Abstract
The traffic capacity of the urban elevated bridge is assessed after it is attacked by a near-field explosion, using the residual bearing capacity of the damaged pier as the assessment index. First, the finite element model of a reinforced concrete slab under near-field explosion is established by ANSYS/LS-DYNA software and compared with the experimental results, which verifies the effectiveness of the ALE (arbitrary Lagrangian–Eulerian) algorithm and the accuracy of the mesh size and material properties. Then, an “explosive-air-pier” coupling analysis model is constructed using the finite element method, and the damage of the reinforced concrete pier under three types of car bombs is evaluated. Furthermore, a response surface model for the residual bearing capacity of the pier is utilized to calculate the failure probabilities of various damage levels of the pier under the three types of car bombs and to assess the traffic capacity of the bridge after near-field explosion. The established assessment method can be used to predict the probability of bridge structural damage at various levels under different types of car bombs and to provide a reference for exploring a probability-based safety assessment method of post-explosion bridges.
1. Introduction
Duwadi and Chase [1] from the Federal Highway Administration (FHWA) pointed out that the bridge is vulnerable to physical, biological, chemical, and radiological attacks in addition to natural hazards. Among these attacks to the bridge, explosion is always a powerful but sudden action. In April 2007, in California of the United States, a tanker truck caught fire and exploded near the San Francisco-Oakland Bay Bridge, destroying the bridge structure and paralyzing the transportation hub. In February 2013, in Henan Province of China, a truck carrying fireworks and firecrackers exploded on the Yichang Bridge, causing the bridge to collapse, and some vehicles fell. The accident resulted in 13 deaths and a direct economic loss of 76.32 million yuan [2]. With the increasingly frequent occurrence of explosion accidents and high potential risk of terrorist activities today, the fast assessment of urban bridge’s traffic capacity after explosion to avoid heavier casualties and greater economic losses becomes more and more important.
At the end of the 20th century, with the rapid development of computers and numerical methods, the research on the bridge structure under explosion by numerical calculation became convenient. Liu et al. [3] introduced the principles and characteristics of explosion simulation and found that the bridge after explosion presented local damage. As one of the main load-bearing parts of the bridge structure, once a pier is attacked by a near-field explosion and has a severe damage, the internal forces of the structural system will be redistributed, which may lead to the collapse of the bridge. Many researchers such as Hwang et al. [4] and Shinozuka et al. [5] directly equated the bridge structure’s damage level with the pier. Therefore, the damage assessment of the pier after near-field explosion is significant to comprehensive protection and antiblast design of urban bridges.
At present, the research about explosion action on the bridge mainly focuses on the deterministic finite element (FE) model for analysis. Shiravand and Parvanehro [6] evaluated the effects of explosion on a posttensioned concrete box bridge using ANSYS-AUTODYN. Al-Smadi [7] investigated the blast response of a bridge located at the vicinity of high explosive charge using FEA software LS-DYNA. Hájek et al. [8] assessed the response of a heterogeneous concrete-based composite bridge under blast, to observe the effect of heterogeneity on the overall response of the structure. However, in these current studies, it is difficult to reflect the contingency of blast load and the randomness of structural parameters, and either to predict the probability of various damage levels to the structure from an overall perspective.
In this paper, a method to assess the traffic capacity of an urban elevated bridge after near-field explosion based on the response surface method is proposed, which can predict the probability of various levels of damage to the bridge under different types of car bombs, to provide a reference for exploring a probability-based method on evaluating the safety of the bridge after explosions.
2. Validation of the Finite Element Model
Wang et al. [9] studied the dynamic response of a reinforced concrete (RC) slab under near-field explosion through a scale model experiment. To verify the validity of the ALE algorithm and the accuracy of the mesh size and material properties used in this paper, the FE model of an RC slab under near-field explosion is established by ANSYS/LS-DYNA software and verified by comparing with the experimental results in [9].
