Failure Mechanism of Progressive Collapse Induced by Hanger Fracture in Through Tied-Arch Bridge: A Comparative Analysis

Abstract

Although through tied-arch bridges exhibit strong structural robustness, collapse incidents triggered by the progressive failure of hangers still occasionally occur. Given that such bridges are unlikely to collapse due to the damage of a single or multiple hangers under the serviceability limit state, this study focuses on the failure safety limit state. Using the Nanfang’ao Bridge with inclined hangers and the Liujiang Bridge with vertical hangers as case studies, this paper investigates the dynamic response and failure modes of the residual structures when single or multiple hangers fail and initiate progressive collapse of all hangers. The results demonstrate that the configuration of hangers significantly influences the distribution of structural importance coefficients and the load transmission paths. Under identical failure scenarios, the Nanfang’ao Bridge with inclined hangers remains stable after the failure of four hangers without experiencing progressive collapse, whereas the Liujiang Bridge with vertical hangers undergoes progressive failure following the loss of only three hangers, which indicates that inclined hanger configurations offer superior resistance to progressive collapse. Based on the aforementioned analysis, the LS-DYNA Simple–Johnson–Cook damage model was employed to simulate the collapse process. The extent of damage and ultimate failure modes of the two bridges differ significantly. In the case of the Nanfang’ao Bridge, following the progressive failure of the hangers, the bridge deck system lost lateral support, leading to excessive downward deflection. The deck subsequently fractured at the mid-span (1/2 position) and collapsed in an inverted “V” shape. This failure then propagated to the tie bar, inducing outward compression at the arch feet and tensile stress in the arch ribs. Stress concentration at the connection between the arch columns and arch rings ultimately triggered global collapse. For the Liujiang Bridge, failure initiated with localized concrete cracking, which propagated to reinforcing bar yielding, resulting in localized damage within the bridge deck system. These observations indicate that progressive stay cable failure serves as the common initial triggering mechanism for both bridges. However, differences in the structural configuration of the bridge deck systems, the geometry of the arch ribs, and the constraint effects of the tie bar result in distinct failure progression patterns and ultimate collapse behaviors between the two structures. Thereby, design recommendations are proposed for through tied-arch bridges, from the aspects of the hanger, arch rib, bridge deck system, and tie bar, to enhance the resistance to progressive collapse.

1. Introduction

Under the influence of impact loads, which are characteristic of extreme events in limit state design, a bridge structure may experience significant deformation. This deformation typically arises from localized damage, leading to the progressive collapse of the remaining structural elements. Within a through tied-arch bridge, the deterioration of the hanger caused by corrosion and fatigue may progressively intensify over time, potentially resulting in sudden fracture [1]. In recent years, the collapse of tied-arch bridges due to hanger fractures has occurred intermittently. Notably, in 2001, the Yibin Nanmen Bridge in Sichuan Province—a half-through reinforced concrete ribbed arch bridge—experienced a catastrophic failure. Some corroded hangers unexpectedly failed under the loads of vehicles, resulting in several beams and bridge panels on both sides near the arch foot collapsing into the river. Two buses traversed this section and subsequently plunged into the water, leading to the tragic loss of four lives [2]. In 2019, the Nanfang’ao Bridge in Taiwan experienced a catastrophic failure due to prolonged neglect of its hangers. This systematic failure led to the collapse of both the bridge deck system and the arch ribs, which subsequently fell into the water. The incident resulted in six fatalities and thirteen injuries [3]. In 2021, the Hangyong Canal Bridge tragically collapsed during its construction phase. Subsequent investigations indicated that inadequate construction management was the fundamental cause, resulting in multiple failures in the anchorage of the hangers. This subsequently initiated the failure of the bridge deck system. Moreover, the instability of the arch rib structure subsequently manifested, ultimately leading to the failure of the bridge structure. On 23 April 2025, the Rainbow Bridge in Shunyi District, Beijing—a concrete-filled steel tube tied-arch bridge designed in the shape of a swallowtail—collapsed as a result of a fire that occurred during cable-laying operations in a trench. The fire caused the bridge’s tie bars to fail, leading to the progressive failure of the hangers. Consequently, the main span of the bridge collapsed due to the loss of load-bearing capacity in the overall structure. The damage observed on-site is illustrated in Figure 1.

Since the terrorist attacks on 11 September 2001, in the United States, researchers have conducted extensive studies on structural resistance to progressive collapse. This includes an examination of the failure mechanisms associated with various types of buildings and an assessment of structural robustness [4]. Progressive collapse refers to the phenomenon in which a structure, whether under normal operating conditions or subjected to localized damage from unforeseen events, may experience partial or even total failure. This occurrence is attributed to the inherent ductility and continuity of the structural system itself [5]. Among the aforementioned bridge collapse accidents, both the Yibin Nanmen Bridge and the Jinsha Bridge experienced failures attributed to inadequate robustness in their bridge deck systems. As illustrated in Figure 2, in the event of hanger failure, there were no force-transmitting components present between the bridge deck system and the arch rib. The bridge deck system transitioned into a suspension system, with the crossbeam primarily responsible for bearing the load. Consequently, the bridge deck sustained dents and damage, which led to the progressive collapse of the remaining structure.

For the deck system of a through suspension bridge, where the crossbeams serve as the primary structural elements, there exists an insufficient longitudinal connection between these crossbeams. This deficiency leads to a compromised integrity of the deck system. Therefore, to enhance the robustness of the suspension bridge deck system, Wu et al. [6] employed a truss-type steel structure with stiffened longitudinal beams to reinforce the through suspension bridge deck. This approach resulted in increased stiffness, improved stability, and reduced deformation under load for the bridge. After the damage occurred at the Jinsha Bridge, as previously mentioned, it was subsequently proposed to replace the original simply supported bridge deck with a continuous system composite bridge deck. This new design would incorporate longitudinal and transverse steel lattice girders, with crossbeams primarily responsible for load-bearing [7]. Zhang et al. [8] enhanced the robustness of the bridge deck system by incorporating steel and concrete longitudinal beams between the crossbeams. They utilized hoop clamps and embedded bars to connect these longitudinal beams with the crossbeams, thereby transforming the suspension system into a longitudinal-transverse beam frame system, which significantly improved its overall robustness. The aforementioned research indicates a growing emphasis on enhancing the progressive collapse resistance of arch bridge structures. By implementing design modifications, engineers aim to improve the overall integrity and structural robustness of bridge systems.

