## 1 Introduction  
The design and assessment of prestressed concrete bridges have evolved significantly over recent decades, driven by the growing need to understand complex dynamic interactions and ensure structural resilience [1]. Railway bridges, in particular, face unique challenges, as they must simultaneously withstand both regular train-induced vibrations and potential seismic events [2]. This multi-hazard scenario becomes increasingly critical in seismically active regions with high-speed rail networks, where the dynamic amplification effects can be substantial [3]. The evolution of research concerning bridge dynamics began with early studies focusing on simple beam models under moving loads [4]. As train speeds increased, researchers discovered that traditional dynamic amplification factors could significantly underestimate actual structural responses [5]. Modern high-speed rail systems have introduced new challenges, particularly regarding resonance phenomena and the potential for train-bridge coupling effects [6]. The interaction between moving masses and prestressed elements has been shown to modify the bridge's natural frequencies and mode shapes in ways that traditional analysis methods fail to capture [7]. Seismic analyses of prestressed bridges have progressed substantially with the advancement of computational capabilities [8]. While early approaches relied heavily on equivalent static analysis methods, modern techniques incorporate sophisticated time-history analyses that consider multiple components of ground motion [9]. The introduction of performance-based seismic design concepts has led to a more nuanced understanding of how prestressed bridges behave under various earthquake scenarios [10]. The recognition that prestressing forces can significantly affect the seismic response through modifications to the structure's initial stiffness, changes in damping characteristics, and alterations to the potential for cracking and damage progression is particularly noteworthy [11]. Considering soil-structure interaction (SSI) further complicates this analysis, as local soil conditions can substantially modify both seismic wave propagation and train-induced vibrations through various mechanisms of energy dissipation and wave scattering [12; 13]. Advanced numerical studies have revealed complex frequency-dependent characteristics of soil-foundation systems, which are particularly relevant for prestressed bridges, where foundation flexibility can affect the distribution of prestressing force [14]. Recent investigations have highlighted the importance of considering nonlinear soil behavior, especially during strong seismic events [15]. The spatial variability in earthquake ground motion, particularly the traveling wave effects, has emerged as a crucial consideration in bridge engineering, especially for long-span structures [16]. These effects can lead to differential ground movements along the bridge length, potentially inducing additional demands on structural elements and support systems [17]. The research has shown that the phase difference in seismic waves arriving at different supports can significantly alter the global response of the bridge system [18]. The interaction between seismic waves and train-induced vibrations creates wave interference patterns that can either amplify or attenuate the structural response [19]. The simultaneous consideration of train loading and seismic excitation represents a relatively new frontier in bridge engineering research [20]. Early attempts to address this problem typically relied on a simple superposition of effects, an approach now recognized as potentially non-conservative [21]. Modern studies have revealed complex interactions between different loading mechanisms, particularly regarding the evolution of prestressing forces under combined dynamic effects [22; 23]. The role of structural damping mechanisms in these complex interactions presents additional challenges [24].