Scour-Dependent Fragility of Railway Bridges: From Component Response to System Reliability Under Seismic Loading

Abstract

Flood-induced scour and earthquake loading jointly govern the seismic performance of river-crossing bridges. Existing conditional fragility assessment frameworks based on static dependence structures do not fully capture the evolving correlations between component failure modes under cumulative hydraulic degradation. This study develops a probabilistic conditional fragility assessment framework for continuous bridges and quantifies the scour-dependent fragility at both the bearing and pier levels, along with the resulting system fragility under series and parallel idealisations. A three-dimensional nonlinear finite element model with scour-dependent soil–structure interaction is constructed in OpenSees, and incremental dynamic analysis is conducted using spectrally compatible ground motions. The results indicate that scour primarily affects the bearing fragility in the moderate to complete regimes, whereas it has a negligible influence on the bearing under minor damage conditions. Unlike bearings, the fragility of piers decreases systematically toward lower PGA values with increasing scour depth, accompanied by a distinct threshold-like sensitivity shift within a specific scour depth range. At the system level, the series model is influenced by the early exceedance probability of the bearings at low PGA, whereas the parallel model is primarily governed by the exceedance probability of the piers at high PGA. Overall, the results demonstrate that scour affects system reliability not only by altering the PGA of the structural components but also by modifying the exceedance probability gap between the bearing and pier. These findings suggest that linear degradation-based management approaches can lead to biases in risk assessment and provide a practical extension and scientific basis for developing bridge system assessments under multi-hazard conditions.

1. Introduction

Transportation networks constitute the operational backbone of modern society, supporting economic activity and post-disaster emergency [1,2,3,4]. Within these transportation networks, bridges are among the most vulnerable assets because of their exposure to hydraulic actions and seismic loading, which can occur in close succession [5,6]. Historical failure analyses indicate that the structural integrity of river-crossing bridges can be disproportionately compromised by the combined influence of flood-induced scour and earthquakes [7]. Global risk assessments have identified scour as a leading contributor to bridge collapse and have highlighted its potential to magnify seismic fragility through the degradation of foundation restraint [8]. Post-event investigations have further demonstrated that scour-degraded foundations may exhibit collapse mechanisms that differ from those associated with intact soil support [9]. Recent earthquakes provide relevant field contexts, such as the 2025 Myanmar event [10] and the 2023 Kahramanmaras earthquake sequence in Turkey [11], where soil–structure interaction influenced the survival or failure of bridges founded on liquefiable or erodible deposits. Consequently, reliance on design provisions that treat earthquake and flood hazards as statistically independent events can lead to non-conservative estimates of failure likelihood.

The interaction between scour and seismic hazards is governed by competing nonlinear mechanisms rather than simple load superposition [12]. Scouring eliminates lateral soil confinement, thereby altering the boundary conditions of deep foundations. This loss of support reduces the lateral stiffness of the pile group and simultaneously increases the unsupported length of the foundation elements [13]. From a static perspective, stiffness loss increases the displacement demands and amplifies the second-order P-Delta (P-∆) effects, which can accelerate instability [14,15]. However, the dynamic consequences of this are more nuanced. The reduction in foundation stiffness lengthens the fundamental vibration period of structures and can reduce the spectral acceleration demand within certain frequency ranges [16,17]. These mixed effects cause uncertainty in performance-based assessments, especially because flood discharge and ground motion intensity fluctuate. Thus, measuring their combined impact at the system level is challenging.

Early academic investigations primarily focused on seismic demands in bridges while neglecting flood-related degradation [18,19,20,21]. Subsequent studies expanded the scope to include fragility assessments that explicitly incorporate scour. Initial studies quantified fragility at the component level and demonstrated that scour depth is a key sensitivity parameter when nonlinear soil–structure interaction (SSI) is considered [22,23]. A probabilistic approach has been introduced to relate scour depth and seismic fragility [24], reporting that scour generally increases foundation vulnerability, while in particular seismic scenarios, it may reduce superstructure demand. This finding was further refined [25] to highlight that the beneficial effect of scour is highly dependent on the spectral characteristics of the ground motion. These studies demonstrated that scour modifies the dynamic behaviour of foundations, in addition to its effect on their static resistance. Recent investigations have further examined the coupled hydroseismic response of bridge systems using advanced numerical simulations. Franciolini et al. [26] investigated the seismic behaviour of a scour-affected isolated viaduct and demonstrated that progressive scour can significantly amplify structural flexibility and seismic demand under severe foundation degradation conditions. However, existing studies remain primarily focused on deterministic structural response amplification and isolated component behaviour rather than probabilistic fragility evolution and system-level conditional exceedance assessment. Consequently, the understanding of how progressive scour modifies the relative vulnerability evolution between displacement-sensitive bearings and ductility-governed pier components at the system level remains limited.

In response to these complexities, methodological developments have progressively shifted the fragility assessment from isolated member-level formulations to integrated probabilistic models. Some studies have emphasised the importance of capturing the performance of scoured bridges under functional and collapse-limit states [22]. More recent contributions have extended the scope to include time-dependent assessments that account for deterioration and progressive scouring [27,28], which focused on investigating long-span bridges and incorporated durability mechanisms, such as chloride-induced corrosion, into seismic fragility. Other studies have integrated life cycle hydrologic uncertainty into seismic performance evaluations to represent infrastructure ageing [29,30,31]. In parallel, multivariate statistical methods, including copula-based models, have been adopted to represent the dependence among component demands and to refine system failure estimates beyond first-order bounds [32,33].

Despite these methodological advances, the understanding of the coupled hydroseismic performance remains constrained by several critical limitations in several respects. Many existing probabilistic frameworks rely on continuous and often linear degradation functions to correlate scour depth with seismic fragility. These models implicitly assume that risk accumulates proportionally with soil removal. In contrast, the soil–pile interaction is inherently nonlinear and governed by the geometric configuration of the pile group. The presence of a critical exposure threshold range at which the foundation behaviour transitions from a rigid base response to a flexible response is rarely addressed. Neglecting this geometric nonlinearity can bias risk estimates, particularly under deep scours, where the foundation behaviour changes markedly. In addition, conventional system reliability models typically adopt a stationary dependence structure when aggregating component failure probabilities. Recent methodologies generally assume that the correlation between pier damage and bearing failure remains constant, independent of the hazard intensity or scour extent [34]. This assumption may be incompatible with the altered failure sequencing, where severe scouring may cause bearing-related limit states to be reached before pier yielding, thereby altering the effective dependence between modes. This alteration has the potential to transform the statistical dependence between failure modes from a highly correlated state to a temporally separated state. Furthermore, relatively few studies have combined high-fidelity three-dimensional (3D) finite element modelling with scour-dependent nonlinear springs within an incremental dynamic analysis (IDA) framework using a large suite of ground motions. The absence of such rigorous modelling limits the ability to identify interaction mechanisms, such as the saturation of failure probability at high seismic intensities.

The specific vulnerability of seismically isolated bridges under scour conditions represents another important knowledge gap. Isolation bearings are designed to reduce the force demand by accommodating relative displacement. However, these components are inherently sensitive to displacements. Current design codes, such as Eurocode 8 [35] and the Chinese code for seismic design of railway engineering and culverts TB 10002-2017 [36], provide limited guidance for adjusting clearance and bearing capacity under deep scours. This shortcoming is consequential for railway bridges because excessive displacement can trigger unseating or track irregularities that may lead to a derailment. Consequently, there is a clear need to link foundation degradation to isolation displacement demand and to system-level reliability.

