Dynamic Response of a Box Multistage Stiffened Beam under the Coupling of Vehicle Load and Air Blast Load

Abstract

If a bridge is subjected to a blast load when there is vehicle traffic, not only its own safety is threatened, but it can also lead to damage to vehicles. In addition, the coupling of a vehicle load and an explosion load may further aggravate the impact of an explosion. To understand the coupling relationship between the two kinds of loads on a bridge, a static load was applied on the bridge using the impact coefficient while a blast load was applied on the outside of the bridge. A numerical simulation was also used to further study the coupling effect of the vehicle load and the explosion load. The results showed that the vehicle load could effectively limit the vertical deformation. The numerical model was accurate in predicting the response process of the stiffened beam. With the coupling of the vehicle load, the equivalent plastic strain of the box multistage stiffened beam was mainly concentrated at the hinge and decreased when the blast loading remained constant. The transverse anti-blast performance of the stiffened beam was mainly provided by the bridge web and the diaphragm under the coupling effect of the vehicle load and the blasting load, but the function of the diaphragm was weakened. Additionally, the hinge used as a connector was able to directly affect the bearing capacity of the bridge. Even if the hinge was only slightly damaged, it could cause the bridge to enter the failure stage, meaning that the strength of the hinge must be greater than that of the bridge.

1. Introduction

Terrorist attacks and natural disasters may cause fatal damage to transportation infrastructure, such as bridges and roads, leading to the paralysis of transportation networks. Box bridges can quickly provide emergency transportation channels to facilitate vehicle mobility and disaster relief under complex road conditions, including enabling personnel passage, vehicle transportation, material transfer, etc. [1,2,3]. At present, the commonly used movable bridges are mainly made of high-strength steel or aluminum alloy, which have the advantages of light weight, high strength, modularization, fast deployment, simple construction, and high reparability [4,5,6]. Most of these movable bridges are less than 30 m in length and are designed in the form of box multistage stiffened beams in order to facilitate rapid installation and transportation, and to avoid restricting the mobility of vehicles due to excessive bridge weight.

In response to the needs of emergency rescue operations, the demand for such box multistage stiffened beams has continued to rise. In recent years, there have been many studies on the maneuverability of such beams in complex environments [7,8,9]. In particular, with the development of computer technology, more and more scholars have begun to use the finite element method to simulate the structural response of bridges, and numerical simulations have also been shown to be effective in predicting structural responses [10,11,12,13]. In battlefield environments, the threat of explosions to stiffened beam structures requires particular attention. Once the structure is damaged, the bearing capacity and service life of the stiffened beam will be reduced. In the analysis of the anti-blast performance of bridges, the coupling effect of a vehicle load is seldom studied, and the effect of a single blast load is still the main focus of research at present [14,15,16]. Chen et al. [17] conducted experimental and numerical simulation studies on the performance of welded steel column joints in bridges. They analyzed the influence of local welding forms on the explosion resistance of connection joints from two aspects—the cross-sectional internal force and the energy absorption—and established a simplified relationship between the plastic rotation angle and the impact energy. Yao et al. [18] used a multi-material fluid–solid coupling algorithm to conduct a numerical simulation analysis on the dynamic mechanical behavior of local multistage stiffened box beam structures during internal explosions. The results indicated that the failure mode of the box beam structure mainly comprised three stages: plastic large deformation, local damage, and load-bearing failure, and the structure was prone to local tearing failure at the component connections. Therefore, it was determined that it is necessary to strengthen the connection between the panel, the reinforcing ribs, and the crossbeam to improve the overall structure. Geng et al. [19] used the principle of similarity to conduct experimental research on the anti-blast performance of a global scaled model of a one-box multistage stiffened bridge and discussed the impact of near-field explosion shock waves on the damage characteristics of different areas of the structure. The results indicated that U-shaped ribs can effectively support the top plate of the structure, concentrating the failure area in the non-stiffened area, and that the deformation of the top plate is mainly caused by local depressions or local damage. When the shock-wave propagates in the structure, opening holes in the diaphragm facilitates the propagation of the wave and prevents multiple reflections of the wave inside the box. Also, the diaphragm can reduce the impact effect of the shock waves on the structural bottom plate, causing the deformation of the bottom plate to be mainly localized uplift. Hashemi et al. [20] and Atakhani et al. [21] used the software LS-DYNA R11.0 to study the local and global dynamic responses of a full-scale model of a cable-stayed bridge under a blast load. The results showed that the damage to the deck for all the blast scenarios was mostly associated with the direct rupture of the steel plates subjected to high-magnitude reflected blast pressures, and that the global response of the pylons was sensitive to the location of the explosion along the deck. The bridge did not experience progressive collapse when subjected to different blast loads resulting from small to large explosions, but structural components, such as the deck and pylons in the vicinity of the explosion, would require significant repairs and rehabilitation. While studying the coupling effect of a vehicle load, Greco et al. [22] proposed a combined model based on a moving mesh methodology and interface modeling to predict the structural behavior in the presence of moving loads, which proved to be effective in predicting the interaction between a vehicle load and the bridge. However, there is relatively little research on the explosion resistance performance of bridges under working conditions, and the impact of explosion loads on bridges is not considered in bridge codes or even in military codes in various countries. Explosion loads are only classified as accidental loads, and there are no specific calculation provisions for structure design [23,24]. Therefore, it is necessary to conduct research on the anti-blast performance of bridge structures subject to loading conditions, which can improve the structural performance and ensure the mobility of vehicles.

