#### Abstract

Most of the recent studies focus on the progressive collapse of ordinary structures due to gravity and blast loads. A few focus on studying progressive collapse due to seismic actions, especially of bridge structures. The past major earthquakes have shown that it is possible to develop improved earthquake-resistant design techniques for new bridges if the process of damage from initial failure to ultimate collapse and its effects on structural failure mechanisms could be analyzed and monitored. This paper presents a simulation and analysis of bridge progressive collapse behavior during seismic actions using the Applied Element Method (AEM) which can take into account the separation of structural components resulted from fracture failure and falling debris contact or impact forces. Simple, continuous, and monolithic bridges’ superstructures were numerically analyzed under the influence of the severe ground motions not considering the live loads. The parameters studied were the superstructure redundancy and the effect of severe ground motion such as Kobe, Chi-Chi, and Northridge ground motions on different bridge structural systems. The effect of reducing the reinforcement ratio on the collapse behavior of RC box girders and the variation of columns height were also studied. The results showed that monolithic bridge models with reduced reinforcement to the minimum reinforcement according to ECP 203/2018 showed a collapse behavior under the effect of severe seismic ground motions. However, changing the bridge structural system from monolithic to continuous or simple on bearing bridge models could prevent the bridge models from collapse.

#### 1. Introduction

Progressive collapse phenomenon is defined as the global damage or collapse behavior of a large part of the structural system that is caused by a failure of a relatively small or localized part of the structure. Structural progressive collapse occurs as a result of failure of one or more structural members or components. The load is transferred in the structural system due to changes in the distribution of stiffness, the pattern of the stress behavior, and/or the structural boundary conditions [1]. This initial failure results in other structural elements being further overloaded and later fail. Studies on the progressive collapse of existing structures have focused primarily on high impact as in blasting or irregular loading. Not so much attention is paid to the vulnerability of structures, especially bridges, with regard to progressive collapse during earthquakes [2].

Seible et al. [3] noted that earthquake loads could result in progressive collapse of bridges influenced by earthquake characteristics such as magnitude, strength, source mechanism, and intensity. This phenomenon is noted during seismic events as with the Northridge and Kobe earthquakes. Continuous vibration after initial failure and repeated stress reversals from cyclic nonlinear behavior can lead to a significant degradation in the stiffness strength of the structural system. There would be a ductility capacity reduction due to cumulative damage effects in the dynamic structural behavior and response during major earthquakes.

Previous studies on the progressive collapse of structures gave insights into the effect of strengthening on improving the structural capacity to prevent progressive collapse due to seismic and blast loads. Those studies have shown that such detailing and strengthening against the earthquake could enhance buildings’ progressive collapse resistance to such load effects [4, 5]. Gurley [6] commented that improving a structure’s ductility through reinforcement details could help improve its progressive mechanism of collapse. Given these results, there is still a lack of information on the progressive collapse and the mechanism of collapse that can be used to enhance the seismic design of new bridges and to establish more successful methodologies for strengthening and retrofitting old bridges.

Wibowo et al. [7] studied the seismic progressive collapse of RC bridges during earthquakes. They modeled only a continuous bridge that was previously experimented with “Guedes, 1997” [8]. The results have shown a good agreement. The separation of structural components resulting from fracture failure and impact forces from falling debris had been taken into consideration. The results have shown a significant influence on the performance of bridges during major earthquakes that were visible in its progressive collapse analysis. These also demonstrate the need to include progressive failure mechanisms in the assessment of seismic design efficiency and bridge evaluation that would not only lead to a better and more robust earthquake-resistant design for new structures but also more efficient retrofitting and reinforcement strategies for older structures.

In a similar vein, Salem et al. [9] analyzed numerically the collapse of Tsuyagawa Bridge damaged by the Tohoku Tsunami in March 2011. The Tohoku Tsunami swept across Japan’s eastern coast, killing over 15,000 people and listing missing over 2,500. The tsunami caused more than 400,000 buildings to collapse and more than 250 coastal bridges to be washed away. The analysis showed accurately the collapse behavior of the bridge, showing that the bridge collapsed at a water velocity of 6.6 m/s caused by its piers’ flexural failure. Tsuyagawa Bridge’s AEM analysis has shown the ability to simulate the 2011 Tohoku Tsunami collapse effectively, although the analytical results showed less ductility when compared to reality.