To address these limitations, this study develops a conditional fragility-based framework for railway bridges subjected to scour-induced foundation degradation and seismic excitation. The proposed methodology integrates scour-dependent soil–structure interaction, nonlinear incremental dynamic analysis, component-level fragility assessment, and idealised system-level exceedance evaluation to investigate how progressive scour modifies the interaction between displacement-sensitive bearings and ductility-governed pier components. To achieve this, a rigorous 3D finite element model of a representative continuous railway bridge is developed, incorporating scour-dependent nonlinear soil springs to represent the degradation of the boundary conditions. Component fragility across scour depths is evaluated using nonlinear incremental dynamic analysis with 26 scaled ground motion records, which generated 1872 unique hazard cases for the bridge. The simulation set is used to identify the threshold range at which the structural response undergoes a pronounced transition. Unlike copula-based approaches that explicitly construct statistical dependence models among component demands, the present study focuses on how scour-induced foundation degradation modifies the relative exceedance evolution between displacement-sensitive bearings and ductility-governed piers within idealised system configurations. Therefore, the term “interaction evolution” is used herein to describe scour-dependent changes in the probabilistic separation of component limit-state exceedance. Subsequently, the reliability of the bridge system is assessed by examining the probabilistic separation margin, which is defined as the probabilistic difference between the series and parallel system-level exceedance probabilities of the limit states. This metric is used to illustrate how deep scours can reconfigure the dependence structure between the bearing and pier limit states.

Additionally, aleatory record-to-record variability is addressed through a bootstrap-based reliability assessment. The ultimate goal of this study is to inform the development of next-generation hazard risk management strategies that explicitly account for the dynamic evolution of structural vulnerability. The remainder of this paper details the probabilistic framework, validates the numerical modelling strategy, reports the fragility and system reliability results, and discusses the implications for conditional fragility assessment under combined scour–earthquake scenarios of bridges.

2. Methods

2.1. Scour Hazard

Unlike transient mechanical actions, such as gravity or seismic inertia, scour is a cumulative hydraulic process that progressively alters the boundary conditions of bridge substructures [15]. The associated loss of soil resistance reduces the lateral stiffness of the pile foundation, thereby modifying the dynamic response to earthquake loading [13]. To represent this evolution in a manner suitable for probabilistic performance assessment, this study adopts a stochastic scour characterisation based on the Federal Highway Administration Hydraulic Engineering Circular No. 18 [37]. The applicability conditions associated with the adopted HEC-18 formulation are verified relative to the investigated bridge site. The analysed bridge is located within a non-cohesive alluvial riverbed environment with clear water scour conditions and flow characteristics consistent with the recommended applicability range of the HEC-18 pier scour equation. Given the potential conservatism of the standard HEC-18 formulation in estimating pier scour, a probabilistic adjustment is introduced to reduce the model bias and represent the epistemic uncertainty [38]. A correction factor, $k_w$, is used to transform the nominal prediction into a probabilistic estimate of scour depth ($y_s$), written as

$$y_s=2.0{k_{w}y}_1{k}_1{k}_2{k}_3{k}_4{\left({\displaystyle \frac{a}{y_1}}\right)}^{0.65}F{r}^{0.43},$$

(1)

In Equation (1), $y_s$ denotes local scour depth, $y_1$ represents approach flow depth, and $a$ corresponds to the pier width. The coefficients $k_1$ to $k_4$ account for the pier nose shape, the flow angle of attack, the streambed condition, and the bed material size, respectively. These parameters are obtained from HEC-18 [37]. For the present study, $k_1$ and $k_2$ are both set to 1.0 for round-ended piers under normal approach flow, $k_3$ is equal to 1.1 for sand and gravel mixtures, and $k_4$ is set to 1.0 for clear water conditions. The Froude number ($Fr$) is computed as

$$Fr={\displaystyle \frac{v}{(g{y}_1{)}^{0.5}}},$$

(2)

where $v$ signifies the mean flow velocity upstream of the pier, and $g$ is the gravitational acceleration, taken as 9.81 m/s². The hydraulic variables $y_1$ and $v$ are obtained from continuity and Manning-based resistance relationships

$$Q=y_1\cdot b\cdot v,$$

(3)

$$Q={\displaystyle \frac{b{y}_1}{n}}{\left({\displaystyle \frac{b{y}_1}{b+y_1}}\right)}^{{\displaystyle \frac{2}{3}}}{s}^{{\displaystyle \frac{1}{2}}},$$

(4)

where Q is the flow discharge rate, and b denotes the waterway width. For the studied bridge, the field measurement results show that the width b on both banks is 260 m. n represents the Manning roughness coefficient, and s indicates the slope of the streambed. The calculated slope s and Manning coefficient n are, respectively, 0.001 and 0.035, determined by field measurement verification and HEC-18 [37], ensuring reliability in bridge scour assessment.

The correction factor $k_w$ depends on flow and sediment regimes. Accordingly, for scenarios characterised by subcritical flow and uniform non-cohesive sediments, this study adopted the specific formulation established in HEC-18. These specific regimes correspond to a flow depth-to-pier width ratio of less than 0.8 and a pier width-to-median particle size ratio $D_{50}$ exceeding 50. The factor follows a piecewise definition relative to the critical velocity ($v_c$) required for sediment transport as follows:

$$k_w=\left\{\begin{array}{ll}2.58{\left({\displaystyle \frac{y_1}{a}}\right)}^{0.34}F{r}^{0.65}& \mathrm{for} v<v_c\\ 1.00{\left({\displaystyle \frac{y_1}{a}}\right)}^{0.13}F{r}^{0.25}& \mathrm{for} v\ge v_c\end{array}\right.,$$

(5)

Here, the critical velocity $v_c$ is computed as

$$v_c=6.19y_1^{{\displaystyle \frac{1}{6}}}{D}_{50}^{{\displaystyle \frac{1}{3}}},$$

(6)

2.2. Fragility Analysis Method

In contrast to deterministic methodologies that depend on specific events, this study uses a statistically meaningful record set to represent the record-to-record variability in the seismic input. The record selection is governed by site conditions and the dynamic sensitivity of railway bridges to guarantee that subsequent nonlinear time history simulations remain physically consistent. Given that the seismic response is inherently stochastic and bridge components can manifest pronounced nonlinear behaviour [39,40,41], a deterministic evaluation alone is inadequate for probabilistic risk quantification. Consequently, a seismic fragility framework is adopted to describe the performance in terms of the conditional probability of exceeding the prescribed limit states. In comparison with empirical approaches, which are constrained by limited damage observations in combined hazard settings, simulation-based approaches provide direct access to the demand-capacity relationship over a wide range of intensity measure levels [42].