The threat of a blast load on a bridge when vehicles pass through cannot be ignored and the loss caused by an explosion can be huge, but there is little research on this issue at present. Therefore, it is very necessary to study the dynamic response of a bridge under the coupling effect of a vehicle load and a blast load. In this paper, a box multistage stiffened beam is used as a scaled-down prototype, and the dynamic behavior of a stiffened beam under the coupling effect of a vehicle load and a blasting load is experimentally studied. The finite element software named Abaqus 2020 is also used to simulate the dynamic response of the stiffened beam. By studying the coupling effect of a vehicle load and a blast load, this can provide a reference for improving the service life and engineering applications of the bridge in complex explosive environments, and help formulate optimization strategies for anti-blast performance and improve the protection performance of the bridge.

2. Model Design

2.1. Scaled Beam Model

In this study, a multistage stiffened bridge is taken as a prototype and a global proportional scaling ratio is chosen based on the experimental requirements and the explosive load characteristics. The structure of the box beam, which has a variable cross-section, is composed of two connected symmetrical half-bridges (named B1 and B2). Folding or expansion between the two halves of the bridge can be achieved through the bottom hinge in the mid-span of the bridge. The structural diagram of the half-bridge B2 is shown in Figure 1.

The beam models in the bridge have a total length of 6.0 m and a span of 5.4 m. Each half-bridge is 3.0 m long along the bridge length (longitudinal) direction, with a maximum width of 0.287 m along the bridge width (transverse) direction, and a maximum height of 0.215 mm along the bridge height (vertical) direction. The bridge deck is located on the flange plate of the bridge web, with a thickness of 0.8 mm. The uniformly distributed circular holes on the bridge deck have a diameter of 10 mm, which can reduce the impact area and the reflection effect of the blast shock wave. These holes can also increase the surface friction of the bridge deck, which is beneficial for the passage of people and vehicles in complex weather conditions. The number of longitudinal stiffeners is three, with a thickness of 1 mm, evenly distributed on the bridge deck and penetrating all the π-shaped diaphragms. The bridge web is an H-shaped structure with a spacing of 250 mm on both sides and a thickness of 2 mm. The bridge web contains three inclination angles from the end of the bridge to the middle of the bridge, which are 46.4°, 17.3°, and 2.4°, respectively. This multi-angle inclination design is conducive to the rapid passage of motor vehicles. The half-bridge B2 consists of 13 diaphragms, numbered from I0 to I12, with a distance of 225 mm between each partition and a plate thickness of 1.0 mm. The diaphragms named I0–I3 and I12 are located at the head and tail ends of the half-bridge, respectively, using traditional rectangular diaphragms with holes. The diaphragms named I4–I11 adopt a π-shaped design. Flange plates are added to the π-shaped diaphragm, which can improve the structural strength. Furthermore, in order to improve the strength of the bridge, a staggered distribution method is adopted to set up the π-shaped diaphragms, namely, I4, I6, I8, and I10 are strong π-shaped diaphragms. The π-shaped diaphragm on the bridge deck can effectively ensure the continuity of the internal space of the multistage stiffened beam, avoiding shock-wave pressurization inside. Moreover, the connection area between the diaphragm and the bridge deck is greater than the connection area between the diaphragm and the bridge web, which can reduce the mass of the partition while providing sufficient support for the bridge deck.

