Response analysis of a pagoda in China under blast loading

Response analysis of a pagoda in China under blast loading | Research Square window.SnipcartSettings = { analytics: { enabled: false } }; (function() { var accessVector = localStorage.getItem('access_vector') || ''; window.dataLayer = window.dataLayer || []; if (accessVector) { window.dataLayer.push({ user: { profile: { profileInfo: { snid: accessVector } } } }); } })(); (function(w,d,s,l,i){w[l]=w[l]||[];w[l].push({'gtm.start':new Date().getTime(),event:'gtm.js'});var f=d.getElementsByTagName(s)[0],j=d.createElement(s),dl=l!='dataLayer'?'&l='+l:'';j.async=true;j.src='https://www.googletagmanager.com/gtm.js?id='+i+dl;f.parentNode.insertBefore(j,f);})(window,document,'script','dataLayer','GTM-K279D39R'); Browse Preprints In Review Journals COVID-19 Preprints AJE Video Bytes Research Tools Research Promotion AJE Professional Editing AJE Rubriq About Preprint Platform In Review Editorial Policies Our Team Advisory Board Help Center Sign In Submit a Preprint Cite Share Download PDF Research Article Response analysis of a pagoda in China under blast loading Guangxing Zhao, Dewen Liu, Wei Wang, Qing He, Yongbing Sun, Yang Liu This is a preprint; it has not been peer reviewed by a journal. https://doi.org/ 10.21203/rs.3.rs-5111761/v1 This work is licensed under a CC BY 4.0 License Status: Posted Version 1 posted You are reading this latest preprint version Abstract Based on a nine-story ancient pagoda with important historical and cultural value in Guangxi, China, this study simulated the collapse process of the ancient pagoda under different explosion points by using the fluid-structure coupling method, which can better explore the explosion-proof performance of the ancient pagoda. By constructing the finite element model of 5 explosion points, it is found that the area closest to the explosion point is the most seriously damaged, and the damage of the front explosion surface is obviously higher than that of the back explosion surface. In addition, the study shows that the response of the pagoda body is more severe and the damage is more serious when the number of blasting layers is 3, 5 and 7. In order to optimize the explosion-proof design, it is suggested to strengthen the structure of the surface of the explosion facing, improve the support system and masonry connection mode, and pay attention to the safety margin in the middle area of the pagoda body, so as to provide an effective scheme for the protection technology of the ancient pagoda. Ancient masonry pagoda Continuous collapse Explosion load Fluid-structure coupling Damage analysis Figures Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7 Figure 8 Figure 9 Figure 10 Figure 11 0 Introduction As an important part of the World Cultural Heritage, ancient pagodas not only carry rich historical and cultural values, but also the crystallization of architectural art and engineering technology [1]. However, in the current environment of rapid technological development, the ancient pagoda, as a symbol of a time-honored civilization, is facing unprecedented challenges. In addition to natural disasters such as earthquakes, mudslides and tsunamis, ancient pagodas may also be deliberately damaged by human factors, such as terrorist attacks, pollution and erosion, among which the use of drones to carry out precision bombing attacks on ancient pagodas is one of the potential dangers threatening the structural safety of ancient pagodas [2-4]. This requires us to adopt stricter protection measures and strengthen monitorings and maintenance of the ancient pagodas. Because the ancient pagoda is tall and constructed with dense masonry materials, it is highly sensitive to vibration, so it is important to evaluate its vibration resistance. Especially in areas of high seismic risk and border areas of neighboring countries, ancient pagodas are exposed to different forms of danger at all times. Therefore, the study of the dynamic characteristics of ancient pagodas and the formulation of effective reinforcement and seismic measures are the key to protecting these cultural heritage. Lu Junlong et al. [5,6] analyzed the dynamic performance and structural damage of Xingjiao Temple Pagoda through ultra-low frequency dynamic testing, and found that the first two frequencies of the pagoda were similar, the first mode bending and the second bending shear, and the equivalent elastic modulus decreased significantly, indicating that the ancient pagoda structure was seriously damaged. Xu Dunfeng et al.[7] based on the scaled model and shaking table test of Xi'an Xiaoyan Pagoda as the prototype, combined with finite element analysis, revealed the failure modes and dynamic characteristics of the brick ancient pagoda under different earthquakes, summarized the main damage characteristics of the ancient pagoda, and proposed refined earthquake damage assessment criteria, which provided scientific basis for the seismic design and protection measures of the ancient pagoda. Lv Yifan et al. [8] found that the blasting vibration response of Xiahe Pagoda was positively correlated with the explosion volume and insensitive to high-frequency vibration. The finite element analysis showed that the underlying compressive strength was insufficient but there was no crushing damage, and the impact of environmental vibration was more significant. Lee et al. [9] studied the natural frequency of the measured stone pagoda and compared it with the finite element model, and found that there was an average difference of 13% between the measured data and the peak ground acceleration (PGA) of the finite element analysis. This showed that the finite element method can effectively predict the dynamic characteristics of such structures. De Iasio et al. [10] established a detailed model of the Dragon Tiger Pagoda in China using ABAQUS, verified the accuracy of the topping analysis results through nonlinear dynamic analysis, and provided useful information about the damage and crack expansion of the pagoda after the earthquake. In addition, they used the dynamics theorem of limit analysis to study the collapse mechanism of the pagoda and its velocity, and explore the ability to predict earthquake damage. Shakya et al. [11] carried out an earthquake vulnerability assessment on 78 pagodas and temples with historical and heritage value in Kathmandu Valley, Nepal, and adopted the simplified method and vulnerability index for analysis. The results of the assessment and its database were integrated into the GIS tool, enabling spatial visualization of the damage scenario, which helped in the retrofit planning of the seismic risk mitigation. Lu J et al. [12] took Xuanzang Pagoda as an example, tested the 1/8 scale model by the shaking table, and compared the results with the finite element simulation results. The test results show that the acceleration amplification coefficient and displacement results are slightly different, but basically the same. This difference was explained by the effect of cracks in masonry itself on the dynamic response of the structure. Angela Ferrante [13] used the LMGC90© and 3DEC© codes to simulate the nonlinear dynamic behavior of the Rotella bell pagoda using discontinuous methods, and found that the shape, size and texture of masonry affect the failure mode. These methods effectively simulated large displacement and complex mechanical behavior. Adolfo Preciado [14] analyzed the impact of the 7 and 19 September 2017 earthquakes on masonry buildings in Puebla and Morellos, Mexico, and noted that the earthquake caused damage to 2,340 buildings, particularly in the arch system and the clock pagoda, mainly due to weaknesses in the masonry structure, inertia forces, lack of maintenance and improper renovation. F. Masi [15] used JWL equation to numerically simulate the explosion process, combining the material nonlinearity of existing cracks in the dome and low tensile strength concrete aggregates. In doing so, the study revealed two key phenomena: first, the shock wave tended to concentrate the explosive energy; second, preexisting cracks played an important role in the evolution of structural damage, resulted in the focus of shock waves inside the dome-like vault structure. Caihong Fan [16] proposed an improved calculation method for building blasting vibration response, which converted time domain data into frequency domain by Fourier transform, selected blasting vibration wave intensity index, analyzed main damage characteristics and sensitive parts, evaluated the natural frequency of buildings, and finally used differential equation to calculate the dynamic response. The experimental results showed that the proposed method was superior to the traditional method in accuracy and met the requirement of dynamic calculation of building blasting vibration response. Hanxu Shi [17] studied the safety of blasting excavation under the tunnel of Wuhan Metro Line 5. He found that the vibration velocity at the top of the pagoda was significantly higher than that at the bottom, and the maximum stress concentration was 0.0442MPa, which verified the reliability of the numerical simulation model and indicated that the ancient pagoda was safe under blasting load. This is of great significance to the safety evaluation of ancient buildings in complex urban environment. Ruiran Li [18] proposed a framework based on proportional distance and structural redundancy to rapidly assess the damage of reinforced concrete frames in explosions, and the research verified the accuracy of the framework, with an evaluation error between 4.11% and 9.52%. Based on the above research literature, in order to prevent the attack of explosion load and better reduce the damage degree of ancient buildings, this study proposed to apply explosion load at different positions of the pagoda body and explore the anti-explosion performance of the explosion on different positions of the pagoda body. In order to ensure the accuracy of the research content, the first, third, fifth, seventh, and ninth layers of the pagoda were detonated one by one by using the fluid-structure coupling method (ALE) 1 Overview of the ancient pagoda Wenchang Pagoda located in Guangxi, China, was built in the 27th year of Wanli in the Ming Dynasty (1599), more than 400 years ago, with a total of 9 layers. It is composed of the base, the pagoda body, and the pagoda brake, 48.94m high. The foundation is square, the pagoda body is hexagonal; from the bottom to the top, layer by layer, the interior is the arch, the exterior is the brick eaves. The pagoda body is made of black brick masonry. The top of the pagoda is made of a copper and iron octagonal spire. See Figure 1 Table 1 shows the main geometric dimensions of the Wenchang Pagoda. Tab. 1 Main geometric dimensions of ancient pagoda Number of floors Pagoda length /m Stacked edge length /m Floor height /m Floor elevation /m 1 4.37 5.56 7.32 7.32 2 4.26 5.38 4.28 11.60 3 4.14 5.27 4.10 15.70 4 4.01 5.09 3.91 19.61 5 3.90 4.96 3.66 23.27 6 3.79 4.80 3.52 26.79 7 3.68 4.71 3.37 30.16 8 3.60 4.57 3.33 33.49 9 3.48 4.38 2.95 36.44 Pagoda top — — 12.50 48.94 The pagoda foundation is made of strip stone with a depth of 1.32 m. The masonry material is a mixed slurry of soil and lime. The lower part of the foundation is the artificially compacted plain soil layer, with a depth of 4.4 to 4.7 m under the stone bar foundation, and the thickness of each layer is between 0.20 to 0.35 m. The range is slightly larger than the base, and the edge is about 1.7 to 2.5 m away from the base (see Figure 2). The foundation is designed to provide stable support for the pagoda, ensuring that it is stable and does not move. By evenly distributing the weight of the pagoda underground, the foundation effectively prevents settling or tilting, thereby maintaining the structure's perpendicularity and long-term stability. This paper focuses on the study of the response of the pagoda to vibration loads, so the details of the foundation are not further discussed. 