Review on probabilistic seismic demand modeling and estimation for highway bridge

Abstract

The probabilistic seismic demand model (PSDM) is essential to identify the seismic demand of the highway bridge during and after an earthquake. This paper aims to review the probabilistic seismic demand estimation and modeling methodology options associated with the procedure, analytical analysis, and mathematical framework for a highway bridge. As a result of the review, different techniques with features, applications, and limitations on highway bridges are reviewed and presented. A review has investigated the current PSDM and provides a comprehensive summary with formulas, tables, figures, and frameworks. PSDM steps are constructed and introduced to how scholars use them. Besides, analytical methods are the best choice for investigating the PSDA and PSDM for critical bridge components when damage data is insufficient. They are determined to predict each component’s seismic response for a given deterministic or random variable. This work helps and motivates the decision-makers and stakeholders to extend the application of the PSDM methodology option for a more informed decision.

Keywords

probabilistic seismic demand model nonlinear time history analysis incremental dynamic analysis probabilistic seismic demand analysis highway bridge

Introduction

The highway bridges have catastrophically collapsed due to the loss of seismic performance during seismic events (Hariri-Ardebili and Saouma, 2016; Mackie and Stojadinović, 2001; Muntasir Billah and Shahria Alam, 2015; Shome et al., 1998; Zakeri et al., 2014, 2015). The seismic isolation system is introduced to improve and enhance seismic performance and reduce the seismic fragility of new and retrofitted highway bridge systems (Bayat, 2017; Nielson Bryant and DesRoches, 2006; Nielson and DesRoches, 2007; Song et al., 2018; Zhou et al., 2018). Seismic performance and fragility are estimated via probabilistic seismic demand analysis (PSDA), which determines the relationship between the engineering demand parameter (EDP) and ground motion intensity (IM) to develop the seismic response graph and PSDM (Gardoni et al., 2003; Hariri-Ardebili and Saouma, 2016; Jeon et al., 2019; Ma et al., 2016; Mackie and Stojadinović, 2001; Pahlavan et al., 2016; Shome et al., 1998). PSDA is also an essential supportive tool to identify the seismic response more significantly than the seismic capacity condition on IM measurements (Gardoni et al., 2003; Song et al., 2018). The ground motion suite, the bridge inventory, and modeling and analyzing software are essential for analyzing the nonlinear time history and formulating the PSDM (Alhan and Öncü-Davas, 2016; Bayat, 2017; Billah et al., 2013; Du et al., 2018; Grigoriu and Radu, 2021; Kibboua et al., 2014; Ma et al., 2016; Mackie and Stojadinović, 2001; Nielson and DesRoches, 2007; Ramanathan, 2012; Song et al., 2018; Zhou et al., 2018). The highway bridge design parameters are significantly considered when selecting the bridge structure inventory (Ghazali et al., 2019; Guo et al., 2015; Li et al., 2020b; Zakeri et al., 2015). The finite element modeling software: OpenSees (Jeon et al., 2019; Li et al., 2020a, 2020b; McKenna et al., 2010; Nielson Bryant and DesRoches, 2006; Nielson and DesRoches, 2007; Pahlavan et al., 2016; Song et al., 2018; Zakeri et al., 2015; Zhou et al., 2018; Zhou and Li, 2019), ANSYS, SAP (Vedhanayaghi et al., 2021), SeismoStruct (Alam et al., 2012), and MIDAS are used for modeling the entire bridge system. However, the OpenSees are practical and powerful for modeling the 3D finite modeling of the bridge components (Li et al., 2020a; McKenna et al., 2010; Shamsabadi et al., 2010; Vedhanayaghi et al., 2021; Zhou and Li, 2019). Selecting the appropriate IM and EDP measurement is vital to developing an approximately linear relationship in logarithmic space, called PSDM (Alhan and Öncü-Davas, 2016; Bayat, 2017; Billah et al., 2013; Grigoriu and Radu, 2021; Hwang et al., 2000; Jeon et al., 2019; Kibboua et al., 2014; Nielson Bryant and DesRoches, 2006; Song et al., 2018; Zakeri et al., 2014; Zhou and Li, 2019).

Besides, PSDM is the most indispensable supporting tool to determine the likelihood of seismic demand during and after the earthquake (Gardoni et al., 2003). PSDM is encompassed by PSDA and seismic hazard analysis (Hariri-Ardebili and Saouma, 2016; Mackie and Stojadinović, 2001; Shome et al., 1998). A highway bridge’s seismic demand is analyzed, and PSDM is developed for critical bridge components by different methods (Hwang et al., 2000; Jeon et al., 2019; Vedhanayaghi et al., 2021). However, suppose bridge damage data is insufficient. In that case, an analytical method is the best choice for nonlinear time history analysis (NLTHA) and incremental dynamic analysis (Mackie and Stojadinović, 2001; Muntasir Billah and Shahria Alam, 2015; Wu et al., 2014; Zhang et al., 2019). The NLTHA offers the flexibility to consider analytical models with nonlinear cyclic material characteristics and geometric nonlinearities such as P-∆ or total nonlinear or large deformations (Nielson Bryant and DesRoches, 2006; Ramanathan et al., 2012). The IDA is a unique nonlinear dynamic analysis that determines the bridge component’s seismic demand and develops the relationship between the IM and EDP (Bayat, 2017; Guo et al., 2015; Jeon et al., 2019; Song et al., 2018). However, developing PSDM using the NLTHA and IDA is a thorough and reliable analytical method.

This paper reviews the probabilistic seismic demand estimation and modeling methodology options associated with the procedure, analytical analysis, and mathematical framework for a highway bridge. Different techniques with impediments, applications, and idiosyncrasies of highway bridge’ PSDMs are presented. A review investigates the current PSDM and provides a comprehensive methodology with formulas, tables, figures, and frameworks. PSDM steps are constructed and introduced to how scholars use them. These are vital to encourage the decision-makers and stakeholders to develop a practical PSDM.

This review paper has six main parts and a conclusion. The first section presents the PSDA. Second, the choice of ground motion is clarified with the bin approach, acceleration response graph, distribution of earthquake moment magnitude (Mw), epicentral distance(R), and peak ground acceleration (PGA). Third, the definition and selection of highway bridge inventory are defined and reviewed. Next to that, the formulation of the analytical highway bridge is defined and investigated. Then, the choice of intensity and demand measurement pair is visualized with the effect on each other for structural demand. Finally, more analytical studies for the PSDM are reviewed and discussed, including the NLTHA and IDA. The conclusion of this work is developed by summarizing all parts.