Domaneschi et al. [10] analyzed numerically the collapse of the viaduct over the Polcevera Valley in Genoa that collapsed in August 2018. This incident left 43 deaths and several injuries caused by a collapse of a portion of the highway connection. The results of the analysis showed that the stay cable was the most important item whose failure caused the collapse. Furthermore, the simulation model indicated that the main girder triggered the collapse, and the large visible displacements involved in their collapse would have warned the authorities of the impending fault.

Bridges could be in several structural forms, i.e., simple on bearings, continuous on bearings, and monolithic with column bridge structures. There are no sufficient studies on the effect of severe seismic ground motions for such different bridges’ structural systems; especially, most researchers studied bridges’ models, either continuous or simple bridges. In the current study, Wibowo’s Bridge is modeled and verified using the applied element method. Also, the effect of the reduction in reinforcement quantity on the collapse of RC box girder bridges under Kobe ground motion is investigated. Several bridges’ models with different structural systems such as simple on bearings, continuous on bearings, and monolithic with column bridges are analyzed under severe seismic ground motions: Kobe, Chi-Chi, and Northridge earthquakes.

#### 2. Applied Element Method

The Extreme Loading for Structures (ELS) program, developed by ASI-2018 [11], is based on the AEM, which was initially developed by Tagel-Din and Meguro [12, 13] at the University of Tokyo in 1998 to solve problems related to two-dimensional plane stresses. It was later expanded to solve three-dimensional problems. The AEM is a novel method of modeling that adopts the discrete cracking concept in AEM. Structures are modeled as an element assembly. The elements are not rigid and connected by normal and shear springs along their joint surfaces. These springs are responsible for normal and shear stresses transfer between adjacent elements. Each spring represents a certain volume of material stresses and deformations (see Figure 1). Once the connecting springs fail, each of the two adjacent elements can be completely separated. The AEM adopts fully nonlinear path-dependent material constitutive models. AEM is a stiffness-based approach in which an overall stiffness matrix is formulated and equilibrium equations for each of the stiffness, mass, and damping matrices for structural deformations (displacements and rotations) are nonlinearly solved. The equilibrium equation solution is an implicit one that takes step-by-step dynamic integration (Newmark-beta time integration procedure) [14, 15]. If the springs connecting the elements are ruptured, two adjacent elements are separated from each other. Elements may separate, recontact, or contact other elements automatically depending on the structural response (see Figure 2).

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#### 3. Material Models

##### 3.1. Modeling Concrete and Reinforcing Steel

Maekawa and Okamura's model [16] is used for concrete in compression, whereas for concrete in tension, the linear stress-strain relationship is adopted. In this stage, concrete is exposed to tension up to cracking where the stresses are set to zero afterward. Furthermore, for concrete in shear, a linear relationship between shear stresses and strain is assumed before the cracking. After cracking, a drop in the value of shear stresses to zero takes place [17]. Springs are also used to define the reinforcement between elements. Ristic’s model [18] is used to model the reinforcement. Newmark-*β* approach is used to solve equations of dynamics. The equilibrium equations are indeed linear for each step and are generally solved, in AEM, by using a direct or an iterative solver (Figure 1).

##### 3.2. Bridge Bearing Material

An interface material is used to model bearings. The interface material model is a precracked element where the material is initially cracked and cannot bear tensile stresses. As for compression, the stress–strain relation is linear up to compression failure stress (Figure 3). The relationship between shear stress and shear strain is linear until the shear stress approaches *μσn* (coefficient of normal friction × normal stress). At this stress level, the shear stress remains the value (*μσn*) as long as there is no change in normal stresses. The compressive stress variation allows the proportional variation in shear stresses (*μσn*). The shear stiffness is set as a minimum, if the crack opens or during active sliding of the bearing (see Figure 3 [9]).

#### 4. Comparison of AEM and FEM

During progressive collapse analysis, the failure, separation, contact, and falling debris of elements must be traced. Using FEM, it is very difficult to model progressive collapse. On the other hand, using AEM, to analyze these processes is made easy and effective taking into consideration all the analysis stages until collision (Figure 4).

#### 5. Verification

##### 5.1. Numerical Study

The selected numerical example was previously investigated experimentally by Guedes [8] and numerically by Wibowo et al. [7]. The bridge has four 20 m spans. The bridge deck is a hollow box girder, which has 5.6 m wide. The bridge has two end abutments at points 1 and 5 and has three piers at points 2, 3, and 5. The pier heights are shown in Figure 4. The piers are fixed at bottom base and the abutments are hinged at points 1 and 5. The connection between piers and the bridge box girder is hinged. The cross section of the bridge piers is a rectangular hollow-core RC section (Figure 5). The longitudinal reinforcement details of the box girder were maintained from Wibowo et al. ([7], see Figure 6). The material properties are also summarized in Table 1.

