Fragility curve development of highway bridges using probabilistic evaluation (case study: Tehran City)

Seismic fragility analysis of bridges can be used to identify the probability of different damage states of bridges under earthquake hazards. This study aims to generate the fragility curve of highway bridges in Tehran, Iran, where these curves are not provided yet. To derive new fragility curves, the fragility curve of highway bridges from studies all around the world has been classified into different categories, based on their structural designs, bridge materials, country, and their analytical methods. A decision-tree analysis has been used to determine new fragility curves. The accuracy of these new curves was then assessed using a case study bridge and incremental dynamic analysis. The assessment analysis indicates that newly developed fragility curves for Tehran city have reasonable accuracy. The average difference between the derived fragility curve by the curves determined using Incremental Dynamic Analysis was acceptable. These new fragility curves helped to identify bridges throughout the Tehran city highways with higher seismic vulnerability. These can also be used for risk assessment and reducing seismic risk studies and future urban planning.


Introduction
One of the most common methods used in recent years to investigate seismic vulnerability is using fragility curve analysis. Fragility curves are conditional probability statements of a bridge's vulnerability as a function of ground motion intensity. Using the fragility analysis of bridges, the probability of different damage states of bridges under earthquake can be obtained. Using this the seismic performance evaluation of the bridge structure system can be investigated. Fragility curves can be used for different purposes including post-earthquake initial assessments, economic assessment of damages, design control, and assessing the type of bridge system. These curves can also be used to optimize bridgeretrofit methods and to develop post-event action plans.
Recently, frequent earthquakes have caused enormous economic loss, and it has adversely affected human life. Bridges, key components of the nations' roadway networks, are extensively distributed and could be easily damaged by earthquakes. Therefore, when earthquakes occur, many highway bridges will be directly damaged, which not only leads to a wide variety of troubles for communities but also creates considerable problems in compensatory damages . Thus, the vulnerability assessment of bridges is valuable for disaster response planning, assessment of direct monetary loss, decision analysis for seismic retrofit of structures, and estimation of loss of functionality of highway systems; and, therefore, it is essential to recognize the degree of damage to the highway bridge structures owing to earthquakes. To evaluate a damage level (no damage, slight damage, moderate damage, extensive damage, and complete damage) to highway bridge structures, fragility curves are found to be a practical tool Mander & Basöz, 1999;Yamazaki et al., (2000).
Fragility curves have been vastly used in probabilistic seismic risk assessment of highway bridges. They are conditional probability statements that present the probability of meeting or exceeding a specific level of damage for a given ground motion intensity measure. The conditioning parameter is generally a single intensity measure, namely peak ground acceleration (PGA) or spectral acceleration at the geometric mean of the longitudinal and transverse periods Ramanathan (2012).
According to Novak and Collins, the geometric characteristics of an existing bridge are inherently probabilistic. These uncertainty specifications include not only the geometry and materials of the bridge but also the site specifications, magnitude, and direction of the earthquake. In this research, the Latin Hypercube method was used to develop the fragility curve and assess the vulnerability of the bridge Ramanathan (2012). The 1989Loma Prieta, 1994Northridge, and 1995 Kobe earthquakes were a turning point for fragility investigation. Several researchers established empirical fragility curves determined by the actual damage of these three earthquakes Basöz et al., (1999;Elnashai et al., (2004). Shinozuka et al. (2001) applied the Maximum Likelihood Method (MLE) to determine the parameters that characterize the curve, i.e., the median θy and the log-standard deviation β. This method presumes that the fragility curves are conveyed by the cumulative lognormal distribution. Basöz and Kiremidjian (1999) developed fragility curves by assembling damage frequency matrices and performing logistic regression analysis. The observations on bridge damage after the Northridge earthquake were used in the research, bridges were classified into 11 classes, and fragility curves were developed for each category. Der Kiureghian (2002) utilized a Bayesian approach and the Likelihood function to create fragility curves. Nevertheless, generating empirical fragility from actual earthquake data is straightforward. Still, it has some limitations and disadvantages, such as the lack of adequate damage data, the discrepancy between the damage level assessments in the aftermath of a seismic event, and the difference in the ground motion intensities depending on who generated them. Therefore, another essential source of uncertainty, analytical fragility curves, has entered into fragility curves Ramanathan (2012).