Research on the progressive collapse mechanism of bridge structures, Fan et al. [9] investigated the feasibility and failure mechanisms of two reinforcement methods: the simply-supported-to-continuous method and the rigid integral tied arch method. This study was conducted under conditions of systematic hanger breakage, revealing the failure mechanism of the tied-arch bridge. However, currently, there is a paucity of research regarding the impact of various structural design parameters on the capacity of the arch-beam combined structure system to withstand progressive collapse. Derseh et al. [10] conducted a study on the design and collapse failure of five distinct types of bridge structures, including arch bridges, continuous girder bridges, cable-stayed bridges, suspension bridges, and steel truss bridges. Based on the findings from case studies and data surveys, this research investigates the impact of various parameters on bridge components that contribute to different types of structural failures. This paper, however, presents a study on the collapse mechanisms of through tied-arch bridges with varying designs. Ran et al. [11] employed a high-fidelity computational approach to investigate the failure of a pedestrian concrete truss bridge during its construction in Miami, Florida, USA. They simulated the behavior of key components throughout the construction phase to identify the primary factors contributing to failure. Nicola et al. [12] categorized the Applied Element Method (AEM) into local and global models to investigate the collapse of a reinforced concrete arch bridge in Massa and Carrara, Italy. Their study elucidated potential failure factors and patterns associated with these accidents. Domaneschi et al. [13] employed the Applied Element Method (AEM) to investigate the asymmetric collapse of a cable-stayed bridge. They assessed its effectiveness by comparing it with the debris distribution of the collapsed bridge. Furthermore, they proposed that incorporating structural redundancy could mitigate asymmetric collapse and enhance overall robustness. Salem et al. [14] employed the Applied Element Method (AEM) to develop a model of the I-35 truss arch bridge in Minneapolis, based on the construction drawings. The research indicated that the failure of the gusset plates was a contributing factor to the collapse. The real-world bridge case study by Whelan et al. [15] further highlights the importance of integrating phased damage characteristics with load–response nonlinear analysis, as traditional linear models exhibit significant limitations in capturing structural behavior during advanced damage stages. To improve the identification of nonlinear damage features, time-frequency analysis techniques have been adopted to effectively characterize non-stationary signals. Gholipour et al. [16] performed a detailed nonlinear finite element analysis under ship collision scenarios, uncovering the progressive collapse mechanism of cable-stayed bridge piers—from localized crushing to global instability—and underscoring the critical role of material nonlinearity and contact interactions in accurately predicting collapse paths.

Current research predominantly focuses on damage paths induced by external sudden loads, such as collisions and explosions [16], while the progressive instability mechanisms resulting from structural self-damage accumulation—such as fatigue and corrosion—are largely overlooked. Investigations into stay cable failure primarily concentrate on flexible systems like cable-stayed and suspension bridges [15], whereas studies addressing the chain reaction between stay cable damage and overall collapse in arch bridges—particularly arch-rigid frame bridges—are significantly limited. Furthermore, existing models fail to account for the force redistribution characteristics among stay cables, arch ribs, and tie bars in arch bridge systems and do not quantify the influence of varying structural configurations on collapse path evolution.

Addressing the aforementioned research limitations, this study proposes a failure-path-based structural redundancy analysis framework and establishes a bridge damage failure model under stay cable degradation. The investigation further elucidates the “valve effect”—how variations in key design parameters, including stay cable distribution, arch rib configuration, deck system layout, and tie beam arrangement, influence the propagation of structural damage. This work bridges the critical research gap in understanding self-induced collapse mechanisms in arch-girder composite bridges triggered by progressive damage accumulation, thereby offering valuable design insights for enhancing bridge performance across its entire lifecycle. The framework is illustrated in Figure 3.

2. The Engineering Background and Finite Element Model Validation

Two tied-arch bridges were selected for comparative analysis, providing the engineering context for this research. The Nanfang’ao Bridge, previously mentioned in the introduction, is a through steel tied arch bridge characterized by a double fork and a single arch, with an overall length of 140 m. At the same time, to investigate the impact of various design forms on the stress and failure mechanisms of tied-arch bridges, the Liujiang Bridge was introduced as a reference case.

2.1. Bridge Design

• Nanfang’ao bridge

There were 13 hangers arranged symmetrically around the central hanger on the Nanfang’ao bridge. The hangers were designated as No. 1 to No. 13, arranged from left to right. Hangers No. 1 and No. 13 utilized a total of 17 bundles of steel wires measuring 7 × Φ15.75 mm, with a cross-sectional area of 150 mm² each. In contrast, the remaining hangers employed 13 bundles of steel strands. The ultimate bearing capacity for hangers No. 1 and No. 13 was determined to be 4715.17 kN, while the ultimate bearing capacity for the other hangers was recorded at 3605.71 kN. The arch rib consisted of an arch ring measuring 2.5 m by 1.8 m, with a total length of 90 m, and double-fork arch columns on both sides, each measuring 1.8 m by 1.8 m and extending for a length of 25 m. Additionally, rigid diaphragms were incorporated within the structure. The arch rib comprised an arch ring with dimensions of 2.5 m by 1.8 m, featuring a total length of 90 m. On both sides, double-fork arch columns were incorporated, each measuring 1.8 m by 1.8 m and extending to a length of 25 m. Furthermore, rigid diaphragms were integrated within the structure.

In the finite element modeling process, beam elements are employed to simulate the primary load-bearing components, such as the main girder and piers; shell elements are utilized for the bridge deck pavement, and link elements are applied to model the hangers. The boundary conditions are defined as follows: full fixation is applied to the z-direction of the tie beam and cap beam; the z-direction of the bridge deck beam is fully constrained; the y- and z-directions of the left bridge cap are fixed; and all three translational degrees of freedom (x, y, and z) at the right bridge cap are completely restrained. To accurately reflect the actual mechanical behavior of the hangers, the translational degrees of freedom in all three directions of the hanger elements are coupled, and the Instate command is used to simulate the prestress application. The entire bridge structure is modeled using Q355 steel, while the suspenders are represented by strand1860 materials. This configuration is illustrated in Figure 4.

2.

Liujiang bridge

The Liujiang Bridge is a through concrete-filled steel tube tied arch bridge, measuring 95.5 m in length. It features 12 pairs of vertical hangers, each composed of 91 × Φ7 mm galvanized high-strength steel wires, arranged with a spacing of 7.1 m. These hangers were designated as D₁ to D₆ from left to right. The ultimate bearing capacity of the hangers was determined to be 4848.57 kN. The arch rib featured a dumbbell-shaped cross-section, with a height of 2.4 m. It consisted of two Φ1000 × 16 mm steel pipes and web plates, which were filled with C50 concrete. The center-to-center distance between the arch ribs measured 22.37 m. Each side was connected by two I-shaped cross braces and one K-shaped cross brace.

The deck system comprised a prestressed T-shaped combined beam as the cross beam, along with a prestressed box girder measuring 2.9 m × 3.22 m and a Π-shaped bridge deck. The tied beam was designed as a prestressed concrete box girder measuring 2 m × 2.75 m, reinforced with sixteen Φ15.24-16 steel strands, as illustrated in Figure 5.