2.2. Model Processing

The multi-stage stiffened beam structure consisted of numerous components, and each component needed to be processed before preparation. However, these components were small and thin, making it difficult to effectively control the accuracy using traditional machining methods. Therefore, laser cutting technology was used to process the sheet metal. After completing the cutting of the plate, different components needed to be processed and then assembled into target specimens. Due to the complex structural form of the target specimen, which was a multi-fold spatial structure, it was difficult to prepare using laser welding technology. Therefore, in order to achieve welding strength, an NBC-350 semi-automatic CO₂ shielded welding machine was selected for manual welding. In Step 1, the components were processed, and the sheet units were assembled into component units, before fixing them by welding, as shown in Figure 2a. In Step 2, the main components were installed, as shown in Figure 2b. The workpiece was first placed on the processing platform, and the diaphragm positioned at the lower flange of the bridge web. After determining the position, a jig was used to clamp the component and fix it using spot welding. Secondly, the jig was removed, and longitudinal stiffeners were inserted into the diaphragm before welding. Thirdly, the bridge web was installed, the position of the weld seam was confirmed, and was then fixed with a fixture. Finally, the instrument parameters were adjusted, the interval dislocation method was used to weld the bridge web and diaphragm symmetrically, and the blowing cooling method was used to control the shrinkage deformation appropriately. In Step 3, the welding of the half-bridge was completed, the bridge deck was fixed with spot welding, and the connection areas of each component were welded using the interval dislocation method. Similarly, a blowing cooling method was used to control shrinkage deformation, as shown in Figure 2c. After passing the inspection, the movable π-shaped multistage stiffened beam structure was subjected to tail cleaning and rust prevention operations.

2.3. Material Mechanical Properties

In this study, mild steel with good ductility was used for processing and preparing the box stiffened beam. In order to obtain the mechanical characteristic parameters of the material, tensile specimens were designed according to the requirements of “Metallic materials—Tensile testing—Part 1: Method of test at room temperature” [25], and quasi-static tensile tests were conducted at room temperature. The sampling location and processing accuracy also complied with the requirements of “Steel and steel products—Location and preparation of samples and test pieces for mechanical testing” [26]. The size of the specimen is shown in Figure 3. A = 80 mm is the length of the parallel section of the tensile specimen, W = 20 mm is the width of the parallel section of the specimen, R = 25 mm is the transition radius of the specimen, B = 48 mm is the clamping length of the specimen, C = 45 mm is the width of the specimen, L = 220 mm is the total length of the specimen, and T = 6 mm is the thickness of the specimen. Three sets of quasi-static material tensile tests were conducted to ensure the reliability of the test results.

In order to accurately measure the strain changes of the specimen during the stretching process, a microcomputer-controlled universal testing machine was used to perform the stretching test, while an extensometer and a VIC-3D non-contact measurement system were used to capture the tensile strain of the specimen. The VIC-3D system consisted of two high-speed cameras and digital analysis software, mainly capturing the deformation of relevant points through DIC (digital image correlation) digital image analysis technology, enabling calculation of the surface displacement and strain distribution of the object [27,28]. Before using this system, it was necessary to first perform spot processing on the surface of the tensile specimen, and then to capture real-time image changes of the spot during the test process. After computation with the software, the full field strain data distribution of the structure could be obtained. Compared with conventional strain measurement methods, the VIC-3D system is simpler to operate, and the calculated strain data are more accurate.

During the experiment, the loading speed was taken to be 1.0 mm/min. The displacement (strain) during the stretching process was recorded using an extensometer and a VIC-3D non-contact full field strain measurement system. The load (stress) during the stretching process was recorded using a universal testing machine. The strain changes on the test equipment and the specimen surface are shown in Figure 4. The experimental data were processed to obtain the material mechanical performance parameters and engineering stress–strain relationship curves, as shown in

Table 1. Mechanical property parameters of the material.

Parameters	Unit	Value
Density ρ	kg/m3	7850
Elastic modulus E	GPa	207.7
Yield strength σs	MPa	301.7
Yield strain ε0/10−3	-	4.63
Tensile strength σb	MPa	453.1
Fracture strain εf	-	0.33

and Figure 5, respectively.