The lack of adequate and actual earthquake damage data has led to the development of fragility curves using analytical and simulation-based methods to assess highway bridges' performance. There has been a great deal of research in which analytical fragility curves for bridges have been used. In this methodology, the structural demand and capacity ought to be modeled to develop analytical fragility curves. Various researchers have utilized several procedures to achieve this goal. The methods they have used range from simplistic to fairly elaborate (Nielson (2005). Yu et al. (1991) evaluated the seismic vulnerability of highway bridges in Kentucky. They applied simple singledegree-of-freedom models and Elastic Response Spectrum Analysis (RSA) to generate seismic fragility curves. Hwang et al. (2000) further developed this methodology by assessing uncertainties in capacity assessment and seismic demand for the major bridges in Memphis. In this procedure, Capacity/Demand (C/D) ratios are produced for varied bridge components. The capacities of the bridge components are determined based on the Federal Highway Administration's (FHWA) Seismic Retrofitting Manual for Highway Bridges. Hwang et al. (2000) concluded that the results of this procedure compare logically well to those created by more complicated methods. Still, this methodology has a few drawbacks. For instance, a notable nonlinearity occurs; this method does not accurately predict the displacement demand and force (Ramanathan, 2012).
A full non-linear time history analysis can be extremely time-consuming and arduous. Therefore, a simplified procedure has been introduced that has the advantages of a nonlinear analysis but does not suffer from the computational cost of a time history analysis. This procedure is a nonlinear static method and is widely known as the Capacity-Spectrum Method (CSM) (FEMA, 2003). Nonlinear static procedures (NSP) are alternative methods that utilize the force-deformation characteristics of structures deriving from pushover analysis. This approach has been used by several researchers to develop analytical fragility curves for bridges Banerjee and Shinozuka (2007;Mander and Basöz (1999). What is significant is that this procedure was adopted in the generation of seismic bridge fragility curves for " Hazards US" or HAZUS FEMA (2003).
Seismic fragility curves can also be developed by a nonlinear time history (NLTH) approach, which is believed to be the most demanding procedure for evaluating the inelastic seismic demands of structures. However, this method tends to be the most time-consuming and computationally expensive, but it is one of the most reliable approaches available Shinozuka et al., (2000). The response could often be highly delicate to the characteristics of the individual ground motion utilized as seismic inputs. Hence, multiple analyses are mandatory, using various ground motion records to attain a reliable estimation of the probabilistic distribution of structural response Ramanathan (2012). This methodology has been used in multiple ways by different researchers to develop fragility curves Hwang et al., (2001;Padgett and DesRoches (2009).
Incremental Dynamic Analysis is a nonlinear dynamic analysis that eases seismic structural demand and capacity comparison by using a series of NLTH for ground motions. This is a parametric method in which one or more seismic records are each scaled to a certain intensity and applied to the structure. IDA not only evaluates the seismic behavior of the structure but also shows the structure's capacity and can be used to determine the seismic performance of structures Ramanathan (2012). Vamvatsikos and Cornell (2002) introduced the overall formulation of this methodology, and it has been utilized in various forms in the work of numerous researchers. Tran et al. (2021) utilized the finite element analysis program, OpenSees, to conduct IDA, and indicated that bridges with shorter piers are less vulnerable to earthquakes than bridges with higher piers. Aldea et al. (2022) analyzed the seismic performance of Chilean skewed highway bridges using nonlinear static and incremental dynamic analyses and fragility curves and concluded that repair measures considered after the 2010 Maule earthquake enhance the seismic performance of skewed bridges and effectively avert span unseating.
Various methodologies for generating the fragility curve of bridges have been developed. Among different approaches, fragility curve methods using analytical approaches have become widely adopted. This is because these analytical approaches, in comparison with more traditional experimental methods, have become widely adopted because they are more readily applied to bridge types and geographical regions where seismic bridge damage records are insufficient. The aim of this study is to quickly extract the fragility curve of highway bridges for a city by analyzing and reusing the history of existing curves from other studies in the past. These fragility curves were created for bridges with different structures.