The finite element modeling of the Liujiang Bridge comprised 6422 nodes and 5520 elements. The arch rib, cross braces, tied beam, and transverse beams were represented using beam elements for simulation purposes. Spatial bar elements were employed to simulate the hangers and tie bars. The Π-shaped bridge deck was modeled using shell elements. For the bridge deck, all degrees of freedom associated with the nodes at both ends of the cross beam and tie bar beam were coupled. In contrast, for the corresponding nodes of the middle cross beam, coupling was applied to translational degrees of freedom in the z direction as well as rotational degrees of freedom; translational degrees of freedom in both x and y directions were also coupled. Regarding the upper and lower nodes of the hanger, all translational degrees of freedom were coupled while retaining their respective rotational degrees of freedom. The supports were simulated by implementing a fixed support on one side and a hinged support on the opposite side.

2.2. Bridge Model Verification

• Nanfang’ao Bridge

On 1 October 2019, the Nanfang’ao Bridge collapsed. The model verification presented in this study is based on data from the “Comparison of Tension Distribution in Chain-Breaking Stay Hangers” section of the accident investigation report, focusing specifically on the intermediate failure process of stay hangers No. 7 to No. 3.

To investigate the cause of the bridge collapse, the National Transportation Safety Committee conducted a tensile test on steel strands on 21 January 2020, at the Yangmei Laboratory of Huaguang Engineering Consultants Co., Ltd. (Taoyuan, Taiwan). The test involved 15 steel strands retrieved from the collapsed bridge and 2 new steel strands (Figure 6). The testing protocol included measurements of the outer diameter of the steel strands, core wire, and outer layer wires, as well as assessments of ultimate tensile load, yield strength, and tensile strength. Given that the 15 steel strands had been in service for many years, the results reflected their residual strength. The data obtained for each stay cable are summarized in

Table 1. Comparison of hanger tension distribution in the failure process (Unit: kN).

Hanger Number	Yield Load	Ultimate Tensile	Ultimate Tensile Load of Anchorage System	Hanger Tension (Measured Value/Finite Element Value)
				Breakage of No. 6 and No. 7 Hangers	Breakage of No. 4 and No. 5 Hangers	Breakage of No. 3 Hanger
1	3748.40	4243.60	3872.37	353.39/356.92	0/0	844.27/852.71
2	2866.50	3245.17	2853.27	452.07/456.59	180.61/182.42	2676.77/2703.54
3	2866.50	3245.17	2853.27	678.45/685.24	1114.36/1125.50	6285.13/6347.98
4	2866.50	3245.17	2853.27	1117.40/1128.56	2873.56/2902.29	/
5	2548.00	2884.53	2853.27	1801.04/1819.05	5063.17/5113.80	/
6	2866.50	3245.17	2853.27	3651.38/3687.90	/	/
7	980.00	1109.46	2853.27	2073.97/2094.71	/	/

. Combined with video recordings and on-site investigation findings, the analysis indicated that severe corrosion of the stay hangers was the primary cause of the accident.

To verify the accuracy of the finite element modeling of the Nanfang’ao Bridge, the axial force values of the stay cables obtained from the model were compared with those measured in the tensile tests. As shown in Table 1, the maximum error between the finite element results and the tensile test data is 1%, indicating that the modeling approach and the dynamic simulation method for hanger breakage are valid. The failure process of hangers No. 7 to No. 3 corresponds to the intermediate failure stage. Prior to this stage, hangers No. 8 to No. 11 had already failed, resulting in axial forces of 0 kN; these values are not further detailed in the table.

2.

Liu Jiang Bridge

In May 2015, the Changda Highway Engineering Inspection and Testing Center carried out a comprehensive bridge inspection on the Liujiang Bridge, with particular emphasis on the condition and performance of its hangers. Based on the theory of string vibration, an explicit mathematical relationship between the lateral vibration frequency of the hangers and their corresponding tension was formulated. By measuring the vibration frequencies of individual hangers, the tensile forces were subsequently calculated. During the field inspection, appropriate sensors were strategically installed to collect vibration data from the hangers. This empirical data were used to validate the accuracy and reliability of the derived tension calculation formula. Given that the slenderness ratio (L/D) of each hanger in this bridge is less than or equal to 1.4 × 10³, Equations (1) through (3) were applied to determine the axial force of the hangers under self-weight loading conditions [17].

$$T=4{\pi }^{2}m{l}^2{\displaystyle \frac{f_n^2}{y_n^2}}-{\displaystyle \frac{EI}{l^2}}{y}_n^2$$

(1)

$$y_n=n\pi +A{\varphi }_n+B{\varphi }_n^2; A=-18.9+26.2n+15.1n^2$$

(2)

$$B=\left\{\begin{array}{l}290\left(n=1\right)\\ 0\left(n>1\right)\end{array}\right.; {\varphi }_n={\displaystyle \frac{1}{x{y}_n}}=\sqrt{{\displaystyle \frac{EI}{m{\omega }_n^2{l}^4}}}; {\omega }_n=2\pi f_n$$

(3)

Figure 7 presents a comparison between the theoretical values obtained from the equations and the data derived from the established finite element model. The maximum deviation observed is 6.8%, indicating that the finite element model accurately represents the actual structural behavior and thus confirming its validity.

3. Comparative Study on the Structural Dynamic Behavior Considering the Breakage of Hangers

In tied-arch bridges, hangers serve as crucial components that establish the connection between the arch ribs and the deck system. The systematic failure of hangers can result in a lack of connection and force transmission components between the deck system and the arch ribs. This deficiency may contribute to the damage and eventual failure of the bridge. Therefore, it is essential to assess the significance and performance of hangers under various design parameters. Given the same failure conditions for the hangers, it is crucial to consider the changes in the dynamic behavior of the remaining structure in order to investigate the stress and failure mechanisms associated with this type of bridge across different structural designs. The ultimate limit state of structural safety refers to the condition in which a structure or structural component attains its maximum load-bearing capacity or experiences deformation that renders it unsuitable for continued load-bearing. This state represents the critical threshold that determines the structural integrity and safety. Among these, the evaluation of bridge performance under the condition of continuous failure of the hangers corresponds to the consideration of the ultimate limit state of structural safety.

3.1. Structural Design and Dynamic Characteristics of Hangers

3.1.1. The Importance Assessment of Hangers

A through tied-arch bridge consists of various components, including hangers, tie bars, longitudinal and transverse beams, bridge deck panels, arch ribs, abutments, and additional elements. The various components exert distinct influences on the collapse mechanism, which is contingent upon their relative importance. The significance of importance is influenced by the force transmission pathway, inherent characteristics, and the manner in which loads are applied. Components that possess a high degree of significance exert considerable influence on the overall structure and are more likely to precipitate failures in the bridge. Regarding this, Zheng et al. [18] employed the energy method and explicit dynamic numerical analysis to investigate the significance of components, the redundancy of the remaining structure, and the failure pathways following initial component damage under heavy loads. Huang et al. [19] utilized a generalized stiffness method to evaluate the significance of components throughout the collapse process.