Metal materials, especially low-carbon steels, are very sensitive to the strain rates in plastic flow. Explosion is a high-speed loading problem. Under explosive shock-wave loading, the metal structure quickly exceeds its elasticity and enters the plastic flow stage. Therefore, the influence of strain rates cannot be ignored. In the simulation calculation of metal thin-walled structures, the Cowper–Symonds (CS) constitutive model [29] is usually used to consider the problem of the material strain rate. In the CS constitutive model, the plastic flow stress of low-carbon steel can be expressed as,

$${\sigma }_{pl}=f({\epsilon }_{pl})\cdot R({\dot{\epsilon }}_{pl}){\epsilon }_{fd}/{\epsilon }_u$$

(1)

where $f({\epsilon }_{pl})$ is the stress–strain behavior of the material under quasi-static conditions, ${\dot{\epsilon }}_{pl}$ is the plastic strain rate, and $R({\dot{\epsilon }}_{pl})$ is the ratio of the dynamic stress to the static stress at any strain rate,

$${\dot{\epsilon }}_{pl}=D{[R({\dot{\epsilon }}_{pl})-1]}^p$$

(2)

where D is the reference strain rate, and p is a constant, both of which are related to the metal material category. For mild steel, Jones [29] gave a recommended value of D = 40.4 s⁻¹ and p = 5 after extensive testing.

Combining Equations (1) and (2), the plastic flow stress of the material in the CS model can be expressed as,

$${\sigma }_{pl}=f({\epsilon }_{pl})\cdot \left[1+{\left({\dot{\epsilon }}_{pl}/D\right)}^{1/p}\right]$$

(3)

3. Explosion and Numerical Results

A stiffened beam bridge is mainly used for pothole roads, and the clearance under the bridge is generally high. To achieve a real application scenario of the bridge, the clearance under the specimen was taken as 1 m in the experiment. However, the cement base was mainly used to provide rigid support, which could be replaced with a simplified boundary to reduce the modeling steps and improve the computational efficiency in the numerical simulation.

3.1. Test Scheme

When vehicles pass over bridges, the dynamic impact effect of the vehicle should be considered in addition to their own weight. Therefore, the impact coefficient α is usually used to replace the vehicle load with the equivalent static load in order to facilitate calculation; its expression is as follows [30],

$$\alpha =1+\frac{1}{2}\frac{15}{37.5+L}$$

(4)

where L represents the span of the bridge. Given that the span of the scaled model of the box beam is L = 5.4 m, the impact coefficient is α = 1.175. The original model of the box multistage stiffened beam mainly supports a vehicle load of 150 kN~500 kN. Scaled to 1:4 and combined with the impact coefficient, α, it can be calculated that the equivalent static load required for the scaled model of the box beam is 2.75 kN~9.18 kN. Hence, a load of 3 kN is regarded as the minimum vehicle load, and a load of 9 kN is regarded as the maximum vehicle load required for the safe operation of the bridge in the subsequent analysis.

Before the test, several cement blocks were designed as loading parts. The size of a single cement block was 1.0 m × 0.25 m × 0.14 m, and the weight was 75 kg. The test conditions, such as the explosive charge (m), the blasting distance^®, the scaled distance (Z), the vehicle load (F), and the field layout are shown in

Table 2. Blast test design of the box beam.

Tests	m/kg	R/m	Z/(m/kg1/3)	F/kg	Loading Conditions
T1	0.1	1.0	2.154	0	No load
T2	0.1	1.0	2.154	150	Unilateral load (B1)
T3	0.1	1.0	2.154	300	Unilateral load (B1)
T4	0.1	1.5	2.154	300	Bilateral load (B1/B2)

and Figure 6a, where the T3 and T4 conditions were used to simulate the effect of the minimum vehicle load. During loading, the cement blocks were placed in the central area of the structure, and only 300 kg was loaded at B1 in T3, while 150 kg was loaded at each of the two halves of the bridge in T4. During the test, the explosives were placed on the outside of the middle of the bridge, while the box beams were erected on the cement supports. Expansion bolts were installed to simulate point constraints, and strain gauges S1–S4 were attached to typical areas to characterize the structural response. The layout of the measurement points is shown in Figure 6b, where the measurement points S1, S2, and S4 were arranged longitudinally, while the measurement point S3 was arranged vertically.