2. Finite element model establishment 2.1 Modeling of Wenchang pagoda In this study, according to the size data of Wenchang Pagoda measured in the field, a finite element model of an ancient pagoda structure with nine floors above ground is established by using ANSYS/LS-DYNA software in accordance with the unit m, as shown in Figure 3(b). In order to guarantee the calculation accuracy and reduce the calculation cost, the grid subdivision strategy of equal proportion for the air domain is adopted. Based on the different responses of different parts of the structure under the action of explosion load, the grid thinning scheme of partition is implemented for the model of the ancient pagoda. Tetrahedral element grid is adopted for the 5 explosion points under the explosion load (as shown in Figure 3(a)), and the size was set at 125 mm. For the rest of the pagoda layers, the size of the grid in the horizontal direction should be consistent with the bottom frame, that is, 125 mm, and set to 200mm in the vertical direction. The specific parameters of the pagoda body are shown in Table 2. In the initial boundary condition setting, the six corners at the bottom of the base are constrained. By default, single point integration and hourglass control are used to speed up the unit column, effectively shorten the calculation time and improve the calculation efficiency. The units are connected through common nodes [19]. ALE fluid-structure coupling method is used to analyze the explosion problem. ALE mesh is used for air and Lagrange mesh is used for other solid structures. By using this method, because the material can flow in the grid, there is no problem of element distortion, and the accuracy and effectiveness of load transfer can be guaranteed [20]. When the explosive exploded close to the Ghouta, the TNT equivalent of 1.64kg, the location of each explosion point is shown in the figure 3 (a). Tab. 2 Pagoda modeling information ET R EX(Pa) PRXY DENS(kg/m 3 ) Solid 186 -- 0.637e9 0.15 1900 Meshing SMRTSIZE MSHAPE MSHKEY Unit Total Pagoda bulk property Precision 6 1(Tetrahedral unit) 0 90230 V:572.69m 3 M:1030.6t Note: ET is a cell type; R is a real constant; EX is elastic modulus; PRXY is Poisson's ratio; DENS represents density; SMRTSIZE indicates the mesh size. MSHAPE is tetrahedral mesh shape; MSHKEY indicates the mesh partitioning method. 2.2 Material model 2.2.1 Masonry material model The concrete parameters of the bricks can be obtained by the test analysis of the debris stones of the nearby ancient pagoda, where the density ρ is 1900 kg/m 3 , the elastic modulus E is 1800 MPa, and the Poisson ratio υ is 0.15. The concrete damage plasticity (CDP) constitutive model is used to simulate the brittle damage properties of bricks. The damage parameters are approximated by the energy equivalent principle to simulate the damage characteristics of the material. Figure. 4 shows how concrete behaves during tension, compression, and stiffness recovery. The expansion Angle of the plastic damage model is set to 30°, the eccentricity is 0.1, the biaxial and uniaxial compressive strength ratio is 1.16, the second stress invariant ratio on the tension meridian is 0.6667, and the viscosity parameter is 0.0005. Based on the measured data, the *MAT_JOHNSON_HOLMQUIST_CONCRETE material model built in LS-DYNA is used for simulation, and the elastic modulus of concrete is modified by weighting method (as shown in Table 3), so as to realize the real simulation of masonry materials [21]. Tab. 3 Material parameters of * MAT_JOHNSON_HOLMQUIST_CONCRETE model р 0 /(g·cm -3 ) f’c/MPa A B C S max G/GPa 1.9 48 0.79 1.6 0.007 7.0 14.86 D 1 D 2 N εf.min T/MPa P c /MPa μ c 0.04 1.0 0.61 0.01 4 16 0.001 P 1 /GPa μ 1 K 1 /MPa K 2 /MPa K 3 /MPa E/GPa ε 0 0.8 0.01 85 -171 208 35.7 1 Note: Density of zero grade concrete; f 'c is the static yield strength; A is the cohesion strength of dimension one; B is the pressure strengthening coefficient of dimension one; C is strain rate sensitivity coefficient; Smax is the maximum strength that concrete can achieve; G is the shear modulus; D 1 and D 2 are the damage constants of concrete. N is the stress hardening coefficient; εf. min is the minimum plastic strain of broken concrete. T is the maximum tensile strength of concrete; P c and μ c are the hydrostatic pressure and volumetric strain of collapse. P 1 and μ 1 are the pressure and volume strain of concrete in the compaction stage, respectively. E is the elastic modulus; K 1 , K 2 and K 3 are the bulk elastic modulus of the HJC model in three stages. ε 0 is the reference strain. 2.2.2 Explosive material model and equation of state The material model MAT_HIGH_EXPLOSIVE_BURN was used to simulate the explo-sion of the explosives, and the dynamic r-elationship between the pressure and the volume during the explosion was described by the equation of state EOS_JWL. The specific parameter settings are shown in Table 4 and Table 5. Tab. 4 Material parameters of MAT_HIGH_EXPLOSIVE_BURN MID ρ 0 /(g·cm -3 ) υ 0 /(g·cm -3 ) P CJ /GPa 1 1.64 6930 21 BETA K/MPa G/MPa SIGY/MPa 0 0 0 0 Note: MID is the material identification number; ρ 0 is the mass density; υ 0 is the detonation velocity. P CJ is Chapman-Jouget pressure; BETA indicates the identification of the pressure calculation formula; K is the volume modulus; G is the shear modulus; SIGY is the yield pressure. Tab. 5 Parameters of JWL state equation EOSID A/GPa B/GPa R 1 R 2 ω E 0 /(kJ·mm -3 ) V 0 1 374 323 4.15 0.95 0.3 7 1.0 Note: EOSID is an equation of state identifier; A, B, R 1 , R 2 and ω are all constants that characterize the characteristics of explosives. E 0 is the initial internal energy per unit volume of explosive. V 0 is the initial relative volume. 2.2.3 Air material model and equation of state The MAT_NULL material model is used to simulate the air, and EOS_LINEAR_POLYNOMIAL is introduced to describe the relationship between the air pressure, volume and internal energy. The air material and its equation of state parameters are detailed in Table 6 and Table 7. Tab. 6 Material parameters of *MAT_NULL EOSID ρ 0 /(g·cm -3 ) P C μ TEROD CEROD E ν 2 1.29 0 0 0 0 0 0 Note: EOSID is the material identification number; ρ 0 is the mass density; P C is the cross section pressure; μ is the dynamic viscosity coefficient. TEROD represents the relative volume of tensile erosion. CEROD is the relative volume of compression erosion. E is Young's modulus; ν is Poisson's ratio. Tab. 7 Parameters of air state equation ESIOD C 0 C 1 C 2 C 3 C 4 C 5 C 6 E 0 /MPa V 0 2 0 0 0 0 0.4 0.4 0 2.5e+05 1.0 Note: EOSD is the equation of state identification number; C 0 ~ C 6 is the coefficient of polynomial state equation: E 0 represents the initial internal energy per unit volume of explosive; V 0 is the initial relative volume. 3 Analysis of continuous collapse of pagoda structure under different explosion points 3.1 Analysis of overpressure time history curve Figure 5(a) shows the ideal time-space profile of the overpressure Ps(t,q) induced by the explosion. When the excitation wave reaches q point, the pressure immediately jumps from the environmental pressure P o to the peak value of Ps, and then decreases exponentially to P o after t > t A , marking the end of the positive phase and entering the negative phase stage, where the pressure decreases relative to P o and recovers to P o after to. The pressure attenuation during the negative phase is much lower than the peak of the positive phase, and although the negative phase is usually negligible, its duration is often much longer than the positive phase, and the actual situation may vary from disturbance to disturbance. Shock wave is the main mechanical effect of explosion on the pagoda, while the dynamic pressure generated by the expansion of hot gas is small and the propagation speed is slow, and the shock wave may reflect on the solid surface and act on other surfaces in the form of reflected waves. The rate at which the overpressure P s at point q decreases with time and distance is much greater than that in space. The explosion overpressure is similar to the local pressure wave propagating at high speed and gradually weakening with distance [22]. The pressure P r on the surface acting on the overpressure P s shock is usually greater than the peak value of P s , assuming that it is measured at the same point and there is no surface interference. The pressure time history curve under the impact of actual explosion load is shown in Figure 5 (b), which shows that it falls to negative value after direct loading from 0 to peak value, and then gradually rises to stability. Figure 5 (a) shows overpressure changes over time and space in a simplified ideal model, emphasizing sharp changes in time and slow changes in space. Figure 5(b) shows the pressure time history curve in the actual explosion, reflecting complex effects such as reflected waves and hot gas expansion [23]. The difference between the two is mainly due to various interference and complex effects in the actual situation, which are ignored in the idealized model. Select the representative unit at the location of the explosion point as the monitoring point, and draw the vibration velocity and stress time history curve of the ancient pagoda under the action of each explosion point. The velocity and stress curves obtained by the finite element numerical calculation model are shown in Figure 6 (a) and Figure 6 (b). The results show that the vibration velocity and stress peak within 1.28 to 1.5 seconds after the blasting starts, and then rapidly decays and becomes stable. Therefore, it is reasonable to evaluate the dynamic response and safety of the pagoda structure under the most dangerous working conditions, that is, based on the maximum vibration speed and stress decays and becomes stable. Therefore, it is reasonable to evaluate the dynamic response and safety of the pagoda structure under the most dangerous working conditions, that is, based on the maximum vibration speed and stress. 3.2 Time history analysis of displacement and velocity Through the simulation analysis of five kinds of explosions (1st layer, 3th layer, 5th layer, 7th layer, and 9th layer), the damage process of explosion shock wave to the structure is studied. The shock wave under each explosion point exerted a strong impact on the explosion location, resulting in rapid destruction of the pagoda body. According to the key position selection principle of the spare load path method in GSA specification, 6 corner points of each explosion layer are selected as the target construction of applying explosion load [24]. In order to facilitate subsequent analysis, each node is marked with working condition number plus node number. For example, working condition 1 at corner point A is denoted as A1; point 2 is denoted as A2 at corner point A. The schematic diagram of the location of explosion load and node location is shown in Figure. 7, and the displacement and velocity time-history curves of these nodes under different explosion points are recorded respectively (shown Figure 8 and Figure 9). The explosion caused the pagoda to respond in the x, y, and z directions, so the extracted data was synthesized in all directions to better analyze the structure as a whole. The explosion facing surface (such as A, B, C points) is directly affected by the high pressure and high-speed airflow of the shock wave, resulting in large velocity and displacement. In contrast, the backside regions (such as points D, E, and F) experience reverse airflow and vortex effects, with lower airflow velocity and dispersed energy, so displacement and velocity are usually lower than on the explosion side. With the increase of the explosion load, the cracks in the pagoda body expand rapidly to the inside and on both sides, and the high pressure detonation gas fills the pagoda layer, causing significant damage to the vault, bond hole, and pagoda brake of the pagoda body, and the continuous action of the explosion load causes the pagoda body to significantly swing. Under the influence of explosion shock wave, the pagoda underwent various physical deformations, which significantly changed its motion characteristics. In response parameters such as strain, acceleration, and displacement angle, strain and acceleration are greatly affected by material distribution due to the inhomogeneity of masonry materials of ancient pagodas, so displacement parameter is particularly important. It directly reflects the structural damage situation, provides basic data for the functional and safety assessment of buildings after damage, and can also be used as a benchmark parameter for numerical simulation model verification. Figure. 