Probabilistic seismic demand analysis

PSDA determines the relationship between EDP and IM and develops the seismic response graph and PSDM. PSDA is vital because it is conditioned on an IM: PGA, PGV, spectral acceleration (Sa), Mw, and R to analyze the demand and capacity of the structure (Hwang et al., 2000; Jeon et al., 2019; Mackie and Stojadinović, 2001; Vedhanayaghi et al., 2021). PSDA is an essential supportive tool to identify the seismic response more significantly than the seismic capacity condition on IM measurements (Gardoni et al., 2003; Song et al., 2018). A postulated seismic hazard environment estimates the probability of exceeding structural demand. The vulnerability of structures is evaluated by a representation series of ground motion time history (GMTHs), which has a more incredible response than the capacity conditioned on IM (Alam et al., 2012). Besides, PSDA is investigated using different seismic analysis methods: expert-based or judgmental, empirical, experimental, analytical, and hybrid (Muntasir Billah and Shahria Alam, 2015). The expert-based method is analyzed based on knowledge about the probability distribution functions representing a particular damage level at various levels of IM. The empirical method examines the damage distribution from post-earthquake field observations or reconnaissance reports. The experimental method is analyzed using the observed laboratory results from the shaking table and the cyclic load test. The analytical method is determined through either elastic spectral or PSDM if there is no adequate damage data of the bridge components (Jeon et al., 2019; Kim and Shinozuka, 2004; Zhou et al., 2018). However, all except analytical have a significant limitation and considerable impediment to full-scale analysis. For instance, judgment is based on the questionnaire and uses the panel’s expertise. For example, the first method is based on the questionnaire. It uses the panel’s experience of certain structural types with typical configurations, detailing and materials. They give biased judgments and involve several uncertainties that are not quantified explicitly in the vulnerable functions. The second encumbrance is a lack of inference associated with substantial uncertainty. In addition, the third is incapable due to inadequate data and a weak correlation between geometry and structural properties. Notwithstanding, an analytical method is a realistic and best choice to investigate PSDA and establish the PSDM when damaged data is inadequate (Muntasir Billah and Shahria Alam, 2015; Vedhanayaghi et al., 2021).

Furthermore, NLTHA is the most stringent and reliable way to analyze the seismic demand for highway bridges (Nielson Bryant and DesRoches, 2006). They are easily applied to ground motion record areas. This method’s damage matrix and distribution function are essential to developing the PSDM. The NLTHA offers the flexibility to consider analytical models with linear or nonlinear cyclic material characteristics and geometric nonlinearities. The P-∆ (total nonlinear deformations) is considered in the analysis to capture the impact of tall columns and relatively heavy bridge decks (Mackie and Stojadinović, 2001). The PSDM is developed from the response data of NLTHA, integrating with the geometric and material variability in finite element modeling software (Ramanathan et al., 2012). The PSDM formulation through the NLTHA procedure is described more in Figure 6. The IDA is a unique nonlinear dynamic analysis that determines the bridge component’s seismic demand and develops the relationship between the IM and EDP (Bayat, 2017; Guo et al., 2015; Jeon et al., 2019; Song et al., 2018). IDA is essential to calculate the bridge’s uncertainties and drift capacities (Mackie and Stojadinović, 2001; Nielson Bryant and DesRoches, 2006). Figure 1 illustrates and proposes a framework for PSDM through the PSDA. During the preparation for PSDA, an analyst should consider the following:

• Firstly, IM and structural demand measures should produce a practical, sufficient, effective, and efficient PSDM.

• Secondly, reduce the dispersion in PSDMs because it directly affects the number of ground motions and time-history analyses required to compute the PSDM parameter.

• Then, ground motion selection should accurately represent the region’s seismicity.

• Next, understanding the capacities and shortcomings of the tool used to model and analyze the structure is essential for adequately interpreting PSDMs.

• Finally, the computational effort required to create PSDMs for a type of structure and the complexity of the database is necessary to track the results of many analyses.

Furthermore, the development process of PSDM is the following steps:

(1) First, select the N ground motions suite either from the NGA-West2 database of PEER (New Generation Attenuation Database of Pacific Earthquake Engineering Research Center) or the ESM-Engineering Strong-Motion Database using the suitable selection criteria.

(2) Assemble a suite of N ground motions that apply to the geographical area of interest for the chosen intensity measure.

(3) The Latin Hypercube sampling technique generates the geometric and material variables of the n-statically significant and typically identical bridge models.

(4) Monte Carlo simulation paired N-bridge models for the N-ground motion, forming the bridge-ground motion sample used as input during the analysis.

(5) Perform a complete NLTHA for each ground motion-bridge pair and record the critical component responses in each case. The peak response versus its corresponding IM is recorded and plotted for each analysis.

(6) Then, peak responses against the ground motion graph became PSDM in lognormal space. At the same time, the PSDM is initially developed on the regular coordinate system and then transferred to a logarithmic coordinate system with EDP and IM.

(7) Finally, determine the unknown regression coefficients, a and b, and the dispersion of the demand $β_{E D P / I M}$ condition upon the IM using the regression analysis method.

NB Step 4 repeats for all major vulnerable components in the bridge.

Generally, PSDA is preceded by selecting a set of ground motions, defining a local and global EDP for the structure, preparing a nonlinear finite element model, performing a nonlinear transient analysis, and finally establishing a PSDM for the system. The logical interrelationships and supplementary flow chart for different aspects of PSDM and estimation are illustrated in Figure 2.

Ground motion selection

The ground motion suite is essential for analyzing the nonlinear time history and formulating the PSDM. There are specific guidelines for ground motion selection and scaling (e.g., ASCE 7-22, ASCE/SEI 41-17, NBC 2020). Regardless, the adaptive intensity measure (ADP-IMs) improves the selection of ground motion suites for highway bridge PSDM portfolios (Du et al., 2018). The ADP-IMs concept includes two spectral IMs adopting the optimal period, two fractional-order IMs of the optimal fractional order, and Sa for the optimal period and damping ratio. Based on the PGA and Sa, the four one-parameter ADP-IMs and two-parameter ADP-IMs are included. In expression $I M (x^{*}; c)$ , where $x^{*} = {x_{1}^{*}, \dots, x_{n}^{*}}$ denote the optimal IM parameter set, and the IM parameter is $C = {c_{1}, \dots, c_{m}}$ . The six non-Adaptive IMs are $P G A$ , $P G V,$ $S a$ , $S a, c$ , $I N p$ and $S_{a, E a d s}$ ; those categories into two: well-known conventional IMs and advanced spectral IMs. The well-known conventional IMs define PGA, PGV, and S_a(T_gm), which are given by PGV(equation (1)), PGA(equation (2)), and S_a(T_gm) are 5% damped Sa at period T_gm. Sa gives the advanced spectral IMs,c(equation (3)), INp(equation (4)), and S_a, _Eads(equation (5)).