Methodology
The fragility curve of highway bridges from studies all around the world has been classified into different categories based on their structural designs, bridge materials, country, and analytical method used to calculate them. By using these existing fragility curves and deploying decision trees analysis, new fragility curves were developed for Tehran, which is a city vulnerable to earthquakes. The accuracy of new curves was then assessed using a case study bridge and incremental dynamic analysis (Nielson, 2005;Ramanatha, (2012). The survey method and chart used in this research are shown in (Fig. 1).

Extracting fragility curve from existing studies
Different methods have been developed to determine the fragility curves of bridges. These methods will be divided into three main categories of experimental fragility curves, analytical seismic fragility, and based on the opinion of the bridge (Fig. 2). 41 well-known and highly cited studies were documented and their fragility curves were extracted. Table 1 summarises the year of research, region of study, and method used to determine the fragility curve. In total 400 fragility curves were derived from these studies.

Classifying bridges of Tehran city
Highways in Tehran city have more than 400 multilevel interchanges. 600 bridge structures were made on these junctions. Figure 3 shows the location and structural framework of these bridges. Table 2 summarizes information about these bridges. Investigating these bridges indicates that there are 72 types of bridge structures. These bridges were then divided into six main categories of structural framework designs including simple, steel, steel girder-slab concrete, slab concrete-box steel, slab concrete, and slab box concrete structures. Bridge structural frameworks are the main parameters that can be used to derive fragility curves. Parameters such as materials of structure, number of bridge spans, number of bridge piers, type of bridge structure, and design method.

Classifying fragility curves of existing bridges
To derive weighting coefficients in decision tree analysis, designs from previous studies that were used to extract fragility curves were compared against the structural designs of Tehran city bridges. These coefficients, which ranged between 0 and 1, were selected based on engineering judgment for each parameter. These coefficients are listed in Table 3. For instance, for steel girder-concrete slab bridges, a weighting coefficient of 0.2 for only concrete or only steel was considered, whereas for composite material weighting of 0.6 was used. For this type of bridge, based on the number of spans for multi-span bridges a coefficient of 0.6 and for single-span 0.4 was used. These bridges were mostly made when earthquake standards were not commonly used in structural designs. Therefore, a coefficient of 0.67 for no seismic design method and 0.33 for the seismic design method was considered. Continuous restraints were considered 0.7 and 0.3 for simple restraints.
Although the design methods were similar, countries may have different standards and executions. This will potentially impact the resistance of the bridges to earthquakes and therefore, derive fragility curves. To select reference studies, we tried to find variable information from different parts of the world. Selecting studies from various countries made our analysis more robust by considering regional execution and structural supervision. Table 4 summarizes weighting coefficients (value), which range between 0 and 1 for each country based on the similarity of bridge structure design of those countries to Tehran 1 3 city bridges. For instance, coefficient 1 was considered for the country of Turkey because of the similarity in the method of design, execution, engineering culture, and details of bridge structures in Turkey to Iran (Tehran is the capital of Iran). Smaller coefficients are for those regions with less similarity. For European countries with less earthquake occurrence, regional coefficients were considered way less than Tehran.
The fragility curve of bridges was also classified based on the analytical method and its development technique. Four analytical groups were used to derive fragility curves including physical-based relations, non-linear static analytical methods, non-linear dynamic analytical methods, and coupled physical-based and non-linear dynamics analytical methods.
Analytical methods to derive fragility curves will have a great impact on their predictive accuracy. Physical-based methods are very accurate techniques for deriving fragility curves since they are based on real previous earthquakes. In addition, the non-linear dynamic analytical method has reasonable accuracy for deriving fragility curves. In Table 5, coefficients between 0 and 1 were considered for each analytical method. Higher coefficients represent analytical methods that have greater accuracy and are closer to reality.

