Safety Performance of a Polygonal Chord Stiffened Double-Deck Continuous Steel Truss Bridge Under Mixed Traffic Loading

Abstract

As a complex structural form capable of simultaneously bearing upper-deck highway traffic, lower-deck highway traffic, and rail transit, the curved chord stiffened double-deck continuous steel truss bridge is distinct from traditional single-deck bridges. The spatial superposition of multiple traffic types within this structure may result in multiple components approaching their critical states concurrently. Despite prior research efforts on this structural type, the failure evolution process from local yielding to global collapse under mixed traffic loads remains ambiguous. This study addresses these questions through systematic numerical investigation of a nine-span bridge with a 300 m main span. A two-stage analytical approach is employed: a Midas/Civil analysis first identifies critically stressed regions, then ABAQUS multi-scale modeling enables refined analysis of critical components while maintaining computational efficiency. Twenty-nine combined traffic loading cases encompassing dual- and triple-category configurations are systematically analyzed. The results show that the ultimate load-carrying capacity coefficients range from approximately 7 to 18, with a minimum of 7.137, and the dual-level highway combinations exert greater influence than road–rail combinations. More importantly, three failure path convergence characteristics were discovered. First, the initial failure position under each working condition tends to be consistent, initiating at the lower chord near the top of the mid-span pier, which confirms that inherent structural defects exist at this location. Second, the gusset plate at the top of pier W6 appears as the second failure location in 48% of cases and ranks within the first four locations across all cases. Third, path similarity progressively increases with traffic diversity. Additionally, Q370qE steel exhibits 5–22% stress exceedance with variable critical locations depending on traffic conditions. Based on these convergence characteristics, a safety monitoring scheme is proposed: monitoring points need to be arranged symmetrically on both sides of the bridge on the top chords, bottom chords, web members, and wedge plates near the tops of the piers.

1. Introduction

Combined road–rail bridges are subjected to vertical loads and lateral sway forces from vehicles and trains, necessitating high vertical and lateral stiffness, as well as favorable dynamic performance [1,2,3]. Over the past few decades, the structural forms of steel truss bridges employed in combined road–rail applications have undergone continuous innovation [4,5,6,7,8], evolving from conventional parallel chord steel truss bridges to diverse configurations, including steel truss arch bridges [9,10,11]. The polygonal chord stiffened double-deck continuous steel truss bridge combines polygonal chord stiffening technology with a double-deck layout. This design not only retains the advantage of uniform load distribution of continuous beams but also possesses the efficient load-bearing characteristics of steel trusses, thereby achieving high section stiffness and lower self-weight. Such bridges have been successfully applied in numerous highway–railway river-crossing projects.

Unlike conventional single-deck bridges, the polygonal chord stiffened double-deck continuous steel truss bridge accommodates highway lanes on the upper deck, while the lower deck carries both highway lanes and rail tracks [12,13,14,15]. During operation, upper-deck vehicles, lower-deck vehicles, and rail trains may act on the structure simultaneously in various combinations, generating mixed traffic loading conditions [16]. Compared with single traffic flow, mixed traffic loading not only increases the total load magnitude but, more critically, the spatial superposition of multiple traffic types may cause several structural components to approach critical states concurrently, potentially altering the structural failure mode. Consequently, investigating the mechanical behavior and failure paths of double-deck steel truss bridges under mixed traffic loading is of considerable significance for the safety assessment of such structures.

Systematic studies have been conducted by scholars worldwide on the ultimate bearing capacity of steel truss girder bridges.

In terms of experimental research, Chen et al. [5] took the Weihe River Bridge of Xi’an Metro Line 10 as the prototype and carried out tests using a 1:30 scaled model. They found that adding arched stiffening chords can alter the load transfer path. Structural failure initiates with the buckling of web members above the bearings, followed by yielding of the bottom chords, while the stiffening chords remain elastic throughout. Luo et al. [17] investigated different fiber types in UHPC hollow bridge decks and proposed an ultimate bearing capacity calculation formula that agrees well with experimental results. The results show that the cracking stress and ultimate bearing capacity of steel fiber specimens are significantly higher than those of basalt fiber specimens. Xue et al. [18] studied the influences of parameters such as bottom chord angle steel size and web member spacing through bending and shear tests on 12 specimens. They found that a larger bottom chord angle steel size leads to higher flexural capacity of the box girder; a smaller shear span ratio and denser web member spacing result in stronger shear capacity. Corresponding bearing capacity calculation formulas were also proposed. In a numerical simulation, Wu et al. [9] focused on Zhuzhou Qingshuitang Bridge and investigated the mechanical performance of a double-layer steel truss arch bridge with a steel-UHPC lightweight composite deck via finite element simulation, verifying the structural stability and economic efficiency of the system. Bai et al. [19] established a multi-scale finite element model incorporating geometric imperfections and residual stresses. The study indicates that local buckling can reduce the bridge’s ultimate bearing capacity by up to 22.3%, revealing that geometric imperfections are the dominant influencing factor. Wang and Ren et al. [20,21,22] combined the finite element method and response surface methodology, with optimization objectives of structural mechanical performance and material consumption. Optimization was performed using the AdaDelta algorithm and verified via the NSGA-II algorithm, ultimately yielding an optimal combination of design parameters balancing structural safety and economy, which provides an effective approach for parameter optimization of similar bridges. In theoretical research on ultimate bearing capacity, Xie [23] proposed a linear elastic iteration method based on the Homogeneous Generalized Yield Function (HGYF). The study demonstrates that this method can accurately predict the ultimate bearing capacity, effectively compensate for the deficiencies of the existing Generalized Yield Function (GYF), and precisely identify high-load-carrying and low-load-carrying members in trusses. Li et al. [24] developed a generalized analytical framework to evaluate the alternate load path redundancy of steel truss bridges when critical members suddenly fail. The suggested modeling approach was found to be computationally efficient in identifying the critical members for steel truss bridges through nonlinear dynamic analysis, and it was also demonstrated to be effective and accurate in simulating both the yielding and buckling behavior of truss members following the sudden removal of a critical member.

Based on the above literature review, existing studies mainly focus on single-functional highway or railway steel truss bridges, while limited attention has been paid to the safety performance of double-layer continuous steel truss bridges strengthened with polygonal chords under mixed traffic loads. Three fundamental research gaps remain regarding the structural safety of such bridges under mixed traffic loading: (1) What is the ultimate bearing capacity under different mixed traffic combinations? (2) Does the failure path exhibit identifiable regular characteristics? (3) Can the corresponding research outcomes provide guidance for structural health monitoring? This study aims to address the above issues through systematic numerical simulations.

To address this research gap, the present study investigates a combined road–rail polygonal chord stiffened double-deck continuous steel truss bridge, building upon established research findings and ultimate load-carrying capacity analysis methodologies. A preliminary full-bridge analysis is first performed using Midas/Civil to identify critically stressed regions, followed by the development of a multi-scale finite element model in ABAQUS 2023 incorporating refined modeling of critical components. Subsequently, 35 loading combinations encompassing single, dual, and triple traffic flow scenarios are designed to systematically examine the effects of traffic flow quantity, transverse load distribution patterns, and traffic type combinations on ultimate load-carrying capacity, yielding the ultimate load factors for each loading case. The failure evolution process is then traced using steel yield strength as the criterion, enabling the summarization of failure path regularity characteristics and the identification of extensive failure regions at the ultimate limit state. Finally, based on the structural characteristics and safety performance analysis results, monitoring recommendations for the operational phase of such bridges are proposed, providing guidance for the safe operation of similar structures.