8 and Figure. 9 are the combined velocity and displacement time-history curves of the recording nodes. As can be seen from the figure, after the explosion, the target position that is most affected by the explosion impact is point A (curve A), which always has the largest velocity and displacement, because point A is almost at the detonation center. Before 1s, no displacement occurred at all points, but with the outward diffusion of the explosion shock wave for 1s, all points began to generate displacement, and the displacement generated by three points A, B, and C on the explosion facing surface is much larger than that generated by D, E, and F on the back side. When the explosion load is applied at the third layer position (explosion 2), the displacement generated by each point is 1.5 ~ 2 times that of the explosion load applied at other positions. Accordingly, before about 0.8s, the speed of each point tends to be stable basically following the law of gradually increasing, and the speed increases significantly after 1s. Among them, the velocity and displacement of point F (curve F) on the back explosion surface increase slowly, indicating that the node closest to the explosion position fails first under each explosion point, and the vertical displacement of the superstructure keeps increasing after the other positions fail successively. The initial damage caused by point F in the symmetric position is much greater than that at position E. Through comparative analysis, the collapse of the structure develops more rapidly on the side of the explosion face. From 1.0 to 1.5s, the velocity and displacement of each point continue to increase, and the pagoda gradually collapses beyond the bearing capacity. Applying the same yield of explosives at different locations, the most violent response is to apply explosive loads at the third layer position. 3.3 Time history analysis of interstory displacement In the above section, we analyzed the displacement effects of explosion loads applied at different locations on the corner column nodes of each explosion layer. The results show that significant displacement occurs in each layer under the action of explosion load. Especially after the failure of the pagoda body structure, each layer may experience a large interstory displacement, which is great significance for the analysis of the whole collapse of the structure. Under the five kinds of explosion, the interstory displacement of each layer is shown in Figure 10. For example, the explosion load applied at different locations shows the maximum interstory displacement of the explosion layer. Under the loading of explosion shock wave, the interlayer displacement of each layer shows a decreasing trend, in which the interstory displacement generated by the middle explosion position is significantly larger than the explosion load imposed on the bottom and top, indicating that the anti-explosion ability of the middle pagoda is lower than that on the bottom and top. In particular, the third layer, whose interstory displacement reaches nearly 3 times that of the other layers, further reveals the weakness of the central region when bearing the blast load. According to the trend of the data, it can be seen that the area near the explosion point shows a large inter-story displacement, while the area far away from the explosion source shows a small inter-story displacement, which also verifies that the damage of the ancient pagoda is the most serious at the explosion point. 3.4 Cloud image of damage As the ancient pagoda is made of masonry splicing and stacking, its vertical collapse resistance is weak, and the fracture and crushing of masonry caused by the explosion make the structure withstand the tensile, bending and torsion stress, which ultimately led to overall collapse (Figure 11). When the blast wave touches the masonry surface, the local pressure increases instantaneously, causing stress concentration. As the shock waves penetrate deep into the pagoda, the masonry undergoes extreme stress, spalling, splintering and sputtering[25-29]. In this process, the impact kinetic energy is gradually transformed into deformation energy and friction heat, resulting in the decline of the strength and toughness of the masonry, deformation and melting phenomenon. At the same time, the masonry is under tensile deformation, the internal pressure increases, the surface micro-cracks expand and intensify, and eventually lead to structural failure. When the explosion load is applied to each layer, the structure failure process as follows: At the beginning, the explosion shock wave quickly spreads throughout the whole pagoda within 0.9s, resulting in stress concentration in the top area of the pagoda. After 1.25s, the explosion load exerted a strong impact on the corner columns of each explosion layer, resulting in the instantaneous failure of the pagoda bottom, and the destruction of the pagoda body and each bond hole. Subsequently, cracks in the bottom of the pagoda spread rapidly, and high-pressure detonation gas filled the entire pagoda layer, exerting great damage to the internal beams and plates. After the failure of the bottom corner column of each layer, the pagoda body, vault and pagoda brake of the superstructure underwent stress redistribution, which caused the failure to spread downward. The explosion shock wave continuously produces negative pressure effect inside the structure, causing the failure of the elements of the explosion facing surface, causing the center of gravity shift of the structure, and finally causes the continuous collapse of the explosion facing surface. When the explosion load is applied to the three layers, the stress at each layer within 0.9s reaches the maximum, and the bond holes of each layer are separated from the pagoda body at the first time, and all components fail successively until collapse. Compared with other explosion points at the same time, the pagoda body of the detonated layer is damaged more severely, resulting in greater overturning and lateral displacement. 4 Conclusions In this study, an ancient masonry structure pagoda in Guangxi, China was used as a prototype to simulate the collapse process of the structure by using the fluid-structure coupling method, which can capture both the collapse phenomenon and the dynamic response of the structure. At the same time, the finite element model of 5 explosion points is established for comparative analysis and parameter expansion, and the collapse mechanism of the ancient masonry pagoda is discussed. Through the analysis and comparison of the numerical simulation, the following conclusions are drawn: 1) Through comparative analysis of the settings of 5 explosion points, the results show that the damage degree of the explosion layer is the most serious in the area closest to the explosion point, and the damage degree of the explosion surface is significantly higher than that of the back explosion surface. 2) The vertical resistance to sequential collapse of ancient masonry pagodas is poor, and the masonry near the explosion zone is fractured and crushed, which is the trigger factor for structural collapse. 3) After the failure of the member, the structure experiences internal force redistribution. The masonry in the damaged area is forced to withstand tension, bending, and twisting at the same time, and the bricks have serious displacement and broken parts, causing the structure to gradually lose stability. The pagoda body obviously tilts towards the explosion face, the support structure of the explosion layer rises, the mortar connection between the bricks and stones will break, resulting in the pagoda body splitting, the perpendicularity of the whole pagoda is destroyed, and finally the whole pagoda collapsed. 4) The parameter analysis shows that compared with the explosion on the 1st and 9th floors of the ancient pagoda, the explosion on the 3rd, 5th, and 7th floors has a more severe response to the pagoda body and causes more significant damage. 5) In order to improve the collapse resistance of ancient masonry pagodas in the case of explosion, the following suggestions are put forward: Optimize the explosion-proof design, and enhance the vertical collapse resistance by strengthening the structural reinforcement of the blast-facing surface of the pagoda. Recommendations included the addition of support systems and improved masonry connections to enhance structural stability, while in dynamic load analysis, special attention is paid to the central region to ensure the safety margin of the various hierarchies. Declarations Declaration of conflicting interests The authors declared no potential conflicts to the research, authorship, and publication of this article. Author contributions GxZ collected pagoda parameters, wrote the main manuscript text and drew Fig. 1(b), 2, 3, 4, 5, 6, 7, 8, 9, 10, 11; Tabs. 2, 3, 4, 5, 6, 7; DwL provides research ideas and financial assistance for Structural response analysis of a pagoda under explosion impact in China; YL provides resources and ideas; WW, QH, and YbS build the pagoda model shown in Fig. 1 (a) and Tab. 1. All authors reviewed the manuscript. Funding This research was funded by National Natural Science Fund of China (Nos. 52168072, 51808467), High-level Talent Support Project of Yunnan Province, China (2020). Data Availability Statement All data included in this study are available upon request by contact with the corresponding author. Declaration of conflicting interests The authors declared no potential conflicts to the research, authorship, and publication of this article. References Zhang, S.; Liang, J.; Su, X.; Chen, Y.; Wei, Q. Research on Global Cultural Heritage Tourism Based on Bibliometric Analysis. Heritage. Science. 2023, 11 (1), 139. Parisi, F. Blast Resistance of Tuff Stone Masonry Walls. Engineering. Structures. 2016. Lu, J. X.; Wu, H.; Cheng, Y. H.; Chen, G. Q. Blast Resistance of Grouting Sleeve Connected Precast Concrete Columns under Close-in Explosions. International Journal of Impact Engineering. 2024, 187, 104908. Invernizzi, S.; Lacidogna, G.; Lozano-Ramírez, N. E.; Carpinteri, A. 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P.; Hu, Y.; Bai, Z.; Zhang, H.; Li, Z.; Wang, X. A Reinforced Concrete Shear Wall Building Structure Subjected to Internal TNT Explosions: Test Results and Numerical Validation. International Journal of Impact Engineering. 2024, 190, 104950. Hajirezaei, R. Overturning Response Analysis of Submerged Free-Standing Blocks and Towers Subjected to Sinusoidal Pulses. 2024. Yan, J.; Liu, Y.; Yan, J.; Yan, Z.; Xu, Y.; Gao, C.; Huang, F. Collapse of Concrete Target Subjected to Embedded Explosion of Shelled Explosive. Engineering Failure Analysis. 2024, 161, 108298. Xiao, Y.; Zhu, W.; Song, J.; Jia, J.; Li, Z. Experimental and Numerical Investigation of Steel–Concrete Composite Beam Subjected to Contact Explosion. International Journal of Impact Engineering. 2024, 187, 104916. Zhang, Q.; Jiao, Y.; Cao, M.; Jin, L. Simulation Analysis On Summer Conditions Of Ancient Architecture of Tower Buildings Based on CFD. Energy Procedia. 2017, 143, 313–319. Du, J. Energy Transfer Ratio Evaluation Method of Explosion and Shock Isolation Efficiency. Journal of Building Structures. 2012, 33 (02), 72–77. Chen, X. Calculation Method of Blast Loadings in Engineering Analysis. Structure & Environment Engineeing. 2020, 47 (01), 26–32. Du, Y. Simulation of Uninterrupted Collapse Isolated Structure under Blast Load in Basement. Journal of Lanzhou University of Technology. 2019, 45 (2), 113–120. Si, D. Analysis of the Dynamic Response of Prestressed Concrete Frame Structures under Blast Load. Explosion and Shock Waves. 