A PSDM describes the seismic demand of a structure or structural component in terms of an approximate IM (Padgett and DesRoches, 2008), P[EDP≥dIM]=1-ϕln⁡(d)-ln⁡EDP^βEDPIM, where the median EDP was estimated as a power model, EDP^=aIMb, or linear logarithm model, ln⁡(EDP)=ln⁡(a)+bln⁡(IM), where a and b are regression coefficients, ϕ is the standard normal cumulative distribution function, EDP^ is the median value of engineering demand, d is the limit state to determine the damage level and βEDPIM (dispersion) is the conditional standard deviation of the regression (Siqueira et al., 2014), βEDPIM≅∑ln⁡di-ln⁡aIMb2N-2.

Component fragility describes the seismic behavior of different components under the same level of damage and allows the weakest bridge component to be determined. Buckling capacities of all members were calculated, and fragility curves for Truss Piers, Trusses, and Stringer members were derived using PSDMs. Truss piers are the most fragile component because of their total length and the slenderness of their elements.

However, the point of system fragility is to determine all possible damage probabilities in the system, since all components must be considered to derive the overall bridge fragility curve. Bridge damage probability for a chosen limit state is the union of probabilities of each component for the same limit state (Nielson and DesRoches, 2007), PFailsystem=⋃i=1nPFailcomponent-i. Correlation coefficients between EDPs and IMs were considered to follow the conditional joint normal distribution, and system fragility curves were derived using Eq. (6) on 106 demand samples generated by Monte Carlo simulation.

The effect of ground motion on the structure was obtained using a linear or nonlinear mathematical model. Nonlinear dynamic time history analysis was used to minimize structural response uncertainties and provide the relationship between ground motion IMs and EDP. These relations can be obtained using cloud (direct) (Shome, 1999), incremental dynamic analysis (IDA) (Vamvatsikos and Allin Cornell, 2002), or stripes (Mackie and Stojadinovic, 2005) methods. This study employed the cloud method, including many real ground motion records, without prior scaling (Mackie et al., 2008).

Earthquake data were selected considering different soil types, moment (4.9–7.4), PGAs (0.01–0.82 g), and central distances (2.5–217.4 km). Figure 3 shows the distribution of moment with central distance. Sixty real earthquake data were chosen for soil types A, B, and C, and unscaled earthquake data were used for time history analysis (Pitilakis et al., 2004).

The bridge elements were modeled by a 2 node beam element. Member support and release were applied according to as-built drawings and site visual inspections. The difference between member center points and connections were all considered as rigid bars account for moments that can occur due to eccentricity.

Sleeper and rail profile weights were calculated and applied to the stringer beams as dead load (mass and load). The bridge material was assumed to be ST37 steel, given the construction time. Dead load calculation was performed using the Sap 2000 finite elements software, incorporating the given member properties (area, length, and density). The finite element models were composed of 1609 frame members, 832 nodes, and 120 link elements. A three-dimensional model of the bridge is shown in Fig. 4.

Time history analyses were applied to the model, considering both material and geometrical nonlinearity. Hinges were defined as steel interacting PMM plastic hinges, as defined in FEMA 356, Eq. (5-4) (FEMA-356, 2000) To detect bridge hazards, plastic hinges were defined at the start, middle, and end points of all frame members. Geometric nonlinearity is defined as Δ - δ, and large displacement Newmark direct integration was employed for the analysis. All three earthquake components (one longitudinal and two horizontal) were defined in the time history process.

PSDMs are traditionally conditioned on a single IM, and the degree of uncertainty depends on the selected IM. Therefore, determining the optimum IM is an important step to derive more realistic fragility curves. There are many different IMs used to characterize seismic behavior, and 9 of the most common, as shown in Table 1, were selected and compared in terms of practicality, efficiency, and proficiency.

The PGA, PGV, Sa-0.2s, and Sa-1.0s IMs are characteristic parameters obtained from earthquake records and related to vector characteristics of ground motion, such as acceleration and velocity. IA, IV, CAV, CAD, and ASI IMs are intensity measures characterizing seismic ground motion and are related to ground motion energy (Hsieh and Lee, 2011; Kayen and Mitchell, 1997; Mackie and Stojadinovic, 2004; Özgür, 2009).

Practicality is defined as the correlation between an IM and demand on a structure or structural component, and the more practical intensity measures tend to produce higher correlations. The practicality can be evaluated with the PDSM regression parameter, b, where larger b implies a more practical IM.

Efficiency is defined as the demand alteration for a given IM and can be measured by dispersion, with smaller dispersion implying a higher efficiency IM. Proficiency includes practicality and efficiency (Padgett et al., 2008), defined as modified dispersion, ζ=βEDPIMb, where smaller ζ implies a more proficient IM. Table 2 shows the demand models and intensity.

Maximum b = 0.67, 0.63, and 0.37 for the longitudinal, transverse, and gravity directions, respectively. Since higher correlation PSDM provide more realistic results, higher correlation IMs are more practical. Thus, ASI is more practical than other IM options.

Minimum βEDPIM = 0.45, 0.50, and 0.77 for the longitudinal, transverse, and gravity directions, respectively. Since smaller dispersion PSDM give more accurate results, smaller dispersion IMs give efficient results. Thus, ASI is more efficient than other IM options. Neilson derive fragility curve for Multi span simply supported (MSSS) steel girder bridge for two suite of ground motions and determine βEDPIM as 0.51 and 0.44 for column curvature, which is related to top displacement of bridge. βEDPIM obtained in these study gives similar results (Nielson, 2005).