In this study, the IDA method is implemented to map the nonlinear structural response against monotonically increasing ground motion intensities and to establish the seismic fragility curve of the bridge under multiple scour depths. This computational strategy establishes the boundary conditions for structural capacity. These responses progressively drive the system from an elastic state to an unstable state. The failure probability is derived using a frequency-count analysis approach. For a particular damage state denoted as ${LS}_I$, the index $i$ takes values from 1 to 4, corresponding to minor, moderate, severe, and complete damage, respectively. The probability of failure $P_f$ is assessed by evaluating the proportion of exceedance events relative to the entire set of simulations. This relationship is expressed numerically as follows:

$$P_f=P\left[DI\ge {LS}_i|IM\right]={\displaystyle \frac{n_i}{N}},$$

(7)

where $N$ is the total number of analyses at the selected intensity level, and $n_i$ is the number of records for which the damage index ($DI$) of the component exceeds the threshold of the $i$-th limit state (${LS}_i$) at a given intensity level. When failure probabilities are plotted against discrete intensity levels, the resulting trend is commonly represented using a log-normal model [43]. The fragility function is written as

$$P_f=P\left[DI\ge L{S}_i|IM\right]=\mathsf{\Phi }\left[{\displaystyle \frac{\ln \left(\frac{IM}{M_{IM}}\right)}{{\xi }_{IM}}}\right],$$

(8)

where $\Phi (·)$ refers to the standard normal cumulative distribution function. The parameter $M_{\mathit{IM}}$ signifies the median seismic capacity. Specifically, the intensity measure level corresponding to a 50% probability of failure on the fragility curve. It is crucial to note that $M_{\mathit{IM}}$ represents the median, not the arithmetic mean, which characterises the central tendency of the seismic demand required to induce a specified damage state. For brevity, M is used in the subsequent expressions. The parameter ${\xi }_{\mathit{IM}}$ quantifies the logarithmic standard deviation, representing the dispersion or uncertainty inherent in the structural response.

Because scour influences foundation exposure and stiffness, a single intensity-based fragility curve is inadequate for representing the joint dependence of seismic intensity and scour depth. To describe the coupled effects, this study constructs a fragility surface in which the exceedance probability is expressed as a function of earthquake intensity measure ($\mathit{IM}$) and scour depth ($H$).

$$P_f=P\left[DI\ge L{S}_i|IM·H\right]=\mathsf{\Phi }\left[{\displaystyle \frac{\ln \left(\frac{IM}{M_{IM\cdot H}}\right)}{{\xi }_{IM\cdot H}}}\right],$$

(9)

where the median capacity $M_{\mathit{IM}·H}$ and logarithmic standard deviation ${\xi }_{\mathit{IM}·H}$ vary with $H$ and quantify the scour-dependent shift in median capacity and dispersion. This representation enables the direct evaluation of how increasing scour depth modifies vulnerability across the intensity range and captures the coupled effects of the multi-hazard environment.

2.3. System-Level Seismic Fragility Assessment

To assess the global seismic performance, it is necessary to strictly define the interaction between individual structural components and the overall system stability. Exclusive reliance on component-level damage states may lead to a non-negligible underestimation or overestimation of the systemic risk. Recent advancements in seismic risk assessment underscore the importance of integrating component fragility into a unified, system-level framework. Neglecting the multivariate dependence among demand parameters introduces substantial bias in the risk estimation of multi-span bridges, as shown in multivariate surrogate demand studies [34]. Furthermore, the application of copula functions to model the joint failure probability highlights that the system failure domain is fundamentally governed by the correlation structure between the critical components [44]. Therefore, for complex structural systems, the transition from local component damage to global instability requires probabilistic integration rather than simplified bounding methods [44,45].

Conventional first-order bounds and conditional marginal product formulations typically assume a dependence [46,47]. To examine how scour modifies component interaction, this study represents a bridge using two idealised configurations associated with distinct performance objectives. The system reliability analysis employs series and parallel idealisations as conceptual bounding exercises to bracket the structural performance limits. The series system defines the operational limit state where the exceedance of any individual component capacity triggers functional loss. This configuration incorporates all critical bearings and pier columns into a unified statistical union. The corresponding system failure probability $P_{sys}^{ser}$ is expressed as follows [48]:

$$P_{sys}^{ser}=P\left({\displaystyle {\bigcup }_{i=1}^n} F_i\right)=1-P\left({\displaystyle {\bigcap }_{i=1}^n} {\overline{S}}_i\right),$$

(10)

where $F_i$ denotes the failure event of the $i$-th component, and ${\overline{S}}_i$ denotes its survival. This formulation represents the lower bound of system reliability, governing immediate post-event functionality. Conversely, a parallel system model is used to represent collapse prevention, in which global instability occurs only when all critical load-carrying mechanisms exceed their limit-state. This principle aligns with established structural reliability theories [49]. Consequently, the intersection of the component failure events is defined by Equation (11) as the failure probability of the collapse prevention limit state $P_{sys}^{par}$.

$$P_{sys}^{par}=P\left({\displaystyle {\bigcap }_{i=1}^n} F_i\right),$$

(11)

Flood-induced scour can alter the stiffness and modal properties of the foundation system, thereby changing the statistical dependence between the pier ductility demand and bearing displacement [50,51]. Existing analytical approaches that rely on pre-assumed correlation matrices or copulas typically treat the structural configuration as static, potentially obscuring the scour-dependent evolution [45,52]. To avoid the pre-specification of dependence, this study uses simulation outputs from the IDA database to estimate the joint failure probabilities. Specifically, a bootstrap resampling procedure with 10,000 iterations is employed to integrate joint exceedance based on paired demand vectors, including bearing displacement and pier ductility demand, obtained directly from the finite element analyses. The estimation indicator function is written as

$$P_{sys}^{ser}={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{k=1}^N} I\left({\bigcup }_i \left\{D_{i,k}\ge C_i\right\}\right),P_{sys}^{par}={\displaystyle \frac{1}{N}}{\displaystyle {\sum }_{k=1}^N} I\left({\bigcap }_i \left\{D_{i,k}\ge C_i\right\}\right),$$

(12)

where $N$ represents the total number of bootstrap samples required to ensure statistical convergence. The term $D_{i,k}$ denotes the seismic demand of the $i$-th component in the $k$-th simulation, and $C_i$ represents the corresponding capacity limit state. The indicator function $I\left(\cdot \right)$ returns one when the union condition is satisfied for the series model or when the intersection condition is satisfied for the parallel model. This simulation-based integration reduces the aggregation error associated with the bound-based methods and provides a direct quantification of the scour-dependent system performance.

3. Numerical Modelling and Analysis Procedures

3.1. Bridge Description

The proposed framework is demonstrated using a representative three-span continuous railway bridge in Southwest China, selected to reflect the hazard exposure of modern rail infrastructure. The bridge is located in a high-seismicity region and crosses a flood-prone, alluvial plain. The site is designed for a PGA of 0.30 g, corresponding to a seismic intensity of VIII. The hydraulic design considers a 100-year return period flood, and a documented discharge (Q) of 4200 m³/s is reported for the 2018 event. Although the bridge incorporates seismic isolation and foundation protection measures, the current provisions in the Chinese code for the design of railway bridges and culverts (TB 10002-2017) and Eurocode 8 do not provide explicit procedures for combined flood and earthquake effects. Consequently, this bridge provides an appropriate case study for investigating how scour-induced foundation exposure modifies seismic demands and component-limit-state exceedance.