##### 5.2. Artificial Earthquake Ground Acceleration

The used earthquake ground motion was artificial; it followed the work of Guedes [8]. Also, the mentioned artificial ground motion is previously used by Chiara and Rui [19], Lau and Wibowo [20], and Seyedkhoei et al. [21]. The peak ground acceleration (PGA) was 1.05 g. Based on the estimated bridge design capacity, the PGA is increased by 1.2. The total time used for the analysis from the artificial earthquake record is 4 seconds, and it is applied in the transverse direction (Figure 7).

Hereby, the analysis is performed twice. The first assumes that the box girder is elastic, and the piers are assumed to be nonlinear, while the second considers the nonlinear behavior of the box girder and the piers. The top displacement of the piers is compared by that obtained from Wibowo et al. [7].

As shown in Figure 8, in the linear model, the average difference percentage of the current study and Guedes [8] was 7.87%, 4.2%, and 4.6% for short, medium, and long piers, respectively. It was found out that the studied model could follow the response of the bridge piers with high precision. Also, the peak displacements of the piers that took place at the time of 3 to 3.5 s were identically obtained.

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Analyzing the nonlinear model, the top displacements of the piers were compared by those obtained by Wibowo et al. [7] (see Figure 9). It was found out that the displacements were similar to a good degree. The difference in the displacements between the two models could be originated as the diameters of transverse reinforcement were not clearly mentioned by Wibowo et al. [7]. This can have a major effect on the difference with the current model.

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Throughout the analysis of the nonlinear model, the collapse mechanism was monitored (Figure 10). Firstly, the box girder cracked at bridge abutments and then cracked at the locations of the piers because of the redistribution of the deck’s own weight on the piers. Subsequently, the bridge deck showed shear failure because of punching on the bearing supports. Finally, the bridge deck progressively collapsed.

Through these results, it can be concluded that ELS can correctly model the bridge linear and nonlinear behavior, including various structural elements such as piers, bridge superstructure, reinforcement details, and material properties, and thus can provide reliable results.

#### 6. Bridge Models

##### 6.1. Bridge Layout

RC box girder bridges were modeled 3 × 25000 millimeters bays (Figure 11). Various bridge configurations were studied; the bridge superstructures are monolithic box girder with columns, continuous box girder on bearings, and simple box girder on bearings. The columns were assumed to be fixed at its bases. The bridge superstructure was rested on elastomeric bearing plates. Each row of the bearing contained five elastomeric bearings. The bridge dimensions and reinforcement details were originally taken from executed multispan box girder bridges in Egypt. The damping ratio is assumed to be 5% during the analysis. The analyzed bridge models and the reinforcement of the box girders are shown in Figure 12 and Table 2. The purpose of analyzing models A1-M-K, A2-M-K, and A3-M-K (Table 2) was to determine the reinforcement reduction effect on the collapse behavior during the dynamic seismic loading. In models A3-C-K and A3-S-K (Table 2), the structural system of the bridge superstructures was changed from the monolithic to simple box girder on bearings and continuous on bearings, respectively. The behavior of such systems was compared with that of the monolithic bridge. Also, a comparison between simple, continuous, and monolithic bridge models was carried out under severe ground motions, i.e., Kobe, Chi-Chi, and Northridge.

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##### 6.2. Material Properties

The material properties adopted in AEM analysis are presented in Table 3. A full bond was assumed between the concrete and the reinforcing steel. The used bearing was composed of a top and bottom steel plates and bearing material in between as in Salem et al.’s work [9]. The dimensions of the steel plates used were 500 × 500 × 50 mm. The dimension of the elastomeric bearing interface was assumed to be 350 × 350 × 130 mm [22]. The interface between the steel plates was given bearing material properties[9]. A relatively high compressive strength was given to the bearing interface so it could not fail in compression and act linearly [23]. The shear modulus of the bearing was assumed to be 2 MPa [22, 24].

#### 7. Ground Acceleration

Kobe, Chi-Chi, and Northridge ground accelerations were used in the progressive collapse analysis of the bridge models, as there was some bridge collapse during these earthquakes [25]−[29]. The ground motions data was obtained from the Pacific Earthquake Engineering Research (PEER), Strong Motion Database [30]. A summary of the earthquake ground motions used in this research is presented in Table 4 and is shown in Figure 13. The time used in the seismic analysis was reduced to the time that contains the largest cycles of seismic accelerations to reduce the ELS analysis time, as the time that would not contain significant values of acceleration could be omitted. The used time step during the analysis was 0.004. Earthquake analysis usually requires Δ of 0.001–0.01 sec. when a collision is expected to occur. The smaller the time step is, the higher the accuracy and the convergence of results becomes.