Derived fragility curves
The decision tree method and coefficients explained in previous sections were used to derive fragility curves. These were discrete curves in which the x-axis was peak ground acceleration (PGA) and the y-axis was the probability of damage state exceedance (PDSCE). Using decision tree methods and considering various uncertainties, different values were derived from fragility curves. Figure 4 indicates the fragility curves of bridges with six different structures and different ages and four scenarios of damage state exceedance including slight, moderate, extensive, and complete. These values which were derived from the fragility curve of previous studies were classified based on the algorithm explained in Fig. 1. These were then employed in the decision tree analysis to derive new fragility curves for Tehran city bridges for different structural systems.
A bridge case study to assess the accuracy of the adopted fragility curve A bridge in a junction between Modarres highway and Motahari Street with more than 10 years of age was used as a case study (Fig. 5) to assess the accuracy and robustness of the method used in this study.

Structural characteristics of case study bridge
This case study bridge has two separate decks with a concrete slab box structure. The lengths of each span are 15 m, where joins expansion between two decks. The longitudinal length of the bridge deck is 31 m with 31 m in width. Each deck has a 12-degree slope with 3 lanes.
The height of the bridge column was varying between 6.7 and 7.3 m. The height of bridge piers was ranging between 4.9 and 5.5 m. The foundation of the middle piers had a length of 33 m and bottom width of 6 m. The foundation of abutments had a width of 5.05 m and a length of 33 m (Fig. 6). The details of the bridge structure are listed in Table 6.

Ground motion suite
To do the analysis, a series of earthquake records that represent earthquake hazards for the study region were used. Earthquakes with magnitudes between 6 and 8 and epicentral distances of more than 8 km were considered for this analysis. Shome (1999) suggested that 10 to 20 records of earthquakes are sufficient for analysis. In this study, 20 earthquakes from Iran and across the world from the PEER earthquake dataset were selected (PEER (2022). Table 7 summarizes the different characteristics of these 20 earthquakes used in this study.  In addition, earthquake ground motion records by comparing periods in seconds versus spectral acceleration (g) for different percentiles were shown in Fig. 7. Different colored curves represent every 20 earthquakes listed in Table 7 which with IDA analysis mean, 25 percentiles, and 75 percentiles were calculated (shown with black color in Fig. 7).