The structural framework comprises a three-span continuous reinforced concrete (RC) box-girder superstructure with span lengths of 80, 100, and 80 m. The substructure consists of single-column piers characterised by hollow circular cross-sections with an external diameter of 4.0 m and a wall thickness of 0.4 m. These pier columns extend to a height of 28 m and transfer loads to a pile group foundation consisting of nine cast-in-place piles. Each pile has a diameter of 2.2 m and a length of 45 m embedded into the underlying moderately weathered rock stratum to provide fixity and overturning resistance. Figure 1 illustrates the cross-sectional geometry and reinforcement details of the pier, pile cap, and pile foundation.

The material and reinforcement specifications ensure the high durability and load-bearing capacity of the structural components. The primary load-carrying members use C50 concrete, and the protective layers employ C40 concrete. The reinforcement strategy incorporates HRB400 steel bars, where 32 mm diameter bars provide longitudinal reinforcement and 16 mm diameter spirals supply transverse confinement. Double-curved spherical isolation bearings with a maximum displacement capacity of 250 mm are employed to control the force demand. Figure 2 presents the global finite element model of the bridge with a severe scour scenario of 6.8 m deep.

3.2. Finite Element Modelling

A numerical model is developed in OpenSees using a fibre-based formulation to represent the material nonlinearity in the substructure. Linear elastic elements represent the superstructure based on the assumption that the inelastic action under the considered hazard levels remains concentrated in the piers. Piers are modelled using displacement-based nonlinear beam-column elements. The cross-sectional discretisation employs fibres for confined and unconfined concrete, as well as for reinforcing steel. The confined concrete behaviour is defined using the Mander model, and the reinforcing steel is represented using Steel 02 material to capture yielding and strain hardening [53]. Second-order effects are included to account for the axial force interaction with lateral deformation. This modelling strategy targets the substructural mechanisms that govern the system-level loss of performance. The unconfined concrete compressive strength is 40 MPa, and the confined core strength reaches 50 MPa due to transverse reinforcement confinement. The longitudinal reinforcing steel features a yield strength of 400 MPa and an initial elastic modulus of 200,000 MPa with a post-yield hardening ratio of 0.01. Structural mass is distributed lumped at the node joints, and the dynamic damping is simulated using a Rayleigh damping model with a constant damping ratio of 0.05 assigned to the first and third modes. The double-curved spherical isolation bearings are simulated using flat slider bearing elements. These elements incorporate a characteristic sliding friction coefficient of 0.03 under design pressure and an initial elastic stiffness of 45 kN/mm before sliding occurs. The pile cap is assumed to be a rigid concrete block, and the pile–soil interaction parameters are calibrated according to the standard American Petroleum Institute (API) guidelines. The fundamental modal properties obtained from the numerical model are also verified against theoretical estimations based on the equivalent structural stiffness and mass distribution of the bridge system, showing satisfactory agreement in the global dynamic characteristics.

The soil–structure interaction (SSI) is represented using distributed nonlinear springs along the pile length to capture the lateral resistance and axial load transfer. P-y elements represent the lateral response, whereas the shaft friction and tip resistance are modelled by t-z and q-z elements, respectively. The stratigraphic and soil parameters, listed in

Table 1. Geotechnical characteristics of soil at bridge site—Summary of geotechnical parameters of soil strata.

Top Depth (m)	Bottom Depth (m)	Soil Type	$\mathit{\gamma }$ ($\mathbf{K}\mathbf{N}/{\mathbf{m}}^3$)	${\mathit{N}}_{60}$
0	10	Silt Sand	18	25
10	20	Clay	19	15
20	35	Medium Sand	20	35
35	45	Weathered Rock	22	45

: Geotechnical characteristics of soil at bridge site—Summary of geotechnical parameters of soil strata, are used to calibrate the relevant modelling parameters, following the procedures outlined in the American Petroleum Institute (API) guidelines. The scour effects are implemented by removing springs within the eroded zone to represent the loss of soil resistance above the scour depth. This treatment increases the unsupported pile length and reduces the overall foundation stiffness, which, in turn, modifies the dynamic impedance and seismic response [54,55,56].

To cover a representative range of flood severities for the case study site, three flood scenarios are considered with return periods of 10, 50, and 200 years. The corresponding discharge rates (Q) values are 2500, 3500, and 5000 m³/s, and the resulting probabilistic scour depths ($y_s$) are summarised in

Table 2. Scour depths for probabilistic and fixed scenarios—Computed and assumed scour depths for analysis.

Scenario Description	Return Period (Years)	Discharge * Q (${\mathbf{m}}^3/\mathbf{s})$	Scour Depth ** ${\mathit{y}}_{\mathit{s}}$ (m)
Probabilistic Flood Events
10-year Flood Event	10	2500	5.75
50-year Flood Event	50	3500	6.26
200-year Flood Event	200	5000	6.80
Fixed Scour Scenarios (Sensitivity)
No Scour (Baseline)			0.00
Fixed Scour Depth (2 m)			2.00
Fixed Scour Depth (4 m)			4.00

* Q denotes the peak discharge flow (${\mathrm{m}}^3/\mathrm{s}).$ ** $y_s$ indicates the final scour depth (m) for the given return period.

: Scour depths for probabilistic and fixed scenarios—Computed and assumed scour depths for analysis. Furthermore, discrete fixed scour depths of 0, 2, and 4 m are included to support a sensitivity-based interpretation of the structural response and provide intermediate exposure conditions within the fragility assessment.

The overall workflow, which integrates hazard inputs, finite element simulations, and probabilistic damage assessments, is illustrated in Figure 3.

Table 3. Fundamental periods of the first to fourth modes at different scour depths.

Scour Depth (m)	Mode 1 (s)	Mode 2 (s)	Mode 3 (s)	Mode 4 (s)
0.00	1.9600	1.8445	1.1832	0.8455
2.00	1.9807	1.8636	1.1983	0.8555
4.00	1.9981	1.8792	1.2127	0.8657
5.75	2.0234	1.9019	1.2351	0.8826
6.26	2.0350	1.9123	1.2457	0.8910
6.80	2.0503	1.9262	1.2602	0.9027

Fundamental periods of the first to fourth modes at different scour depths. presents the first four modal periods as a function of the scour depth. The fundamental period of mode 1 increases from 1.9600 s without scour to 2.0503 s at a scour depth of 6.8 m. This period increase reflects reduced lateral stiffness and motivates the demand shifts discussed in Section 4.

3.3. Ground Motion Selection

An intensity measure characterises the seismic hazard input for incremental dynamic analysis (IDA) (IM) that supports an efficient prediction of structural demand. Common choices include the peak ground acceleration (PGA), spectral acceleration (SA), and peak ground velocity (PGV) [24,32,57]. In this study, PGA is selected as the primary IM to provide a consistent basis for scaling records over response levels ranging from elastic behaviour to near collapse, and ensures direct compatibility with the regional railway engineering design codes (GB 50111-2006).

To represent the record-to-record variability while maintaining compatibility with the bridge site conditions, a candidate set of 50 ground motion records is assembled from the Pacific Earthquake Engineering Research Centre Database [58]. The screening targets events with moment magnitudes between 6.5 and 7.6 and source-to-site distances ranging from 10 to 30 km. The site conditions are constrained using the expected shear wave velocity in the upper 30 m, which ranges from 250 to 600 m/s, consistent with the case study classification. The resulting candidate pool includes records from 14 earthquakes that occurred between 1976 and 2008.