2.1. Description of the Numerical Model
The dimension of the RC slab is 1250 mm × 1250 mm × 50 mm, with fixed constraints on the two symmetric sides and no constraints on the others, as shown in Figure 1. The equivalent mass of the explosive is 0.94 kg, and it is placed 500 mm above the slab. The dimension of the air area is 1250 mm × 1250 mm × 1000 mm. The experimental model in [9] and the finite element model in this paper are shown in Figure 2.
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The reinforcement and concrete models are established separately, and the joints between them are the complete bond in simulation. The interaction between explosive, air, and RC slab is computed by the ALE algorithm. The boundary of the air is nonreflective boundary condition.
The reinforcement is calculated using the beam element (BEAM161), and the explosives, concrete, and air are all calculated using the solid element (SOLID164). The reinforcement and concrete are simulated by the Lagrange mesh (12.5 mm), and the air and explosive are simulated by the Euler mesh (25 mm).
The air is assumed to be an ideal gas by using the MAT_NULL model to describe its material properties and using the EOS_LINEAR_POLYNOMIAL model to characterize the state equation. For TNT explosive, the MAT_HIGH_EXPLOSION_BURN model is adopted to describe the relationship between the stress and strain, and the EOS_JWJ model is adopted to characterize the relationship between the pressure and volume deformation. The kinematic plastic material model MAT_PLASTIC_KINEMATIC is used to describe the constitutive relation of the reinforcement. This material has its strain failure criterion: once the strain of the reinforcement element reaches the failure strain (FS = 0.12) [10], the failure element will be automatically deleted from the calculation model. The MAT_CONCRETE_DAMAGE_REL3 model is adopted in concrete simulation, which has been proved successful to reflect the damage effect, strain rate effect, strain strengthening, and softening effect of concrete, and it can effectively simulate the energy dissipation mechanism of concrete structures under explosion and impact load [11]. The failure of the concrete material is defined by the keyword MAT_ADD_EROSION, which is controlled by the maximum principal strain failure criterion (MXEPS = 0.06) [12].
2.2. Analysis of Numerical Simulation
The results of FE simulation are compared with the experimental results in [9], as shown in Figure 3.
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The results show that the RC slab under the explosive is seriously damaged, and the average diameter of the damaged area is about 150 mm, but none of the reinforcement is broken. Besides, there are a lot of radial cracks on the slab back. By comparison, the local damage in the numerical model of RC components is in good agreement with the experimental result, and the damage characteristics of them are similar. The calculation results also indicate that the mesh size and material properties used in the model are reasonable, and the ALE algorithm can simulate the interaction between the explosion shock wave and structure effectively. The proposed numerical model of the RC slab under explosion load can predict the structure’s dynamic behavior accurately.
3. Damage Assessment of the Urban Elevated Bridge Pier under Explosion Load
3.1. Explosion Risk Sources
The main risk sources of the urban elevated bridge subjected to explosion are classified into three types of car bombs [13], according to the carrying capacities of various cars. Shown in Table 1 are their corresponding TNT equivalent mass and the heights from the explosion centers to the ground.
3.2. Damage Assessment of the Urban Elevated Bridge Pier under Explosion Load
The analysis model of the urban elevated bridge pier under explosion load is composed of air, explosive, and pier. The pier is a single-column RC structure with a rectangular cross section. The height of the pier is 4 m, its cross section size is 1.5 m × 1.5 m, and the concrete grade is C30. According to the principle of ductile seismic design, the stirrup spacing is 10 cm in the vertical direction within 1 m from the top end and 1.1 m from the bottom end, and the rest of the stirrup spacing is 20 cm. The diameter of the stirrup is 1.6 cm, and its material is HRB335. The diameter of longitudinal reinforcement is 3.2 cm, and its material is HRB400. The reinforcement arrangement is shown in Figure 4.
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In the FE model, the explosive is placed on the central axis with a distance R from the side surface of the pier, the height from the ground is recorded as H, and the bottom of the pier is set to a fixed constraint, as shown in Figure 5. The ground boundary is set to reflective, and other boundaries of air are set to nonreflective. The mesh sizes and material properties of each part in the model are shown in Table 2, in which the mesh sizes for different elements are identical to the previously calibrated ones.