2. Engineering Background and Finite Element Modeling

2.1. Project Overview

The bridge investigated in this study is a polygonal chord stiffened double-deck continuous steel truss bridge constructed using full falsework with single-stage de-centering. The span arrangement is 124 + 132 + 132 + 168 + 300 + 168 + 132 + 132 + 124 m, forming a nine-span continuous structure. The steel truss adopts a triangular configuration with a truss depth of 12 m and a transverse spacing of 30.5 m between main truss centerlines. The upper deck accommodates a six-lane highway, while the lower deck carries a four-lane highway, two rapid transit rail lines, and cantilevered pedestrian walkways on both sides. The general arrangement is illustrated in Figure 1 and Figure 2.

A differentiated steel grade strategy is employed for the steel girder. The primary structural members utilize Q370qE steel (yield strength of 370 MPa), the stiffening chords employ Q420qE steel (yield strength of 420 MPa), and the main truss members in the vicinity of the mid-span pier tops adopt Q500qE steel (yield strength of 500 MPa). To facilitate subsequent analysis, the members of the main truss and polygonal chord stiffening system are numbered as shown in Figure 3. Given the structural symmetry of the bridge, only member designations for half of the span are provided. Members in the symmetric half about the bridge centerline are denoted with prime notation (e.g., members A58 and A58′ are symmetric about the span centerline).

2.2. Midas/Civil Preliminary Analysis Model

To identify the critically stressed regions throughout the bridge and establish a foundation for subsequent refined modeling, a full-bridge finite element model was first developed using Midas/Civil for preliminary analysis. Although truss members theoretically carry only axial forces, gusset plate constraints in actual structures inevitably induce bending moments in the members. Consequently, beam elements were adopted to model the main truss members, while the orthotropic steel deck was modeled using plate elements. The complete model comprises 7107 nodes and 18,780 elements, as illustrated in Figure 4. The bearing configuration and material properties in the model are consistent with the design documentation. The short linking members on both sides of the stiffening chord verticals are pin-connected to the stiffening chords and main trusses, which is implemented in the model through the release of beam-end moment restraints.

The axial force distribution and stress distribution throughout the bridge under dead load is shown in Figure 5 and Figure 6. The stress distribution pattern is generally consistent with the axial force distribution, with higher stresses concentrated near the mid-span pier tops. The maximum tensile stress in the upper chord members at the mid-span pier tops reaches 269 MPa, while the maximum compressive stress at the mid-span pier top nodes attains −445 MPa.

For upper chord members under live loading, the axial force variations induced by the three live-load types (upper-deck vehicles, lower-deck vehicles, and trains) are comparable in magnitude, with notable differences occurring only at individual pier tops and within the mid-span segment. Upper-deck vehicles produce the largest axial force variation, reaching 6077.24 kN in the upper chord member A46A47 at the mid-span pier top. Lower-deck vehicles generate the maximum axial force variation in the same member, with a peak value of 5556.61 kN. For train loading, the maximum axial force variation occurs in upper chord member A54A55, which connects to the shortest stiffening chord vertical at mid-span, with a value of 4895.36 kN. The axial force distribution is presented in Figure 7.

For lower chord members under live loading, the axial force variations induced by lower-deck vehicles in the side span regions are considerably larger than those caused by the other two live-load types. The differences in lower chord axial force variations among the three live-load types remain most pronounced at the mid-span pier tops, where lower chord member E45E46 exhibits the maximum axial force variation for all three load types. The maximum lower chord axial force variations are 8107.8 kN for upper-deck vehicles, 7221.49 kN for lower-deck vehicles, and 4895.36 kN for trains. The axial force distribution is presented in Figure 8.

The web member axial forces under live loading are illustrated in Figure 9. All three live-load types produce peak tensile and compressive forces in the web members at the top of pier W5; however, the members exhibiting the maximum axial force variation are not exclusively located at pier W5. Under upper-deck vehicle loading, web member A33E32 at the top of pier W4 exhibits the maximum axial force variation of 5389.7 kN. Under lower-deck vehicle loading, web member A47E46 at the top of pier W5 exhibits the maximum variation of 4818.7 kN. Under train loading, web member A33E33 at the top of pier W4 demonstrates the maximum variation of 5116.23 kN.

For stiffening chord members under live loading, these members remain predominantly in tension. Across all three live-load types, stiffening chord member S40S41 exhibits the maximum axial force variation, with values of 5748.52 kN under upper-deck vehicle loading, 4789.18 kN under lower-deck vehicle loading, and 5159.74 kN under train loading. The axial force distribution is presented in Figure 10.

The comprehensive analysis of dead-load and live-load effects yields the following findings. Dead load constitutes a substantial proportion of the total loading and must not be neglected in subsequent ultimate load-carrying capacity analysis. The effects induced by the three live-load types are comparable in magnitude; consequently, the influence of each load type must be considered in structural performance assessment. Upper-deck vehicles predominantly affect member behavior in the mid-span region, whereas lower-deck vehicles exert greater influence on web members at side span mid-sections and lower chord members near side span pier tops. Although trains impose the largest load magnitude, their limited consist length prevents full coverage of the bridge mid-span, resulting in significant effects only on selected members within the side span segments. The main truss members in the vicinity of each pier top exhibit elevated axial forces and stresses, while the mid-span main truss undergoes substantial deformation. These regions are identified as the critically stressed zones of the bridge and will be modeled with refined shell elements in the subsequent ABAQUS analysis.

2.3. ABAQUS Multi-Scale Finite Element Model

2.3.1. Modeling and Meshing

Based on the preliminary analysis results from the Midas/Civil model, the main truss members at each pier top and within the mid-span segment constitute the critically stressed regions of the bridge—where structural members are most vulnerable to failure. A multi-scale hybrid modeling approach is adopted in this study: the critically stressed regions are modeled using refined S4R four-node doubly curved shell elements with reduced integration in ABAQUS, whereas the remaining regions (including the stiffening chord system and other main girder segments) are modeled using B31 two-node linear spatial beam elements.

The B31 beam elements and S4R shell elements of the main truss are coupled using the Coupling method to eliminate incompatible degrees of freedom. The bridge deck mesh elements are tied to the S4R elements to reflect the actual stress state of the deck. The deck mesh elements are connected to the B31 beam elements of the main truss using an MPC beam connection to achieve simultaneous transfer of forces and deformations. To avoid Saint-Venant’s effect, the modeling range of S4R elements near the main truss joints is appropriately expanded to achieve smooth transfer of internal forces and deformations between different element types.

The stiffening chords, stiffening chord verticals, and stiffening chord cross braces are all modeled as B31 elements, which are merged into the same segment through Boolean operations. The B31 elements of the stiffening chords and stiffening chord verticals are coupled to the S4R elements of the main truss using the Coupling method. To focus on the mechanical performance of the connection between the curved chord stiffening system and the main truss, the connections of the stiffening chords and stiffening chord verticals to the main truss are refined using S4R elements, also avoiding Saint-Venant’s effect. The short linking members in the stiffening chord verticals are pin-connected to the stiffening chords and main trusses. In the ABAQUS model, the pin connection between the B31 elements of the stiffening chord verticals and the S4R elements of the main truss is achieved using an MPC pin connection, while the pin connection with the B31 elements of the stiffening chords is realized by editing the model keywords and adding the RELEASE command to release the relevant degrees of freedom.