Since the investigated bridge site is not located in the seismic environment dominated by near fault fracture, the near-fault pulse-like motions are excluded to avoid a response regime dominated by large velocity pulses that may produce demand mechanisms outside the scope of the present assessment [59,60,61]. Specifically, high-velocity pulses release more than 60% of the total energy, and the PGV/PGA ratio typically exceeds 0.25, thereby causing a risk of derailment [62]. Consistent with railway performance assessment provisions in the Chinese code for seismic design of railway engineering (GB 50111-2006) [63] and Eurocode 8, non-pulse motions are retained by enforcing thresholds on pulse indicators and the peak ground velocity to peak ground acceleration (PGV/PGA) ratio. Figure 4b,c illustrate the distributions of the PGV-to-PGA ratios and pulse energy ratios for the selected suite, respectively, confirming that all retained records satisfied the non-pulse criterion of a ratio of less than 0.2.

Table 4. Summary of selected earthquake ground motion records.

No	RSN	Earthquake Name	Year	Magnitude	Rjb (km)	Rrup (km)	Vs30 (m/s)	PGA (m/s2)	PGV (m/s)
1	125	Friuli, Italy-01	1976	6.5	14.97	15.82	505.23	0.45	0.32
2	167	Imperial Valley-06	1979	6.53	13.52	15.3	259.86	0.50	0.37
3	288	Irpinia, Italy-01	1980	6.9	22.54	22.56	561.04	0.57	0.35
4	313	Corinth, Greece	1981	6.6	10.27	10.27	361.4	0.28	0.27
5	587	New Zealand-02	1987	6.6	16.09	16.09	551.3	0.39	0.36
6	739	Loma Prieta	1989	6.93	19.9	20.26	488.77	0.32	0.29
7	754	Loma Prieta	1989	6.93	20.44	20.8	295.01	0.36	0.31
8	850	Landers	1992	7.28	21.78	21.78	359	0.34	0.38
9	882	Landers	1992	7.28	26.84	26.84	344.67	0.33	0.27
10	952	Northridge-01	1994	6.69	12.39	18.36	545.66	0.59	0.28
11	963	Northridge-01	1994	6.69	20.11	20.72	450.28	0.36	0.33
12	985	Northridge-01	1994	6.69	23.5	29.88	297.07	0.47	0.29
13	988	Northridge-01	1994	6.69	15.53	23.41	277.98	0.36	0.30
14	1000	Northridge-01	1994	6.69	27.82	31.33	304.68	0.30	0.36
15	1034	Northridge-01	1994	6.69	26.77	33.67	349.54	0.48	0.31
16	1077	Northridge-01	1994	6.69	17.28	26.45	336.2	0.88	0.42
17	1082	Northridge-01	1994	6.69	5.59	10.05	320.93	0.28	0.26
18	1083	Northridge-01	1994	6.69	12.38	13.35	402.16	0.81	0.33
19	1086	Northridge-01	1994	6.69	1.74	5.3	440.54	0.33	0.42
20	1107	Kobe, Japan	1995	6.9	22.5	22.5	312	0.38	0.33
21	1615	Duzce, Turkey	1999	7.14	9.14	9.14	338	0.45	0.39
22	4031	San Simeon, CA, USA	2003	6.52	5.07	6.22	410.66	0.44	0.41
23	4207	Niigata, Japan	2004	6.63	4.22	12.81	274.17	0.46	0.47
24	4883	Chuetsu-oki, Japan	2007	6.8	27.83	29.91	254.68	0.27	0.29
25	5678	Iwate, Japan	2008	6.9	5.09	11.12	398.59	0.50	0.38
26	5776	Iwate, Japan	2008	6.9	25.15	25.16	477.55	0.34	0.47

summarises the detailed metadata for these 26 records, and Figure 5a shows the original PGA distribution of the data.

Prior to the dynamic analysis, each record is normalised with respect to the PGA to reduce the dispersion associated with differences in magnitude, distance, and source characteristics while preserving record-specific variability. Figure 5b presents a comparison between the mean response spectrum of the 26 normalised records and the target design spectrum. The corresponding acceleration time histories for the entire suite are presented in Figure 5c. Subsequently, based on the SeismoMatch model, an incremental dynamic analysis dataset is constructed by scaling the amplitudes of 26 records according to PGA from 0.1 g to 1.2 g in 0.1 g increments, resulting in 312 scaled input cases. The matching procedure covers a comprehensive period interval between 0.1 s and 4.0 s with a fixed damping ratio of 0.05 to minimise spectrum mismatch while preserving the non-stationary duration characteristics of the historical events. The effectiveness of scaling is illustrated in Figure 4a by comparing the PGA distributions before and after modulation. Additional verification for a representative ground motion record scaled to a PGA of 0.3 g is provided in Appendix A, and the complete set of scaled accelerograms is listed in Appendix B.

3.4. Limit State Definition

Fragility assessment requires an explicit definition of the component limit states and consistent mapping to the engineering demand parameters. The determination of these limit states is primarily based on experimental data or analytical mechanics to evaluate the degree of performance degradation caused by seismic activity. The literature substantiates that structural capacity indicators can effectively quantify structural damage [64]. For reinforced concrete piers, commonly used demand measures include displacement ductility, curvature ductility, and drift ratio [20,47,54,65,66,67,68,69,70]. The bearing performance is generally evaluated using displacement-related thresholds associated with sliding and unseating [20,21,43,54,55,60,68,71]. In this study, pier damage states are defined in terms of curvature ductility, and bearing damage states are defined in terms of relative bearing displacement, consistent with prior bridge fragility studies [20,47].

Table 5. Summary of the different damage index thresholds.

Component_Demand_Parameter	Minor	Moderate	Severe	Complete	Reference
Pier Curvature ductility	1.29	2.1	3.52	5.24	[47]
	1	5.11	7.5	9	[68]
	1	2	4	7	[20]
	1.29	2.1	3.52	5.24	[67]
	1	2.73	4.54	6.5	[66]
	1	1.08	1.76	3	[54]
	1	2	4	7	[69]
Bearing Displacement (mm)	30	60	150	300	[21]
	30	100	150	255	[68]
	50	100	150	255	[20]
	100	150	200	250	[67]
	30	60	150	300	[54]
	0	50	100	150	[43]
	90	150	200	300	[71]
	100	130	160	200	[60]

summarises the threshold values used for the fragility assessment.

To ensure that these thresholds reflect the mechanical behaviour of the specific case study bridge, the limit state thresholds for the piers are derived from a moment-curvature analysis of the critical cross-sections. Figure 6 presents the computed and idealised bilinear moment–curvature envelopes obtained using the Xtract fibre section analysis method. The values of ${\varphi }_1$ and ${\varphi }_{\mathrm{m}}$ are given in the figure.

• First Yield (${\varphi }_1$): The curvature associated with the first yield of the longitudinal reinforcement.
• Equivalent Yield (${\varphi }_y$): The curvature marking the formation of the plastic hinge.
• Ultimate Capacity (${\varphi }_m$): The curvature at which the concrete reaches its ultimate compressive strain limit.
• Intermediate States (${\varphi }_2,{\varphi }_4$): Curvatures corresponding to concrete compressive strains of 0.002 and 0.004, respectively.