Generally, in both bridge classes, the peak fractional ground response (PGA(a*-RRB) is the best from one parameter, ADP-IMs. Although two-parameter ADP-IM, Sa at the optimal and damping ratio $(S_{a} (T^{*}, ζ^{*}))$ needs considerable improvement. However, Mw and R are currently used to select ground motion from the earthquake database (Muntasir Billah and Shahria Alam, 2015; Song et al., 2018; Zakeri et al., 2014; Zhou et al., 2018; Zhou and Li, 2019). The ground motion bin approach divides the ground motion suite into N-bin numbers with small and large earthquakes at different epicentral lengths (Ma et al., 2016; Mackie and Stojadinović, 2001; Nielson and DesRoches, 2007; Song et al., 2018). The bins approach abstracts individual earthquakes and provides a way to limit the number of ground motion suites (Mackie and Stojadinović, 2001; Song et al., 2018). The proper scaling of IM can reduce dispersion and the number of ground motions required for confident analysis (Ramanathan, 2012; Shome et al., 1998). According to various research on earthquake records, scaling suggests that appropriate scaling depends on factors like soil and near-field conditions, and genetic algorithms can improve results and reduce variance. For example, Kappos and Kyriakakis (2000) conducted a study in the early 2000s on the impact of scaling on elastic and inelastic spectra for strength and displacement. They used various scaling methodologies, including Peak Ground Acceleration, Peak Ground Velocity, Arias Intensity, Root-Mean-Square, Spectrum Intensity, and Housner Intensity. They found that spectrum Intensity scaling led to low scatter and reasonable scatter along the building height. Kurama and Farrow (2003) study examined seven scaling methods for reducing peak lateral displacement demands using nonlinear SDOF and MDOF models. They found that proper scaling depends on multiple factors, as methods effective for stiff soil and far-field conditions lose effectiveness for soft soil and near-field conditions. Naeim et al. (2004) used genetic algorithms to select earthquake records that better match a design spectrum and determine appropriate scaling factors for fitting response spectra against site-specific design spectrums. Shome et al. (1998) studied the impact of bin records of magnitude Mw at distance R on a 5-story steel structure’s nonlinear response. They found that scaling records to the 5% damped spectral acceleration at the structure’s fundamental frequency yielded better results and reduced variance. Kottke and Rathje (2008) developed a method for selecting and scaling ground motions to fit a target acceleration response spectrum, controlling variability, and fitting the target spectral shape, amplitude, and standard deviation. Iervolino et al. (2008) found unscaled records from the European Strong-Motion database, fulfilling EC8 requirements, but found large variability in spectral ordinates within the same set. Hancock et al. (2008) studied the number of scaled/matched accelerograms needed for inelastic dynamic analysis using five techniques. They found that more constraints decrease the required records and bias. Some of the techniques that are used to scale ground motion are geometric mean scaling of pairs of ground motion, spectrum matching of ground motions, first-mode-based scaling to a target spectral acceleration, maximum-minimum orientations, and modal-pushover-based scaling approaches. The geometric-mean scaling approach involves amplitude scaling a pair of seed motions with a single scaling factor in order to reduce the total squared error between the geometric mean (square root of the product) and the target spectral values of the pair’s spectral ordinates. The spectrum-matching approach has been widely utilized in the seismic design of structures. For the first mode-based (Sa (T1)) scaling, ground motion records are scaled in terms of their amplitude to a certain acceleration during the structure’s first mode period. Generally, the scaling approach has proceeded with the same level of intensity at the same fundamental period of a structure. In this case, the individual near- and far-field earthquake types are distinguished rather than near- and far-field records.

P G V = \max ({\dot{U}}_{g} (t))

(1)

P G A = \max ({\ddot{U}}_{g} (t))

(2)

S_{a, c} = S_{a} (T_{g m}) {[\frac{S_{a} (2 T_{g m})}{S_{a} (T_{g m})}]}^{0.5}

(3)

I N_{P} = S_{a} (T_{g m}) {[\frac{S_{a, a v g} (T_{g m}, . . . ., 2 T_{g m})}{S_{a} (T_{g m})}]}^{0.4}

(4)

S_{a, E a d s} = S_{a, a v g} (0.2 T_{g m}, . . ., 3 T_{g m})

(5)

The number of ground motion records used in the NLTHA is explicitly specified in different codes based on various criteria (De Felice and Giannini, 2010; Hariri-Ardebili and Saouma, 2016; Jeon et al., 2019; Li et al., 2020b; Ma et al., 2016; Mackie and Stojadinović, 2001; Nielson and DesRoches, 2007; Song et al., 2018). Besides, Mw, R, and the average shear velocity (Vs₃₀) are significantly considered when selecting the ground motion suite (Ramanathan, 2012). Generally, according to the Chinese seismic design code design, the site is categorized into four types based on Vs₃₀: type I (Vs₃₀ > 510 m/s), type II (260 m/s < Vs₃₀ < 510 m/s), type III (150 m/s < Vs₃₀ < 260 m/s and type IV (Vs₃₀ < 150 m/s). For example, to illustrate the selection of the ground motion suites, 154 ground motion suites are extracted from the NGA-West2 database of PEER (New Generation Attenuation Database of Pacific Earthquake Engineering Research Center). The Mw, R, and average shear velocity (V₃₀) are considered to visualize the selection of the ground motion suite. According to the Chinese seismic design code, the selected ground motion suites are categorized under

t y p e I I (260 m / s < V s_{30} < 510 m / s)

and have far-fault field (R >10 km) characteristics. The general properties of these ground motion suites are (1) the Mw range from 5.5 to 7.5, (2) the R range from 10 km to 90 km, and (3) the average shear wave velocity of the upper 30 m Table 1 describes the selected ground motion data characteristics with their features. Figure 3 illustrates the choice of ground motion based on the bin approach for PSDM, and Figure 3(a) describes the bin approach for selected ground motion. Figure 3(b) demonstrates the response spectra of the ground motion records. The Sa’s mean and mean ± standard deviation indicated the variability inherent in seismic ground motion (see Figure 3(b)). Figure 4 provides essential information related to the selected ground motion distribution. In addition, they provide the relationship between the Mw, R, and Sa to visualize the importance of comprehensive coverage and broad representation during the selection of the ground motion suite. Therefore, during the selection and preparation of the ground motion suite for PSDA and PSDM, considering the ground motion parameter is essential.

Table 1.

The characteristics of the selected ground motion data.