ABAQUS requires mesh generation for the model before solution. The mesh size must be set considering both computational accuracy and efficiency. For B31 elements, which are of lower importance, the global mesh size is set to 1200 mm. For S4R elements, which are the primary focus, the global mesh size is controlled at 600 mm, with local refinement to accommodate structural dimensions. To facilitate convergence, the S4R elements are meshed using the quadrilateral mesh advancing front algorithm. The meshing result is shown in Figure 11. The entire bridge model comprises 928,783 elements, including 824,059 S4R elements and 104,724 B31 elements.

This methodology enables detailed analysis of critical structural components while maintaining accurate boundary condition representation, thereby achieving substantial computational efficiency. The completed ABAQUS multi-scale full-bridge finite element model is illustrated in Figure 12. The bearing configuration and material properties employed in the model are consistent with those described in the preceding sections. As pedestrian loading is excluded from consideration in this study, the pedestrian walkway decking is omitted from the ABAQUS model to reduce computational complexity.

2.3.2. Load Cases

For load application, the structural self-weight component of dead load is implemented by specifying gravitational acceleration, whereas superimposed dead load is applied as surface pressure at the corresponding deck locations. Live-load positioning is determined using the moving load analysis function in Midas/Civil to calculate influence lines and identify the critical loading positions that produce the most unfavorable structural response. The preliminary analysis results indicate that the maximum compressive force in the bridge exceeds the maximum tensile force; accordingly, the maximum compression criterion is adopted for determining the most unfavorable loading state. The moving load analysis reveals that the maximum axial force location under all loading cases occurs at the top of pier W6. Under the most unfavorable state, both upper- and lower-deck vehicle loads are fully applied across spans 1, 3, 5, 6, and 8, while train loading is positioned on the mid-span segment adjacent to pier W6. Based on these results, the corresponding traffic live loads are converted to equivalent surface pressures and applied to the bridge deck in ABAQUS. The critical load positioning results and the ABAQUS load application for Load Case 7 (comprising upper-deck vehicles, lower-deck vehicles, and trains) are presented in Figure 13 and Figure 14.

As a double-deck combined road–rail bridge, the traffic flow combinations are categorized into three types for loading case selection: upper-deck vehicles, lower-deck vehicles, and trains. Due to the segregated arrangement of traffic lanes and rail tracks, the loading positions for upper- and lower-deck vehicles are subdivided into left carriageway, right carriageway, and both carriageways, whereas train loading positions are subdivided into left track, right track, and both tracks, as illustrated in Figure 15. Complete permutations of the three traffic flow categories, combined with dead load, yield 35 loading cases for ultimate load-carrying capacity analysis, as summarized in

Table 1. Loading cases for ultimate load-carrying capacity analysis under combined traffic flow.

Load Case	Upper Deck		Lower Deck		Trains
	L	R	L	R	L	R
1	●	●	○	○	○	○
2	○	○	●	●	○	○
3	○	○	○	○	●	●
4	●	●	●	●	○	○
5	●	●	○	○	●	●
6	○	○	●	●	●	●
7	●	●	●	●	●	●
8	●	○	○	○	○	○
9	○	○	●	○	○	○
10	○	○	○	○	●	○
11	●	○	●	○	○	○
12	●	○	○	●	○	○
13	●	○	○	○	●	○
14	●	○	○	○	○	●
15	○	○	●	○	●	○
16	○	○	●	○	○	●
17	●	○	●	○	●	○
18	●	○	●	○	○	●
19	●	○	○	●	●	○
20	●	○	○	●	○	●
21	●	●	●	○	○	○
22	●	●	○	○	●	○
23	●	○	●	●	○	○
24	○	○	●	●	●	○
25	●	○	○	○	●	●
26	○	○	●	○	●	●
27	●	●	●	○	●	○
28	●	●	●	○	○	●
29	●	○	●	●	●	○
30	●	○	●	●	○	●
31	●	○	●	○	●	●
32	●	○	○	●	●	●
33	●	●	●	●	●	○
34	●	●	●	○	●	●
35	●	○	●	●	●	●

Note: ● denotes load applied; ○ denotes load not applied.

.

2.3.3. Nonlinear Analysis

The ultimate load-carrying capacity analysis of the polygonal chord stiffened double-deck continuous steel truss bridge investigates the entire process from progressive loading to overall collapse of the bridge structure, involving both geometric nonlinearity and material nonlinearity.

For the material constitutive relationship, the stress-strain model for nonlinear steel analysis adopts the five-segment piecewise linear model proposed by Han [25]. The true plastic stress-strain curve for steel is presented in Figure 16.

To account for the influence of initial geometric imperfections on the structure, the geometric nonlinearity of the steel material must be analyzed. First, a linear eigenvalue buckling analysis is performed using the BUCKLE analysis step to extract the first-order buckling mode. The buckling mode obtained in the previous step is then introduced as the initial geometric imperfection according to the following equation:

$$X^{imp}=X^{perfect}+{\displaystyle \sum _{i=1}^m}{\zeta }_i{\varphi }_i$$

(1)

• $X^{perfect}$—nodal coordinates of the perfect model;
• ${\varphi }_i$—eigenvector of the i-th buckling mode (normalized displacement);
• ${\zeta }_i$—imperfection amplitude scaling factor of the i-th mode.

The model coordinates are then modified by incorporating an imperfection amplitude equal to 1% of the characteristic structural dimension. Using the modified finite element model, the NLGEOM option is enabled in the analysis step to account for large-displacement effects, thereby accurately capturing the influence of geometric changes on the structural response during the loading process.

Based on the ultimate load-carrying capacity calculation under the most unfavorable live-load cases, the ultimate load factors and the maximum structural deflections considering material nonlinearity only, geometric nonlinearity only, and both material and geometric nonlinearities are presented in

Table 2. Ultimate load factors and maximum structural deflections under different nonlinearities.

Analysis Type	Ultimate Load Factor	Maximum Structural Deflection (mm)
Dual nonlinearity	7.134	1729
Material nonlinearity	9.14	5208
Geometric nonlinearity	8.46	1793

. The analysis results show that the safety factor obtained after considering nonlinear factors is relatively small. This is inconsistent with the conclusions of linear elastic analysis, demonstrating that nonlinear effects must be taken into account when analyzing the ultimate bearing capacity of bridge structures using finite element methods. The safety factor from the double nonlinear analysis is the smallest, indicating that when both geometric and material nonlinearities are considered simultaneously, the structure reaches its ultimate bearing capacity under a smaller load, which should be a key focus in research. This paper focuses on further analysis of the double nonlinear calculation results.

2.4. Model Validation

To verify the reliability of the ABAQUS multi-scale finite element model, its computational results under dead load conditions are compared with those obtained from the Midas/Civil model. Critically loaded members near the mid-span pier top, including lower chord members E45E46 and E46E47, upper chord member A46A47, diagonal member A33E32, and vertical member S40S41, are selected for comparative analysis. The comparison of dead load axial forces between the two models is presented in Figure 17.