This test was conducted at a field test site in Jiangsu Province, using the DH8302 dynamic signal acquisition instrument produced by Jiangsu Donghua Test Technology Co., LTD., Jingjiang, China to record the test data. The sampling frequency was set to 100 kHz, and a grounding connection was made to reduce external signal interference. The type of explosive used in the experiment was trinitrotoluene (TNT) explosive, with a measured density of 1630 kg/m³, a detonation heat of 4150 kJ/kg, and a detonation pressure of 19 GPa. A free field pressure sensor was installed at a distance of 3.0 m from the explosion center to check the effectiveness of the TNT explosion.

3.2. Explosion Results

The free field pressure, as measured after the test by the sensor, is shown in Figure 7. In order to facilitate the subsequent simulation calculation, the calculation curve in the classical CONWEP method was compared with the test measurement data. The CONWEP method improves the accuracy of the calculation results by fitting a large quantity of explosion test data to the US military specification UFC3-340-02 [31], and it can predict the incident pressure and reflection pressure on the structural surface. This method has been proven to be effective in evaluating the incident pressure of explosion shock-waves in free field air [31].

It can be seen from Figure 7 that the variation trend in the incident pressure amplitude curves measured by the four groups of explosion tests was basically the same, and the period of positive pressure almost coincided with the CONWEP calculation curve. The peak incident pressure values measured in the four groups were 22.2 kPa, 21.6 kPa, 21.3 kPa, and 20.6 kPa, respectively. The peak value calculated by CONWEP was 21.0 kPa, and the difference between the experimental average value and the calculated value was only 2.02%. This indicates that the experimental data were highly consistent with the CONWEP calculation results, and the data were true and effective.

It can be seen from Figure 5 that the elastic limit strain of the mild steel used in the construction of the box beam was 1452.6 με. When the structural deformation is small, it can be characterized by strain. The strain–time curve was obtained after filtering the strain data measured in four working conditions, and the maximum positive strain was recorded as the total strain, as shown in Figure 8 and

Table 3. Measuring total strain of the box beam in the tests.

Working Conditions	Total Strain/με
	S1	S2	S3	S4
T1	91.2	750.4	176.9	389.9
T2	120.6	646.6	211.3	220.2
T3	110.5	790.2	172.2	316.1
T4	119.2	840.4	141.2	289.1

. It can be seen that the strain value of each measuring point in the four working conditions did not exceed the elastic limit strain, but that the strain fluctuation was large, which may have been related to the high vibration frequency of the structure. Under the coupling effect of the vehicle load and the explosion load, the strain fluctuation on the bridge deck increased, and when the box beam was under unilateral load, the strain fluctuation increased with increase in the vehicle load. Compared with the working conditions T2 and T4, it can be seen that although the external load increased when the box beam was under a bilateral load, the change in strain fluctuation was not significant, indicating that the impact of the bilateral load on the hinged box beam bridge deck was relatively small.

For the bridge web, the strain fluctuation at the measuring point S2 was the most obvious, and the strain fluctuation at the measuring point S3 was the weakest, which indicates that when the explosive exploded at the side of the box beam, the bridge web deformation was mainly transverse and longitudinal. Comparing the working conditions T1 and T2, it can be seen that when the box beam was subjected to a unilateral load, the total strain at the measuring points S2 and S4 decreased, while at the measuring point S3 it increased. Comparing the working conditions T2 and T4, it can be seen that when the unilateral load on the box girder half-bridge B1 remained unchanged and the same load was applied to the half-bridge B2, the total strain at the measuring points S2 and S4 increased, while the total strain at the measuring point S3 decreased. This indicates that the vertical deformation of the bridge web can be suppressed by a vehicle load to a certain extent, and the longitudinal deformation of the bridge web can also be suppressed when the vehicle load is small. Compared with a unilateral load, a bilateral load had more influence on the bridge web of the box stiffened beam. In addition, in the unilateral load (B1) mode, the total strain increase rates at points S1 to S4 from T1 to T2 were 32.24%, −13.83%, 19.45% and −43.52% respectively, and the total strain increase rates from T2 to T3 were −8.37%, 22.21%, −18.50% and 43.55%, indicating that the elastic strain on the bridge deck and the web was very sensitive to the change in load under the unilateral load mode. In the bilateral load (B1/B2) mode, the total strain increase rates at points S1 to S4 from T3 to T4 were 7.87%, 6.35%, −18.00% and −8.54%, respectively, showing that the load mode mainly affected the strain along the height of the bridge, but had little effect on the strain along the length of the bridge.