M	Year	Earthquake name	Earthquake station	Mechanism
6.36	1983	Coalinga-01	Parkfield-fault zone 2	Reverse
6.19	1984	Morgan hill	Fremont -mission SanJose	Strike-slip
6.19	1984	Morgan hill	San Juan Bautista 24 Polk St	Strike-slip
6.2	1999	Chi-Chi Taiwan 03	TCU106	Reverse
6.2	1999	Chi-Chi Taiwan 04	TCU123	Strike-slip
6.3	1999	Chi-Chi Taiwan 06	HWA036	Reverse
6.3	1999	Chi-Chi Taiwan 06	HWA037	Reverse
6.3	1999	Chi-Chi Taiwan 06	TCU061	Reverse
6.3	1999	Chi-Chi Taiwan 06	TCU107	Reverse
6.3	1999	Chi-Chi Taiwan 06	TCU108	Reverse
6.3	1999	Chi-Chi Taiwan 06	TCU123	Reverse
5.6	1997	Umbria Marche Italy	Gubbio-Piana	Normal
6.3	2009	L'Aquila Italy	Chieti	Normal
6.2	2011	Christchurch New Zealand	AMBC	Reverse Oblique
6	2004	Parkfield-02 CA	Monarch peak	Strike-slip
6.36	1983	Coalinga-01	Parkfield-Cholame 12W	Reverse
6.19	1984	Morgan hill	Los Banos	Strike-slip
6.46	1992	Big Bear-01	Phelan-Wilson Ranch	Strike-slip
6.4	1995	Kozani Greece-01	Larisa	Normal
5.65	1992	Little Skull Mtn NV	Station #4-Pahrump 2	Normal
5.7	2001	Gulf of California	Imperial valley-midway well	Strike-slip
5.9	1999	Chi-Chi Taiwan 02	TCU034	Reverse
6.2	1999	Chi-Chi Taiwan 03	CHY058	Reverse
6.2	1999	Chi-Chi Taiwan 03	HWA013	Reverse
6.2	1999	Chi-Chi Taiwan 03	HWA028	Reverse
6.2	1999	Chi-Chi Taiwan 03	HWA048	Reverse
6.2	1999	Chi-Chi Taiwan 03	HWA049	Reverse
6.2	1999	Chi-Chi Taiwan 03	TCU034	Reverse
6.2	1999	Chi-Chi Taiwan 03	TTN022	Reverse
6.2	1999	Chi-Chi Taiwan 03	TTN044	Reverse
6.2	1999	Chi-Chi Taiwan 04	CHY070	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	CHY096	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	HWA005	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	HWA011	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	HWA013	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	HWA048	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	HWA049	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	KAU020	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	KAU085	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU040	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU051	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU053	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU059	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU061	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU068	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU070	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU104	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU106	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TCU107	Strike-slip
6.2	1999	Chi-Chi Taiwan 04	TTN048	Strike-slip
6.3	1999	Chi-Chi Taiwan 06	CHY058	Reverse
6.3	1999	Chi-Chi Taiwan 06	HWA014	Reverse
6.3	1999	Chi-Chi Taiwan 06	TCU032	Reverse
6.3	1999	Chi-Chi Taiwan 06	TCU095	Reverse
6.3	1999	Chi-Chi Taiwan 06	TTN001	Reverse
6.3	1999	Chi-Chi Taiwan 06	TTN044	Reverse
6.3	1999	Chi-Chi Taiwan 06	CHY004	Reverse
6	2004	Parkfield 02 CA	King city canal & Reich	Strike-slip
5.7	1997	Umbria Marche Italy	Aquilpark Citta	Normal
5.7	1997	Umbria Marche Italy	Aquilpark Galleria	Normal
5.7	1997	Umbria Marche Italy	Aquilpark Parcheggio	Normal
5.7	1997	Umbria Marche Italy	Rieti	Normal
6	1997	Umbria Marche Italy	Aquilpark Citta	Normal
6	1997	Umbria Marche Italy	Aquilpark Galleria	Normal
6	1997	Umbria Marche Italy	Aquilpark Parcheggio	Normal
6	1997	Umbria Marche Italy	Rieti	Normal
5.5	1997	Umbria Marche Italy	Aquilpark citta	Normal
5.5	1997	Umbria Marche Italy	Aquilpark Galleria	Normal
5.5	1997	Umbria Marche Italy	Aquilpark Parcheggio	Normal
5.5	1997	Umbria Marche Italy	Rieti	Normal
5.6	1997	Umbria Marche Italy	Rieti	Normal
6.3	2009	L’Aquila Italy	Castel di Sangro	Normal
6.3	2009	L’Aquila Italy	Spoleto cantina	Normal
5.6	2009	L’Aquila Italy	Chieti	Normal Oblique
6.2	2011	Christchurch New Zealand	CECS	Reverse Oblique
7.36	1952	Kern county	Taft Lincoln School	Reverse
6.53	1979	Imperial valley 06	Cerro Prieto	Strike-slip
7.28	1992	Landers	Desert hot springs	Strike-slip
7.28	1992	Landers	Mission creek fault	Strike-slip
7.28	1992	Landers	Morongo valley fire Station	Strike-slip
7.28	1992	Landers	North Palm springs	Strike-slip
7.28	1992	Landers	Palm springs airport	Strike-slip
6.69	1994	Northridge 01	Camarillo	Reverse
6.9	1995	Kobe Japan	Tadoka	Strike-slip
7.28	1992	Landers	Indio Jackson Road	Strike-slip
7.28	1992	Landers	Morongo valley Hall	Strike-slip
7.28	1992	Landers	North Palm springs fire sta	Strike-slip
7.28	1992	Landers	Thousand Palms post Office	Strike-slip
7.28	1992	Landers	Whitewater Trout farm	Strike-slip
6.63	2004	Niigata Japan	NIG015	Reverse
6.63	2004	Niigata Japan	NIGH13	Reverse
6.8	2007	Chuetsu Oki Japan	Joetsu Kita	Reverse
6.8	2007	Chuetsu Oki Japan	Kamo Kouiti Town	Reverse
6.8	2007	Chuetsu Oki Japan	Hinodecho Yoshida Tsubame	Reverse
6.8	2007	Chuetsu Oki Japan	NIG015	Reverse
6.8	2007	Chuetsu Oki Japan	NIG028	Reverse
6.8	2007	Chuetsu Oki Japan	NIGH01	Reverse
6.8	2007	Chuetsu Oki Japan	NIGH11	Reverse
6.9	2008	Iwate Japan	AKTH17	Reverse
6.9	2008	Iwate Japan	IWT012	Reverse
6.9	2008	Iwate Japan	IWT014	Reverse
6.9	2008	Iwate Japan	IWTH15	Reverse
6.9	2008	Iwate Japan	IWTH20	Reverse
6.9	2008	Iwate Japan	MYG005	Reverse
6.9	2008	Iwate Japan	YMT002	Reverse
6.9	2008	Iwate Japan	YMT017	Reverse
6.9	2008	Iwate Japan	YMTH09	Reverse
6.9	2008	Iwate Japan	Sake city”	Reverse
6.9	2008	Iwate Japan	Shinmachi Wakayu	Reverse
6.9	2008	Iwate Japan	Tome city Nakadacho	Reverse
7	2010	Darfield New Zealand	Canterbury aero club	Strike-slip
7	2010	Darfield New Zealand	Papanui high School	Strike-slip
6.61	1971	San fernando	2516 via Tejon PV	Reverse
7.3	1986	Taiwan SMART1 (45)	SMART1 O08	Reverse
7.28	1992	Landers	Hemet fire Station	Strike-slip
7.28	1992	Landers	Indio Coachella canal	Strike-slip
7.28	1992	Landers	San Bernardino E& hospitality	Strike-slip
6.69	1994	Northridge 01	LB city Hall	Reverse
6.69	1994	Northridge 01	Ventura harbor & California	Reverse
7.13	1999	Hector mine	Indio Riverside Co fair Grnds	Strike-slip
7.13	1999	Hector mine	Beaumont 6th & maple	Strike-slip
6.52	2003	San Simeon CA	Coalinga fire Station 39	Reverse
6.52	2003	San Simeon CA	Greenfield Police Station	Reverse
6.63	2004	Niigata Japan	GNM005	Reverse
6.63	2004	Niigata Japan	GNMH08	Reverse
6.63	2004	Niigata Japan	NGN001	Reverse
6.8	2007	Chuetsu Oki Japan	FKS030	Reverse
6.8	2007	Chuetsu Oki Japan	GNM003	Reverse
6.8	2007	Chuetsu Oki Japan	GNM004	Reverse
6.8	2007	Chuetsu Oki Japan	GNM005	Reverse
6.8	2007	Chuetsu Oki Japan	GNM006	Reverse
6.8	2007	Chuetsu Oki Japan	NGN004	Reverse
6.8	2007	Chuetsu Oki Japan	NIGH17	Reverse
6.9	2008	Iwate Japan	AKTH03	Reverse
6.9	2008	Iwate Japan	AKTH16	Reverse
6.9	2008	Iwate Japan	IWT021	Reverse
6.9	2008	Iwate Japan	IWTH02	Reverse
6.9	2008	Iwate Japan	MYG010	Reverse
6.9	2008	Iwate Japan	YMT003	Reverse
6.9	2008	Iwate Japan	YMT007	Reverse
6.9	2008	Iwate Japan	YMT008	Reverse
6.9	2008	Iwate Japan	YMT016	Reverse
6.9	2008	Iwate Japan	YMTH08	Reverse
6.9	2008	Iwate Japan	YMTH14	Reverse
6.9	2008	Iwate Japan	YMTH15	Reverse
6.9	2008	Iwate Japan	Sendai airport	Reverse
7.2	2010	El mayor cucapah Mexico	Heroes of the Revolution	Strike-slip
7.2	2010	El mayor cucapah Mexico	Valle de la Trinidad	Strike slip
7.2	2010	El mayor cucapah Mexico	North Shore Durmid	Strike-slip
7.2	2010	El mayor cucapah Mexico	Bombay Beach Bertram	Strike-slip
7.2	2010	El mayor cucapah Mexico	Frink	Strike-slip
7.2	2010	El mayor cucapah Mexico	Borrego springs Scripps clinic	Strike-slip
7.1	1979	Montenegro Yugoslavia	Berane OPST	Reverse
7.2	2010	El mayor cucapah Mexico	SANTA ISABEL VIEJO	Strike slip