As shown in Figure 17, the relative errors in axial force calculations between the two models under dead load conditions are all within 3%, confirming that the ABAQUS multi-scale finite element model exhibits adequate computational accuracy for the subsequent ultimate load-carrying capacity analysis under combined traffic flow conditions.

3. Ultimate Load-Carrying Capacity Analysis Under Combined Traffic Flow

3.1. Criteria and Evaluation Indices for Ultimate Load-Carrying Capacity

When conducting ultimate load-carrying capacity analysis using the finite element method, whether the bridge has reached its ultimate limit state can be determined by examining the convergence of the iterative process, based on the solution principles governing nonlinear equations [26,27]. Two commonly employed convergence criteria are the displacement convergence criterion and the unbalanced force convergence criterion. The displacement convergence criterion requires that the ratio of the displacement increment $\mathsf{\Delta }{\delta }_i$ to the total displacement ${\delta }_{i-1}$ under the same load level be less than the prescribed displacement convergence tolerance ${\epsilon }_d$:

$${\displaystyle \frac{\mathsf{\Delta }{\delta }_i}{‖{\delta }_{i-1}+\mathsf{\Delta }{\delta }_i‖}}<{\epsilon }_d$$

(2)

The unbalanced force convergence criterion requires that the ratio of the nodal unbalanced force $∆F_i$ to the external load $F$ be less than the prescribed unbalanced force convergence tolerance ${\epsilon }_f$:

$${\displaystyle \frac{‖∆F_i‖}{‖F‖}}<{\epsilon }_f$$

(3)

During the iterative process, if both criteria cannot be satisfied simultaneously, the iteration is considered non-convergent, indicating that the structure has reached its ultimate limit state of load-carrying capacity.

Quantitative indices are essential for evaluating structural load-carrying capacity when investigating bridge safety performance. The ultimate load-carrying capacity coefficient ${\lambda }_u$ is the most widely adopted index for this purpose. In this study, dead load is treated as constant, and the live-load coefficient $\lambda$ is defined as the ratio of the applied live load to the design live load, where partial factors  ${\gamma }_0=1.1$, ${\gamma }_G=1.2,$ and ${\gamma }_Q=1.4$, along with the dynamic amplification factor of 1.066, have already been incorporated. Nonlinear implicit analysis is conducted using ABAQUS, with iteration non-convergence (minimum increment step less than $1\times {10}^{-7}$ adopted as the criterion for determining the structural ultimate state. The live-load coefficient at this threshold is designated as the ultimate load-carrying capacity coefficient ${\lambda }_u$. The expression for the structure reaching its ultimate load-carrying capacity is given by

$$P_{cr}=P_G+{\lambda }_u{P}_Q$$

(4)

where $P_{cr}$ denotes the critical load, $P_G$ represents the structural dead load, and $P_Q$ is the live load sustained by the structure. The ultimate load-carrying capacity coefficient ${\lambda }_u$ serves as the ultimate load factor, with its physical interpretation being the multiple of the design live load that the structure can withstand.

3.2. Ultimate Load-Carrying Capacity Under Dual-Category Traffic Flow

The ultimate load-carrying capacity coefficients under dual-category traffic flow combinations are presented in Figure 18. As illustrated, the influence of dual-category combined traffic flow on bridge ultimate load-carrying capacity, in descending order of significance, follows the sequence “upper-deck vehicles + lower-deck vehicles,” “upper-deck vehicles + trains,” and “lower-deck vehicles + trains.” This finding indicates that dual-deck highway traffic combinations exert a greater influence on bridge ultimate load-carrying capacity than highway–railway traffic combinations. Furthermore, the more extensively the traffic is distributed across the transverse direction, the lower the bridge ultimate load-carrying capacity coefficient becomes. A comparison of Load Cases 13 and 14, as well as Load Cases 15 and 16, reveals that whether the train operates on the left or right track has relatively minor influence on the structural ultimate load-carrying capacity.

The location exhibiting maximum true strain across the entire bridge is identified as the failure location when the bridge reaches its ultimate state.

Table 3. Failure locations under dual-category traffic flow.

Load Case No.	Ultimate State Failure Location
Case 4	Gusset plate A21′, left truss, pier W8 top
Case 5	Gusset plate A50′, right truss
Case 6	Gusset plate A49′, right truss
Case 11	Gusset plate A21′, left truss, pier W8 top
Case 12	Floor beam near lower chord E46′E47′, at concentrated load location of lower-deck vehicles on left truss
Case 13	Floor beam near upper chord A52′A53′, at concentrated load location of upper-deck vehicles on left truss
Case 14	Floor beam near upper chord A52′A53′, at concentrated load location of upper-deck vehicles on left truss
Case 15	Floor beam near lower chord E46′E47′, at concentrated load location of lower-deck vehicles on left truss
Case 16	Floor beam near lower chord E46′E47′, at concentrated load location of lower-deck vehicles on left truss
Case 21	Gusset plate A21′, left truss, pier W8 top
Case 22	Gusset plate A48′, left truss
Case 23	Gusset plate A21′, left truss, pier W8 top
Case 24	Gusset plate E47′, left truss
Case 25	Floor beam near upper chord A52′A53′, at concentrated load location of upper-deck vehicles on left truss
Case 26	Floor beam near lower chord E46′E47′, at concentrated load location of lower-deck vehicles on left truss

summarizes the bridge failure locations at ultimate state under dual-category traffic flow, with the corresponding true strain line chart at these failure locations presented in Figure 19. Combining Figure 19 and Table 3, the corresponding true strain contour map at the failure location can be plotted (Figure 20).

For the “upper-deck vehicles + lower-deck vehicles” combination, with the exception of Case 12, the structural failure locations at ultimate state are all situated at gusset plate A21′ on the left truss at the top of pier W8. The overall strain values across the entire bridge remain relatively small, with a maximum strain value of 0.062.

For the “upper-deck vehicles + trains” combination, cases with full-width loading of upper-deck vehicles (Cases 5 and 22) yield a maximum true strain value of 0.022, with structural failure locations at ultimate state occurring at the upper-deck gusset plates. In contrast, cases with single-truss loading of upper-deck vehicles (Cases 13, 14, and 25) produce a maximum true strain value of 0.106, with structural failure locations at ultimate state occurring at the floor beam near upper chord A52′A53′ at the concentrated load location of upper-deck vehicles.

For the “lower-deck vehicles + trains” combination, cases with full-width loading of lower-deck vehicles (Cases 6 and 24) yield a maximum true strain value of 0.021. The structural failure location at ultimate state for Case 6 is at the upper-deck gusset plate, whereas that for Case 24 is at the lower-deck gusset plate. Cases with single-truss loading of lower-deck vehicles (Cases 15, 16, and 26) produce a maximum true strain value of 0.128, with structural failure locations at ultimate state occurring at the floor beam near lower chord E46′E47′ at the concentrated load location of lower-deck vehicles.

The analysis results demonstrate that when train loads are included in the combined traffic flow, the maximum strain values across the entire bridge exhibit a significantly increasing trend compared with dual-deck vehicle combinations. Among these, the “lower-deck vehicles + trains” cases show the greatest increase in true strain values. Within the same category of load cases, combined traffic flow with “full-width loading of one category + single-truss loading of another category” yields relatively smaller true strain values, whereas combined traffic flow with “single-truss loading of both categories” produces relatively larger true strain values.