3.3. Numerical Results

3.3.1. Numerical Model

Due to the high cost of explosion tests and the difficulty in capturing the dynamic response process of a structure under explosive loads, it was necessary to conduct further research using numerical simulation methods. CONWEP, as a simulation method that can convert complex explosive environments into simple calculations, has shown good consistency with experimental results when studying the damage characteristics of large-sized frame structures [32]. This method mainly considers the influence of the incident angle, the incident pressure, and the reflection pressure of the explosion shock wave. The pressure P(t) applied to the structure can be expressed as,

$$P(t)=P_{ref}\times {\cos }^2\theta +P_{in}\times (1+{\cos }^2\theta -2\cos \theta )$$

(5)

where P_ref is the reflected pressure, P_in is the incident pressure, and θ is the angle between the line from the detonation center to the loading point and the normal of the loading point.

Therefore, the CONWEP plug-in in Abaqus was selected to simulate the response characteristics of the box multistage stiffened bridge under explosive loads. The CONWEP plug-in relies on changing the explosive charge, the blasting distance, and the blasting position to adjust the explosion effect. The finite element model is shown in Figure 9. The shell element was used to model the multistage stiffened beam. Based on the symmetry principle, the half-bridge B2 was used for simulation calculation, and the calculation time was 0.1 s. Two patch plates with a length of 130 mm, a width of 2.5 mm, and a thickness of 2 mm were used to connect the half-bridge B1 and the half-bridge B2, and a rigid plate was placed at the connection to simulate the interaction between the two half-bridges. This rigid plate did not serve as the contact surface for shock-wave incidence [33]. Due to the large number of nodes in this model, all the components in the model were connected by common nodes, and the advanced algorithm Advancing Front was selected for meshing. The element type was S4R, a four-node reducing element, and the maximum mesh size was 5 mm. After meshing, the total number of nodes in the B2 model of the half-bridge was 140,771, and the total number of elements was 142,768. In the test, the cement supports were mainly used to provide rigid constraints, so, in the numerical model, the modeling of the supports could be omitted and replaced by simplified boundaries. Additionally, to simulate the influence of the pressure exerted by the explosion over the surface of the loading blocks, the equivalent static loads were added directly to the dynamic/explicit module.

To simulate the response characteristics of the box multistage stiffened beam under explosive loads, especially the failure mode of the structure, a ductile damage failure model was selected to define the failure mode of the structure, which can be expressed by,

$${\epsilon }_f/{\epsilon }_u=\{\begin{array}{l}1.13 T^{\sigma }\le -1/3\\ 0.04+0.86\exp (-0.7T^{\sigma }) -1/3<T^{\sigma }\le 1/3\end{array}$$

(6)

where ${\epsilon }_u$ is the ultimate strain of mild steel, ${\epsilon }_f$ is the initial fracture plastic strain for ductile failure, and $T^{\sigma }$ is the stress triaxiality. In the ductile damage failure model, the damage requires to be assessed using the relationship between the fracture strain and the stress triaxiality. For the mild steel, the ultimate plastic failure strain, ${\epsilon }_u$, in the numerical simulation was set as 0.29 and the failure strain was set as ${\epsilon }_f$ = 0.21 [33].

3.3.2. Critical Scaled Distance

In order to analyze the impact of the vehicle load on the dynamic shock response of the box beam model, the condition of no load was analyzed first, and the critical proportional blasting distance of the model was calculated when the detonation center was located on the outside of the bridge. Since the proportional blasting distance is influenced by both the charge amount and the blasting distance, the random combination of different blasting positions, explosive charge amounts, and blasting distances can seriously restrict the evaluation of the anti-blast performance of bridges for the same proportional blasting distance. Therefore, in the simulation analysis, the detonation core was placed 1.0 m outside the middle of the bridge, and the different charge amounts were selected using the CONWEP method for calculation to obtain the equivalent plastic strain of the box beam model, as shown in Figure 10. As can be seen from the figure, with increase in the TNT charge, the equivalent plastic strain of the box beam model also increased, and when the charge was 500 g, the plastic strain was 0.22, which is basically equivalent to ε_f, indicating that when the blasting distance is 1 m, the critical charge of the box beam in the plastic state is 500 g, and the critical proportional blasting distance is Z₀ = 1.26 m/kg^1/3.