2023, 43 (11), 3–13. Zheng, M. Determination of Mechanical Property Constants for the Mooney-Rivlin Model of Rubber. China Rubber Industry. 2003, 50 (08), 62–465. Kopuz, A. D.; Bal, A. The Conservation of Modern Architectural Heritage Buildings in Turkey: İstanbul Hilton and İstanbul Çınar Hotel as a Case Study. Ain Shams Engineering Journal. 2023, 14 (4), 101918. Additional Declarations No competing interests reported. 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Liu","email":"data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAZAAAAAyAQMAAABI0h/eAAAABlBMVEX///8AAABVwtN+AAAACXBIWXMAAA7EAAAOxAGVKw4bAAAA40lEQVRIiWNgGAWjYDACZgaGAw8YLHgYGBJAXBsGNqK0JDBIwLSkEaEFBIBaGKBaDhNWzd/OYwiyRcbgePIxiY87ztvzSTc/YPhRsQ2nFonDPAZghxmceZYmOfPM7cQ2mWMGjD1nbuPUYsDMlgDRciPH7DZv2+0ENokEA2bGNqK05H+7/bftnD2bRPoHAlqYD8BsYbvN2HaAsU0iB78tEoehWiTPPDP/2duWnAjUUnAQn1/4+w82f/jAYGPPdzz5scHPNjt7+RnpGx/8qMCtBQwY/6EJHMCvfhSMglEwCkYBIQAAgvZS0GwYXAMAAAAASUVORK5CYII=","orcid":"","institution":"Southwest Forestry University","correspondingAuthor":true,"prefix":"","firstName":"Dewen","middleName":"","lastName":"Liu","suffix":""},{"id":376394229,"identity":"3c00b81b-4734-475e-9a34-71c1875283b5","order_by":2,"name":"Wei Wang","email":"","orcid":"","institution":"Southwest Forestry University","correspondingAuthor":false,"prefix":"","firstName":"Wei","middleName":"","lastName":"Wang","suffix":""},{"id":376394230,"identity":"07ebf03d-d6f1-4bc8-8335-8f1b53bc2201","order_by":3,"name":"Qing 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18:17:44","currentVersionCode":1,"declarations":"","doi":"10.21203/rs.3.rs-5111761/v1","doiUrl":"https://doi.org/10.21203/rs.3.rs-5111761/v1","draftVersion":[],"editorialEvents":[],"editorialNote":"","failedWorkflow":false,"files":[{"id":68986068,"identity":"b7224f7b-2480-4e0f-ac4e-d54d350881c4","added_by":"auto","created_at":"2024-11-14 08:36:13","extension":"png","order_by":1,"title":"Figure 1","display":"","copyAsset":false,"role":"figure","size":690083,"visible":true,"origin":"","legend":"\u003cp\u003eWenchang Ancient Pagoda picture\u003c/p\u003e","description":"","filename":"1.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/50a21825502eff970a53eaf9.png"},{"id":68985877,"identity":"55bdb704-2b4e-47dc-98bd-9be9a0fcdbb3","added_by":"auto","created_at":"2024-11-14 08:28:13","extension":"png","order_by":2,"title":"Figure 2","display":"","copyAsset":false,"role":"figure","size":271231,"visible":true,"origin":"","legend":"\u003cp\u003eFoundation of the ancient pagoda\u003c/p\u003e","description":"","filename":"2.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/9541c551f6111abcf8c2508d.png"},{"id":68984725,"identity":"2d0ed906-ed3b-4db0-8610-3dc0c49c772c","added_by":"auto","created_at":"2024-11-14 08:12:12","extension":"png","order_by":3,"title":"Figure 3","display":"","copyAsset":false,"role":"figure","size":505153,"visible":true,"origin":"","legend":"\u003cp\u003eStructure model of ancient pagoda\u003c/p\u003e","description":"","filename":"3.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/a65121d88baa4afc78d9dfbc.png"},{"id":68984994,"identity":"0c249dd3-4c87-429c-9650-89cbc3a7daf6","added_by":"auto","created_at":"2024-11-14 08:20:13","extension":"png","order_by":4,"title":"Figure 4","display":"","copyAsset":false,"role":"figure","size":78832,"visible":true,"origin":"","legend":"\u003cp\u003eMasonry adopts uniaxial tension-compressive stress-strain relationship in CDP model\u003c/p\u003e","description":"","filename":"4.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/38f97c4ebc49a8993b3d4770.png"},{"id":68984731,"identity":"eeb28eeb-2497-43ea-b65b-cc3dd1630928","added_by":"auto","created_at":"2024-11-14 08:12:13","extension":"png","order_by":5,"title":"Figure 5","display":"","copyAsset":false,"role":"figure","size":71799,"visible":true,"origin":"","legend":"\u003cp\u003eOverpressure time history curve of explosion with time\u003c/p\u003e","description":"","filename":"5.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/04c031f616e2fe80ee68c8cf.png"},{"id":68984991,"identity":"e5292ac1-1aac-46d2-a145-16c0cc8867c8","added_by":"auto","created_at":"2024-11-14 08:20:13","extension":"png","order_by":6,"title":"Figure 6","display":"","copyAsset":false,"role":"figure","size":525066,"visible":true,"origin":"","legend":"\u003cp\u003eComparison of velocity and stress time-history curves\u003c/p\u003e","description":"","filename":"6.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/71b1ebc86909057cb6bd3983.png"},{"id":68984737,"identity":"c557cac6-e463-41d6-8cda-dc45b866f7e2","added_by":"auto","created_at":"2024-11-14 08:12:13","extension":"png","order_by":7,"title":"Figure 7","display":"","copyAsset":false,"role":"figure","size":180544,"visible":true,"origin":"","legend":"\u003cp\u003eSchematic diagram of nodes under each explosion point\u003c/p\u003e","description":"","filename":"7.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/9ef5d272ec6bdd554188c195.png"},{"id":68985880,"identity":"30530d33-725b-4c92-8eb5-e80d60474a78","added_by":"auto","created_at":"2024-11-14 08:28:13","extension":"png","order_by":8,"title":"Figure 8","display":"","copyAsset":false,"role":"figure","size":419374,"visible":true,"origin":"","legend":"\u003cp\u003eCombined displacement diagram of 6 positions under each explosion point\u003c/p\u003e","description":"","filename":"8.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/b95ad185e0ebfd1e6cadb049.png"},{"id":68984740,"identity":"73f1d403-67a4-4857-89fc-eae4b6cf690c","added_by":"auto","created_at":"2024-11-14 08:12:14","extension":"png","order_by":9,"title":"Figure 9","display":"","copyAsset":false,"role":"figure","size":397903,"visible":true,"origin":"","legend":"\u003cp\u003eVelocity diagram of 6 locations under each explosion point\u003c/p\u003e","description":"","filename":"9.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/55ce3af8d111bbdf103deb03.png"},{"id":68984732,"identity":"bd116747-01fa-4102-b716-7f263fc21ddc","added_by":"auto","created_at":"2024-11-14 08:12:13","extension":"png","order_by":10,"title":"Figure 10","display":"","copyAsset":false,"role":"figure","size":454296,"visible":true,"origin":"","legend":"\u003cp\u003eInterstory displacement of the ancient pagoda under each explosion point\u003c/p\u003e","description":"","filename":"10.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/f388f4b2c1c89c8522af8c5e.png"},{"id":68987273,"identity":"1651e72e-54ba-4acc-9076-d668fe0dab15","added_by":"auto","created_at":"2024-11-14 08:44:13","extension":"png","order_by":11,"title":"Figure 11","display":"","copyAsset":false,"role":"figure","size":2642413,"visible":true,"origin":"","legend":"\u003cp\u003eCloud images of the damage for each explosion\u003c/p\u003e","description":"","filename":"11.png","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/95a6e1c4f94fe0c4d8c52101.png"},{"id":70145350,"identity":"4f714325-1836-43f2-a9cf-f03abce4a3b9","added_by":"auto","created_at":"2024-11-28 23:16:32","extension":"pdf","order_by":0,"title":"","display":"","copyAsset":false,"role":"manuscript-pdf","size":6188198,"visible":true,"origin":"","legend":"","description":"","filename":"manuscript.pdf","url":"https://assets-eu.researchsquare.com/files/rs-5111761/v1/99fea332-b4a3-4c65-ae6d-03ca08c6cfc3.pdf"}],"financialInterests":"No competing interests reported.","formattedTitle":"Response analysis of a pagoda in China under blast loading","fulltext":[{"header":"0 Introduction","content":"\u003cp\u003eAs an important part of the World Cultural Heritage, ancient pagodas not only carry rich historical and cultural values, but also the crystallization of architectural art and engineering technology\u0026nbsp;[1]. However, in the current environment of rapid technological development, the ancient pagoda, as a symbol of a time-honored civilization, is facing unprecedented challenges. In addition to natural disasters such as earthquakes, mudslides and tsunamis, ancient pagodas may also be deliberately damaged by human factors, such as terrorist attacks, pollution and erosion, among which the use of drones to carry out precision bombing attacks on ancient pagodas is one of the potential dangers threatening the structural safety of ancient pagodas\u0026nbsp;[2-4]. This requires us to adopt stricter protection measures and strengthen monitorings and maintenance of the ancient pagodas. Because the ancient pagoda is tall and constructed with dense masonry materials, it is highly sensitive to vibration, so it is important to evaluate its vibration resistance. Especially in areas of high seismic risk and border areas of neighboring countries, ancient pagodas are exposed to different forms of danger at all times. Therefore, the study of the dynamic characteristics of ancient pagodas and the formulation of effective reinforcement and seismic measures are the key to protecting these cultural heritage. Lu Junlong et al.\u0026nbsp;[5,6]\u0026nbsp;analyzed the dynamic performance and structural damage of Xingjiao Temple Pagoda through ultra-low frequency dynamic testing, and found that the first two frequencies of the pagoda were similar, the first mode bending and the second bending shear, and the equivalent elastic modulus decreased significantly, indicating that the ancient pagoda structure was seriously damaged. Xu Dunfeng et al.[7]\u0026nbsp;based on the scaled model and shaking table test of Xi\u0026apos;an Xiaoyan Pagoda as the prototype, combined with finite element analysis, revealed the failure modes and dynamic characteristics of the brick ancient pagoda under different earthquakes, summarized the main damage characteristics of the ancient pagoda, and proposed refined earthquake damage assessment criteria, which provided scientific basis for the seismic design and protection measures of the ancient pagoda. Lv Yifan et al.\u0026nbsp;[8]\u0026nbsp;found that the blasting vibration response of Xiahe Pagoda was positively correlated with the explosion volume and insensitive to high-frequency vibration. The finite element analysis showed that the underlying compressive strength was insufficient but there was no crushing damage, and the impact of environmental vibration was more significant. Lee et al.\u0026nbsp;[9]\u0026nbsp;studied the natural frequency of the measured stone pagoda and compared it with the finite element model, and found that there was an average difference of 13% between the measured data and the peak ground acceleration (PGA) of the finite element analysis. This showed that the finite element method can effectively predict the dynamic characteristics of such structures. De Iasio et al.\u0026nbsp;[10]\u0026nbsp;established a detailed model of the Dragon Tiger Pagoda in China using ABAQUS, verified the accuracy of the topping analysis results through nonlinear dynamic analysis, and provided useful information about the damage and crack expansion of the pagoda after the earthquake. In addition, they used the dynamics theorem of limit analysis to study the collapse mechanism of the pagoda and its velocity, and explore the ability to predict earthquake damage. Shakya et al.\u0026nbsp;[11]\u0026nbsp;carried out an earthquake vulnerability assessment on 78 pagodas and temples with historical and heritage value in Kathmandu Valley, Nepal, and adopted the simplified method and vulnerability index for analysis. The results of the assessment and its database were integrated into the GIS tool, enabling spatial visualization of the damage scenario, which helped in the retrofit planning of the seismic risk mitigation. Lu J et al.