The generalized stiffness method assesses the contribution of individual components to the overall structural stiffness, thereby evaluating their significance. In contrast, the strain energy method examines the distribution of strain energy in accordance with the principles of energy conservation. Both methods fundamentally pertain to the structural response under applied loads and the mechanical properties of the components, as understood through the principles of mechanics. Therefore, the variations in the strain energy of the remaining structure following the removal of a component serve as a criterion for assessing the significance of that structure, as illustrated in Equation (4) [20].

$${\eta }_i={\displaystyle \frac{U_i-U}{U_i}}=1-{\displaystyle \frac{U}{U_i}}$$

(4)

where η_i = the importance coefficient of component i, U_i = the strain energy of the remaining structure after removing component i, and U = the strain energy of the entire structure when the structure is intact (without removing component i).

Different hanger designs can significantly influence the mechanical properties of bridges, including stiffness, strength, and deformation. Research on hanger design enables engineers to comprehend the advantages and disadvantages associated with various structural forms. This understanding facilitates the selection of the most appropriate design scheme tailored to specific engineering requirements, thereby optimizing bridge performance and enhancing service efficiency and lifespan. Wang et al. [21] conducted a computational simulation to investigate the impact of dynamic responses in long-span suspension bridges following hanger failure. Taking into account the number, location, and failure mechanism of hangers as the primary research focus, single or multiple hangers were systematically removed one at a time until a progressive collapse of the bridge was observed. The results indicate that as the number of removed hangers increases, the degree of damage to the bridge becomes more severe, with the middle position being identified as the most critical area. On the other hand, the outcomes of sequentially breaking hangers one by one are fundamentally consistent with those obtained from simultaneously breaking the same group of hangers. In this section, the component removal method was employed [22]. Specifically, elements were “killed” to simulate the sudden failure of certain components in the through tied-arch bridge under accidental loads. This approach served as the foundation for structural collapse analysis, and investigations were conducted on various hanger design configurations. Given the symmetry of the structure, only half of it is considered for analysis in the subsequent sections.

For the Nanfang’ao Bridge, hangers No. 1 to No. 7 were selected for analysis. Following the failure of each hanger under the self-weight condition of the structure, the variations in strain energy of the remaining structure and the importance distribution coefficients of its components are illustrated in Figure 8a. According to Equation (1), the component importance coefficients for hangers No. 1 through No. 7 were calculated to be 0.2 × 10⁻³, 0.012, 0.036, 0.032, 0.019, 0.012, and 0.93 × 10⁻², respectively.

The Liujiang Bridge features a vertical hanger structure. In contrast, the Nanfang’ao Bridge employs an inclined hanger structure characterized by varying spacings between the deck system and the arch rib. After the failure of each hanger due to self-weight, the variations in strain energy within the remaining structure and the distribution coefficient of component importance are illustrated in Figure 8b. When the D₁ hanger was broken, the strain energy of the remaining structure changed from 33.73 kJ to 33.74 kJ. When the D₂ hanger was broken, the strain energy measured at 35.26 kJ. Upon breaking the D₃ hanger, the strain energy recorded was 39.68 kJ. Following the breakage of the D₄ hanger, the strain energy increased to 54.43 kJ. When the D₅ hanger failed, a further increase in strain energy was observed at 59.45 kJ. Finally, when the D₆ hanger broke, the strain energy reached a value of 60.71 kJ. Therefore, the importance distribution coefficients of the D₁ to D₆ hangers were 0.26 × 10⁻³, 0.043, 0.15, 0.38, 0.43, and 0.44, respectively.

When η_i approaches 1.0, it signifies that the component holds greater significance. Conversely, when η_i equals 0, it indicates that the component has no influence on the force transmission process of the bridge. Furthermore, when η_i is equal to 1.0, it suggests that the failure of this component could potentially lead to a catastrophic bridge collapse. Therefore, among the hangers of the Nanfang’ao Bridge, Hanger No. 3 is identified as the most critical component, while Hanger No. 1 is deemed less significant. Consequently, the failure of Hanger No. 3 represents the most adverse condition for structural integrity. In contrast, within the Liujiang Bridge framework, the D₆ hanger (the long hanger) exerts a more substantial influence on overall stability. The significance of hangers from D₅ to D₁ diminishes progressively in importance; thus, the failure of the D₆ hanger constitutes the most detrimental scenario for this structure. In conclusion, as the design undergoes modifications, the significance of hangers at various positions and the most unfavorable conditions also evolve. Ultimately, this leads to a change in the force transmission path.

3.1.2. Dynamic Response and Redundancy Evaluation of Remaining Hangers

The redundancy of a structure pertains to its capacity to sustain loads following the failure of one of its components. Different conditions of breakage are associated with distinct damage models. The bearing capacity of the residual structure reflects the redundancy inherent in this damage model. When the redundancy of a structure is low, its capacity to resist progressive collapse is limited, resulting in diminished robustness. When the redundancy of a structure is substantial, its capacity to resist progressive collapse is enhanced, resulting in improved robustness. When assessing the robustness of a structure, the minimum value of the remaining structural redundancy across various damage models can serve as a quantitative index, as demonstrated in Equation (5) [23].

$$DC{R}_i={\displaystyle \raisebox{1ex}{${QUD}_{LIM}$}\!\left/ \!\raisebox{-1ex}{$Q_{CE}$}\right.}$$

(5)

where QUD_LIM = maximum load effect of the component during the dynamic damage analysis, and Q_CE = ultimate bearing capacity of members.

The analysis indicates that when the Demand-Capacity Ratio (DCR) is lower, the bearing capacity of the component exceeds its demand, resulting in a stronger safety reserve capacity and a reduced likelihood of failure. Conversely, when the DCR is higher, this situation is reversed.

During the operation of the bridge, damage to the hangers occurs in a random and unpredictable manner. To facilitate the study of the bridge’s damage conditions, hangers were systematically removed from both sides, starting from the central hanger until failure was observed. In the case of Nanfang’ao Bridge, after hanger No. 7 failed, the axial force in hanger No. 6 increased from an initial value of 992.86 kN to 1376.9 kN, as illustrated by the dynamic time history curve shown in Figure 9a. Following this, when hanger No. 6 also failed, there was a further increase in axial force for hanger No. 5 from its initial value of 1159.6 kN to 2089.2 kN; this is depicted in Figure 9b. Subsequently, after hanger No. 5 broke, hanger No. 4 experienced an increase in axial force from an initial measurement of 1410.2 kN to a significant level of 3202.7 kN (see Figure 9c). When hanger No. 4 subsequently failed, it caused an increase in axial force for hanger No. 3 that rose dramatically from its original value of 1548.1 kN to reach a peak of 4523.5 kN—exceeding its ultimate bearing capacity of approximately 3605.71 kN—resulting in failure for hanger No. 3 as well and triggering a cascade effect leading to continuous breakage among the remaining hangers (Figure 9d).