Definition and selection highway bridge inventory

Selecting and defining an appropriate highway bridge structure is crucial for developing the PSDA and PSDM. Also, the number and types of civil structure inventors are selected based on various highway bridge design parameters and conditions for different studies. For example, Mackie and Stojadinović (2001) selected a bridge inventory with ten different highway bridge design parameters, whereas degree of skew, span length, span-to-column height ratio, steel and concrete nominal strengths, amount of longitudinal and transverse column reinforcement, column diameter to superstructure depth ratio, soil properties at pile shafts, and bridge weight. They have recommended the parameter variation ranges for a two-span overpass highway bridge on Table 2.

Table 2.

Recommended parameter variation ranges for a two-span overpass highway bridge (Grigoriu and Radu, 2021; Mackie and Stojadinović, 2001).

Description	Parameter	Range
Degree of skew	$a$	$0 - π / 3 r a d$
Span length	L	18–55 m (60–180 ft)
Span-to-column height ratio	L/H	1.2–2.4
Column-to-superstructure dimension ratio	D_C/D_S	0.67–1.33
Reinforcement nominal yield strength	$f_{y}^{'}$	470–655 MPa (68–95 ksi)
Concrete nominal strength	$f_{c}^{'}$	20–55 MPa (3–8 ksi)
Column longitudinal reinforcement ratio	$ρ_{s, l}$	1%–4%
Column transverse reinforcement ratio	$ρ_{s, t}$	0.4%–1.0%
Soil stiffness based on NEHRP groups	$K_{s o i l}$	A B, C, D
Additional bridge deadload	$W_{d a}$	10%–100% self-weight

For example, Nielson Bryant et al. (Nielson Bryant and DesRoches, 2006) selected eight multi-span simple supported concrete slab bridges common in the central and southeastern United States. The bridge deck part was either reinforced or prestressed concrete slab and solid or hollow core, depending on the span length. The skewed pre-1971 two-span prestressed concrete box girder was selected by Zakeri et al. (Zakeri et al., 2015) with abutment seat type and four different skew angles (0°, 15°, 30°, and 45°) to explore the effect of skewness on the performance of retrofitted bridges. In addition to that, different types of bridges: multi-span-simply-supported (MSSS) concrete slab bridges, three-span continuous steel girder bridges, three-span continuous concrete girder bridges, three-span concrete box girder bridges, multi-span simply supported concrete girder (MSSSCG) bridge class, multi-span continuous slab (MSCS) bridge class, multi-span reinforced concrete continuous girder (MSRCCG), curved concrete box-girder bridges, multi-span continuous RC bridge, were selected based on different highway bridge design parameters for different studies condition on highway bridge design parameter (Du et al., 2018; Ghazali et al., 2019; Guo et al., 2015; Jeon et al., 2019; Li et al., 2020a; Nielson Bryant and DesRoches, 2006; Song et al., 2018; Wang et al., 2018). However, the MSRCCG is used for most small or medium-span highway bridges in China (Li et al., 2020b). Jeon et al. (Jeon et al., 2019) also selected concrete box-girder bridges commonly constructed at an interchange for the intersection purposes of a highway bridge. This bridge type also allows thermal expansion and contraction on a multi-frame within a span.

Furthermore, previous studies on sensitivity to earthquakes have primarily focused on regular bridges with typical configurations (Dukes et al., 2012; Padgett Jamie and DesRoches, 2007). They are focused on regular bridges and selected numerical parameters: restrainer cable, elastomeric bearing, steel jackets, Shear key, longitudinal and transversal reinforcement ratio, column height to column dimension ratio, column height to column dimensions ratio, span length to column height ratio, and superstructure depth to column dimension ratio. However, bridges with irregular configurations are more susceptible to severe damage or collapse during earthquakes, with various parameters affecting response, as studied by Nielson & DesRoches (Nielson and DesRoches, 2006). They are crucial factors in determining the bridge’s seismic response. The seismic response of irregular bridges is modeled using categorical and numerical parameters. Some categorical parameters include the type of bridge, design era, number of columns per bent, soil type, superstructure box type, number of box cells, and directional of applied excitation. On other hands, numerical input parameters include ground motion intensity measure, span length, column height, deck width, girder spacing, top flange thickness, bottom flange thickness, wall thickness, depth of superstructure, column diameter, longitudinal reinforcement ratio, confinement spacing, abutment back wall height, pile spacing, foundation rotational stiffness, restrainer length, initial slack in restrainer cable, restrainer stuffiness, restrainer yield deformation, number of restrainers, concrete compressive strength, reinforcing steel yield strength, shear key capacity, shear modulus, transversal gap between deck and shear keys, longitudinal gap between deck and abutment, pile stiffness, mass factor, damping ground motion time step, and skew angle.

In general, the highway bridge design parameter and conditions used for selection structure inventory are degree of skew, span length, span-to-column height ratio, reinforcement nominal yield strength, concrete nominal strength, amount of longitudinal column reinforcement, amount of transversal column reinforcement, column diameter to superstructure depth ratio, soil properties at pile shafts and bridge weight, restrainer cable, elastomeric bearing, steel jackets, Shear key, longitudinal and transversal reinforcement ratio, column height to column dimension ratio, column height to column dimensions ratio, span length to column height ratio, superstructure depth to column dimension ratio, design era, number of columns per bent, soil type, superstructure box type, number of box cells, directional of applied excitation, ground motion intensity measure, column height, deck width, girder spacing, top flange thickness, bottom flange thickness, wall thickness, depth of superstructure, column diameter, longitudinal reinforcement ratio, confinement spacing, abutment back wall height, pile spacing, foundation rotational stiffness, restrainer length, restrainer stuffiness, restrainer yield deformation, number of restrainers, shear modulus, transversal gap between deck and shear keys, longitudinal gap between deck and abutment, tall ratio, pile stiffness, mass factor, and damping ground motion time step (Alam et al., 2012; Billah et al., 2013; Jeon et al., 2019; Mackie and Stojadinović, 2001; Song et al., 2018; Zakeri et al., 2014, 2015). The highway concrete bridge structure is also classified into reinforced and prestressed concrete bridges, further classified as open or solid (Nielson Bryant and DesRoches, 2006). However, The CALTRANS training manual categorizes bridges into three eras based on their construction and seismic characteristics: pre-1971, 1971–1994, and post-1994 (Pahlavan et al., 2016; Ramanathan, 2012). Across these eras, the study reveals that diaphragm abutments are less vulnerable than seat abutments. For example, the integral pile columns are less vulnerable than traditional multi-column bents (MCBs) in the pre-1971 design era due to better confinement.