\u0026nbsp;[12]\u0026nbsp;took Xuanzang Pagoda as an example, tested the 1/8 scale model by the shaking table, and compared the results with the finite element simulation results. The test results show that the acceleration amplification coefficient and displacement results are slightly different, but basically the same. This difference was explained by the effect of cracks in masonry itself on the dynamic response of the structure. Angela Ferrante\u0026nbsp;[13]\u0026nbsp;used the LMGC90\u0026copy; and 3DEC\u0026copy; codes to simulate the nonlinear dynamic behavior of the Rotella bell pagoda using discontinuous methods, and found that the shape, size and texture of masonry affect the failure mode. These methods effectively simulated large displacement and complex mechanical behavior. Adolfo Preciado\u0026nbsp;[14]\u0026nbsp;analyzed the impact of the 7 and 19 September 2017 earthquakes on masonry buildings in Puebla and Morellos, Mexico, and noted that the earthquake caused damage to 2,340 buildings, particularly in the arch system and the clock pagoda, mainly due to weaknesses in the masonry structure, inertia forces, lack of maintenance and improper renovation. F. Masi\u0026nbsp;[15]\u0026nbsp;used JWL equation to numerically simulate the explosion process, combining the material nonlinearity of existing cracks in the dome and low tensile strength concrete aggregates. In doing so, the study revealed two key phenomena: first, the shock wave tended to concentrate the explosive energy; second, preexisting cracks played an important role in the evolution of structural damage, resulted in the focus of shock waves inside the dome-like vault structure. Caihong Fan\u0026nbsp;[16]\u0026nbsp;proposed an improved calculation method for building blasting vibration response, which converted time domain data into frequency domain by Fourier transform, selected blasting vibration wave intensity index, analyzed main damage characteristics and sensitive parts, evaluated the natural frequency of buildings, and finally used differential equation to calculate the dynamic response. The experimental results showed that the proposed method was superior to the traditional method in accuracy and met the requirement of dynamic calculation of building blasting vibration response. Hanxu Shi\u0026nbsp;[17]\u0026nbsp;studied the safety of blasting excavation under the tunnel of Wuhan Metro Line 5. He found that the vibration velocity at the top of the pagoda was significantly higher than that at the bottom, and the maximum stress concentration was 0.0442MPa, which verified the reliability of the numerical simulation model and indicated that the ancient pagoda was safe under blasting load. This is of great significance to the safety evaluation of ancient buildings in complex urban environment. Ruiran Li\u0026nbsp;[18]\u0026nbsp;proposed a framework based on proportional distance and structural redundancy to rapidly assess the damage of reinforced concrete frames in explosions, and the research verified the accuracy of the framework, with an evaluation error between 4.11% and 9.52%.\u003c/p\u003e\n\u003cp\u003eBased on the above research literature, in order to prevent the attack of explosion load and better reduce the damage degree of ancient buildings, this study proposed to apply explosion load at different positions of the pagoda body and explore the anti-explosion performance of the explosion on different positions of the pagoda body. In order to ensure the accuracy of the research content, the first, third, fifth, seventh, and ninth layers of the pagoda were detonated one by one by using the fluid-structure coupling method (ALE)\u003cstrong\u003e\u003cbr\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e"},{"header":"1 Overview of the ancient pagoda","content":"\u003cp\u003eWenchang Pagoda located in Guangxi, China, was built in the 27th year of Wanli in the Ming Dynasty (1599), more than 400 years ago, with a total of 9 layers. It is composed of the base, the pagoda body, and the pagoda brake, 48.94m high. The foundation is square, the pagoda body is hexagonal; from the bottom to the top, layer by layer, the interior is the arch, the exterior is the brick eaves. The pagoda body is made of black brick masonry. The top of the pagoda is made of a copper and iron octagonal spire. See Figure 1 Table 1 shows the main geometric dimensions of the Wenchang Pagoda.\u003c/p\u003e\n\u003cp\u003eTab.\u0026nbsp;1\u0026nbsp;Main geometric dimensions of ancient pagoda\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003eNumber of floors\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003ePagoda length\u0026nbsp;/m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003eStacked edge length\u0026nbsp;/m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003eFloor height /m\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003eFloor elevation /m\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e4.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e5.56\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e7.32\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e7.32\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e4.26\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e5.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e4.28\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e11.60\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e4.14\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e5.27\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e4.10\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e15.70\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e4.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e5.09\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e3.91\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e19.61\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e5\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e3.90\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e4.96\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e3.66\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e23.27\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e3.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e4.80\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e3.52\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e26.79\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e3.68\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e4.71\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e3.37\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e30.16\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e3.60\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e4.57\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e3.33\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e33.49\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e3.48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e4.38\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e2.95\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e36.44\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003ePagoda top\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e\u0026mdash;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 26.3451%;\"\u003e\n \u003cp\u003e\u0026mdash;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 15.7699%;\"\u003e\n \u003cp\u003e12.50\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 19.295%;\"\u003e\n \u003cp\u003e48.94\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eThe pagoda foundation is made of strip stone with a depth of 1.32 m. The masonry material is a mixed slurry of soil and lime. The lower part of the foundation is the artificially compacted plain soil layer, with a depth of 4.4 to 4.7 m under the stone bar foundation, and the thickness of each layer is between 0.20 to 0.35 m. The range is slightly larger than the base, and the edge is about 1.7 to 2.5 m away from the base (see Figure 2). The foundation is designed to provide stable support for the pagoda, ensuring that it is stable and does not move. By evenly distributing the weight of the pagoda underground, the foundation effectively prevents settling or tilting, thereby maintaining the structure\u0026apos;s perpendicularity and long-term stability. This paper focuses on the study of the response of the pagoda to vibration loads, so the details of the foundation are not further discussed.\u003c/p\u003e"},{"header":"2. Finite element model establishment","content":"\u003cp\u003e\u003cstrong\u003e2.1 Modeling of Wenchang pagoda\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIn this study, according to the size data of Wenchang Pagoda measured in the field, a finite element model of an ancient pagoda structure with nine floors above ground is established by using ANSYS/LS-DYNA software in accordance with the unit m, as shown in Figure 3(b). In order to guarantee the calculation accuracy and reduce the calculation cost, the grid subdivision strategy of equal proportion for the air domain is adopted. Based on the different responses of different parts of the structure under the action of explosion load, the grid thinning scheme of partition is implemented for the model of the ancient pagoda. Tetrahedral element grid is adopted for the 5 explosion points under the explosion load (as shown in Figure 3(a)), and the size was set at 125 mm. For the rest of the pagoda layers, the size of the grid in the horizontal direction should be consistent with the bottom frame, that is, 125 mm, and set to 200mm in the vertical direction. The specific parameters of the pagoda body are shown in Table 2. In the initial boundary condition setting, the six corners at the bottom of the base are constrained. By default, single point integration and hourglass control are used to speed up the unit column, effectively shorten the calculation time and improve the calculation efficiency. The units are connected through common nodes [19]. ALE fluid-structure coupling method is used to analyze the explosion problem. ALE mesh is used for air and Lagrange mesh is used for other solid structures. By using this method, because the material can flow in the grid, there is no problem of element distortion, and the accuracy and effectiveness of load transfer can be guaranteed [20]. When the explosive exploded close to the Ghouta, the TNT equivalent of 1.64kg, the location of each explosion point is shown in the figure 3 (a).\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\" width=\"595\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"5\" valign=\"top\" style=\"width: 100%;\"\u003e\n \u003cp\u003eTab.\u0026nbsp;2\u0026nbsp;Pagoda modeling information\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 23.3613%;\"\u003e\n \u003cp\u003eET\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23.6975%;\"\u003e\n \u003cp\u003eR\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 13.1092%;\"\u003e\n \u003cp\u003eEX(Pa)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 14.6218%;\"\u003e\n \u003cp\u003ePRXY\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 25.2101%;\"\u003e\n \u003cp\u003eDENS(kg/m\u003csup\u003e3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 23.3613%;\"\u003e\n \u003cp\u003eSolid 186\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23.6975%;\"\u003e\n \u003cp\u003e--\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 13.1092%;\"\u003e\n \u003cp\u003e0.637e9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 14.6218%;\"\u003e\n \u003cp\u003e0.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 25.2101%;\"\u003e\n \u003cp\u003e1900\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd colspan=\"5\" valign=\"top\" style=\"width: 100%;\"\u003e\n \u003cp\u003eMeshing\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 23.3613%;\"\u003e\n \u003cp\u003eSMRTSIZE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23.6975%;\"\u003e\n \u003cp\u003eMSHAPE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 13.1092%;\"\u003e\n \u003cp\u003eMSHKEY\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 14.6218%;\"\u003e\n \u003cp\u003eUnit Total\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 25.2101%;\"\u003e\n \u003cp\u003ePagoda bulk property\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd style=\"width: 23.3613%;\"\u003e\n \u003cp\u003ePrecision 6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 23.6975%;\"\u003e\n \u003cp\u003e1(Tetrahedral unit)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 13.1092%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 14.6218%;\"\u003e\n \u003cp\u003e90230\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd style=\"width: 25.2101%;\"\u003e\n \u003cp\u003eV:572.69m\u003csup\u003e3\u003c/sup\u003e\u003c/p\u003e\n \u003cp\u003eM:1030.6t\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003e\u003cbr\u003e\u003c/p\u003e\n\u003cp\u003eNote: ET is a cell type; R is a real constant; EX is elastic modulus; PRXY is Poisson\u0026apos;s ratio; DENS represents density; SMRTSIZE indicates the mesh size. MSHAPE is tetrahedral mesh shape; MSHKEY indicates the mesh partitioning method.