The lateral deformation of the structure (reference point D1) with different charge amounts was analyzed to understand the model behavior in the plastic state, as shown in Figure 11. It can be seen that when the explosion center was located on the outer side of the middle of bridge, the lateral deformation trend of the box beam was basically consistent, reaching the maximum deformation in a short period of time, and then exhibiting approximately periodic vibration. Moreover, the larger the charge amount, the greater the maximum lateral deformation of the structure. When the charging amount increased from 200 g to 500 g, the maximum lateral deformation increase rates of the structure were 32.77%, 22.98%, and 16.61%, respectively, indicating that the influence of the charging amount on the deformation was gradually reduced before plastic failure occurred in the model. Unlike the maximum deformation, the time required for the model to reach the maximum deformation was basically the same under the different charge amounts, indicating that the displacement vibration period of the box beam model before failure was not affected by the explosion shock wave.

4. Discussion of Vehicle Load

Under a blast load, the scaled distance can be used as an index to measure the power of an explosion and to evaluate the safety performance of the bridge by defining the critical scaled distance. Hence, it is necessary to discuss the effect of the vehicle load on the critical scaled distance. According to the simulation results, it can be concluded that when the charge is 500 g and the blasting distance is 1.0 m, the box beam model will enter a plastic failure state under no load conditions. In order to determine the impact of vehicle loads on the failure state of the model, vehicle loads were applied simultaneously on the two half-bridges of the model, with loads of 9 kN (the maximum load required), 12 kN, 13 kN, and 14 kN, respectively, maintaining the same charge amount and blasting distance. The equivalent plastic strain variation of the model under the vehicle loads and explosion loads is shown in Figure 12. In order to facilitate observation, the region where the plastic strain distribution concentrates was amplified, and the results show that these regions were mainly distributed at the connection nodes of the components.

Comparing Figure 10 and Figure 12, it can be seen that when the box beam is subjected to a vehicle load, the equivalent plastic strain of the structure changes significantly, and it is not until the vehicle load increases to 13 kN that the plastic strain is equivalent to the failure strain, indicating that the vehicle load can improve the anti-blast performance of the box beam under explosive loads. In addition, when the vehicle load is small, the plastic strain is mainly concentrated on the diaphragm, and when the vehicle load is large, the plastic strain is mainly concentrated on the hinge joint inside the model. The reason is that, on the one hand, the diaphragm is connected to both the bridge deck and the bridge web, and when the explosives explode on the outer side of the bridge, the bridge web is subjected to lateral impact forces. Vehicle loads cause the bridge deck to be subjected to external forces along the vertical direction. Both lateral and vertical external forces will act on the diaphragm, causing plastic deformation to occur more easily in the local areas of the diaphragm. On the other hand, when the vehicle load is applied to the bridge deck, the model will be deformed along the vertical, resulting in the longitudinal tensile stress on the bottom of the bridge web. When the model deforms laterally under the action of external shock-waves, the bridge web is also subjected to longitudinal tensile stress, which is mainly concentrated at the hinge, making the hinge prone to plastic deformation. Moreover, compared to the diaphragm, the risk of plastic failure of the hinge increases with increase in the vehicle load.

In order to determine the impact of the vehicle load on the dynamic response of the model, the lateral deformation of the model was analyzed, as shown in Figure 13. It can be seen that when the model is subjected to vehicle load, the maximum lateral deformation of the model changes. When the model re-enters the failure state, the maximum deformation is still less than the deformation value in the unloaded state. Moreover, the deformation vibration period of the model also increases with the loading of the vehicle, and the deformation vibration period between different vehicle loads is also basically equal. This further indicates that the vehicle load can improve the lateral anti-blast performance of the structure to a certain extent.