Minimum ζ = 0.67, 0.79, and 2.04 for the longitudinal, transverse, and gravity directions, respectively. Modified dispersion describes IM practicality and efficiency, where smaller ζ implies higher correlation and less dispersion between IMs and EDPs. Thus, ASI is more proficient than other options.

PSDMs were constructed from the peak transverse displacement on the top of the middle pier of the bridge. Sixty nonlinear time history analyses were employed for the selected IM, ASI, following Sect. 5.1. Figures 5 and 6 show there is a good correlation between ASI and transverse displacement of the bridge-middle pier, and ASI and PGA have good correlation, respectively.

There is limited information about damaged steel truss bridges in the literature, but it shows that the main damage causes are buckling of the upper and lower brace elements, and shear failure of the transverse element; with corrosion hastening quickens the phenomena (Kawashima, 2012; Bruneau et al., 1996). Stewart et al. (2009) generated a reliability assessment of a steel truss bridge considering steel element tension and compression capacity as the limit state, calculating the tension capacity as the effective net area of steel members. However, the calculated buckling capacity was found to be smaller than the tension capacity of the members, so the buckling limit is overcome for the compression members during the calculations.

Since there were no specimens tested for the Alasehir bridge, material properties were chosen from previous studies in the literature. Yield and ultimate strength of European railway and roadway bridges are fy = 200 MPA and Fu = 360 MPa, respectively (Larsson and Lagerqvist, 2009). Structural element capacities for the Alasehir bridge were subsequently calculated depending on geometrical and material properties.

Seismic action may cause train derailment or even overturn, e.g. 1999 Kocaeli Turkey earthquake (Byers, 2004). Therefore, this study also considered serviceability, considering lateral displacement as the serviceability limit state. EN1990-Annex A2 includes lateral displacement limits for railway bridges (EN1990-prANNEX A2, 2001), including maximum angular variation and minimum radius of curvature to limit lateral displacement for different velocities, as shown in Table 3.

Derailment and overturning can occur under seismic conditions, with 3 such events observed for the Kocaeli, Turkey earthquake (Byers, 2004). Proscribed serviceability limits to minimize such events were used as the limit state in this study (EN1990-prANNEX A2, 2001), considering speed ranges (km h-1) V < 120, 120 < V < 200, and 200 < V. Railway transport has become more important for both goods and humans, and service speed affects transport capacity and quality (Lindfeldt, 2015)

Figures 7 and 8 show the probability of exceeding of serviceability limit states. The 50 % probability of exceeding the serviceability limits occurs for V > 200 km h-1 at ASI = 250 cm s-1 (Fig. 7) and PGA = 0.11 g (Fig. 8) for V > 200 km h-1, ASI = 850 cm s-1 and PGA = 0.5 g for PGA for 120 < V < 200 km h-1, and ASI = 6250 cm s-1 and PGA = 4.8 g for V < 120 km h-1. There is good correlation between ASI and PGA (Sect. 5.2). Median values for V > 200 km h-1 at PGA = 0.3 g, for 120 < V < 200 km h-1 at PGA = 0.70 g and for V < 120 km h-1 at PGA = 2.36 g. Nielson determined median values for MSSS steel girder bridge as 0.38 g at slight damage which was close to V > 200 km h-1 median value (Nielson, 2005) A PGA value of 10 % probability of exceeding in 50 year is 0.51 g according to Turkish seismic risk map. Probability of exceeding serviceability limit state for these values are 78, 29 and 0.007 % for V > 200 km h-1, 200 > V > 120 km h-1 and V < 120 km h-1 respectively. Probability of exceeding for the same hazard level for MSSS steel bridge are 24, 45, 58 and 85 % for slight, moderate, extensive and collapse damage level respectively. Serviceability damage level is assumed to slight damage (Tsionis and Fardis, 2014). For 200 > V > 120 km h-1 serves velocity fragility values are close to MSSS steel bridge slight damage fragility values.

Element buckling capacities were calculated using AISC 360-2010 specifications, considering axial forces and moments acting on the bridge components, and were used to specify whether damage occurred to the component. Component fragility curves were derived based on the two-parameter log-normal distribution, as shown in Fig. 9.

Truss piers are the most vulnerable bridge elements. There are 5 piers in the Alasehir bridge, from 12–40 m long and constituted of steel truss elements, i.e., truss pier elements are slender and have relatively low buckling capacities. On the other hand, bridge superstructures are the safest components, with significantly lower buckling probabilities. Seismic risk analysis is required to obtain more information about overall bridge reliability.

The entire bridge fragility was derived using a joint probabilistic seismic demand model (JPSDM) and limit state models. Demands for all components were generated using Monte Carlo simulation (106 samples), based on the PDSM median and standard deviation, with fragility curve calculated from Eq. (6).

Bridge overall fragility was also calculated from the system upper and lower bounds. For a serial system, the bounds can be expressed as max⁡i=1nPFi≤PFsystem≤1-∏i=1m1-PFi, where P(Fi) is the failure probability of component i, and P(Fsystem) is the failure probability of the system. Maximum of component failure probability provides the lower bound (Nielson and DesRoches, 2007). Since there is some correlation between component demands, this provides a non-conservative result. In contrast, the upper bound assumes no component correlation and hence provides a conservative result. The actual system fragility curve is expected to lie between the upper and lower bound curves. If only one component significantly affects system fragility, the bounds become close, whereas if many components affect the system, the bounds can become wider (Nielson and DesRoches, 2007). Figure 10 shows the upper and lower bound, and the final Alasehir bridge fragility curves.