The adopted pier ductility thresholds are selected based on the moment–curvature response of the investigated reinforced concrete pier section together with commonly adopted bridge seismic performance classifications reported in previous literature. For the reinforced concrete pier columns, the moment curvature analysis performed via fibre discretization indicates that the first yield curvature of the outer longitudinal steel equals approximately 0.0007 rad/m. The equivalent yield curvature representing plastic hinge formation is 0.0014 rad/m, and the ultimate curvature bounded by concrete crushing is approximately 0.005 rad/m. These mechanical benchmarks verify that the selected curvature ductility ratios of 1 and 2 and 4 and 7 represent realistic thresholds for minor yielding and moderate cracking and severe spalling and complete cross-sectional collapse. For bearings, the displacement thresholds represent progressive failure stages, ranging from the initial unseating risk to the complete loss of support. The displacement thresholds defined for the double-curved spherical bearings are calibrated based on the physical dimensions of the isolation system. The minor damage threshold of 100 mm marks the clearance limit, while the complete damage threshold of 250 mm represents the ultimate sliding capacity before unseating occurs.

Table 6. Classification standards for the damage status of bridge bearings and piers.

Damage State	Displacement dx (mm) (Bearings)	Curvature Ductility Ratio (Piers)	Curvature Ductility Coefficient (Piers)
No damage	dx ≤ 100	${\beta }_{\phi } \le {\beta }_{\phi 1}$	${\beta }_{\phi }\le 1$
Minor damage	100 < dx ≤ 150	${\beta }_{\phi 1}<{\beta }_{\phi } \le {\beta }_y$	${1<\beta }_{\phi }\le 2$
Moderate damage	150 < dx ≤ 200	${\beta }_{\phi y}<{\beta }_{\phi } \le {\beta }_{\phi 2 }{(\beta }_{\phi 4})$	${2<\beta }_{\phi }\le 4$
Severe damage	200 < dx ≤ 250	${\beta }_{\phi 2 }{(\beta }_{\phi 4})<{\beta }_{\phi } \le {\beta }_{\phi max }$	${4<\beta }_{\phi }\le 7$
Complete damage	dx > 250	${\beta }_{\phi }>{\beta }_{\phi max }$	${\beta }_{\phi }>7$

summarises the calibrated thresholds for both types of components in this study. For a given ground motion, the component damage state is identified by comparing the computed demands with the corresponding thresholds. Pier curvature ductility ratio, ${\beta }_{\phi }$, is defined as

$${\beta }_{\phi }={\displaystyle \frac{{\kappa }_{\phi }}{{\kappa }_{\phi 1}}},$$

(13)

In Equation (13), the variable ${\kappa }_{\phi }$ denotes the peak curvature demand at the critical plastic hinge regions of the pier under seismic excitation, and ${\kappa }_{\phi 1}$ represents the reference curvature at first yield of longitudinal reinforcement. Cascading failure logic serves to bound the structural reliability for system-level assessment. For system assessment, series logic corresponds to the weakest link interpretation, in which system failure occurs when any critical component exceeds its limit state. Parallel logic represents a redundancy-based interpretation in which system failure requires the simultaneous exceedance of limit states by multiple critical mechanisms. This multi-metric definition supports the consistent evaluation of component and system reliability under coupled failure modes.

4. Numerical Results

4.1. Bearing Fragility Results

The quantification of component reliability relies on the generation of scour-dependent fragility curves. As illustrated in Figure 7, the probabilistic performance of the bearings is evaluated across six discrete scour depths ranging from 0.0 m to 6.8 m. The solid data points represent the failure probabilities derived from the IDA of 26 spectrally matched ground motions scaled between 0.1 g and 1.2 g. The close agreement between these discrete points and the fitted curves supports the log-normal representation of bearing fragility under coupled scour and seismic loadings.

The results indicate that bearing fragility exhibits scour sensitivity related to the damage states. A modest degradation in the bearing performance is observed as the scour depth increases. Figure 8 summarises the fragility curves for the four damage limit states defined as minor, moderate, severe, and complete damage, corresponding to displacement thresholds of 100, 150, 200, and 250 mm, respectively. The median PGA (M) provides a consistent quantitative measure corresponding to a 50% exceedance probability of the component under a specified damage state. For minor damage, M remains nearly invariant across the scour range, varying only from 0.20 g at 0.0, 2.0, and 4.0 m to 0.21 g at 5.75 and 6.8 m, with a slight increase to 0.22 g at 6.26 m. For the moderate damage state shown in Figure 8b, the M value decreases from 0.34 g under non-scoured conditions to 0.30 g at the maximum scour depth of 6.8 m. The same trend persists for the higher bearing-limit states. The severe state shows a reduction from 0.44 g (0.0 to 4.0 m) to 0.42 g (5.75 and 6.26 m) and 0.41 g at 6.8 m. The complete state exhibits a smaller but still consistent shift, with M ranging from 0.55 g at 0.0 m and 4.0 m to 0.52 g at 6.26 m and 6.8 m. Collectively, these results indicate that scour primarily affects bearing fragility in moderate to complete regimes, whereas the onset of minor bearing damage is essentially insensitive to foundation exposure within the investigated scour range.

The curve indicates that the exceedance probability approaches saturation at high PGA levels. As shown in Figure 7 and Figure 8, when PGA exceeds approximately 0.6 g, the bearing exceedance probabilities reach 1.0 across all scour depths for the minor, moderate, and severe damage states. By contrast, the complete damage state retains a modest dependence on scour depth, although the separation between curves diminishes as PGA increases. These trends indicate that, once exceedance is almost certain for a given bearing limit state, additional soil removal produces only a secondary effect on the exceedance probability.

Figure 9 presents the corresponding fragility surfaces in the joint domain of the PGA and scour depth. The surfaces indicate that the bearing sensitivity to scour is non-uniform. The probability gradient along the scour axis becomes more noticeable beyond approximately 4.0 m, which is consistent with the more pronounced reductions in M for the moderate to complete states once the foundation exposure exceeds this level. Consequently, this depth range acts as a practical transition threshold for the bearing vulnerability within the studied configuration. This finding underscores the significance of incorporating site-specific scour hazard curves into the seismic design of bridges.

4.2. Pier Fragility Results

The probabilistic fragility of the bridge piers is evaluated using scour-dependent fragility curves for the pier columns, as shown in Figure 10. In contrast to bearings, piers exhibit a continuous and pronounced sensitivity to scour, as reflected by a systematic leftward shift of the fragility curves and a general reduction in M with increasing scour depth.