For the Liujiang Bridge, following the failure of hangers D₆ and d₆, the axial force in hangers D₅ and d₅ increased from an initial value of 1984.4 kN to 2697.8 kN. Upon the failure of D₅ and d₅, the axial force in hangers D₄ and d₄ surged from 1457 kN to 4474 kN, placing them in a precarious condition. Subsequently, after the failure of D₄ and d₄, the axial force in hangers D₃ and d₃ escalated from 1401.1 kN to 5891.5 kN, surpassing their ultimate bearing capacity of 4848.57 kN and leading to systematic failures among adjacent hangers. The dynamic time history curves illustrating the breakage events for each hanger are presented in Figure 10.

To summarize, in the context of specific breakage paths, the Nanfang’ao Bridge (with 4 hangers fractured) and the Liujiang Bridge (with 3 pairs of hangers fractured) would lead to a continuous failure of the remaining hangers, as illustrated in

Table 2. Redundancy of adjacent hangers during the breaking process.

Name	Number of Broken Hangers	Breaking the Sequence of Hangers	Extreme Values of the Dynamic Response of Adjacent Hangers	Redundancy of Adjacent Hangers
Liujiang bridge	3	D6 and d6 were broken	D5, d5: 2697.8 kN	0.556
		D5 and d5 were broken	D4, d4: 4474 kN	0.923
		D4 and d4 were broken	D3, d3: 5891.5 kN	1.215
Nanfang’ao bridge	4	No. 7 was broken	No. 6: 1376.9 kN	0.382
		No. 6 was broken	No. 5: 2089.2 kN	0.579
		No. 5 was broken	No. 4: 3202.7 kN	0.888
		No. 4 was broken	No. 3: 4523.5 kN	1.255

. Comparative analysis revealed that under identical breakage conditions, the likelihood of ongoing hanger failures for the Nanfang’ao Bridge was lower than that observed for the Liujiang Bridge.

Liu et al. [24] conducted a study on the hangers utilized in three design configurations of tied arch bridges, specifically vertical hangers, non-crossing inclined hangers, and reticulated inclined hangers. The findings indicated that, in terms of mechanical properties, reticulated inclined hangers exhibit superior performance, followed by non-crossing inclined hangers in second place, while vertical hangers demonstrate relatively weaker characteristics. Consequently, during the design phase of long-span tied arch bridges, it is essential to consider the advantages offered by an inclined hanger layout with respect to mechanical properties.

Therefore, it implies that in the event of an accident, the configuration of inclined hangers enhances the force distribution on the bridge, allowing for a more rational load dispersion and transfer. This design approach not only demonstrates increased stability but also mitigates localized damage caused by uneven stress distributions. Designers can further optimize mechanical properties by adjusting factors such as the angle, quantity, and arrangement of the hangers, thereby providing improved options for enhancing both safety and stability in bridge construction.

3.2. Force Analysis of Remaining Structure Under Systematic Failure of Hangers

The through tied-arch bridge demonstrates the most dynamic response in both the arch rib and bridge deck system following the progressive failure of hangers [25]. The designs of the arch rib and bridge deck systems for both Liujiang Bridge and Nanfangao Bridge differ, resulting in variations in their stress distribution and force transmission mechanisms. This study investigates the alterations in axial force, bending moment, and displacement within the arch rib and bridge deck systems of these two bridges under design load conditions, as well as during a uniform hanger failure sequence. The findings summarize how differing structural designs of the arch rib and bridge deck systems influence their dynamic behavior and force transmission characteristics. Based on their design structures, these two bridges are categorized into two distinct structural systems: a continuous tied-arch bridge (Nanfang’ao Bridge) and a simply supported tied-arch bridge (Liujiang Bridge).

3.2.1. Arch Rib

The arch rib of the Nanfang’ao Bridge features a single arch and a double fork structure. In the event of systematic failure of the hangers, the arch rib structure was responsible for bearing the primary load. The maximum pressure observed at the junction between the arch ring and the arch column was −112.32 kN, indicating significant stress concentration at this location due to variations in cross-section. Additionally, a vertical maximum displacement of 0.036 m occurred at this connection point.

Considerable stress and deformation were noted at the double fork section of the arch rib, rendering it susceptible to damage. Beyond supporting vertical loads and horizontal thrusts, the foot of the arch also experiences bending moments induced by continuous beams under various factors such as temperature fluctuations, concrete shrinkage, and creep effects. The maximum bending moment recorded in this area was 6425.9 kN·m. The forces acting on the arch rib are illustrated in Figure 11.

The arch rib of the Liujiang Bridge is characterized by a double-arch rib structure. K-shaped transverse bracing connects the arch ribs, with the arch rib primarily bearing compressive forces while the transverse bracing predominantly accommodates tensile forces, reaching a maximum tensile force of 10.99 kN. At the foot of the arch, vertical loads and horizontal thrust are mainly supported, allowing for the neglect of bending moments in this region. The connection nodes of the K-shaped transverse bracing experience significant bending moments and deformations, leading to stress concentrations quantified at 931.28 kN·m for moment and 0.0097 m for displacement, respectively. Consequently, it is observed that damage to the transverse bracing occurs first under these conditions. The distribution of bending moments and deformations along the arch axis reveals relative uniformity within the arch rib’s structural response. Figure 12 illustrates the force distribution experienced by the arch rib.

Figure 11 and Figure 12 present the analysis results under stable data conditions. The peak values of the dynamic time-history curves for the two bridges are illustrated in Figure 13.

After the systematic failure of the hangers, it became essential to assess the spatial stability of the arch rib to ensure that it would not experience lateral instability. According to Section 5.3.1 of the “Technical Specifications for Concrete-Filled Steel Tube Arch Bridges” [26], concrete-filled steel tube arch bridges must be evaluated for stability, with an eigenvalue of elastic instability required to be greater than or equal to 4. The buckling analysis conducted on the Nanfang’ao Bridge under two conditions revealed that when the bridge was in good condition, the first-order instability eigenvalue was found to be 13.63, which is greater than 4. Thus, it is confirmed that the spatial elastic stability requirements for this bridge were satisfied. The corresponding buckling mode is illustrated in Figure 14a. However, when there was a continuous failure of hangers, the buckling eigenvalue for the hanger-less state of the bridge dropped significantly to −47.02, which is less than 4. At this point, it became evident that compliance with spatial elastic stability analysis requirements was no longer achievable; consequently, there exists a potential risk for instability in the arch rib of this bridge. The associated buckling mode is depicted in Figure 14b.