In order to further understand the explosion resistance characteristics between different components of the box beam under a vehicle load, the energy absorption characteristics of the model and each component in a test with a charge of 500 g and a blasting distance of 1.0 m were analyzed, as shown in Figure 14. It can be seen that for the whole structure, when the model is loaded with a 9 kN vehicle load, its energy absorption effect is weaker than that of the unloaded state, and with increase in the vehicle load, the structural energy absorption increases, indicating that the vehicle load can improve the stiffness of the structure to a certain extent, and the overall deformation of the structure can be effectively inhibited when the value is small. For the bridge deck and the bridge web, under the vehicle loads, the energy absorption effect both significantly increase, but the energy absorption stability time also prolongs. However, the energy absorption effect of the diaphragm is significantly weakened, and the energy absorption stability time is basically consistent with the unloaded state. This indicates that when the box beam is subjected to vehicle loads, the bridge deck and the bridge web as load-bearing components are the most significantly affected. As a multistage stiffened component connecting the bridge deck and the bridge web, the diaphragm can effectively suppress the coupling effect of vehicle loads and explosion loads, improve the strength and stiffness of the box beam, and, thus, improve the overall anti-blast performance of the structure.

For the energy-absorbing components, the specific energy absorption (energy absorption per unit mass) is considered to be an important indicator to measure the energy absorption characteristics of the structure, as shown in

Table 4. Specific energy absorption of the box beam and the main components.

Type	Mass/kg	Specific Energy Absorption/J·kg−1
		No Load	F = 9 kN	F = 12 kN	F = 13 kN	F = 14 kN
Whole model	43.76	6.41	5.41	6.98	7.80	8.81
Bridge deck	10.35	0.36	1.19	1.61	1.80	2.04
Bridge web	18.69	3.06	6.95	9.67	10.93	12.38
Diaphragm	3.82	30.15	17.08	18.46	19.22	20.19

. It can be seen that the specific energy absorption of the diaphragm is the largest in the loaded state, but the efficiency decreases significantly with a reduction rate of 43.35% under the impact of a vehicle load (from no load to the load of 9 kN). Using the load of 9 kN as a reference, when the vertical vehicle load increases by 33.33%, 44.44%, and 55.56%, the specific energy absorption rate of the bridge deck increases by 35.29%, 51.26%, and 71.43%, respectively; the specific energy absorption rate of the bridge web increases by 39.14%, 57.27%, and 78.13%, respectively; and the specific energy absorption rate of the diaphragm increases by 8.08%, 12.53%, and 18.21%, respectively. The results show that the increase rate of the specific energy absorption of the bridge deck and the bridge web is greater than that of the vehicle load, while the increase rate of the specific energy absorption of the diaphragm is smaller. The average increase rate of the two components compared with the load is 8.22% and 13.73%, respectively, and the average reduction rate of the specific energy absorption of the diaphragm is 31.51%. Hence, under the coupling effect of the blasting load and the vehicle load, the specific energy absorption efficiency of the bridge deck and the bridge web will be increased by approximately 8% and 14%, while the specific energy absorption efficiency of the diaphragm will be reduced by approximately 32%.

5. Conclusions

Based on explosion tests and numerical simulation, this paper investigated the dynamic response characteristics of a box beam under the coupling effect of a lateral explosion and a vertical vehicle load, and further analyzed the effect of a vehicle load on the failure mode of the box beam. This research can provide a reference for the subsequent analysis of the anti-blast performance of a bridge under vehicle traffic conditions and facilitate understanding of the dynamic response characteristics of bridges under the coupling of a vehicle load and a blast load. The main conclusions are as follows:

(a)

When subjected to lateral explosive loads, the bridge deck is more susceptible under the unilateral load of a vehicle load, while the bridge web is more susceptible in the bilateral load state. Under a vertical vehicle load, the top of the box girder bridge web is subjected to longitudinal in-plane compressive stress, and the bottom is subjected to longitudinal in-plane tensile stress. The coupling of the in-plane stress and the transverse shock-waves can improve the anti-blast performance of box beam structures to a certain extent.

(b)

As a multistage stiffened component, the diaphragm can reduce the separation between the bridge deck and the bridge web, thereby improving the overall strength of the bridge structure. It is the main energy-absorbing component of the box beam under load conditions. Under the vehicle load, plastic deformation is more likely to occur in the local areas of the diaphragm and the hinge. With the coupling effect of the blasting load and the vehicle load, the specific energy absorption efficiency of the bridge deck and the bridge web was increased by approximately 8% and 14%, while the specific energy absorption efficiency of the diaphragm was reduced by approximately 32%.