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#### 8. Mesh Sensitivity Analysis

A mesh sensitivity analysis was carried out to obtain a suitable mesh size that would be used in all the analysis cases for columns and bridge superstructure. Horizontal and vertical concentrated loads were used for the column and the box girder, respectively. Figure 14 shows the relationship between the mesh elements and the displacement of the column and the deflection of the box girder. 22 elements per column’s height and 5 × 12 elements per columns’ cross section were used. The maximum dimensions for the columns’ elements were 200 × 200 mm per element cross section and was 38 cm per element height. Each surface area of the box girder (i.e., the deck, soffit, and webs) was divided into 5 × 1 elements with 50 elements per 25 m length (span) in the box girders’ longitudinal direction. This mesh size was found to give accurate results. An analysis using a finer mesh has been carried out without any noticeable difference in the displacement and deformation. The total number of elements used for monolithic, continuous, and simple bridges was 10,000, 13,800, and 13,200, respectively. The AEM mesh used was accurate enough during the elastic region and in the small deformation range of the inelastic region [12, 13].

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#### 9. Analysis Results

The analysis was carried out on two stages; the first was static to take into account the gravity loads and original deformations of the bridge, whereas the second was a dynamic analysis, implementing the different seismic ground motions. For more details about how the analysis was carried out, please refer to ELS theoretical manual, ASI [11]. The principal strain of the bridge models was used to analyze crack locations and collapse patterns.

##### 9.1. Reinforcement Reduction Effect

Figure 15 shows the displacement time history for the right column of A1-M-K, A2-M-K, and A3-M-K models. The three models showed a relatively identical behavior, as the reduction in the reinforcement ratio did not produce any failure in the structural elements to the analysis termination. Once the reinforcement ratio is reduced to 25%, the box girder of model A3-M-K collapsed; it showed a deviation in the displacement–time history. The right bay of the bridge box girder started to collapse. And once it hit the ground, the column exhibited large displacement. The right column maximum horizontal displacement for the three models before failure was 200 mm. It was extended further to 300 mm after failure in model A3-M-K.

Figure 16 shows the straining actions. The straining actions of A1-M-K, A2-M-K, and A3-M-K models were nearly identical excluding model A3-M-K. Model A3-M-K showed a deviation at the 8th second as the right bay of the bridge box girder showed cracks, and it developed further to collapse. Once the collapse initiated, the 50% reduction in the axial column load occurred. Other straining actions behavior for the same model showed a nonuniform behavior to the end of the analysis. The maximum magnitude of moment and shear during the time history analysis was 1250 m·ton and −350 tons, respectively.

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##### 9.2. Effect of Kobe Ground Motion on Different Bridge Structural Configurations

Figure 17 shows the displacement–time history of the right column of the bridge’s different configurations. Displacement time history of models A3-M-K, A3-C-K, and A3-S-K was identical. However, after the failure of the bridge box girder of the monolithic model at the 15th second, the displacement curve showed a deviation from the other models to the end of the analysis. The maximum displacement amplitudes were measured at the 5th second and equal 300, 275, and 200 mm for simple, continuous, and monolithic bridge models, respectively. The simple and continuous models showed large displacement amplitudes, while the monolithic bridge models showed a less displacement amplitude. This could be attributed to the rigid connection between the bridge superstructure of the monolithic bridge and the columns, as it resisted the displacement movement of the bridge superstructure. The less the internal restrains in the bridge are, the more increase in displacements takes place.

Figure 18 shows a comparison between the straining actions of monolithic with columns, continuous on bearings, and simple on bearings bridge models. As mentioned before, the monolithic bridge model, A3-M-K, showed a collapse behavior. However, when analyzing simple and continuous models, A3-S-K, and A3-C-K, it was found out that the displacements are similar and they did not show any collapse behavior to the end of the analysis.