From the buckling analysis of the Liujiang Bridge conducted under two different conditions, it was observed that when the bridge was in good condition, the eigenvalue for first-order instability was 5.49, which exceeds the threshold of 4. Consequently, this indicates that the spatial elastic stability analysis of the bridge satisfied the necessary requirements. The corresponding buckling mode is illustrated in Figure 15a. In contrast, when multiple hangers were compromised, the buckling eigenvalue for the hanger-less configuration of the bridge dropped to 2.625, falling below 4. At this juncture, it became evident that the criteria for spatial elastic stability analysis were no longer fulfilled, thereby introducing a potential risk of instability for this structure. The associated buckling mode is depicted in Figure 15b.

As shown in

Table 3. Comparison of the arch rib structures of the two bridges.

Name	Axial Force		Displacement		Bending Moment		Stability
	Location	Maximum	Location	Maximum	Location	Maximum
Nanfang’ao bridge	The connection point between the arch ring and the arch column	−112.32 kN	The connection point between the arch ring and the arch column	0.02 m	Arch springing	6425.9 kN·m	Instability
Liujiang bridge	Cross brace	10.99 kN	Cross brace	0.02 m	Cross brace	931.28 kN·m	Stability

, regarding load-bearing performance, the double-arch rib structure incorporates two arch ribs that jointly carry the load from the upper part of the bridge. This design enables a more uniform distribution and transfer of the load to the substructure, thereby enhancing the overall load-bearing capacity and making it particularly suitable for bridges with larger spans and heavier loading conditions. In terms of structural stability, compared to the single-arch rib double-fork configuration, the double-arch rib structure demonstrates superior resistance to lateral forces and horizontal thrusts. This improved performance contributes to better maintenance of structural integrity, reduced deformation and displacement, and enhanced overall bridge safety.

3.2.2. Bridge Deck System

The bridge deck system of the Nanfang’ao Bridge consists of a steel box girder supported by a bifurcated tie beam structure beneath. This configuration forms a continuous steel box girder structure, with each end connected to adjacent piers or abutments. The ends of the bridge deck system experience tensile forces, while the central section is subjected to compressive forces. The maximum tension recorded was 149.19 kN. Under vertical loading conditions, the continuous beam not only withstands bending moments and shear forces but also generates an area characterized by negative bending moments.

The continuous box girder structure exhibits significant stiffness, which enhances load distribution and mitigates the mid-span bending moment. The location of the maximum bending moment was identified near hanger No. 3, measuring 2.08 × 10⁵ kN·m. The horizontal thrust exerted by the arch rib is transmitted to adjacent piers via the beam body. Additionally, the tensile force distribution within the tie bar is more uniform. The axial force acting on the tie beam opposes the direction of the steel box girder; thus, while both sides experience compression, the center undergoes tension—this aligns with established interaction force relationships between these two structural components. The maximum deformation observed in the box girder structure reached 0.47 m, which may influence stress states in both the arch rib and tie bar systems. Figure 16 illustrates the forces acting on the bridge deck system.

The bridge deck system of the Liujiang Bridge consists of tie beams and tie bars on both sides, as well as longitudinal and transverse beams along with the upper bridge deck slab, which collectively bear loads. The beam structure is simply supported, resting on piers at both ends. The tie bars primarily resist tension to counterbalance the horizontal thrust induced by the load on the bridge deck. The maximum axial force recorded was 26,533 kN. Under vertical loading conditions, the beam mainly experiences bending moments and shear forces; specifically, the peak bending moment reached 2.7549 × 10⁵ kN·m. The loading conditions acting upon the beam are relatively straightforward and well-defined. The primary function of the beam is to support direct loads from the bridge deck while transferring these loads to hangers and arch ribs. Notably, the maximum vertical displacement observed in response to this loading within the bridge deck system was 0.37 m. A detailed representation of forces acting on the bridge deck system can be found in Figure 17.

Figure 16 and Figure 17 present the analysis results under stable data conditions. The peak values of the dynamic time-history curves for the two bridges are illustrated in Figure 18.

As previously discussed and illustrated in

Table 4. Comparison of the deck systems of the two bridges.

Name	Axial Force		Displacement		Bending Moment
	Location	Maximum	Location	Maximum	Location	Maximum
Nanfang’ao bridge	At both ends of the bridge deck	149.19 kN	At the 1/2 mark	0.47 m	No. 3 hanger	2.08 × 105 kN·m
Liujiang bridge	At the 1/2 mark	26,533 kN	At the 1/2 mark	0.37 m	Arch springing	275,490 kN·m

, simply supported beam tied-arch bridges and continuous beam tied-arch bridges exhibit distinct differences in their force transmission mechanisms. In practical engineering applications, the selection of an appropriate bridge structural form should be systematically determined by specific site conditions and design requirements to ensure structural performance, safety, and cost-effectiveness.

4. Comparative Analysis of Stress Mechanism and Failure Mode

Based on the preceding analysis, the stress mechanism was examined, and the Simple–Johnson–Cook damage and failure model was employed in LS-DYNA to conduct a comprehensive discussion of the failure states associated with these two types of bridges, respectively.

4.1. Stress Mechanism

For the Nanfang’ao Bridge, the damage to hangers No. 4 to No. 7 resulted in a progressive failure of the remaining hangers, leading to both arch feet experiencing tension forces denoted as F1 on either side. Additionally, the two sides of the steel box girder were subjected to tension forces represented by F₂. The tie beams located beneath the box girder experienced compressive forces. In accordance with the stress characteristics inherent in through tied-arch bridges, it can be observed that the compression within the tie beams serves to balance out the tensile forces acting on both the arch feet and the steel box girder. Notably, during this process, pressure was exerted on the middle section of the double-fork branch of the steel box girder, resulting in a downward displacement designated as Y₁. This displacement value corresponded with that generated by its underlying tie beam. The highest stress concentration occurred at the junction between the arch ring and its corresponding arch column. The compression experienced by this arch ring induced a downward displacement labeled Y₂ at its double fork connection point. The stress mechanism is illustrated in Figure 19a.

When the hangers experience continuous damage, the arch ribs of the Liujiang Bridge primarily bear the pressure denoted as F₂. The arch feet are subjected to horizontal tension represented by F₁. The tie beams on both sides of the bridge deck system serve to balance the horizontal forces acting on the arch feet. Due to the compressive characteristics inherent in the arch ribs, K-shaped and circular transverse braces positioned between these ribs endure significant tension to counterbalance the pressures exerted by the arch ribs. For analytical purposes, consider a micro-element of the bridge deck system; this element experiences a bending moment M₁ and pressure F₃, resulting in compression within both upper concrete and steel mesh components while inducing tension in their lower counterparts. The stress mechanism is illustrated in Figure 19b.