3.3. Ultimate Load-Carrying Capacity Under Triple-Category Traffic Flow

The ultimate load-carrying capacity coefficients under triple-category traffic flow are presented in Figure 21. As illustrated, with an increasing number of combined traffic categories, the live load acting on the bridge increases while the ultimate load-carrying capacity coefficient decreases. Additionally, the more extensively the traffic is distributed across the transverse direction, the smaller the bridge ultimate load-carrying capacity coefficient becomes. The ultimate load-carrying capacity coefficients for triple-category traffic flow are concentrated within the range of 7 to 10, with the most unfavorable load case yielding an ultimate load-carrying capacity coefficient of 7.137. Among the triple-category traffic flow cases, when the loading positions of two traffic categories are fixed, whether the remaining category is loaded on the left truss (left track) or right truss (right track) has minimal influence on the structural ultimate load-carrying capacity.

The location exhibiting maximum true strain across the entire bridge is identified as the failure location when the bridge reaches its ultimate state.

Table 4. Failure locations under triple-category traffic flow.

Load Case No.	Ultimate State Failure Location
Case 7	Gusset plate A49′, right truss
Case 17	Gusset plate A21′, left truss, pier W8 top
Case 18	Gusset plate A21′, left truss, pier W8 top
Case 19	Floor beam near upper chord A52′A53′, at concentrated load location of upper-deck vehicles on left truss
Case 20	Floor beam near upper chord A52′A53′, at concentrated load location of upper-deck vehicles on left truss
Case 27	Gusset plate A21′, left truss, pier W8 top
Case 28	Gusset plate A21′, left truss, pier W8 top
Case 29	Gusset plate A48′, left truss
Case 30	Gusset plate A21′, left truss, pier W8 top
Case 31	Gusset plate A48′, left truss
Case 32	Gusset plate A48′, left truss
Case 33	Gusset plate A21′, left truss, pier W8 top
Case 34	Gusset plate A48′, left truss
Case 35	Gusset plate A48′, left truss

summarizes the bridge failure locations at ultimate state under triple-category traffic flow, with the corresponding true strain presented in Figure 22. Combining Figure 22 and Table 4, we can plot the corresponding true strain contour map at the failure location (Figure 23).

As evident from Figure 23, the bridge failure locations at ultimate state for all cases occur at gusset plates, with the exception of Cases 19 and 20, where failure occurs at the floor beam near upper chord A52′A53′ at the concentrated load location of upper-deck vehicles on the left truss.

For the triple-category full-width loading case (Case 7), the bridge failure location at ultimate state is at gusset plate A49′ on the right truss. For cases with two-category full-width loading combined with one-category single-truss loading (Cases 17, 18, 27, 28, 30, and 33), the bridge failure location at ultimate state is at gusset plate A21′ on the left truss at the top of pier W8. For cases with two-category single-truss loading combined with one-category full-width loading (Cases 29, 31, 32, 34, and 35), the bridge failure location at ultimate state is at gusset plate A48′ on the left truss. In this latter category of cases, trains are predominantly loaded on both tracks.

The preceding analysis has established the ultimate load-carrying capacity coefficients under various traffic combinations. However, the numerical values alone do not reveal the structural failure mechanism. To understand how the bridge transitions from local yielding to global failure and to identify critical vulnerable locations for monitoring purposes, a detailed failure path analysis is conducted in the following section.

4. Failure Mode and Path Analysis at Ultimate State

4.1. Mechanical Mechanism of Failure Paths

During continuous load increase, bridge structural members sequentially undergo elastic behavior, yielding, and failure stages, forming specific failure evolution paths [28]. The investigation of failure paths aims to reveal the dynamic evolution patterns from local damage to global collapse, identify structural vulnerabilities, and provide a theoretical foundation for bridge safety operation and structural health monitoring with early warning capabilities.

In this section, the attainment of yield strength by steel is adopted as the criterion for member failure. During ABAQUS nonlinear analysis, the structural failure path is determined by tracking the stress level variations in each member as a function of the live-load coefficient and recording the sequence in which members reach yield strength. The failure path description refers to the progressive failure process of bridge component members under continuously increasing loads, which is derived from ultimate load-carrying capacity analysis.

For the polygonal-chord stiffened double-deck continuous steel truss bridge, failure evolution follows the mechanical mechanism of “stress concentration → local yielding → internal force redistribution → progressive yielding.” The mid-span pier top region sustains significant negative bending moments and shear forces, with lower chord members subjected to high compressive stress states, frequently serving as the initiation point of the failure path. Gusset plates, functioning as connection nodes for main truss members, experience combined multidirectional stress states and become stress concentration transfer zones following lower chord member failure. The stiffening chord system transmits partial internal forces to the main truss members at the mid-span pier top section through vertical members. After the structure enters the plastic stage, the junction between the stiffening chord and the upper chord also becomes a critical node along the failure path. It should be emphasized that the “failure path” in this paper only refers to the sequential order in which different members reach their yield strength for the first time (initial yield) under gradually increased live loads. This sequence is adopted to identify the members most susceptible to initial plasticization and track the propagation process of plasticity throughout the entire bridge. Initial yield does not mean an immediate loss of load-bearing capacity of the member. After yielding, owing to strain hardening, the member can still sustain additional loads unless local buckling or connection failure occurs.

To facilitate failure path description, the primary failure locations and their corresponding abbreviations are summarized in

Table 5. Failure locations and corresponding abbreviations.

Abbreviation	Failure Location	Abbreviation	Failure Location
LC1	Lower chord at pier W1 top	LC5/6	Lower chord at pier W5/W6 top
LCT	Lower chord in train loading segment	GP2	Gusset plate at pier W2 top
GP3	Gusset plate at pier W3 top	GP4	Gusset plate at pier W4 top
GP5	Gusset plate at pier W5 top	GP6	Gusset plate at pier W6 top
GP8	Gusset plate at pier W8 top	GP9	Gusset plate at pier W9 top
GPT	Gusset plate in train loading segment	WR6	Web member at pier W6 top
WRUV	Upper flange of web member at upper-deck vehicle concentrated load location	WRT	Lower flange of web member in train loading segment
DHL6	Lower-deck floor beam at pier W6 top	DHT	Floor beam in train loading segment
DZT	Deck stringer in train loading segment	UU1	Junction of stiffening chord and upper chord
BDL6	Lower-deck plate at pier W6 top

. Given the symmetric configuration of the bridge, failure locations are not differentiated between left and right trusses.

4.2. Local Plastic Failure Under Dual-Category Traffic Flow

Dual-category traffic flow load cases comprise three combination types: “upper-deck vehicles + lower-deck vehicles,” “upper-deck vehicles + trains,” and “lower-deck vehicles + trains.” The bridge failure paths under dual-category traffic flow load cases and the distribution of the first ten failure locations across the entire bridge for each load case are illustrated in Figure 24.

As illustrated in Figure 20 and Figure 21, with respect to failure initiation location, the bridge consistently begins failing from the lower chord at the mid-span pier top (LC5/6) under all dual-category traffic flow conditions. This is attributed to the mid-span pier top being situated at the junction between the 168 m span and the 300 m span, where it sustains significant negative bending moments. The compressive stress level in the lower chord is the highest across the entire bridge, having already approached relatively high stress levels under dead load alone. The superposition of live load causes this member to reach yield strength first.