To perform damage assessment of the pier after explosion, the loading procedure on the FE model is divided into three stages, as shown in Figure 6.
First stage: the self-weight of the girders and other vertical loads are applied on the pier to simulate the initial stress state of the pier, which is realized by applying the loads at a slow rate on the top of the pier until the structure reaches a static equilibrium state.
Second stage: after stress initialization of the pier, the air and explosive parts are added in the model. The explosion load is applied on the pier by the ALE algorithm, and then the dynamic response and damage of the pier are analyzed.
According to the scale of three types of car bombs, 10 analytical cases are designed for each car-bomb type, with the explosion distances of 1.2 m, 2 m, 2.5 m, 3 m, 3.5 m, 4 m, 4.5 m, 5 m, 5.5 m, and 6 m. After attacked by three types of car bombs, the damage states of the pier under explosion distances of 1.2 m, 3.5 m, and 6 m are shown in Figure 7 and described in Table 3.
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Third stage: after the post-explosion structure reaches a new state of static equilibrium, the calculation is continued by applying a displacement-controlled load at the pier top until the pier is destroyed, to determine the pier’s residual bearing capacity [14].
To explain how to calculate the pier’s residual bearing capacity, the residual axial load-time history curve of the pier is extracted at the third stage after the 3# car bomb (M = 200 kg, H = 0.5 m) attacks on the pier at 1.2 m as an analytical case, as shown in Figure 8. The value of residual bearing capacity of the pier can be determined according to the characteristics of the curve.
It is clearly found from Figure 8 that when t = 1061.69 ms, the axial load of the pier reaches its peak, which is defined as the residual bearing capacity of the pier (Pres = 59.80 MN). The same method is used to obtain the residual bearing capacity of the pier in the other 29 cases. Similarly, the pier without the explosion effect is calculated by applying a displacement-controlled load at the pier top, and its initial bearing capacity (Pini = 70.35 MN) is extracted.
Based on the residual bearing capacity of the pier, the damage index D can be calculated, which is defined as [15]where is the residual bearing capacity of the pier after explosion and is the initial bearing capacity of the pier. The corresponding damage indices D related to different explosion distances are calculated, respectively, and their relations are shown through the curves in Figure 9. The level classification of the damage indices D in the figure is quoted from [15].
As shown in Figure 9, the curve characteristics of the pier under the three types of car bombs are similar. In the initial stage of increasing the explosion distance, the damage index decreases rapidly, and then the decrease becomes slow. Under the attack of 1# car bomb, the most considerable variation of the damage index can be found than other cases, and all damage levels may appear. Under the attack of the 3# car bomb, the curve looks like a horizontal line, and the pier is only slightly damaged. It can be concluded that, under the attack of 1# car bomb, the damage index of the pier is most sensitive to the explosion distance, while under the 3# car bomb, the explosion distance has little effect.
Normally, the scaled distance is usually introduced to describe the intensity of explosive shock wave by combining explosive equivalent mass and explosion distance. The expression to calculate the scaled distance iswhere R is the explosion distance and is the explosive equivalent mass. The corresponding scaled distances are calculated under all analytical cases, and the damage index-scaled distance curve of the pier is shown in Figure 10.
From Figure 10, it is clearly seen that the dispersion of the damage index is severe when the scaled distance is less than 0.6 m/kg1/3. Therefore, it is less suitable than direct use of explosion distance as an independent variable when rating the damage index for the pier after near-field explosion. At the same time, most of the similar studies [16–18] established the relationship between the explosion distance and the structural damage; the following sections of this paper will mainly study the influence of explosion distance on the damage level of the pier.
4. Traffic Capacity Assessment of the Urban Elevated Bridge after Near-Field Explosion Based on the Response Surface Method
In recent years, traffic capacity assessment of bridges after disasters has attracted the attention of some scholars. Zhou [19] summarized the calculation methods of the axial residual bearing capacity of the pier after the earthquake and assessed the bridge’s traffic capacity based on the vulnerability curve of the pier. Gu [20] used the damage index to quantitatively evaluate the damage level of the pier under rockfall impact and assessed the bridge’s traffic capacity by calculating the probability of the pier at various damage levels.