Assumptions and modelling approach
For non-linear incremental dynamic analysis, OpenSEES software was used. For this analysis, fiber element of the column-beam with concentrated plasticity was used. In concentrated plasticity modeling, a plastic hinge was considered as a point in a center of a longitudinal element (Fig. 8). Therefore, non-linear incremental dynamic analysis was considered plastic in the hinge and elastic in the rest of the pier lengths.
The bridge under study was modeled in 3D in OpenSEES software in a completely accurate manner and following valid papers. In the following, the modeling technique used for important components of the bridge in the OpenSEES software is briefly explained.
The modeling of the bridge deck has a great effect on the response and behavior of the bridge in dynamic analysis. In this research, considering that the behavior of the deck is not expected to enter the nonlinear phase, the bridge deck is modeled linearly Naseri et al., 2020b;Pahlavan et al., (2018). To model the deck behavior to be linear, elastic beam-column linear elements are used.
Bridge poles are sub-structure components that act as intermediate supports for the bridge deck in horizontal and vertical directions. Non-linear inelastic beam-column elements, based on displacement, have been used to model the columns.
Considering the existence of various methods in modeling the abutments, in this study, to model bridge abutments, the method proposed as the result of the work of Aviram et al. (2008) is used. In this model, a series of nonlinear springs were considered to model the behavior of the abutment. To model the abutment, two parts, including pile and soil, should be modeled.
Hyperbolic gap materials were used to model the soil behind the abutment. This model was done using the latest studies, namely the pseudo-hyperbolic model proposed   by Shamsabadi and Yan (2008). Using the results of the experiments conducted on real abutments, they used two numerical models (pseudo-hyperbolic logarithmic spiral model and finite element model) to construct the pseudohyperbolic force-deformation relationship, which is stated in Eq. (1): In this equation, F is the lateral force per unit of wall width and in lateral displacement y, H is the height of the abutment, a, b, c, and n are constants of the equation and have different values for granular and cohesive soils. The numerical values of these constants are presented in Table 1. After calculating F ult , the value of K is also easily calculated.
To model the piles, the trilinear model of Choi (2002) has been used. The piles and the soil act as two parallel springs in the longitudinal direction of the passive state, so the values of the forces in their force-deformation diagram are added in this direction and in the same deformations. In the transverse direction as well as in the longitudinal direction of the active state, only the piles work. The proposed equations of this model are presented in relation 2. Moreover, the trilinear model provided by Choi is shown in Fig. 8. where K eff is the strength of a pile, K 1 and K 2 are the parameters in the diagram, and Δ 1 and Δ 2 are allowable displacements. After calculating the bearing capacity of the piles, the characteristics of the piles were assigned to the section using zero-length elements Naseri et al., (2021).   Elastomeric pads are a common type of concrete bridge supports. These supports transmit the horizontal force with the help of friction, and their behavior depends a lot on the initial stiffness. In the investigated bridge, elastomeric pads have been used in the abutment and pier cap area.
With the increase of friction, the stiffness of the support pad reaches zero, so the response of these members can be modeled with fully elastic-plastic material. In OpenSEES, Steel01 material has been used to model this behavior Naseri et al., (2021). The ultimate strength of the elastomer F y and the initial stiffness of this material K pad are determined by Eq. (3) where G is the shear modulus, A is the cross-sectional area (length in width) and h is the thickness of the pad. Figure 8 shows the behavior of this material.
The corresponding yield force, Fy, is obtained by multiplying the vertical force on the support by the coefficient of friction between elastomer and concrete ( μ ). In 1981, Scharge presented Eq. (4) for the friction coefficient between concrete and elastomer based on experimental observations, where σ m is the normal stress on the pad in MPa (Schrage 1981).
Again, zero-length elements are used for modeling elastomeric pads. It should be noted that the mentioned materials are assigned to the element in both longitudinal and transverse directions. In other words, it is assumed that supports act in both longitudinal and transverse directions (Naseri et al., (2020c).
Shear keys are often used in bridges with medium or low spans and to provide a lateral support for the bridge deck. These members do not bear gravity load, but during an earthquake, they transfer the reaction of the superstructure to the abutment and pile cap. Then the piles and the side walls of the abutment as well as the piles transmit the mentioned  reaction to the ground as a shear force. The design of these members is such that their final capacity does not exceed 75% of the shear capacity of the piles plus one of the side walls. Megally et al., in collaboration with Caltrans, conducted almost extensive field tests for both types of shear keys and presented separate analytical models for each. In this research, only external shear keys are considered. Megally et al. (2001) assumed the maximum displacement for a shear key to be constantly equal to 10 cm. In OpenSEES, hysteretic materials are used to detect the above-mentioned behavior of the shear key. The parameters related to these materials are defined in such a way that the function of the keys is like a fuse, that is, after failure, they do not have any more strength. It should be noted that these materials are applied only in the transverse direction. Figure 8 shows the cyclic behavior of an external shear key, which is the result of studies by Megally et al (2001).
Collisions between bridge elements (deck and abutment wall) are among those that have been proven to have a large effect on the seismic response of bridges. Therefore, it is necessary to consider them in the analytical model. One of the usual methods in modeling this effect is the use of elements known as impact elements, which are activated only when the seam is closed. The analytical model of the impact element is shown in Fig. 8. Ramanatan (2012) obtained the values of the impact element parameters for a width of 1 m, which were modified according to the distance of the nodal points of the abutment (Eq. (5)). Table 8 brifely shows how to model each of the different components of the bridge along with its reference. Figure 8 shows a view of the studied bridge along with the non-linear behavior of its various components Table 9.  Modeling uncertainties related to the structure and materials used in the bridge and design of the deck and foundation for different damage levels were determined using Ramantan estimatesFig. 9. Therefore Table 7, the analytical results of Modarres-Motahari case study bridge were compared based on the Ramantan study values to derive its fragility curve (Fig. 10). These results were listed in Table 10. In addition, the Period for the first two modes of this bridge is shown in Fig. 11.  (Naseri et al., 2022)