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.2 Material model\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.2.1 Masonry material model\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe concrete parameters of the bricks can be obtained by the test analysis of the debris stones of the nearby ancient pagoda, where the density \u0026rho; is 1900 kg/m\u003csup\u003e3\u003c/sup\u003e, the elastic modulus E is 1800 MPa, and the Poisson ratio \u0026upsilon; is 0.15. The concrete damage plasticity (CDP) constitutive model is used to simulate the brittle damage properties of bricks. The damage parameters are approximated by the energy equivalent principle to simulate the damage characteristics of the material.\u0026nbsp;Figure. 4\u0026nbsp;shows how concrete behaves during tension, compression, and stiffness recovery. The expansion Angle of the plastic damage model is set to 30\u0026deg;, the eccentricity is 0.1, the biaxial and uniaxial compressive strength ratio is 1.16, the second stress invariant ratio on the tension meridian is 0.6667, and the viscosity parameter is 0.0005. Based on the measured data, the *MAT_JOHNSON_HOLMQUIST_CONCRETE material model built in LS-DYNA is used for simulation, and the elastic modulus of concrete is modified by weighting method (as shown in\u0026nbsp;Table 3), so as to realize the real simulation of masonry materials\u0026nbsp;[21].\u003c/p\u003e\n\u003cp\u003eTab. 3 Material parameters of * MAT_JOHNSON_HOLMQUIST_CONCRETE model\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 15.8845%;\"\u003e\n \u003cp\u003e\u003cem\u003eр\u003c/em\u003e\u003csub\u003e0\u003c/sub\u003e/(g\u0026middot;cm\u003csup\u003e-3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003ef\u0026rsquo;c/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eB\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eC\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eS\u003csub\u003emax\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003eG/GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 15.8845%;\"\u003e\n \u003cp\u003e1.9\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e48\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e0.79\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e1.6\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e0.007\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e7.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e14.86\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 15.8845%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 15.8845%;\"\u003e\n \u003cp\u003eD\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003eD\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eN\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026epsilon;f.min\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eT/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eP\u003csub\u003ec\u003c/sub\u003e/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e\u0026mu;\u003csub\u003ec\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 15.8845%;\"\u003e\n \u003cp\u003e0.04\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e1.0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e0.61\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e0.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e16\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e0.001\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 15.8845%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 15.8845%;\"\u003e\n \u003cp\u003eP\u003csub\u003e1\u003c/sub\u003e/GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e\u0026mu;\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eK\u003csub\u003e1\u003c/sub\u003e/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eK\u003csub\u003e2\u003c/sub\u003e/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eK\u003csub\u003e3\u003c/sub\u003e/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003eE/GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e\u0026epsilon;\u003csub\u003e0\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 15.8845%;\"\u003e\n \u003cp\u003e0.8\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e0.01\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e85\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e-171\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e208\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.0794%;\"\u003e\n \u003cp\u003e35.7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 13.8989%;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eNote: Density of zero grade concrete; f \u0026apos;c is the static yield strength; A is the cohesion strength of dimension one; B is the pressure strengthening coefficient of dimension one; C is strain rate sensitivity coefficient; Smax is the maximum strength that concrete can achieve; G is the shear modulus; D\u003csub\u003e1\u003c/sub\u003e and D\u003csub\u003e2\u003c/sub\u003e are the damage constants of concrete. N is the stress hardening coefficient; \u0026epsilon;f. min is the minimum plastic strain of broken concrete. T is the maximum tensile strength of concrete; P\u003csub\u003ec\u003c/sub\u003e and \u0026mu;\u003csub\u003ec\u003c/sub\u003e are the hydrostatic pressure and volumetric strain of collapse. P\u003csub\u003e1\u003c/sub\u003e and \u0026mu;\u003csub\u003e1\u003c/sub\u003e are the pressure and volume strain of concrete in the compaction stage, respectively. E is the elastic modulus; K\u003csub\u003e1\u003c/sub\u003e, K\u003csub\u003e2\u003c/sub\u003e and K\u003csub\u003e3\u003c/sub\u003e are the bulk elastic modulus of the HJC model in three stages. \u0026epsilon;\u003csub\u003e0\u003c/sub\u003e is the reference strain.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.2.2 Explosive material model and equation of state\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe material model MAT_HIGH_EXPLOSIVE_BURN was used to simulate the explo-sion of the explosives, and the dynamic r-elationship between the pressure and the volume during the explosion was described by the equation of state EOS_JWL. The specific parameter settings are shown in Table 4 and Table 5.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTab.\u0026nbsp;4\u0026nbsp;Material parameters of MAT_HIGH_EXPLOSIVE_BURN\u003c/p\u003e\n\u003cdiv align=\"\"\u003e\n \u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 30.7692%;\"\u003e\n \u003cp\u003eMID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003e\u0026rho;\u003csub\u003e0\u003c/sub\u003e/(g\u0026middot;cm\u003csup\u003e-3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 24.1758%;\"\u003e\n \u003cp\u003e\u0026upsilon;\u003csub\u003e0\u003c/sub\u003e/(g\u0026middot;cm\u003csup\u003e-3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003eP\u003csub\u003eCJ\u003c/sub\u003e/GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 30.7692%;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003e1.64\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 24.1758%;\"\u003e\n \u003cp\u003e6930\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003e21\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 30.7692%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 24.1758%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003e\u0026nbsp;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 30.7692%;\"\u003e\n \u003cp\u003eBETA\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003eK/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 24.1758%;\"\u003e\n \u003cp\u003eG/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003eSIGY/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 30.7692%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 24.1758%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 22.5275%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n \u003c/table\u003e\n\u003c/div\u003e\n\u003cp\u003eNote: MID is the material identification number; \u0026rho;\u003csub\u003e0\u003c/sub\u003e is the mass density; \u0026upsilon;\u003csub\u003e0\u003c/sub\u003e is the detonation velocity. P\u003csub\u003eCJ\u003c/sub\u003e is Chapman-Jouget pressure; BETA indicates the identification of the pressure calculation formula; K is the volume modulus; G is the shear modulus; SIGY is the yield pressure.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTab.\u0026nbsp;5\u0026nbsp;Parameters of JWL state equation\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 12.4549%;\"\u003e\n \u003cp\u003eEOSID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.4549%;\"\u003e\n \u003cp\u003eA/GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.4549%;\"\u003e\n \u003cp\u003eB/GPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.0939%;\"\u003e\n \u003cp\u003eR\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.0939%;\"\u003e\n \u003cp\u003eR\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.0939%;\"\u003e\n \u003cp\u003e\u0026omega;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.8014%;\"\u003e\n \u003cp\u003eE\u003csub\u003e0\u003c/sub\u003e/(kJ\u0026middot;mm\u003csup\u003e-3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 11.5523%;\"\u003e\n \u003cp\u003eV\u003csub\u003e0\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 12.4549%;\"\u003e\n \u003cp\u003e1\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.4549%;\"\u003e\n \u003cp\u003e374\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.4549%;\"\u003e\n \u003cp\u003e323\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.0939%;\"\u003e\n \u003cp\u003e4.15\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.0939%;\"\u003e\n \u003cp\u003e0.95\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.0939%;\"\u003e\n \u003cp\u003e0.3\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.8014%;\"\u003e\n \u003cp\u003e7\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 11.5523%;\"\u003e\n \u003cp\u003e1.0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eNote: EOSID is an equation of state identifier; A, B, R\u003csub\u003e1\u003c/sub\u003e, R\u003csub\u003e2\u003c/sub\u003e and \u0026omega; are all constants that characterize the characteristics of explosives. E\u003csub\u003e0\u003c/sub\u003e is the initial internal energy per unit volume of explosive. V\u003csub\u003e0\u0026nbsp;\u003c/sub\u003eis the initial relative volume.\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e2.2.3 Air material model and equation of state\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe MAT_NULL material model is used to simulate the air, and EOS_LINEAR_POLYNOMIAL is introduced to describe the relationship between the air pressure, volume and internal energy. The air material and its equation of state parameters are detailed in Table 6 and Table 7.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003eTab.\u0026nbsp;6\u0026nbsp;Material parameters of *MAT_NULL\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 12.6582%;\"\u003e\n \u003cp\u003eEOSID\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.6474%;\"\u003e\n \u003cp\u003e\u0026rho;\u003csub\u003e0\u003c/sub\u003e/(g\u0026middot;cm\u003csup\u003e-3\u003c/sup\u003e)\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 10.4882%;\"\u003e\n \u003cp\u003eP\u003csub\u003eC\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.2966%;\"\u003e\n \u003cp\u003e\u0026mu;\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.6582%;\"\u003e\n \u003cp\u003eTEROD\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.6582%;\"\u003e\n \u003cp\u003eCEROD\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.2966%;\"\u003e\n \u003cp\u003eE\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.2966%;\"\u003e\n \u003cp\u003e\u0026nu;\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 12.6582%;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 14.6474%;\"\u003e\n \u003cp\u003e1.29\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 10.4882%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.2966%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.6582%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.6582%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.2966%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 12.2966%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eNote: EOSID is the material identification number;\u0026nbsp;\u0026rho;\u003csub\u003e0\u003c/sub\u003e is the mass density; P\u003csub\u003eC\u003c/sub\u003e is the cross section pressure; \u0026mu; is the dynamic viscosity coefficient. TEROD represents the relative volume of tensile erosion. CEROD is the relative volume of compression erosion. E is Young\u0026apos;s modulus; \u0026nu; is Poisson\u0026apos;s ratio.