The response surface method is a combination of statistics and mathematics, and it performs a fitting analysis for the response problem of the objective function under the influence of multiple factors, to obtain the optimal response model. Bucher and Bourgund [21] found that the response surface model can well replace the complicated functional relationship between the structural basic parameters and responses and analyzed the reliability of the fitted response surface model. However, at present, the research on bridge’s traffic capacity after explosion is still insufficient.
Referring to the assessment method of bridge’s traffic capacity after the earthquake, in this section, a simple response surface mathematical model is used to replace the individual analytical case by FE model analysis.
4.1. Assessment Method of Traffic Capacity of the Urban Elevated Bridge after Near-Field Explosion
The processes to assess the traffic capacity of the urban elevated bridge after near-field explosion are shown in Figure 11. First, the main control parameters that affect the residual bearing capacity of the pier are analyzed. Then, a series of parameter combinations are obtained through experimental design, and these parameter combinations are substituted into the FE model for calculation, based on which the response surface model is established by fitting the results. Finally, the failure probabilities at various damage levels of the pier under the attack of different car bombs are calculated by the Monte Carlo sampling method, and the vulnerability curves of the pier are plotted, through which the traffic capacity of the bridge after the near-field explosion is assessed.
To determine the traffic capacity of the bridge after explosion, the failure probability of the pier at various damage levels should be known in advance. Referring to the seismic vulnerability analysis method of the structure [22], the failure probability of the pier under explosion is defined as follows: when an explosion occurs at a distance of R = i from the surface of the pier, the structure will reach a certain limit state (Lj) probability.where is the failure probability of the structure, S is the response of the structure under explosion load, and Lj is the damage limit state. As previously mentioned, the damage level of the pier has been graded according to the damage index D so that different damage limit states (Lj) can be determined. Among them, j = 1, 2, 3, and 4 represents slightly damaged, moderately damaged, severely damaged, and completely collapsed of the pier, respectively.
The traffic indicator value TA is used to evaluate the traffic status of the bridge after explosion [19], which is calculated bywhere is the failure probability at each damage level; Kj is the weight factor corresponding to each damage level, and its value is determined by experience as 0.80 for slightly damaged, 0.65 for moderately damaged, 0.30 for severely damaged, and 0 for completely collapsed.
Considering certain allowance, the traffic of the bridge after explosion can be controlled according to different TA ranges as traffic prohibition, traffic restriction, and normal traffic operation, and the corresponding traffic rates are, respectively, specified as 0%, 50%, and 100% of the original one, as shown in Table 4.
4.2. Analysis of the Main Control Parameters
In the response surface model analysis, whether the calculation model can fit the response of the actual structure quickly and effectively depends on the selection of design parameters. If the sensitivity of the selected design parameters is low, the efficiency to predict structural damage will be reduced. Therefore, the main control parameters that affect the pier’s residual bearing capacity should be analyzed quantitatively before the experimental design.
When the pier is attacked by an explosion load, its response and damage are not only influenced by the explosion distance R but also by the variations of material properties and geometric dimensions of the structure itself. The main parameters studied in this paper include concrete compressive strength fc, longitudinal reinforcement ratio ρl, stirrup ratio ρs (proportion of stirrup volume in concrete volume), longitudinal reinforcement yield strength fy, and stirrup yield strength fys. According to related studies [23], each main parameter is assumed to be a random variable obeying a certain distribution, as shown in Table 5.
With reference to the probability density distributions of the parameters in Table 5, the single-variable method is used to analyze the explosion of the 1# car bomb (M = 1000 kg, H = 1.0 m) at a distance of 3.5 m from the pier, to obtain its residual bearing capacity and the initial bearing capacity under different analytical cases. Then, the relationship between the cases is analyzed by linear regression analysis, as shown in Table 6 and Figure 12, in which the slope k fitted for each parameter indicates the sensitivity of the residual bearing capacity of the pier to this parameter.