Meanwhile, seat abutments show no significant reduction in bridge system vulnerability for either type. Besides, in the 1971–1990 and post-1990 design eras, traditional MCBs were less vulnerable due to enhanced energy dissipation and ductile characteristics. However, the vulnerability of traditional MCBs decreases with an increase in seat width, as the abutment seat contributes more to overall vulnerability in the latter design era. In contrast, the percentage reduction in vulnerability between diaphragm and seat abutments is not consistent across the design era.

Formulation of highway bridge model

The analytical highway bridge model is developed by using different types of finite element modeling software: OpenSees (Jeon et al., 2019; Li et al., 2020a, 2020b; McKenna et al., 2010; Nielson Bryant and DesRoches, 2006; Nielson and DesRoches, 2007; Pahlavan et al., 2016; Song et al., 2018; Zakeri et al., 2015; Zhou et al., 2018; Zhou and Li, 2019), ANSYS, SAP (Vedhanayaghi et al., 2021; Wang et al., 2012, 2018; Wang and Wu, 2018), SeismoStruct (Alam et al., 2012), MIDAS. However, the OpenSees are famous, practical, and powerful for modeling the 3D finite modeling of the bridge components (Jeon et al., 2019; Li et al., 2020a, 2020b; McKenna et al., 2010; Nielson Bryant and DesRoches, 2006; Nielson and DesRoches, 2007; Pahlavan et al., 2016; Song et al., 2018; Zakeri et al., 2015; Zhou et al., 2018; Zhou and Li, 2019). The superstructure is always considered flexible during seismic excitation if the pounding impact is insignificant (Jeon et al., 2019; Li et al., 2020b). Therefore, deck and cap beams are modeled with linear beam-column elements.

The highway bridge has been demanded to seismic excitation events (Alam et al., 2012; Billah et al., 2013; Li et al., 2020a; Zakeri et al., 2015; Zhou and Li, 2019). However, the seismic isolation system is introduced to mitigate the load transfer to the substructure for both new and retrofitted highway bridge systems (Bayat, 2017; Nielson Bryant and DesRoches, 2006; Nielson and DesRoches, 2007; Song et al., 2018; Zhou et al., 2018). Generally, there are different types of seismic isolation systems. However, laminated rubber and sliding bearings are commonly used in highway bridge seismic isolation systems to transmit earthquake-induced forces (Alam et al., 2012; Li et al., 2020a, 2020b). The laminated rubber bearing shifts the natural bearing period to prevent excitation resonance filled with damping properties to prevent excessive displacement of the isolated bridges (Alam et al., 2012). The laminated elastomeric bearing is modeled by a nonlinear translational spring, which is determined based on the geometry with a perfect elastic-plastic, considering the degradation of bearing stiffness (Song et al., 2018). In our case, the laminated bearing is modeled with nonlinear translational springs, assuming it will slide after reaching its ultimate capacity (Song et al., 2018). The column cross-section and reinforcing steel are modeled by nonlinear beam-column elements, concrete 02, and Steel 02 in OpenSees (Li et al., 2020b). A nonlinear translational spring models the abutment’s passive, active and transverse action (Nielson Bryant and DesRoches, 2006; Xie et al., 2019). The hyperbolic backbone curve is essential to model the passive soil pressure (Shamsabadi et al., 2010). The linear translational and rotational springs are imperative to mimic the pile foundations to catch the translation and rotation behaviors of the foundation system (Li et al., 2020a). The elastomeric bearings, shear key, and pounding between the adjacent decks simulate an in-span hinge response (Guo et al., 2015; Jeon et al., 2019; Li et al., 2020a; Nielson Bryant and DesRoches, 2006; Shamsabadi et al., 2010; Song et al., 2018; Zakeri et al., 2015; Zhou et al., 2018). The pounding effect between the deck and abutments can be modeled using the gap and contact elements, considering the impact of hysteretic energy loss in OpenSees (Song et al., 2018). At the same time, the cross-section uses the fibers with the corresponding material stress-strain relationship (Jeon et al., 2019). Abutment behaviors significantly impact bridge system response under dynamic excitation, especially in short spans with high superstructure stiffness. Aviram et al. (2008) studied bridge seismic response sensitivity using three abutment modeling approaches: roller, simplified, and spring. They use the OpenSees finite element modeling platform to capture critical components and modes of abutment response. The simplified model is a compromise between the efficient roller and comprehensive spring models. A realistic abutment model should accurately consider all significant components of the abutment system, including mass, stiffness, and nonlinear hysteretic behaviors, to capture the bridge’s seismic response accurately.

A set of numerical bridge models in OpenSees accounts for material and geometric uncertainties based on a bridge inventory analysis (Billah et al., 2013; Jeon et al., 2019). Figure 5 defines and visualizes the bridge system’s finite element model and the modeled material properties for each bridge component. In addition, the Latin hypercube sampling considers the sampling approach instead of the random selection based on the static distribution of the uncertainty parameters (Jeon et al., 2019; Zhou and Li, 2019). Furthermore, the Monte Carlo method is essential to pair analytical bridge models with a chosen suite of ground motions (Billah and Alam, 2013; Jeon et al., 2019; Li et al., 2020b; Nielson Bryant and DesRoches, 2006; Nielson and DesRoches, 2007).

Choice of intensity and demand measure pairs

Appropriate IM and EDP measures are essential to developing an approximately linear relationship in logarithmic space. For each beam and column, Gauss-Lobatto integration points range from 4 to 7 to avoid slowing down the integration algorithm (Tavares et al., 2013). The IM is selected based on the ground motion parameter (PGA, PGV, and peak ground displacement (PGD)) (Alhan and Öncü-Davas, 2016; Bayat, 2017; Billah et al., 2013; Grigoriu and Radu, 2021; Hwang et al., 2000; Jeon et al., 2019; Kibboua et al., 2014; Nielson Bryant and DesRoches, 2006; Song et al., 2018; Zakeri et al., 2014; Zhou and Li, 2019) and maximum earthquake response (Ma et al., 2016). Table 3 describes the IM measurement for PSDA, defined by the symbolic representation and equations.

Table 3.

Symbolic representation and equations of IM measurement for PSDA.