With respect to failure propagation patterns, following lower chord failure, stress transfers to adjacent gusset plates. Among the first ten failure locations, common failure locations shared across all dual-category traffic flow load cases include the lower chord at the tops of piers W5/W6 (LC5/6), the gusset plates at the tops of piers W5/W6/W8 (GP5/GP6/GP8), and the junction between the mid-span stiffening chord and upper chord (UU1). The gusset plate at the top of pier W6 (GP6) appears within the first four failure locations for all load cases, demonstrating that this gusset plate constitutes a critical structural vulnerability second only to the lower chord at the mid-span pier top.

With respect to combined traffic flow effects, compared with single-category traffic flow, the dual-category combination causes early-failing locations to progressively shift toward a bridge-wide distribution. Relative to single-category traffic flow, the first ten failure locations under dual-category traffic flow load cases exhibit reduced occurrence of local members, including the gusset plates in the train loading segment (GPT), the deck stringers (DZT), the upper flanges of the web members at upper-deck vehicle concentrated load locations (WRUV), the web members at the pier W6 top (WR6), and the lower-deck plates (BDL6). Failure locations become more concentrated within critical main truss regions.

With respect to the influence of transverse loading configuration, although the ultimate load-carrying capacity coefficients for Cases 13 and 14, as well as Cases 15 and 16, are similar, their failure paths are not entirely identical. This indicates that whether trains are loaded on the left track or right track exerts a certain influence on the failure propagation sequence. In contrast, Cases 4 and 12 represent full-width and single-truss loading configurations of “upper-deck vehicles + lower-deck vehicles,” respectively, and their first ten failure locations follow an identical failure sequence. This demonstrates that transverse loading configuration has minimal influence on the failure path for the same traffic flow combination.

4.3. Local Plastic Failure Under Triple-Category Traffic Flow

Triple-category traffic flow load cases comprise various combinations of upper-deck vehicles, lower-deck vehicles, and trains. The bridge failure paths under triple-category traffic flow load cases and the distribution of the first ten failure locations across the entire bridge for each load case are illustrated in Figure 25.

As illustrated in Figure 24 and Figure 25, with respect to failure initiation location, the bridge under triple-category traffic flow similarly begins failing from the lower chord at the mid-span pier top (LC5/6), consistent with dual-category traffic flow load cases. The immediately subsequent failure locations in most load cases are gusset plates at the pier W2/W6/W8 top. However, under conditions of significant vehicle load eccentricity, such as Cases 17, 18, and 31, the gusset plates near the pier W5 top fail earlier. This indicates that eccentric loading effects alter the stress distribution in local regions, thereby influencing the failure propagation sequence.

With respect to failure propagation patterns, common failure locations shared across all triple-category traffic flow load cases include the lower chord at the tops of piers W5/W6 (LC5/6), the gusset plates at tops of piers W2/W3/W5/W6/W8 (GP2/GP3/GP5/GP6/GP8), the junction between the mid-span stiffening chord and upper chord (UU1), and the lower chord at the top of pier W1 (LC1). Compared with dual-category traffic flow, the common failure locations under triple-category traffic flow load cases additionally include gusset plates at the pier W2/W3 top and the lower chord at the pier W1 top, indicating that as load intensity increases, the failure range extends toward the side-span regions.

With respect to combined traffic flow effects, similar to patterns observed under dual-category traffic flow combinations, triple-category traffic flow combinations cause early-failing locations to progressively extend further toward a bridge-wide distribution. The first ten failure locations show further reduction compared with dual-category traffic flow, with floor beams in the train loading segment (DHT) no longer appearing. Failure locations become more concentrated on main truss members and gusset plates.

With respect to failure path similarity, load case groups with highly similar failure paths exist among triple-category traffic flow load cases. The failure paths for Cases 19 and 20, Cases 27 and 28, and Cases 29 and 30 are completely identical. Cases 17 and 18 exhibit identical failure sequences, except for the reversed order of GP6 and GP8. The analysis reveals that when vehicle loading configurations remain unchanged, different train loading arrangements (left track, right track, or dual track) essentially do not influence the failure path. This is attributed to the relatively small transverse eccentricity of train loads, which results in limited influence on the overall structural stress state.

With respect to lower-deck loading effects, when loading on the lower deck of the bridge is more substantial, the floor beam near the top of pier W6 (DHL6) and locations near the mid-span train loading segment fail earlier. This occurs because lower-deck vehicles and trains act directly on the lower-deck system, thereby elevating the stress levels in lower-deck members.

4.4. Convergence Patterns of Local Plastic Failure Paths

Based on comprehensive analysis of local plastic failure under dual-category and triple-category traffic flow, the following convergence patterns can be identified:

(1)

Convergence of failure initiation location. Across all 29 combined traffic flow load cases, the bridge invariably begins failing from the lower chord at the mid-span pier top (LC5/6) without exception. This demonstrates that the lower chord at the mid-span pier top constitutes an inherent vulnerability of the polygonal-chord stiffened double-deck continuous steel truss bridge, with its failure initiation location unaffected by traffic flow combination types or transverse loading configurations.

(2)

Convergence of the second failure location. Among the 29 load cases, 14 cases exhibit the gusset plate at the top of pier W6 (GP6) as the second failure location, representing approximately 48%. Furthermore, GP6 appears within the first four failure locations for all load cases, demonstrating that the gusset plate at the top of pier W6 is a critical vulnerability second only to the lower chord at the mid-span pier top and warrants focused attention in structural health monitoring.

(3)

Concentration tendency of failure locations. As traffic flow progressively increases, the early-failing locations of the bridge gradually converge. The first ten failure locations under dual-category traffic flow load cases encompass a greater variety of member types, whereas the first ten failure locations under triple-category traffic flow load cases are more concentrated on critical main truss components, including lower chords, gusset plates, and the junction between the stiffening chord and upper chord. This pattern indicates that under high-intensity combined loading, the structural failure mode tends toward stabilization, with main truss members becoming the critical factors governing the overall structural load-carrying capacity.

(4)

Failure path similarity. The degree of failure path similarity among triple-category traffic flow load cases is notably higher than that among dual-category traffic flow load cases. Within triple-category traffic flow load cases, when vehicle loading configurations remain identical, failure paths corresponding to different train loading arrangements are essentially consistent. This pattern provides a basis for simplification in bridge safety assessment: when conducting ultimate state analysis, failure characteristics of similar load cases can be inferred through analysis of representative load cases.

4.5. Extensive Failure Regions

To further comprehensively reveal the failure characteristics of the entire bridge at the ultimate limit state, this section analyzes the extensive failure regions when the structure reaches its ultimate load-carrying capacity. Based on ultimate load-carrying capacity analysis results, load cases with the minimum ultimate load-carrying capacity from dual-category and triple-category traffic flow are selected as representative cases for investigation.

For dual-category traffic flow load cases, Case 4 (upper-deck vehicles + lower-deck vehicles), Case 5 (upper-deck vehicles + trains), and Case 6 (lower-deck vehicles + trains) are selected as representative cases. The extensive failure regions under dual-category traffic flow are presented in

Table 6. Extensive failure regions under dual-category traffic flow.