Table 6 also shows that the concrete compressive strength fc, stirrup ratio ρs, and longitudinal reinforcement ratio ρl have great influence on the residual bearing capacity of the pier within the random distribution range, while the longitudinal reinforcement yield strength fy and stirrup yield strength fys have little influence. Therefore, in the following calculation of the response surface model, only fc, ρs, and ρl are considered as random parameters, while fy and fys are set as fixed values (fy = 400 MPa and fys = 335 MPa).
4.3. Experimental Design
The key step in the assessment of bridge’s traffic capacity after near-field explosion is to design the experimental schemes considering the randomness of structural parameters. In this section, the central composite design (CCD) method which is the most commonly used experimental design method in the response surface method is used. This method is easy and simple in estimation of the first and second orders of the response surface numerical model, with accurate result and reasonable experiment amount [24]. The pier under the attack of each car bomb is experimented 25 times, of which the numbers of center point, axis point, and quadrant point experiments are 1 time, 8 times, and 16 times, respectively.
Referring to the random distribution characteristics of concrete compressive strength fc, stirrup ratio ρs, and longitudinal reinforcement ratio ρl in Table 5, the probability density function curves of the parameters are drawn in Figure 13. According to the probability density distributions, in the following analysis, the range of fc is set to 20 MPa∼40 MPa, ρs is set to 1.06%∼1.72%, and ρl is set to 1.42%∼2.34%. Using various combinations of these random parameters, the analytical cases of the pier under the attack of three types of car bombs based on the CCD method are shown in Table 7.
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4.4. Establishment of the Response Surface Model
The polynomial response surface model is commonly used in response surface analysis. The above-calculated results are fitted with a quadratic polynomial to form the residual bearing capacity response surface model of the pier. Its polynomial form iswhere fc is the concrete compressive strength, ρs is the stirrup ratio, ρl is the longitudinal reinforcement ratio, R is the explosion distance, and a1 to a15 are the partial coefficients to be determined.
According to the calculation results, a1 to a15 are obtained by multivariate nonlinear fitting, as shown in Table 8.
The fitting accuracy is usually expressed by the square of the correlation coefficient. The closer the value to 1 is, the better the fitting accuracy of the response surface model is. The correlation coefficients’ square values of the model under the attack of the three types of car bombs are 0.99, 0.997, and 0.977, respectively, which indicates a high fitting accuracy.
In order to compare the influence of stirrup ratio ρs and explosion distance R on the residual bearing capacity of the pier after the explosion, the response surface diagram under the 1# car bomb (fc = 30 MPa and ρl = 1.88%) is drawn in Figure 14. It can be found that, in the diagram, the response surface inclined toward smaller R and ds, indicating that the closer the car bomb to the pier and the smaller the stirrup ratio ds, the lower the residual bearing capacity of the pier and the more likely the structure will be damaged or collapsed.
According to the analysis of Figures 12 and 14, the influence level of the parameters on the residual bearing capacity of the pier is ranked as explosion distance R > stirrup ratio ρs > concrete compressive strength fc > longitudinal reinforcement ratio ρl > stirrup yield strength fys > longitudinal reinforcement yield strength fy.
4.5. Traffic Capacity Assessment of the Bridge after Explosion
With reference to the incremental dynamic analysis method in seismic vulnerability analysis [25], the range of the explosion distance R is set from 6 m to 1 m, and the calculation is performed at an interval of 0.02 m. The Monte Carlo method is used to sample the random variables fc, ρs, and ρl obeying the normal distribution, and the sample size is N = 1 × 107. Under a certain analytical case, the failure probability of the pier reaching the corresponding damage index can be expressed aswhere n1, n2, n3, and n4 are numbers of the samples, whose damage index D reaches slightly damaged, moderately damaged, severely damaged, and completely collapsed states at each explosion distance R.
Shown in Figure 15 are the failure probability curves of the pier under three types of car bombs.