4.2. Failure Mode

The Simple–Johnson–Cook model is an empirical constitutive model widely used to characterize the mechanical behavior of metallic materials under extreme conditions, including high strain rates, large strains, and elevated temperatures. By incorporating three fundamental effects—strain hardening, strain rate hardening, and thermal softening—it effectively captures the complete deformation process from elastic response to plastic flow and eventual failure. This model is especially applicable for dynamic response analysis of engineering metals such as steel, aluminum, and titanium alloys.

This model features a relatively short computational time and is widely applied in engineering scenarios involving dynamic impact and high-speed collisions. It is particularly suitable for capturing the transient impact effects associated with hanger failure in this study. Furthermore, as a damage failure model, it enables direct simulation of material fracture processes and provides predictive capability for structural failure behavior. Given that this research focuses on simulating and predicting bridge collapse, where fracture-induced damage can be effectively modeled using this approach, the Johnson-Cook model is deemed appropriate. During the collapse process, bridges experience significant plastic deformation, and the failure duration is extremely brief—typically lasting only a few seconds—which aligns well with the model’s applicability. Therefore, the selection of the Simple–Johnson–Cook model is justified.

Based on the discretization of the structure into a series of interacting elements, the Simple-Johnson-Cook material damage failure model is employed for damage simulation. This model is specifically applicable to integrated beam elements and large-scale bridge structures. Flow stress (${\sigma }_{flow}$) refers to the stress value at which a material resists further deformation during plastic deformation. It varies with factors such as deformation degree, deformation rate, and temperature, reflecting the mechanical behavior of the material in the plastic flow stage. If the material has no work hardening, the flow stress is equal to the yield strength and remains constant. It is presented in Equation (6).

This simplified model ignores thermal effects and damage, directly limiting the maximum stress without considering the thermal softening that plays a crucial role in reducing the yield stress under adiabatic loading. To compensate for the absence of thermal softening, a limiting stress value is used to maintain the stress within a reasonable range. The maximum stress (${\sigma }_y$) should satisfy the limitation of Equation (7), and the calculation equation is as follows:

$${\sigma }_{flow}=\left(A+B{\overline{\epsilon }}^{p^n}\right)\left(1 + c\ln {\dot{\epsilon }}^{\ast }\right)$$

(6)

$${\sigma }_y=\min \left\{\min \left[A+B{\overline{\epsilon }}^{p^n},{\sigma }_{\max }\right]\left(1+c\ln {\dot{\epsilon }}^{\ast }\right),{\sigma }_{\mathrm{sat}}\right\}$$

(7)

where A, B, C, and n are the input constants related to the material properties; ${\overline{\epsilon }}^p$ = effective plastic strain; and ${\dot{\epsilon }}^{\ast }$ = normalized effective plastic strain rate. Where ${\sigma }_{\max }$ = maximum stress before the structure with the initial force reaches the extreme value of the physical property, and ${\sigma }_{\mathrm{sat}}$ = maximum effective stress limit of the effective stress. When the plastic strain of a structural member exceeds its effective value, it indicates that the structural member is damaged and fails.

To further investigate the failure process of each component of the bridge, particularly in light of the systematic failure of the hangers, we employed the Simple-Johnson-Cook material damage and failure model. This model was specifically designed to simulate the material parameters relevant to the bridge’s failure process. The various settings for these material parameters are presented in

Table 5. The parameters of the material model.

Material	Material Model	Parameters	Value
Prestressed steel bar	MAT 098	Density/(kg·m−3)	7.8 × 103
		Elastic modulus/Mpa	2.06 × 105
		Poisson’s ratio	0.3
		Failure stress/Mpa	420
		Bulk modulus/Mpa	1.67 × 105
		Failure strain	0.002
Concrete		Density/(kg·m−3)	2.6 × 103
		Elastic modulus/Mpa	3.55 × 105
		Poisson’s ratio	0.1667
		Failure stress/Mpa	420
		Bulk modulus/Mpa	1.67 × 105
		Failure strain	0.002

.

The analysis of the stress stages reveals that the working performance of the Nanfang’ao Bridge can be categorized into four distinct phases. The stress mechanisms associated with each phase are detailed below, and the failure stage is illustrated in Figure 20.

Stage 1: Human activities resulted in damage to hangers No. 4 through No. 7.

Stage 2: The initial failure of the hangers initiates a redistribution of internal forces among the remaining hangers. The adjacent hangers experience corresponding axial forces and are subjected to continuous failure due to the impact from the broken hangers, leading to an ongoing sequence of failures in Hangers No. 1 through No. 3. The breakage process for the hangers on the opposite side follows a similar pattern, ultimately resulting in a bridge that lacks functional support from its hangers.

Stage 3: Based on the stress characteristics, the midpoint of the bridge deck system experienced an internal force F₃, resulting in a downward displacement Y₁. The loss of support from the hangers led to a significant value of Y₁, which ultimately caused failure at this midpoint due to excessive displacement. The deflection of the integral continuous box girder bridge deck system surpassed its ultimate strength and failed at the midpoint, exhibiting an inverted “V”-shaped damage pattern before collapsing.

Stage 4: The arch feet were extruded outward by F₂. The stress concentration at the junction between the arch ring and the arch column resulted in a vertical displacement, denoted as Y₂. Damage to the bridge deck system subsequently led to deterioration of the tie beams located beneath. The significant displacement of the arch feet on both sides of the bridge was subjected to tensile forces. Consequently, the arch rib experienced internal tensile stresses, leading to excessive stress at the connection point between the arch column and the arch ring, which ultimately resulted in structural damage.

In conjunction with the analysis of the stress stages, it can be observed that the variation in the operational performance of the Liujiang Bridge can be categorized into three distinct stages. The stress mechanisms associated with each stage are detailed below, with the failure stages illustrated in Figure 21.

Stage 1: Under the most adverse failure conditions, the D(d)₄~D(d)’_4,6 hangers of the bridge sustained damage.

Stage 2: The initial failure of the hangers initiated a redistribution of internal forces among the remaining hangers. The adjacent hangers experienced corresponding axial forces and were subjected to continuous failure due to the impact from the broken hangers. This process ultimately led to a systematic collapse of the remaining hangers once the adjacent ones exceeded their redundancy limits.