The M values in Figure 10 quantify the scour-induced capacity loss for each limit state. For minor damage, M remains 0.35 g from 0.0 to 4.0 m and then drops to 0.32 g at 5.75, 6.26, and 6.8 m, indicating that the onset of pier yielding becomes more likely once scour exceeds the moderate exposure level. For moderate damage, M decreases from 0.59 g at 0.0 m to 0.54 g at 6.8 m, with intermediate values of 0.60 g at 2.0 m, 0.58 g at 4.0 m, 0.57 g at 5.75 m, and 0.56 g at 6.26 m. For severe damage, M increases slightly from 0.84 g at 0.0 m to 0.87 g at 2.0 m and then decreases to 0.86 g at 4.0 m, 0.83 g at 5.75 m, and 0.82 g at 6.26 and 6.8 m. This pattern suggests that moderate scour can alter the demand distribution without immediately degrading the M value in the severe state in a strictly monotonic manner, whereas deep scour results in a consistent reduction. For the complete damage state, M declines from 1.29 g at 0.0 m to 1.22 g at 6.8 m, passing through 1.27 g at 2.0 m, 1.24 g at 4.0 m, 1.21 g at 5.75 m, and 1.19 g at 6.26 m. The small rebound from 1.19 g at 6.26 m to 1.22 g at 6.8 m indicates that M of the collapse limit state is influenced by record set variability at a high intensity measure, even though the overall trend remains a reduction in collapse capacity with deep scour. It is worth noting that for several complete damage states, the estimated median exceedance intensity approached or slightly exceeded the upper bound of the analysed PGA scaling range. These capacity predictions represent statistical extrapolations derived from the lognormal cumulative distribution assumption. Consequently, the corresponding complete damage state estimates may involve increased extrapolation uncertainty relative to lower damage states.

Consistent with the median (M) shifts, the exceedance probabilities at fixed PGA levels increase markedly with scour depth. For the severe damage state, the exceedance probability at a PGA of 0.9 g increases from 69% at a scour depth of 0.0 m to 88% at a scour depth of 6.8 m. For the complete damage state, the exceedance probability at 1.1 g increases from 19% to 31% over the same scour range. These results confirm that the loss of lateral confinement disproportionately degrades the pier ductility capacity and stability resistance under high demand levels.

To quantify the effect of scour on the distribution of failure outcomes, Figure 11 presents the comparative damage state distributions at PGA of 0.4, 0.8, and 1.2 g. The distributions demonstrate that once scour exceeds approximately 4.0 m, the proportion of severe and complete damage outcomes increases disproportionately, indicating a nonlinear degradation process rather than uniform progression. This observation is further supported by the fragility of the surfaces, as depicted in Figure 12. The minor and moderate surfaces exhibit limited scour sensitivity below 4.0 m, while the severe and complete surfaces show a stronger dependence on scour depth and a broader region of elevated exceedance probability under deep scour. Collectively, Figure 10, Figure 11 and Figure 12 indicate that deep scours shift the pier response into a more vulnerable regime across the PGA domain, with the effect becoming most consequential near the severe and collapse-related limit states.

5. Reliability Analysis and Discussion

5.1. Bootstrap Uncertainty Quantification

To ensure the statistical reliability of the vulnerability estimates and quantify record-to-record variability, this study implements a non-parametric bootstrap resampling method based on the IDA dataset. The adopted bootstrap procedure primarily captures record-to-record variability and fragility fitting uncertainty conditional on the selected numerical model and scour scenarios. For each scour depth and damage state, 10,000 bootstrap replicates are generated to construct a 95% confidence interval (CI) for the fitted fragility functions. Figure 13 and Figure 14 show the confidence bands for the fitted fragility curves of the bearings and piers, respectively.

The structure of the confidence bands provides direct evidence of the reliability of the prediction model. As indicated by the plotted curves, the 95% CI remains notably narrow in the lower damage states, indicating high stability in identifying the onset of the component exceedance probability. In comparison, the band widens near the upper limit state, reflecting the increased dispersion of the nonlinear dynamic response under high PGA excitation. Despite this increase, the fitted function remains within an acceptable tolerance across all scour depth scenarios. This result supports the robustness of the fitted fragility representation and its suitability for subsequent system-level assessments.

5.2. System Fragility

The system performance depends on how the component exceedance events accumulate under a given intensity measure and scour depth. Figure 15 presents the system-level fragility curves under series and parallel idealisations for both minor and severe damage. These profiles span the full spectrum of scour depths, ranging from 0.0 m to 6.8 m. For minor damage, the series exceedance probability (gold curve) increases rapidly and approaches 1.0 at low PGA across all scour depths, indicating that the bearings govern the early loss of serviceability. This result is consistent with the low median capacities observed for the bearing limit states in Section 4.1.

In contrast, the parallel-system fragility, indicated by the green curves and controlled by piers, exhibits a more gradual upward trend and remains well below 1.0 over the same PGA range. In the case of severe damage, the separation between the series and parallel system curves becomes more pronounced. It is due to the fact that the exceedance probability of bearing saturates at moderate PGA range, while the exceedance probability of pier continues to evolve toward the upper PGA range. As a result, a clear probabilistic separation between widespread component distress and global instability is produced.

This separation is quantified through the probabilistic separation margin $M_R$, defined as the difference between the exceedance probabilities of the series and parallel systems at the same PGA and scour depth, that is, the exceedance probability gap between the two system idealisations, as shown in Figure 16 and Figure 17. It should be emphasised that the proposed separation margin is not intended to represent structural redundancy in a strict reliability-theory sense. Instead, it provides a probabilistic indicator describing the separation between early component-level exceedance and global system-level deterioration under the adopted series and parallel idealisations. Accordingly, the metric should be interpreted as a qualitative descriptor of component interaction evolution rather than a direct measurement of structural redundancy capacity. For the minor damage stage, $M_R$ forms a sharp peak concentrated within a narrow PGA range roughly between 0.20 g and 0.40 g. The peak value of $M_R$ decreases with scour depth, from approximately 0.93 in the no-scour case to approximately 0.79 at the 6.8 m scour condition, while the peak location remains nearly unchanged around PGA 0.30 g. This observation indicates that scour primarily reduces the probabilistic separation between the bearing-governed minor exceedance probability and pier-governed system stability, rather than shifting both mechanisms uniformly along the intensity axis (PGA). For the severe damage stage, $M_R$ remains nonzero across a substantially wider PGA range, approximately 0.35 g to beyond 1.0 g, and approaches 1.0 near PGA values of approximately 0.55 g to 0.65 g. This trend suggests that the bearing and pier exceedance probabilities become separated over a broader set of PGA, which is consistent with the larger separation between the component median PGA ($M$) at higher damage states. Therefore, the failure modes of the two components are less synchronised in terms of higher PGA.

5.3. Discussion on the Interpretation of Scour-Induced Failure

The results indicate that the observed changes in component interaction is primarily driven by scour-induced modification of the foundation restraint, which reshapes both the demand distribution and the failure sequencing. The simplified spring removal approach captures the primary structural effect of increased unsupported pile length but introduces limitations by neglecting the overburden stress relief and the exact geometry of the scour hole. Although these secondary mechanisms may influence the absolute seismic response magnitude, the adopted modelling strategy is intended to isolate the dominant stiffness degradation mechanism associated with progressive loss of lateral soil restraint under increasing scour depth.

Stiffness degradation lengthens the dominant vibration period and increases the deformation compatibility demand for displacement-sensitive components, which concentrates deformation within the isolation layer rather than reducing it. While previous studies by [24] suggested that scour might reduce superstructure demands in certain configurations, the findings in Section 4.1 demonstrate that isolation systems may exhibit a distinct response. This observation is consistent with the recent findings reported by [72], which suggest that isolation bearings are sensitive to displacement demand. For the analysed bridge configuration, increased flexibility results in larger relative displacements. Consequently, the probability of bearing exceedance and potential unseating increases significantly.