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##### 9.3. Collapse Analysis of Different Bridge Models during Kobe Ground Motion

Cracks are represented through the principal strains. A comparison between models A1-M-K, A2-M-K, and A3-M-K is presented in Figures 19–21, respectively. The less the bridge superstructure reinforcement, the more cracks or collapse observed in the box girder. At the end of the analysis time, A1-M-K and A2-M-K did not collapse. Reinforcement reduction beyond the minimum reinforcement ratio, model A3-M-K according to the ECP203-2018 [31], allowed the bridge model to show a collapse behavior. At failure initiation after 4 seconds, the webs of the box girder showed excessive cracks. At 7.25 seconds, the web of the right box girder failed mainly in shear. At 8 seconds, the right bay of the box girder hit the ground. At the 15th second, the left box girder initiated failure in shear and settled on the ground. At analysis termination reaching the 20th second, the middle bay failed, and the bay weight dragged the left column down.

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In the monolithic bridge models, the seismic ground motions brought about cracks at the upper and lower ends of the columns in the plastic hinge zones, since they are the areas of stress concentration. It should be noted that the cracks did not spread along with the column height, as the column dimensions were very large (1 × 2.5 meters), and the amount of reinforcing steel, either stirrups or longitudinal reinforcement, was very high. Also, in Figure 21, at the end of the seismic analysis, the middle bay of the box girder collapsed and dragged the left column down to the ground and the left column showed large strain values along with its height.

Figures 22 and 23 show principal normal strain during the time of models A3-C-K and A3-S-K, respectively. Models A3-C-K and A3-S-K did not show any collapse behavior even though they contain the same reinforcement amount as model A3-M-K. The reason is that the less the internal restrains in the bridge, a decrease in collapse possibility takes place. Also, the columns did not show enough strain values as the tops of these columns are free to move and they are not constrained with the bridge superstructure.

##### 9.4. Collapse Analysis of the Different Bridge Models under Different Severe Ground Motions

Kobe, Chi-Chi, and Northridge ground motions were used to analyze monolithic, simple, and continuous bridge models to determine the effect of severe ground motions on the collapse behavior of these bridge models. From models A3-M-K, A3-M-C, and A3-M-N, it is noted that the monolithic bridge model always had a collapse behavior once the reinforcement ratio of the superstructure has been reduced below the minimum reinforcement ratio (Figures 21, 24, and 25) On the other hand, when analyzing simple and continuous on bearing bridge models with the minimum reinforcement ratios by using the different ground motions, they did not show any collapse behavior (Figures 22, 23, and 26–29).

In model A3-M-K, it is noted that the monolithic bridge model collapsed by Kobe ground acceleration, while by analyzing the same bridge configuration it showed a partial collapse by Chi-Chi and Northridge ground accelerations, as in models A3-M-C and A3-M-N. Thus, the collapse of the structures did not depend on the largest value of the peak ground acceleration; yet it depended on the number of cyclic reversal accelerations.

#### 10. Conclusions

In the current study, the seismic progressive collapse behavior and analysis of reinforced concrete bridges were analyzed. Various bridge configurations: monolithic with columns, continuous on bearings, and simple on bearings bridge models were analyzed. The bridge models and selected earthquake excitations used in the study were discussed. A summary of the findings is presented herein.(i)Progressive collapse will not occur unless the reinforcement ratio of the bridge superstructure is reduced to the minimum reinforcement ratio according to the ECP 203-2018.(ii)Monolithic bridge models with reduced reinforcement ratio always show a collapse behavior during the analysis using severe ground motions. However, changing the bridge structural system from monolithic to continuous or simple on bearing bridge models prevented the bridge models from collapse.(iii)The total collapse of the structures does not depend on the largest value of the peak ground acceleration as in Northridge, and it depends on the number of reversal cycles of reversal waves as in Kobe.(iv)In reinforced bridge models that are designed according to “The Egyptian Code of Practice 203-2018,” the reinforcement ratio in the bridge prevents any bridges from collapsing even under the influence of the severe ground motions, but it may show some cracks both at the bridge superstructure and at the column bases.(v)Similar progressive collapse phenomenon is observed in the collapse of bridge structures caused by blast or abnormal loads and is also observed in the failures of structures during earthquakes even although the damage and propagation characteristics are more universal in nature, as can be detected in the analysis and simulation results in this study.(vi)ELS program can be a means to predict the behavior of ordinary and special structures against abnormal events during the design, construction, and service loads.(vii)Sufficient ductility capacity in bridge columns or piers can help ensure ductile behavior and higher deformation capability of the structure that leads to better performance against seismic progressive collapse.

#### Data Availability

The datasets used to support the findings of this study are incorporated into the article.

#### Conflicts of Interest

The authors declare that that there are no conflicts of interest regarding the publication of this paper.