Load Case No.	Extensive Failure Regions at Ultimate State
Case 4	Main truss members at mid-span pier top and concentrated live-load segment; deck plate at mid-span pier top
Case 5	Main truss members at mid-span pier top and concentrated live-load segment; deck plate at upper-deck vehicle concentrated load location and at mid-span pier top
Case 6	Main truss members at mid-span pier top and concentrated live-load segment; deck plate at junction between stiffening chord and upper chord in mid-span region and at mid-span pier top; floor beams and stringers in train loading segment

, with corresponding ABAQUS stress contour plots shown in Figure 26.

As evident from Table 6 and Figure 26, the extensive failure regions of the bridge at ultimate state under dual-category traffic flow are influenced by the combined traffic flow configuration. Failure initially occurs at the loading locations of individual traffic categories contained within each combined traffic flow load case, then progressively concentrates from the concentrated force loading locations of individual traffic categories toward the main truss members within the combined concentrated live-load segment. With increasing traffic flow, the deck plate at the junction between the stiffening chord and upper chord in the mid-span region also undergoes extensive failure.

For triple-category traffic flow load cases, Case 7 (triple-category full-width loading), Case 18 (two-category full-width + one-category single-truss loading), Case 32 (one-category full-width + two-category single-truss loading), and Case 34 (triple-category single-truss loading) are selected as representative cases. The extensive failure regions under triple-category traffic flow are presented in

Table 7. Extensive failure regions under triple-category traffic flow.

Load Case No.	Extensive Failure Regions at Ultimate State
Case 7	Main truss members at mid-span pier top and in concentrated live-load segment; deck plate at upper-deck vehicle concentrated load location and at mid-span pier top
Case 18	Main truss members at mid-span pier top and in concentrated live-load segment; deck plate, floor beams, and stringers at mid-span pier top
Case 32	Main truss members at mid-span pier top and in concentrated live-load segment; deck plate at upper-deck vehicle concentrated load location and at mid-span pier top
Case 34	Main truss members at mid-span pier top and in concentrated live-load segment; deck plate at upper-deck vehicle concentrated load location and at mid-span pier top

, with the corresponding ABAQUS stress contour plots shown in Figure 27.

As evident from Table 7 and Figure 27, the extensive failure regions of the bridge at ultimate state under triple-category traffic flow are concentrated at main truss members at the mid-span pier top and in the concentrated live-load segment, with the deck plate at the mid-span pier top also undergoing extensive failure. Compared with dual-category traffic flow, the distribution of extensive failure regions under triple-category traffic flow is more uniform, indicating that under high-intensity combined loading, the ultimate failure mode of the structure tends toward stabilization.

An analysis of stress levels for different steel grades at ultimate state was conducted. The stress levels of various steel grades at ultimate state under representative load cases of dual-category and triple-category traffic flow are presented in

Table 8. Stress levels of various steel grades at ultimate state under dual-category traffic flow.

Load Case No.	Steel Grade	Maximum Stress (MPa)	Location of Maximum Stress
Case 4	Q500qE	529.9	Lower chord at mid-span pier top bearing
	Q420qE	421.1	South stiffening chord node S47
	Q370qE	429.2	Upper deck gusset plate at pier W8 top
Case 5	Q500qE	531.6	Lower chord at mid-span pier top bearing
	Q420qE	416.8	South stiffening chord node S47
	Q370qE	392.0	Upper deck gusset plate near pier W6
Case 6	Q500qE	549.4	Lower chord at mid-span pier top bearing
	Q420qE	421.3	South stiffening chord node S47
	Q370qE	428.1	Lower chord in concentrated live-load segment

and

Table 9. Stress levels of various steel grades at ultimate state under triple-category traffic flow.

Load Case No.	Steel Grade	Maximum Stress (MPa)	Location of Maximum Stress
Case 7	Q500qE	549.8	Lower chord at mid-span pier top bearing
	Q420qE	421.5	South stiffening chord node S47
	Q370qE	412.0	Upper deck gusset plate in concentrated live-load segment
Case 18	Q500qE	539.0	Lower chord at mid-span pier top bearing
	Q420qE	421.2	South stiffening chord node S47
	Q370qE	450.7	Upper deck gusset plate at pier W8 top
Case 32	Q500qE	535.2	Lower chord at mid-span pier top bearing
	Q420qE	421.1	South stiffening chord node S47
	Q370qE	388.1	Upper deck gusset plate in concentrated live-load segment
Case 34	Q500qE	557.2	Lower chord at mid-span pier top bearing
	Q420qE	421.4	South stiffening chord node S47
	Q370qE	420.8	Lower chord in concentrated live-load segment

, respectively.

Based on comprehensive analysis of Table 8 and Table 9, since Q500qE and Q420qE grade steels have limited application ranges across the entire bridge (used for main truss members near the mid-span pier top and for stiffening chords, respectively), their locations of maximum stress at ultimate state are relatively fixed: the maximum stress location for Q500qE grade steel is consistently at the lower chord at the mid-span pier top bearing, whereas that for Q420qE grade steel is consistently at the south stiffening chord node S47. The stress level of Q420qE grade steel at ultimate state remains near the yield stress (420 MPa) without significant exceedance beyond the yield limit.

Q370qE grade steel has extensive application across the entire bridge, with stress exceedance beyond yield ranging from 5% to 22%. The maximum stress location is significantly influenced by traffic flow load cases: under “upper-deck vehicles + lower-deck vehicles” load cases, maximum stress locations are primarily at pier top gusset plates; under load cases involving trains, maximum stress locations are distributed between pier top gusset plates and lower chords in the live-load segment, varying with different load cases. Consequently, Q370qE grade steel members warrant focused attention in bridge safety monitoring, particularly gusset plates at various pier tops and lower chords in the concentrated live-load segment.

5. Bridge Safety Monitoring Scheme

From the static performance analysis results of the bridge presented above, it is known that under dead or live loads, the lower chord members near the pier tops on both sides of the mid-span experience significant stresses. Further analysis using the multi-scale model identified the specific failure locations, while the failure path analysis yielded the early-failure locations of the bridge under various loading cases. The stresses and strains at these locations require focused attention.

This chapter monitors the stress and strain of the bridge in real time to determine whether the structural stress exceeds the allowable limits and to assess the structural safety condition. The monitoring of structural stresses is achieved through the conversion of strain data. During bridge operation, stress and strain gauges must be installed at locations where the structural stresses are high, and additional gauges should be placed at the vulnerable failure locations. Strain sensors that can be used for monitoring include fibre-optic sensors, foil strain gauges, semiconductor strain gauges, and vibrating-wire strain gauges, among others.

From the static analysis, the chord members with high stresses across the entire bridge are those at the mid-span of each span, as well as the pier-top members of all piers except piers W1/W10. The web members with high stresses are also the pier-top members of all piers except piers W1/W10. Under most combined traffic flow cases, the location of maximum stress in the stiffening chords at the ultimate limit state is near node S47 of the stiffening chord. The stresses in the stiffening chord verticals and cross braces are very small, so they may be excluded from monitoring.

