Earthquake Engineering and Structural Dynamics 37, 711–725.

Shome, N., and Cornell, C. A., 1999. Probabilistic seismic demand analysis of non-linear structures, Reliability of Marine Structures Report No. RMS-35, Stanford University, Stanford.

Stewart, J. P., Chiou, S-J., Bray, J. D., Graves, R. W., Somerville, P. G., and Abraham-son, N. A., 2002. Ground motion evaluation procedures for performance-based design, Soil Dynamics and Earthquake Engineering 22, 765–722.

Tothong, P., and Cornell, C. A., 2006. Probabilistic seismic demand analysis using advanced ground motion intensity measures, attenuation relationships, and near-fault effects. Pacific Earthquake Engineering Research Center, University of Cali-fornia, Berkeley, PEER Report 2006–11.

Watson-Lamprey, J., and Abrahamson, N. A., 2006. Selection of ground motion time series and limits on scaling. Soil Dynamics and Earthquake Engineering 26, 477–482.


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G D ( )DM EDP G ( )EDP IM I - Department of Civil Engineering, METUcourses.ce.metu.edu.tr/ce7013/wp-content/uploads/sites/35/2015/05… · Tesfamariam—Handbook of seismic risk analysis