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The traffic indicator value of each analytical case can be calculated according to equation (4), and the explosion distance-traffic indicator value curves are drawn in Figure 16.
It can be seen from Figure 16 and Table 4 that, for safe traffic of the bridge after the attack of the 1# car bomb (M = 1000 kg, H = 1.0 m), when the explosion distance R is less than 2.84 m, the traffic indicator TA is less than 0.5, so no vehicle passage is allowed; when R is greater than 2.84 m and less than 6 m, TA is between 0.75 and 0.5, so vehicle passage should be restricted with a 50% traffic rate. It is noted that when the 1# car bomb explodes at a distance of 2.84 m from the pier, TA is 0.509, which is very close to the limit value 0.5 of the traffic prohibition state. Therefore, the potential hazard distance of 1# car bomb is 2.84 m.
For the bridge after the attack of 2# car bomb (M = 500 kg, H = 0.8 m), when the explosion distance R is less than 1.24 m, the traffic indicator TA is less than 0.5, so the traffic on the bridge should be prohibited; when R is greater than 1.24 m and less than 3.06 m, TA is between 0.75 and 0.5, so 50% of the original traffic rate is allowed; when R is greater than 3.06 m, TA is bigger than 0.75, so there is no influence on bridge capacity, and original traffic rate is feasible. Therefore, the potential hazard distance of 2# car bomb is 1.24 m.
For the bridge after the attack of 3# car bomb (M = 200 kg, H = 0.5 m), when the explosion distance R is greater than 1 m, the traffic indicator TA remains greater than 0.75, so the bridge can be opened with the original traffic rate.
5. Conclusions
In this paper, an “explosive-air-pier” coupling model by the FE analysis method is established. By using the residual bearing capacity of the damaged pier as the assessment index, a method for assessing the traffic capacity of the urban elevated bridge after near-field explosion based on the response surface method is proposed. The main conclusions are as follows:(1)The damage index of the pier after near-field explosion is closely related to TNT equivalent mass and explosion distance: the greater the TNT equivalent mass of the bomb and the shorter the explosion distance are, the more seriously the pier is damaged. The pier’s damage is obvious under the minibus carrying medium bomb (500 kg TNT equivalent mass) and the cargo carrying big bomb (1000 kg TNT equivalent mass). When they explode at 6.0 m away from the pier, the pier is only slightly damaged. When the explosion distance is shortened to 1.2 m, the pier is, respectively, in moderately damaged and completely collapsed states, while for the normal car carrying small bomb (200 kg TNT equivalent mass), the explosion distance has little effect on the damage index of the pier after near-field explosion.(2)Through multiparameter analysis by the response surface model on the pier’s residual bearing capacity under the three types of car bombs, the influence levels of the parameters are ranked as explosion distance R > stirrup ratio ρs > concrete compressive strength fc > longitudinal reinforcement ratio ρl > stirrup yield strength fys > longitudinal reinforcement yield strength fy. It shows that appropriately increasing the stirrup ratio is more conducive to improving the antiexplosion ability of the pier, and increasing the protective distance between the car bombs and the pier will significantly reduce the risk of structural collapse.(3)For different types of car bombs, the explosion distance has a significant effect on the traffic capacity of the urban elevated bridge. For the minibus carrying 500 kg (TNT equivalent mass) bomb and the cargo carrying 1000 kg (TNT equivalent mass) bomb, the potential hazard distances that may cause severe damage or complete collapse of the structure are 1.24 m and 2.84 m, respectively. Therefore, reasonable protective measures should be taken for important piers to isolate vehicles carrying bombs beyond the potential hazard distances, in order to ensure the traffic capacity of urban elevated bridges and the safe operation of urban traffic. Once the car bomb explodes within the potential hazard distances, the traffic on the bridge should be effectively controlled.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This paper was supported by the Fundamental Research Funds for the Central Universities of China (2020JBM042) and Introducing Talents Base of Mitigating Wind-induced Disaster of Wind-sensitive Infrastructure (B13002).