Descriptions	Symbolic representation	Equations
Arias intensity	$I_{s}$	$I_{s} = (π / 2 g) \int_{0}^{\infty} U_{g}^{2} (t) d t$
Velocity intensity	$I_{v}$	$I_{v} = \int_{0}^{\infty} [{\ddot{U}}_{g}^{2} (t) / {\dot{U}}_{g} (t)] d t$
Cumulative absolute velocity	$I_{\| A \|}$	$I_{\| A \|} = \int_{0}^{\infty} \| {\ddot{U}}_{g} (t) \| d t$
Cumulative absolute displacement	$I_{\| v \|}$	$I_{\| v \|} = \int_{0}^{\infty} \| {\dot{U}}_{g} (t) \| d t$
Frequency ration 1	FR ₁	$F R_{1} = U_{g, \max} / {\ddot{U}}_{g, \max}$
Frequency ration 2	FR ₂	$F R_{2} = U_{g, \max} / U_{g, \max}$
Strong duration motion	T _d	$T_{d} = 0.02 e^{0.74 M} + 0.3 R$
RMS acceleration	$α_{R M S}$	$α_{R M S} = \sqrt{(\frac{1}{T_{d}})} \int_{0}^{d} {\ddot{U}}_{g}^{2} (t) d t$
Characteristics intensity	$I_{c}$	$I_{c} = α_{R M S}^{1.5} T_{d}^{1.5}$
Effective peak acceleration	EPA	$E P A = {0.4 S_{a, a v g} \|}_{0.1}^{0.5}$
Effective peak velocity	EPV	$E P V = {0.4 S_{v, a v g} \|}_{0.1}^{0.5}$
Effective peak displacement	EPD	$E P D = {0.4 S_{d, a v g} \|}_{0.1}^{0.5}$
Spectral quantity	S _d	$S_{d} = S_{v} / ω = S_{a} / ω^{2}$
Epicentral distance	R	—
Earthquake moment magnitude	M	—

Besides, the bridge components (e.g., superstructure, piers, columns, bearing, and abutments) are vital to estimating highway bridge PSDM (Jeon et al., 2019). Also, different EDPs are considered for highway bridge seismic demand analysis (see Table 4). On the other hand, the EDPs are often described with critical bridge components’ seismic demands (Du et al., 2018; Hariri-Ardebili and Saouma, 2016; Jeon et al., 2019; Nielson and DesRoches, 2007; Zhou and Li, 2019). The EDP is studied in various seismic vulnerability analyses (Li et al., 2020b; Ma et al., 2016; Nielson Bryant and DesRoches, 2006; Ramanathan et al., 2012; Song et al., 2018). Table 5 summarizes the description and symbolic parameter details used in various seismic vulnerability analyses.

Table 4.

Bridge performance and design parameter.

Parameters descriptions	Symbolic representations	Equations
Peak steel strain	$ε_{s}$
Peak concrete strain	$ε_{c}$
Curvature ductility factor	$μ_{ϕ}$	$μ_{ϕ} = ϕ_{\max} / ϕ_{y}$
Plastic rotation	$θ_{p}$	$θ_{p} = (u_{\max} - u_{y}) / H$
Principal compression	$P_{c}$
Principal Tension stresses	$P_{T}$
Drift ratio	$γ$	$γ = u_{\max} / H$
Displacement ductility factor	$μ_{Δ}$	$μ_{Δ} = {u_{\max} / u}_{y}$
Residual deformation index	RDI	$R D I = u_{r e s i d u a l} / u_{y}$
Equivalent viscous damping ratio	$ξ_{e q}$
Normalize effective stiffness	$n_{k}$
Hysteretic energy	HE	$H E = \oint F d u$
Normalized hysteretic energy	NHE	$N H E = H E / (F_{y} u_{y})$
Peak column curvature	$ϕ_{m} a x$

Table 5.

The summarized description and symbolic parameter details used in various seismic vulnerability analyses.

Parameter description	Parametric symbol	Reference
Peak column curvature ductility	$μ_{ϕ}$	(Du et al., 2018; Jeon et al., 2019; Zhou and Li, 2019)
Peak unseating deformation	$δ_{u}, i n m m$
Peak bearing shear strain	$γ_{b}, i n m m$
Peak radial (passive) abutment deformation	$δ_{p}, i n m m$
Peak radial (active) abutment deformation	$δ_{a}, i n m m$
Peak tangential abutment deformation	$δ_{t}, i n m m$
Column curvature ductility	$μ$	(Li et al., 2020b; Nielson Bryant and DesRoches, 2006; Ramanathan et al., 2012)
Longitudinal deformation in the fixed bearings	$δ, i n m m$
Transverse deformation in the fixed bearings	$δ, i n m m$
Longitudinal deformation in the expansion bearings	$δ, i n m m$
Transverse deformation in the expansion bearings	$δ, i n m m$
Peak Radial (passive) abutment deformation	$δ_{p}, i n m m$
Peak Radial (active) abutment deformation	$δ_{a}, i n m m$
Peak tangential abutment deformation	$δ_{t}, i n m m$
Displacement of bearings		(Ma et al., 2016; Nielson and DesRoches, 2007; Song et al., 2018)
Abutments
Displacement ductility of a column
Displacement drift ratio
Column curvature ductility	$μ$
Longitudinal deformation in the fixed bearings	$δ, i n m m$
Transverse deformation in the fixed bearings	$δ, i n m m$
Longitudinal deformation in the expansion bearings	$δ, i n m m$
Transverse deformation in the expansion bearings	$δ, i n m m$
Passive abutment deformation	$δ_{p}, i n m m$
Active abutment deformation	$δ_{a}, i n m m$
Tangential (transverse) abutment deformation	$δ_{t}, i n m m$
Horizontal displacement		(Hariri-Ardebili and Saouma, 2016)
Maximum crack length ratio
Maximum joint opening and maximum joint sliding of concrete dam
Transverse deformation of elastomeric bearing	$δ_{e t}, m m$
Peak unseating deformation	$δ_{u}, i n m m$
Transverse deformation of elastomeric bearing	$δ_{e l}, m m$
Deformation of abutment wing wall	$δ_{w w}, m m$
Deformation of abutment back wall	$δ_{b w}, m m$
Column drift displacement of expansion bearings
Displacement of fixed direction

Probabilistic seismic demand models

The PSDM is a mathematical expression that relates the IM and demand to predict a particular value for designing structures based on seismic performance (Gardoni et al., 2003; Hariri-Ardebili and Saouma, 2016). The PSDM is the response graph of the highway bridge during and after an earthquake due to seismic events. The likelihood of a critical bridge component experiencing a certain demand level for a selected IM is vital to developing the EDP and IM relationship. The IM measures such as PGA, PGD, Sa, spectral velocity (Sv), Arias Intensity (AI), and spectral acceleration at the first-mode period (Sa (T₁)) are used to analyze seismic demand and develop the PSDM. The core purpose of PSDM is to predict the seismic response on each component for a given deterministic or random variable. The seismic demand for bridges is mainly affected by steel yield strength (f_Y), concrete strength (f_C), and mass (m) (De Felice and Giannini, 2010). The first three external random variables have structural properties. At the same time, the last one is related to seismic input centered at the first natural period of the structure.

Besides, the multivariate probabilistic seismic demand model (PSDM) is developed using a multivariate logarithmic normal distribution, considering structural failure domains and failure mechanisms and focusing on the pier as the most vulnerable component (Wang et al., 2012, 2018; Wang and Wu, 2018). The structural failure domain is identified by constructing the multi-dimensional performance limit state (PLS) formula with considering PLS correlation. There are five common structural failure mechanisms: formation of column plastic hinges, restrainer failure at expansion joints or bridge abutments, pounding between bridge decks, column premature shear failure, and surrounding soil liquefaction. The whole structural performance is overestimated when the multiple components of the structure are not considered contributors to the structural failure during the seismic performance analysis. Generally, the bridge pier components are usually viewed as the most vulnerable component, which is developed to represent the vulnerability of the whole bridge.