Incremental dynamic analysis
To do the incremental dynamic analysis, firstly, records of the earthquake were normalized by a maximum acceleration of 0.35 g. Based on the period mode of those bridges, each record was then scaled with 0.1 g time steps. In the dynamic analysis, the displacement of top piers was considered as a damage parameter, and spectral acceleration was considered as the intensity of earthquakes' ground motion. Incremental dynamic analysis was then conducted at each time step and at the end of each analysis step and under each scale record, a structural response curve was drawn against seismic intensity. Based on the IDA curves, the seismic behavior of piers under each ground motion was determined. Using these curves corresponding yield points, collapse threshold, and instability of bridge piers were given. Figure 12 shows IDA curves for example bridge pier under transverse and longitudinal tensions. As shown in Fig. 12, bridge piers have elastic behavior and then enter a level of plasticity until the bridge collapses. In the instability stage, bridge piers do not show any tolerance and with a slight increase in the intensity of the earthquake, structural displacement will significantly increase.
Based on the analyses, the fragility curves of the case study bridge for Modarres-Motahari Highway junction were determined. As shown in Fig. 13, fragility curves developed using statistical and analytical methods were compared. For a bridge with a concrete slab, by comparing developed fragility curves using the method explained in this paper and fragility curves derived from the IDA method, for the slight level, there is an agreement between these two approaches.
The results of two tree analysis models (model number1) and IDA analysis model (model number2) indicate that for the slight level for PGA less than 0.4 g, it almost matches, but for PGA greater than 0.4 g, average difference between these two methods was 8%. For moderate level, differences were negligible for PGA of less than 0.8 g and were about 10% and for PGA above 0.8 g, this discrepancy comes to 15%. For the complete level, only for PGA less than 0.3 g, the resulting values were close to each other, but for PGA greater than 0.3 g, a greater difference was obtained and it reached 18% and for extensive level, only up to the PGA less than 0.3, the results were close, but for PGA greater than 0.3, difference goes to 22%.

Conclusion
Earthquake is one of the natural disasters that threaten major cities around the world. To reduce human, economic, or even political and social losses, modeling of urban structures and their analysis for possible earthquakes need to be carried out so that it can help city officials make decisions more efficiently. One of the required tools that will be effective in the modeling of structures is the fragility curves for each  structure, which can be expressed according to the probability of an earthquake, and the damage percentage of that structure for different situations. Therefore, in this research, a fast method with an acceptable approximation was introduced to generate the fragility curves of bridges. In this research, the fragility curves of Tehran bridges were fully extracted by using the available fragility curves information obtained from previous studies, by inserting the weighting coefficients of these studies in the decision tree analysis. The more data available about the bridges and the more experts' opinions used, the higher accuracy can be achieved.
The evaluation using a case study showed that the newly developed fragility curves for the city of Tehran have 1 3 reasonable accuracy. So that the results of two tree analysis models (model no. 1) and IDA analysis model (model no. 2) show for the slight level, the difference between these two methods is on average 8% and for the moderate level this difference is 15% And it reaches 18% for the complete level and 22% for the extensive level.
Using these newly developed fragility curves can be used for large-scale planning and modeling with high analysis speed.
The outputs obtained from the modeling and analysis of the structures or elements of the vital arteries of a city for a possible seismic risk can be a great help to urban risk management to take the necessary decisions to face and deal with this crisis, which will reduce casualties and make a city more resilient. It can be used normally or even in industries such as earthquake insurance.