The system response exhibits a threshold-type evolution with scour that is consistent with the geometric transition of the pile group from a relatively stiff base behaviour to a more flexible foundation-controlled regime. This transition changes not only the median PGA values of the components but also the relative ordering of exceedance probability under limit states across bearings and piers. These changes are directly reflected in the depth-dependent reduction in the peak $M_R$ at the minor stage and the reshaping of the $M_R$ surface at the severe stage. Collectively, these trends demonstrate that representing scour effects using a simple linear degradation of the fragility parameters is insufficient for system reliability. It is due to the fact that scour effects modify the rate at which the exceedance probability of series and parallel systems changes with PGA, and where this separation reaches its maximum.

5.4. Discussion on the System Level

Flood-induced scour exhibits distinct differences in the seismic fragility of bridge components. The bearing results suggest that the fragility profile is dependent on the damage state. Within the studied scour depth range, scour is essentially insensitive to the fragility of the bearings under minor-damage conditions. However, deeper scour increases the exceedance probability of the bearing for moderate-to-complete damage states. It indicates that scour primarily amplifies the displacement demand at the isolation layer, thereby compromising the bearing capacity under large deformations rather than inducing a uniform shift in vulnerability across all limit states.

In contrast to bearings, the fragility of piers is governed by a robust scour dependency, shifting systematically toward higher vulnerability. Across all limit states, the fragility of the piers shifts toward lower PGA values as scouring increases. A quantitative assessment reveals that for a given PGA value, the exceedance probability increases significantly with increasing scour depth, particularly under conditions of severe and complete damage. For instance, at a PGA value of 1.1 g, the probability of reaching the collapse limit state increases from 19% to 31% within the same scour depth range. Crucially, the fragility surfaces demonstrate a threshold-like sensitivity transition in the scour depth range of 4.00 to 5.75 m. Within this interval, the rate of vulnerability accumulation progresses more rapidly, which is consistent with the geometric transition in the foundation restraint. This behaviour marks a transition in the pile group response, producing a rapid increase in the collapse probability once a specific scour depth range is exceeded by the scour depth.

To further investigate the physical mechanism associated with the identified scour-sensitive transition range, additional modal analyses are conducted for all scour conditions. As shown in Figure 18, the results indicate that the normalized fundamental period, defined as the ratio between the fundamental vibration period under a given scour condition T and the corresponding no-scour period T0, increases progressively with scour depth, while a noticeably accelerated period elongation occurs between approximately 4.00 m and 5.75 m of scour depth. Table 3 also confirms this result, where the fundamental vibration period nonlinearly transitions from 1.9981 s to 2.0234 s. This behaviour suggests that the pile foundation system gradually transitions from a relatively stiff foundation-controlled response toward a more flexible deformation-dominated regime as scour deepens. Consequently, seismic demand redistribution becomes increasingly pronounced within this transition range, leading to the accelerated increase in fragility observed in both component and system-level responses. This trend indicates that progressive scour substantially reduces lateral soil restraint and foundation stiffness, thereby increasing the dynamic flexibility of the bridge system. Unlike the previous framework by [34], which quantifies fragility sensitivity using continuous linear degradation functions, this study provides evidence that significant risk increases are suppressed until the scour depth exceeds the critical geometric threshold. Accordingly, this nonlinearity supports the conclusion that linear scour degradation assumptions can bias collapse-related estimates once the critical scour range is exceeded.

At the system level, series and parallel idealisations provide complementary reliability metrics that distinguish early component failures from overall load-carrying degradation. The series system, which represents the loss of functionality under the weakest link assumption, is dominated by bearing displacement limit states at low to moderate PGA. The parallel system experiences a loss of overall load-carrying capacity controlled by the pier limit states, which primarily evolve at higher PGA levels. The separation between these system responses is quantified by the probabilistic separation margin ($M_R$), which is defined as the difference between the exceedance probabilities of the series and parallel systems evaluated at the same PGA and scour depth. For the minor damage, this probability gap ($M_R$) forms a narrow peak with lower PGA values, and its maximum peak value decreases with deeper scour. In the case of severe damage stages, the probability gap remains nonzero over a substantially wider PGA range, with the upper limit of the PGA attaining high values. These trends clarify the governing mechanisms of the scour dependent interaction evolution. Results indicate that scour affects system reliability not only by reducing the median PGA capacities ($M$), but also by altering the PGA-dependent relationship between the exceedance probabilities of bearing and pier under limit states, which governs the variation of the exceedance probability difference ($M_R$).

6. Conclusions

This study develops a detailed probabilistic framework to quantify the component interaction evolution of failure mechanisms of a railway bridge subjected to flood-induced scouring and seismic excitation. A three-dimensional nonlinear model with scour-dependent pile–soil interaction is integrated with incremental dynamic analysis (IDA) of a series of matched records across six discrete scour depths, producing 1872 response histories. Record-to-record variability is quantified using a non-parametric bootstrap with 10,000 resamples to construct 95% confidence intervals (CI) for the component and system fragility. The system formulation combines component fragility with series and parallel idealisations, which enables a direct interpretation of how scour reshapes failure sequencing and the dependence structure between bearings and piers. The main findings are summarised as follows.

• Scour has minimal impact on bearing fragility under minor damage but markedly increases exceedance probabilities for moderate-to-complete damage states by amplifying displacement demands at the isolation layer.
• Scour consistently increases pier vulnerability across all limit states, with fragility shifting to lower PGA values and exhibiting a nonlinear escalation. The identified transition range between approximately 4.00 m and 5.75 m corresponds to a noticeable acceleration in modal-period elongation caused by scour-induced foundation flexibility. This behaviour suggests that the pile-group system gradually transitions from a relatively stiff foundation-controlled response toward a more flexible deformation-dominated regime. Consequently, seismic demand redistribution becomes more pronounced beyond this scour range, leading to the accelerated increase in component exceedance probability observed in the fragility surfaces.
• The results indicate that scour primarily alters the interaction hierarchy between bearings and piers by modifying the relative progression of component exceedance probabilities under increasing seismic intensity. The mechanism involves shifting dominance from bearings at low-to-moderate PGA (series response) to piers at higher PGA (parallel response), while decreasing probabilistic separation margin and altering component interaction effects. This interaction evolution becomes increasingly pronounced under severe scour conditions, leading to larger probabilistic separation between early functionality loss and global collapse-related exceedance.

The proposed probabilistic framework evaluates the conditional seismic fragility of the railway bridge system under discrete scour depths. The adopted scour scenarios are considered pre-existing hydraulic degradation states prior to seismic excitation. The system reliability analysis employs series and parallel idealisations as conceptual bounding exercises to bracket the structural performance limits. This study focuses on the interactions between piers and bearings in multi-span continuous bridges, assuming time-invariant material properties. Some conclusions may depend on bridge-specific parameters such as span arrangement, pile geometry, and bearing configuration. This engineering case study establishes an operational conditional fragility assessment framework that can be generalized to evaluate other continuous railway bridge classes exposed to cumulative hydraulic degradation. Future investigations should incorporate the contribution of abutment-soil interactions and three-dimensional modal coupling effects present in skewed or curved bridge geometries. Furthermore, integrating deterioration mechanisms, such as reinforcement corrosion and concrete carbonation, into the fragility surface formulation would strengthen the life-cycle risk assessments for aged bridges in aggressive hydraulic environments.