Furthermore, the relation between EDP and IM gives the seismic demand response graphs called the PSDMs (Gardoni et al., 2003; Hariri-Ardebili and Saouma, 2016; Jeon et al., 2019; Ma et al., 2016; Mackie and Stojadinović, 2001; Pahlavan et al., 2016; Shome et al., 1998). Different seismic fragility analysis methods determine the PSDM through PSDA or PSHA (Hariri-Ardebili and Saouma, 2016; Shome et al., 1998). On the other hand, the IM measurement to the structure-class-specific demand measurement is conveyed with the cloud and scaling approach. The cloud approach is a powerful way to estimate the seismic demand and capacity for multiple intensity levels. In the “cloud,” the median structural seismic demand $(S_{d})$ and IMs follow a lognormal correlation and distribution according to the power-law function (Billah and Alam, 2013; Li et al., 2020b; Ma et al., 2016; Zhou and Li, 2019). Besides, regression analysis can obtain the mean and dispersion from analytical analysis assuming the power-law function, which gives a logarithmic correlation between median EDP and selected IM.

S_{d} = E D P = α {(I M)}^{b}

(6)

\ln (S_{d}) = \ln (E D P) = \ln (a) + b \ln (I M)

(7)

These a and b are unknown coefficients or regression parameters determined by regression conditioned on the IM from the bridge seismic response data. Also, the dispersion of the demand $β_{E D P / I M}$ condition upon the IM is estimated using equation (8) (Billah and Alam, 2013).

β_{E D P / I M} = \sqrt{\frac{\sum_{i = 1}^{N} {[\ln (E D P) - \ln (a I M^{b})]}^{2}}{N - 2}} = \sqrt{\frac{\sum_{i = 1}^{N} {[Y - (b x + a)]}^{2}}{N - 2}} = \sqrt{\frac{\sum_{i = 1}^{N} {[Y - Y^{*}]}^{2}}{N - 2}}

(8)

N is the total number of the ground motion suit. The efficient seismic demand model has a small dispersion and uses a small number of NLTHA to compute coefficients a and b with the same confidence. Scaling is another approach to developing the PSDM (Alam et al., 2012; Zhang and Huo, 2009). The scaling approach is comparatively precise and has more computational effort than the cloud approach. However, the cloud and scaling approaches are the most frequently used methods to develop bridge PSDMs. In general, seismic hazards, demand, and capacity are the uncertain components of the structure during the seismic design that the overall behavior of a bridge under earthquakes, including performance-based engineering. Figure 6 visualizes and assesses the schematic representation of the NLTHA procedure used to develop PSDMs of civil structures like highway bridges (Billah and Alam, 2013; Nielson Bryant and DesRoches, 2006; Nielson and DesRoches, 2007; Pahlavan et al., 2016; Ramanathan et al., 2012; Zhang et al., 2019). Different parameters are included with the ground motion suite, analytical bridge models with NLTHA, and bridge component response. On the other hand, Figure 7 visualizes and assesses the schematic representation of the IDA procedure used to develop PSDMs of civil structures like highway bridges (Billah and Alam, 2013; Nielson Bryant and DesRoches, 2006; Nielson and DesRoches, 2007; Pahlavan et al., 2016; Ramanathan et al., 2012; Zhang et al., 2019). Different parameters such as ground motion suite, analytical bridge models with incremental dynamic analyses, and bridge component response are included and described.

In general, the proposed methodology for developing the PSDM using different options is visualized in Figure 8. Furthermore, there are also different steps used to develop the PSDM through NLTHA (Nielson and DesRoches, 2007):

(1) The number of ground motion suites is selected from the earthquake database.

(2) Assemble a suite of N ground motions that apply to the geographical area of interest for the chosen intensity measure.

(3) The Latin Hypercube sampling technique generates the geometric and material variables of the N-statically significant and typically identical bridge models.

(4) Pairing steps of N-bridge models randomly with the N-ground motion pair

(5) Perform a complete NLTHA for each ground motion-bridge pair and record the critical component responses in each case. For each analysis, peak responses versus the peak value of the intensity are recorded and plotted.

(6) Perform the cloud and scaling approach for the analytical bridge-ground motion pairs to investigate the process of the PSDA and PSDM. In the cloud approach, the median structural seismic demand $(S_{d})$ and IMs follow a lognormal correlation and distribution according to the power-law function given by equations (6) and (7).

(7) Then, the graph of the peak responses against the ground motion gives PSDM.

(8) Finally, regression analysis estimates the dispersion β_D/IM and coefficients a and b from analytical analysis assuming the power-law function, which gives a logarithmic correlation between median EDP and selected IM using equations (6)–(8).

NB Step 4 repeats for all major vulnerable components in the bridge.

Research outlook for probabilistic seismic demand modeling and estimation method

Although a wide variety of methodologies exist for probabilistic seismic demand modeling and estimation for highway bridge development, there is still scope for significant improvement in the probabilistic seismic demand modeling and estimation methodology for highway bridges. Key features of the studies described the gradual development of probabilistic seismic demand modeling and estimation methodology. In this study, the proposed methodology is visualized in Figure 8 for developing the PSDM using different options. They help the decision-makers to develop and make decisions on the PSDM of the highway bridge. However, there remains scope to improve the existing probabilistic demand modeling and estimation development methodology for highway bridges.

On the other hand, the PSDA is an essential supportive tool to identify the seismic response more significantly than the seismic capacity condition on IM measurements. They evaluated by a representation series of ground motion time history (GMTHS), which has a more incredible response than the capacity condition on IM. In addition to that, different seismic analysis methods, including expert-based, empirical, experimental, analytical, and hybrid, are used to investigate PSDA and PSDM, but analytical methods are the most realistic, effective and practical when damaged data is insufficient. Besides, NLTHA provides a reliable and flexible analytical method for analyzing highway bridges’ seismic demand, allowing for consideration of linear or nonlinear cyclic material characteristics and geometric nonlinearities. Furthermore, the PSDM formulation through the NLTHA procedure is described more in Figure 6. Although this section briefly describes the possible future development of the seismic analysis method, further study along with detailed examples is required to check the adequacy of the proposed method.

Conclusions

This study reviewed the probabilistic seismic demand estimation and modeling methodology options associated with the procedure, analytical analysis, and mathematical framework for a highway bridge. In this case study, first, different seismic demand analysis methods, like analytical methods, and the current knowledge on the PSDM of the highway bridge are reviewed and presented in different techniques with its features, application, advantages, and limitations of a highway bridge. Various methods for PSDM are discussed, including their idiosyncrasy, application, and impediments. The steps to build PSDM are constructed and introduce how the scholars use it to analyze the PSDA and establish highway bridge’s PSDM. Comprehensive methodology options are provided with tables, figures, and a framework to analyze PSDA and develop PSDM for a highway bridge. Finally, this paper contributed insightful methodology options to help the decision-makers develop and make decisions on the PSDM of a highway bridge. Future work establishes PSDM methodology and more options for PSDA of a highway bridge structure.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by Chinese Polar Environment Comprehensive Investigation and Assessment Programmes (2023YFB2604402).

ORCID iD

Kirubel Tefera Gesho

Appendix

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