\u003c/p\u003e\n\u003cp\u003eTab.\u0026nbsp;7\u0026nbsp;Parameters of air state equation\u003c/p\u003e\n\u003ctable border=\"1\" cellspacing=\"0\" cellpadding=\"0\"\u003e\n \u003ctbody\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 10.1266%;\"\u003e\n \u003cp\u003eESIOD\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003eC\u003csub\u003e0\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003eC\u003csub\u003e1\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003eC\u003csub\u003e2\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003eC\u003csub\u003e3\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003eC\u003csub\u003e4\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003eC\u003csub\u003e5\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003eC\u003csub\u003e6\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 10.3074%;\"\u003e\n \u003cp\u003eE\u003csub\u003e0\u003c/sub\u003e/MPa\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003eV\u003csub\u003e0\u003c/sub\u003e\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003ctr\u003e\n \u003ctd valign=\"top\" style=\"width: 10.1266%;\"\u003e\n \u003cp\u003e2\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003e0.4\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003e0\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 10.3074%;\"\u003e\n \u003cp\u003e2.5e+05\u003c/p\u003e\n \u003c/td\u003e\n \u003ctd valign=\"top\" style=\"width: 9.94575%;\"\u003e\n \u003cp\u003e1.0\u003c/p\u003e\n \u003c/td\u003e\n \u003c/tr\u003e\n \u003c/tbody\u003e\n\u003c/table\u003e\n\u003cp\u003eNote: EOSD is the equation of state identification number; C\u003csub\u003e0\u003c/sub\u003e ~ C\u003csub\u003e6\u003c/sub\u003e is the coefficient of polynomial state equation: E\u003csub\u003e0\u003c/sub\u003e represents the initial internal energy per unit volume of explosive; V\u003csub\u003e0\u003c/sub\u003e is the initial relative volume.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e"},{"header":"3 Analysis of continuous collapse of pagoda structure under different explosion points","content":"\u003cp\u003e\u003cstrong\u003e3.1 Analysis of overpressure time history curve\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eFigure 5(a)\u0026nbsp;shows the ideal time-space profile of the overpressure Ps(t,q) induced by the explosion. When the excitation wave reaches q point, the pressure immediately jumps from the environmental pressure P\u003csub\u003eo\u003c/sub\u003e to the peak value of Ps, and then decreases exponentially to P\u003csub\u003eo\u003c/sub\u003e after t \u0026gt; t\u003csub\u003eA\u003c/sub\u003e, marking the end of the positive phase and entering the negative phase stage, where the pressure decreases relative to P\u003csub\u003eo\u003c/sub\u003e and recovers to P\u003csub\u003eo\u003c/sub\u003e after to. The pressure attenuation during the negative phase is much lower than the peak of the positive phase, and although the negative phase is usually negligible, its duration is often much longer than the positive phase, and the actual situation may vary from disturbance to disturbance. Shock wave is the main mechanical effect of explosion on the pagoda, while the dynamic pressure generated by the expansion of hot gas is small and the propagation speed is slow, and the shock wave may reflect on the solid surface and act on other surfaces in the form of reflected waves. The rate at which the overpressure P\u003csub\u003es\u003c/sub\u003e at point q decreases with time and distance is much greater than that in space. The explosion overpressure is similar to the local pressure wave propagating at high speed and gradually weakening with distance [22]. The pressure P\u003csub\u003er\u003c/sub\u003e on the surface acting on the overpressure P\u003csub\u003es\u003c/sub\u003e shock is usually greater than the peak value of P\u003csub\u003es\u003c/sub\u003e, assuming that it is measured at the same point and there is no surface interference. The pressure time history curve under the impact of actual explosion load is shown in\u0026nbsp;Figure 5\u0026nbsp;(b), which shows that it falls to negative value after direct loading from 0 to peak value, and then gradually rises to stability.\u0026nbsp;Figure 5\u0026nbsp;(a) shows overpressure changes over time and space in a simplified ideal model, emphasizing sharp changes in time and slow changes in space.\u0026nbsp;Figure 5(b) shows the pressure time history curve in the actual explosion, reflecting complex effects such as reflected waves and hot gas expansion\u0026nbsp;[23]. The difference between the two is mainly due to various interference and complex effects in the actual situation, which are ignored in the idealized model.\u003c/p\u003e\n\u003cp\u003eSelect the representative unit at the location of the explosion point as the monitoring point, and draw the vibration velocity and stress time history curve of the ancient pagoda under the action of each explosion point. The velocity and stress curves obtained by the finite element numerical calculation model are shown in Figure 6 (a) and Figure 6 (b). The results show that the vibration velocity and stress peak within 1.28 to 1.5 seconds after the blasting starts, and then rapidly decays and becomes stable. Therefore, it is reasonable to evaluate the dynamic response and safety of the pagoda structure under the most dangerous working conditions, that is, based on the maximum vibration speed and stress decays and becomes stable. Therefore, it is reasonable to evaluate the dynamic response and safety of the pagoda structure under the most dangerous working conditions, that is, based on the maximum vibration speed and stress.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.2 Time history analysis of displacement and velocity\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThrough the simulation analysis of five kinds of explosions (1st layer, 3th layer, 5th layer, 7th layer, and 9th layer), the damage process of explosion shock wave to the structure is studied. The shock wave under each explosion point exerted a strong impact on the explosion location, resulting in rapid destruction of the pagoda body. According to the key position selection principle of the spare load path method in GSA specification, 6 corner points of each explosion layer are selected as the target construction of applying explosion load\u0026nbsp;[24]. In order to facilitate subsequent analysis, each node is marked with working condition number plus node number. For example, working condition 1 at corner point A is denoted as A1; point 2 is denoted as A2 at corner point A. The schematic diagram of the location of explosion load and node location is shown in\u0026nbsp;Figure. 7, and the displacement and velocity time-history curves of these nodes under different explosion points are recorded respectively (shown\u0026nbsp;Figure 8\u0026nbsp;and\u0026nbsp;Figure 9). The explosion caused the pagoda to respond in the x, y, and z directions, so the extracted data was synthesized in all directions to better analyze the structure as a whole. The explosion facing surface (such as A, B, C points) is directly affected by the high pressure and high-speed airflow of the shock wave, resulting in large velocity and displacement. In contrast, the backside regions (such as points D, E, and F) experience reverse airflow and vortex effects, with lower airflow velocity and dispersed energy, so displacement and velocity are usually lower than on the explosion side. With the increase of the explosion load, the cracks in the pagoda body expand rapidly to the inside and on both sides, and the high pressure detonation gas fills the pagoda layer, causing significant damage to the vault, bond hole, and pagoda brake of the pagoda body, and the continuous action of the explosion load causes the pagoda body to significantly swing. Under the influence of explosion shock wave, the pagoda underwent various physical deformations, which significantly changed its motion characteristics. In response parameters such as strain, acceleration, and displacement angle, strain and acceleration are greatly affected by material distribution due to the inhomogeneity of masonry materials of ancient pagodas, so displacement parameter is particularly important. It directly reflects the structural damage situation, provides basic data for the functional and safety assessment of buildings after damage, and can also be used as a benchmark parameter for numerical simulation model verification.\u0026nbsp;Figure. 8\u0026nbsp;and\u0026nbsp;Figure. 9\u0026nbsp;are the combined velocity and displacement time-history curves of the recording nodes. As can be seen from the figure, after the explosion, the target position that is most affected by the explosion impact is point A (curve A), which always has the largest velocity and displacement, because point A is almost at the detonation center. Before 1s, no displacement occurred at all points, but with the outward diffusion of the explosion shock wave for 1s, all points began to generate displacement, and the displacement generated by three points A, B, and C on the explosion facing surface is much larger than that generated by D, E, and F on the back side. When the explosion load is applied at the third layer position (explosion 2), the displacement generated by each point is 1.5 ~ 2 times that of the explosion load applied at other positions. Accordingly, before about 0.8s, the speed of each point tends to be stable basically following the law of gradually increasing, and the speed increases significantly after 1s. Among them, the velocity and displacement of point F (curve F) on the back explosion surface increase slowly, indicating that the node closest to the explosion position fails first under each explosion point, and the vertical displacement of the superstructure keeps increasing after the other positions fail successively. The initial damage caused by point F in the symmetric position is much greater than that at position E. Through comparative analysis, the collapse of the structure develops more rapidly on the side of the explosion face. From 1.0 to 1.5s, the velocity and displacement of each point continue to increase, and the pagoda gradually collapses beyond the bearing capacity. Applying the same yield of explosives at different locations, the most violent response is to apply explosive loads at the third layer position.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.3 Time history analysis of interstory displacement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eIn the above section, we analyzed the displacement effects of explosion loads applied at different locations on the corner column nodes of each explosion layer. The results show that significant displacement occurs in each layer under the action of explosion load. Especially after the failure of the pagoda body structure, each layer may experience a large interstory displacement, which is great significance for the analysis of the whole collapse of the structure. Under the five kinds of explosion, the interstory displacement of each layer is shown in\u0026nbsp;Figure 10. For example, the explosion load applied at different locations shows the maximum interstory displacement of the explosion layer. Under the loading of explosion shock wave, the interlayer displacement of each layer shows a decreasing trend, in which the interstory displacement generated by the middle explosion position is significantly larger than the explosion load imposed on the bottom and top, indicating that the anti-explosion ability of the middle pagoda is lower than that on the bottom and top. In particular, the third layer, whose interstory displacement reaches nearly 3 times that of the other layers, further reveals the weakness of the central region when bearing the blast load. According to the\u0026nbsp;trend of the data, it can be seen that the area near the explosion point shows a large inter-story displacement, while the area far away from the explosion source shows a small inter-story displacement, which also verifies that the damage of the ancient pagoda is the most serious at the explosion point.