In addition to the high-stress locations, the vulnerable failure locations of the bridge can be obtained from the failure path analysis. Because there are many combined traffic flow cases, the early-failure locations statistically derived from the failure paths are numerous, and some locations fail early only under isolated cases. Adding gauges at all these locations would be inefficient and costly; therefore, a selection must be made. The failure order of bridge components varies across different loading cases, exhibiting different vulnerability characteristics. Consequently, this paper adopts a scoring method to perform a vulnerability analysis on the failure locations listed in Table 5, and selects those with higher vulnerability scores as the vulnerable failure locations where additional gauges should be installed.

In the vulnerability analysis, the failure locations are first scored according to their failure order under each loading case. Then the scores under different loading cases are normalized using the ultimate load-carrying capacity coefficient. Finally, the normalized scores of each bridge component are summed across all loading cases to obtain the vulnerability index of each component. The scoring criteria are presented in

Table 10. Scoring criteria for bridge components.

Failure Order	1	2	3	4	5	6	7	8	9	10	>10
Score	10	9	8	7	6	5	4	3	2	1	0

.

According to the scoring criteria in Table 10, the scores $p_i$ of the bridge failure locations listed in Table 5 under each loading case are obtained, where $i$ denotes the loading case number. The ultimate load-carrying capacity coefficient ${\lambda }_{ui}$ is introduced as a normalization parameter to adjust the score is the ultimate load-carrying capacity coefficient corresponding to loading case $i$, and its specific values are given in Figure 18 and Figure 21.

The final formula for calculating the vulnerability index $P$ of each bridge component is as follows:

$$P=\sum {\displaystyle \frac{p_i}{{\lambda }_{ui}}}$$

(5)

Using Equation (5), the vulnerability indices of the bridge failure locations are obtained, as shown in

Table 11. Vulnerability indices of bridge failure locations.

Rank	Abbreviation	Failure Location	Vulnerability Index
1	LC5/6	Lower chord at pier W5/W6 top	32.31
2	GP6	Gusset plate at pier W6 top	26.68
3	GP8	Gusset plate at pier W8 top	24.21
4	GP2	Gusset plate at pier W2 top	21.39
5	GP5	Gusset plate at pier W5 top	18.12
6	GP3	Gusset plate at pier W3 top	13.77
7	LC1	Lower chord at pier W1 top	13.68
8	UU1	Junction of stiffening chord and upper chord	11.45
9	GP9	Gusset plate at pier W9 top	4.87
10	WRT	Lower flange of web member in train loading segment	3.48
11	GP4	Gusset plate at pier W4 top	3.25
12	DHL6	Lower-deck floor beam at pier W6 top	1.92
13	LCT	Lower chord in train loading segment	1.32
14	DHT	Floor beam in train loading segment	0.52
15	GPT	Gusset plate in train loading segment	0.44
16	DZT	Deck stringer in train loading segment	0.11
17	WR6	Web member at pier W6 top	0.08
18	BDL6	Lower-deck plate at pier W6 top	0.06
19	WRUV	Upper flange of web member at upper-deck vehicle concentrated load location	0.05

.

Based on the vulnerability analysis results in Table 11, the components with a vulnerability index are identified as vulnerable failure locations, where additional stress-strain monitoring points should be installed. These additional points are concentrated at the gusset plates of the chords near the pier tops.

Synthesizing the above analysis and considering the symmetry of the bridge, the proposed layout scheme for stress-strain monitoring points is shown in Figure 28. Within the red-framed areas at each pier top in the figure, monitoring points shall be arranged on the upper chord, lower chord, web members, and gusset plates. Monitoring points are to be installed on both sides of the bridge, one on each side.

6. Conclusions

This study investigates the safety performance of a polygonal-chord stiffened double-deck continuous steel truss bridge under mixed traffic loading, addressing three fundamental questions: ultimate load-carrying capacity under various traffic combinations, failure path regularities, and implications for structural health monitoring. A two-stage analytical approach was employed: preliminary full-bridge analysis using Midas/Civil to identify critically stressed regions, followed by ABAQUS multi-scale finite element modeling to examine 35 loading combinations. The main conclusions are as follows:

(1)

Ultimate load-carrying capacity. The ultimate load-carrying capacity coefficients under various mixed traffic loading conditions range from 8 to 18, with a minimum value of 7.137 under the most unfavorable condition. Among dual-category traffic combinations, the influence on ultimate load-carrying capacity follows the descending order of “upper-deck vehicles + lower-deck vehicles,” “upper-deck vehicles + trains,” and “lower-deck vehicles + trains.” This demonstrates that dual-level highway traffic combinations exert a greater influence on ultimate load-carrying capacity than road–rail traffic combinations. Additionally, more extensive transverse loading coverage results in lower ultimate load-carrying capacity coefficients, whereas the distinction between left and right carriageway (or track) loading has minimal influence on structural capacity.

(2)

Failure path regularities. Three convergence characteristics are identified. First, failure initiation location convergence: For all thirty-five load conditions analyzed, the critical failure zone of the bridge is highly consistent, primarily concentrating on the lower chord members at the upper region of mid-span piers (LC5/6). Second, secondary failure location convergence: the gusset plate at the top of pier W6 appears as the second failure location in approximately 48% of the combined traffic loading cases and ranks within the first four failure locations in all cases, identifying it as a critical vulnerability secondary only to the lower chord. Third, failure path similarity convergence: as the diversity of traffic types increases, the similarity among failure paths progressively rises, with triple-category traffic combinations exhibiting notably higher path similarity than dual-category combinations. Furthermore, when vehicle loading configurations remain identical, variations in train positioning (left track, right track, or both tracks) exert minimal influence on the failure sequence.

(3)

Ultimate state characteristics. The extensive failure regions are concentrated on the main truss members near the mid-span pier top and within the concentrated live-load segment, with the deck plate at the mid-span pier top also undergoing extensive failure. Under triple-category traffic loading, the distribution of extensive failure regions becomes more uniform, indicating that the structural failure mode tends to stabilize under high-intensity combined loading. Regarding material performance, Q370qE steel exhibits stress levels exceeding the yield strength by 5% to 22% at the ultimate state, and the location of maximum stress varies significantly with traffic loading conditions, whereas the locations of maximum stress for Q500qE and Q420qE steels remain fixed. Consequently, monitoring efforts should primarily focus on the lower chord members near the mid-span pier top, secondarily on the gusset plate at pier W6 top, and special attention should be given to Q370qE steel members due to the high variability of their stress distribution patterns.

(4)

Monitoring scheme and vulnerability quantification. Based on the failure path statistics under 35 mixed traffic loading cases, a vulnerability scoring method (scoring according to the local failure order, normalized by the ultimate load-carrying capacity coefficient) was used to quantitatively rank the potential failure locations of the entire bridge. Accordingly, a layout scheme for stress-strain monitoring points is proposed: monitoring points are required on the upper chord, lower chord, web members, and gusset plates near each pier top, arranged symmetrically on both sides of the bridge. This scheme can effectively cover the weakest links of the structure and provides some assistance for operational-period health monitoring.

(5)

This study reveals the convergence characteristics of the failure path of double-layer steel truss bridges strengthened with polygonal chord members under mixed traffic loads, providing a theoretical basis for the safety evaluation and health monitoring of similar structures. For complex structural forms of rail-cum-road bridges, future research may incorporate the uncertainty of random traffic flow to explore the vehicle–bridge coupling effect under transient dynamic conditions.