Stage 3: Based on the stress characteristics, the arch rib primarily supported the pressure F₂ and induced a tension F₁ at both sides of the arch feet. The presence of the tie bar served to balance the displacements generated at the arch feet while also allowing for mutual restriction between the displacements caused by the arch rib and those produced by the transverse brace. At mid-span (1/2 position) of the bridge deck system, a downward displacement Y₁ occurred due to the combined effects of F₃ and M₁. The bridge deck system was constructed as a reinforced concrete grillage structure. Initial failure within this system manifested as localized cracking in the concrete, which progressively expanded outward until reaching a point where steel reinforcement bars attained their yield strength, leading to the potential falling of cross beams. However, it is noteworthy that none of the main bars in the tie beam had yielded during this process; thus, no catastrophic collapse of the bridge deck system transpired. Consequently, there was minimal damage sustained by the arch rib structure.

5. Discussion

At present, there are no clear specification requirements for the design of bridge structures. Influenced by factors such as the construction environment, aesthetic considerations from designers, and various design requirements, there is a notable absence of quantitative evaluation criteria to ascertain which structural designs better meet these needs. Therefore, based on the aforementioned simulation analysis, the following design recommendations are proposed for through tied-arch bridges.

• Design of hangers

Different designs of hanger structures exhibit varying performance when subjected to different loads and unforeseen circumstances. The study revealed that inclined hangers demonstrate a superior ability to withstand unexpected events, such as hanger failure, compared to vertical hangers. Furthermore, inclined hangers contribute to enhanced stability of the bridge, thereby improving overall safety.

2.

Design of arch rib

The double arch rib structure exhibits enhanced stability. The two arch ribs collaboratively support the load, resulting in a higher bearing capacity and effective stress dispersion. When subjected to significant external forces, such as strong winds and earthquakes, this design demonstrates superior resistance to deformation. Furthermore, the double arch ribs provide improved control over structural deflection, thereby mitigating the risk of deformation. Consequently, it is advisable to implement a double arch rib structure for through tied-arch bridges.

3.

Design of bridge deck system

The mid-span section of the bridge deck system of the Nanfang’ao Bridge exhibits insufficient flexural rigidity. Due to the restraint provided by the stiffness of the arch rib at the arch foot joint, this bridge deck system behaves similarly to a simply supported beam structure with horizontal spring restraints at both ends. Under load, it resembles a simply supported beam subjected to force, resulting in maximum bending moment occurring at mid-span. However, while the flexural rigidity of the cross-section is substantial near both supports, it is relatively low in the mid-span region. This disparity leads to a sharp increase in mid-span bending moment values, forming a plastic hinge and consequently increasing deflection in the main beam. This process may result in cracking at its lower end until complete failure occurs and ultimately causes the collapse of the bridge deck system. The structure thus transitions into a geometrically variable system; this entire process is rapid and characterized as a brittle failure mode.

In contrast, failure within the bridge deck system of Liujiang Bridge initiates from localized cracking within its compressed concrete slab. Over time, these cracks will progressively expand laterally on either side. The initial cracked regions will continue to deepen along their longitudinal axis. Concurrently, the internal steel reinforcement transitions from an elastic state to the yielding phase, ultimately resulting in structural failure and beam drop-off.

Compared with Nanfang’ao Bridge’s performance under stress conditions, Liujiang Bridge’s deck system experiences ductile failure, which affords additional time for personnel evacuation and escape during emergencies. Therefore, it is recommended that future designs adopt a bridge deck system structured with longitudinal and transverse beams for through tied-arch bridges.

4.

Design of the tie bar

The arch rib and tie beam of the Nanfang’ao Bridge are integrated, with the tie beam positioned beneath the bridge deck system. The downward movement of the bridge deck system induces displacement in the tie beam, leading to significant deformation and displacement of the arch rib. In contrast, the tie beam of Liujiang Bridge consists of a concrete box girder complemented by tie bars. When damage occurs to the hangers, it primarily affects the outer concrete component while exerting minimal impact on the tie cables; consequently, this ensures that the arch rib remains relatively secure. Therefore, selecting an appropriate structure for the tie beam is crucial for enhancing resistance to progressive collapse in through tied-arch bridges.

The Johnson–Cook model adopted in this study was originally designed for metallic materials. Although it can be extended through parameter adjustment, it is difficult to accurately describe the non-homogeneous characteristics of concrete, such as brittle fracture and aggregate-mortar interface debonding. It also cannot directly simulate the bond-slip behavior between steel bars and concrete and does not consider the influence of long-term deterioration on the dynamic response of materials. Therefore, it is only applicable to the instantaneous behavior under short-term extreme loads. Bridges often contain concrete, steel bars, composite materials, etc. Due to the limitations of this model, in the next step of the research, a better research method will be sought for the collapse and failure of bridges, and the visualization of the failure can be achieved.

6. Conclusions

This study investigates how variations in structural design affect the load-bearing mechanisms and failure modes of through-type arch-girder composite bridges. The key findings are summarized as follows:

(1)

Mechanical performance comparisons demonstrate that the non-cross diagonal hanger configuration outperforms the vertical hanger design. Specifically, under identical hanger failure paths, the Nanfang’ao Bridge with diagonal hangers remains stable after the failure of four hangers without experiencing progressive collapse, whereas the Liujiang Bridge with vertical hangers undergoes continuous failure following the rupture of only three hangers. This comparison confirms the significant advantage of diagonal hangers in enhancing resistance against progressive collapse. Therefore, in the design of long-span tied-arch bridges, the mechanical benefits of diagonal hanger layouts should be fully incorporated to achieve improved structural safety and stability.

(2)

Mechanism analysis of progressive collapse reveals that, when the hanger system undergoes failure, the Nanfang’ao Bridge experiences a V-shaped collapse pattern. This is attributed to insufficient flexural stiffness in both the mid-span section of the bridge deck system and the single-rib double-fork arch rib, combined with the absence of anti-drop beam measures at the bearings. These deficiencies lead to a dramatic alteration in the structural system and load transfer path, leaving no alternative load-carrying pathways to prevent collapse. In the case of the Liujiang Bridge, upon disruption of hanger force transmission, compressive stresses develop in the bridge deck system, initiating localized concrete cracking. The cracks propagate laterally toward both sides and longitudinally through the bridge deck. As a result, the reinforcing steel transitions from the elastic stage to plastic yielding, ultimately leading to crack coalescence and beam drop, exhibiting characteristics of progressive collapse. Therefore, it is critical to enhance the maintenance and inspection of hangers and associated structural components, preventing such progressive failure modes in similar bridge structures.

(3)

For arch-girder composite structures, the structural layout should be optimized during the design phase to mitigate stress concentrations and eliminate potential weak links. The use of double-fork arch ribs is discouraged, and critical vulnerable components such as girder supports and joints should be reinforced to enhance structural integrity. Furthermore, redundancy principles should be integrated into the design process by increasing structural redundancy, thereby ensuring that the system retains a certain load-bearing capacity even in the event of sudden component failure. This approach would effectively prevent rapid structural collapse within a short timeframe.