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003e3.4 Cloud image of damage\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAs the ancient pagoda is made of masonry splicing and stacking, its vertical collapse resistance is weak, and the fracture and crushing of masonry caused by the explosion make the structure withstand the tensile, bending and torsion stress, which ultimately led to overall collapse (Figure 11). When the blast wave touches the masonry surface, the local pressure increases instantaneously, causing stress concentration. As the shock waves penetrate deep into the pagoda, the masonry undergoes extreme stress, spalling, splintering and sputtering[25-29]. In this process, the impact kinetic energy is gradually transformed into deformation energy and friction heat, resulting in the decline of the strength and toughness of the masonry, deformation and melting phenomenon. At the same time, the masonry is under tensile deformation, the internal pressure increases, the surface micro-cracks expand and intensify, and eventually lead to structural failure. When the explosion load is applied to each layer, the structure failure process as follows: At the beginning, the explosion shock wave quickly spreads throughout the whole pagoda within 0.9s, resulting in stress concentration in the top area of the pagoda. After 1.25s, the explosion load exerted a strong impact on the corner columns of each explosion layer, resulting in the instantaneous failure of the pagoda bottom, and the destruction of the pagoda body and each bond hole. Subsequently, cracks in the bottom of the pagoda spread rapidly, and high-pressure detonation gas filled the entire pagoda layer, exerting great damage to the internal beams and plates. After the failure of the bottom corner column of each layer, the pagoda body, vault and pagoda brake of the superstructure underwent stress redistribution, which caused the failure to spread downward. The explosion shock wave continuously produces negative pressure effect inside the structure, causing the failure of the elements of the explosion facing surface, causing the center of gravity shift of the structure, and finally causes the continuous collapse of the explosion facing surface. When the explosion load is applied to the three layers, the stress at each layer within 0.9s reaches the maximum, and the bond holes of each layer are separated from the pagoda body at the first time, and all components fail successively until collapse. Compared with other explosion points at the same time, the pagoda body of the detonated layer is damaged more severely, resulting in greater overturning and lateral displacement.\u003c/p\u003e"},{"header":"4 Conclusions","content":"\u003cp\u003eIn this study, an ancient masonry structure pagoda in Guangxi, China was used as a prototype to simulate the collapse process of the structure by using the fluid-structure coupling method, which can capture both the collapse phenomenon and the dynamic response of the structure. At the same time, the finite element model of 5 explosion points is established for comparative analysis and parameter expansion, and the collapse mechanism of the ancient masonry pagoda is discussed. Through the analysis and comparison of the numerical simulation, the following conclusions are drawn:\u003c/p\u003e\n\u003cp\u003e1) Through comparative analysis of the settings of 5 explosion points, the results show that the damage degree of the explosion layer is the most serious in the area closest to the explosion point, and the damage degree of the explosion surface is significantly higher than that of the back explosion surface.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e2) The vertical resistance to sequential collapse of ancient masonry pagodas is poor, and the masonry near the explosion zone is fractured and crushed, which is the trigger factor for structural collapse.\u003c/p\u003e\n\u003cp\u003e3) After the failure of the member, the structure experiences internal force redistribution. The masonry in the damaged area is forced to withstand tension, bending, and twisting at the same time, and the bricks have serious displacement and broken parts, causing the structure to gradually lose stability. The pagoda body obviously tilts towards the explosion face, the support structure of the explosion layer rises, the mortar connection between the bricks and stones will break, resulting in the pagoda body splitting, the perpendicularity of the whole pagoda is destroyed, and finally the whole pagoda collapsed.\u003c/p\u003e\n\u003cp\u003e4) The parameter analysis shows that compared with the explosion on the 1st and 9th floors of the ancient pagoda, the explosion on the 3rd, 5th, and 7th floors has a more severe response to the pagoda body and causes more significant damage.\u003c/p\u003e\n\u003cp\u003e5) In order to improve the collapse resistance of ancient masonry pagodas in the case of explosion, the following suggestions are put forward: Optimize the explosion-proof design, and enhance the vertical collapse resistance by strengthening the structural reinforcement of the blast-facing surface of the pagoda. Recommendations included the addition of support systems and improved masonry connections to enhance structural stability, while in dynamic load analysis, special attention is paid to the central region to ensure the safety margin of the various hierarchies.\u003c/p\u003e"},{"header":"Declarations","content":"\u003cp\u003e\u003cstrong\u003eDeclaration of conflicting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declared no potential conflicts to the research, authorship, and publication of this article.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eAuthor contributions\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eGxZ collected pagoda parameters, wrote the main manuscript text and drew Fig. 1(b), 2, 3, 4, 5, 6, 7, 8, 9, 10, 11; Tabs. 2, 3, 4, 5, 6, 7; DwL provides research ideas and financial assistance for Structural response analysis of a pagoda under explosion impact in China; YL provides resources and ideas; WW, QH, and YbS build the pagoda model shown in Fig. 1 (a) and Tab. 1. All authors reviewed the manuscript.\u0026nbsp;\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eFunding\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThis research was funded by National Natural Science Fund of China (Nos. 52168072, 51808467), High-level Talent Support Project of Yunnan Province, China (2020).\u003cstrong\u003e\u0026nbsp;\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eData Availability Statement\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eAll data included in this study are available upon request by contact with the corresponding author.\u003c/p\u003e\n\u003cp\u003e\u003cstrong\u003eDeclaration of conflicting interests\u003c/strong\u003e\u003c/p\u003e\n\u003cp\u003eThe authors declared no potential conflicts to the research, authorship, and publication of this article.\u003c/p\u003e"},{"header":"References","content":"\u003col\u003e\n \u003cli\u003eZhang, S.; Liang, J.; Su, X.; Chen, Y.; Wei, Q. 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Evaluation of Seismic Performance of Masonry Stone Pagoda: Dynamic Centrifuge Test and Numerical Simulation Analysis. 2023.\u003c/li\u003e\n \u003cli\u003eDe Iasio, A.; Wang, P.; Scacco, J.; Milani, G.; Li, S. Longhu Pagoda: Advanced Numerical Investigations for Assessing Performance at Failure under Horizontal Loads. Engineering. Structures. 2021, 244, 112715.\u003c/li\u003e\n \u003cli\u003eShakya, M.; Varum, H.; Vicente, R.; Costa, A. Seismic Vulnerability and Loss Assessment of the Nepalese Pagoda Temples. Bull. Earthquake Engineering \u0026amp; Structural Dynamics. 2015, 13 (7), 2197\u0026ndash;2223.\u0026nbsp;\u003c/li\u003e\n \u003cli\u003eLu, J.; Han, X.; Wang, Z.; Li, C. 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Ain Shams Engineering Journal. 2023, 14 (4), 101918.\u0026nbsp;\u003c/li\u003e\n\u003c/ol\u003e"}],"fulltextSource":"","fullText":"","funders":[],"hasAdminPriorityOnWorkflow":false,"hasManuscriptDocX":true,"hasOptedInToPreprint":true,"hasPassedJournalQc":"","hasAnyPriority":false,"hideJournal":true,"highlight":"","institution":"","isAcceptedByJournal":false,"isAuthorSuppliedPdf":false,"isDeskRejected":"","isHiddenFromSearch":false,"isInQc":false,"isInWorkflow":false,"isPdf":false,"isPdfUpToDate":true,"isWithdrawnOrRetracted":false,"journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true},"keywords":"Ancient masonry pagoda, Continuous collapse, Explosion load, Fluid-structure coupling, Damage analysis","lastPublishedDoi":"10.21203/rs.3.rs-5111761/v1","lastPublishedDoiUrl":"https://doi.org/10.21203/rs.3.rs-5111761/v1","license":{"name":"CC BY 4.0","url":"https://creativecommons.org/licenses/by/4.0/"},"manuscriptAbstract":"\u003cp\u003eBased on a nine-story ancient pagoda with important historical and cultural value in Guangxi, China, this study simulated the collapse process of the ancient pagoda under different explosion points by using the fluid-structure coupling method, which can better explore the explosion-proof performance of the ancient pagoda. By constructing the finite element model of 5 explosion points, it is found that the area closest to the explosion point is the most seriously damaged, and the damage of the front explosion surface is obviously higher than that of the back explosion surface. In addition, the study shows that the response of the pagoda body is more severe and the damage is more serious when the number of blasting layers is 3, 5 and 7. In order to optimize the explosion-proof design, it is suggested to strengthen the structure of the surface of the explosion facing, improve the support system and masonry connection mode, and pay attention to the safety margin in the middle area of the pagoda body, so as to provide an effective scheme for the protection technology of the ancient pagoda.\u003c/p\u003e","manuscriptTitle":"Response analysis of a pagoda in China under blast loading","msid":"","msnumber":"","nonDraftVersions":[{"code":1,"date":"2024-11-14 08:12:07","doi":"10.21203/rs.3.rs-5111761/v1","editorialEvents":[{"type":"communityComments","content":0}],"status":"published","journal":{"display":true,"email":"[email protected]","identity":"researchsquare","isNatureJournal":false,"hasQc":true,"allowDirectSubmit":true,"externalIdentity":"","sideBox":"","snPcode":"","submissionUrl":"/submission","title":"Research Square","twitterHandle":"researchsquare","acdcEnabled":true,"dfaEnabled":false,"editorialSystem":"","reportingPortfolio":"","inReviewEnabled":false,"inReviewRevisionsEnabled":true}}],"origin":"","ownerIdentity":"210c701a-5cc1-4f5c-8384-370213db37a0","owner":[],"postedDate":"November 14th, 2024","published":true,"recentEditorialEvents":[],"rejectedJournal":[],"revision":"","amendment":"","status":"posted","subjectAreas":[],"tags":[],"updatedAt":"2024-11-28T23:08:22+00:00","versionOfRecord":[],"versionCreatedAt":"2024-11-14 08:12:07","video":"","vorDoi":"","vorDoiUrl":"","workflowStages":[]},"version":"v1","identity":"rs-5111761","journalConfig":"researchsquare"},"__N_SSP":true},"page":"/article/[identity]/[[...version]]","query":{"redirect":"/article/rs-5111761","identity":"rs-5111761","version":["v1"]},"buildId":"qtupq5eGEP_6zYnWcrvyt","isFallback":false,"isExperimentalCompile":false,"dynamicIds":[84888],"gssp":true,"scriptLoader":[]}