Analysis of Damage Factors of Reinforced Concrete Frame Under Internal Explosion

Abstract

To explore the mechanisms of the damage to reinforced concrete (RC) frame structures subjected to internal explosions, this paper establishes a precise finite element model (FEM) of an RC frame utilizing ANSYS/LS-DYNA software 14.5. The influence of four important damage factors on the degree of structural damage is systematically analyzed. Specifically, the vertical displacement at the top center of the frame serves as the primary evaluation metric, while the four damage factors are treated as independent variables. An empty column is incorporated as an error term, facilitating a five-factor, four-level orthogonal optimization design for the simulation experiments. Based on this design, a variance analysis of the simulation outcomes is conducted. The results show that by increasing the reinforcement ratio of the beam section and reducing the charge weight, when the explosion point is located at the higher part of the building floor and near the external window, the vertical displacement of the building after the internal explosion can be reduced. The order of the influence degree of each damage factor on the damage to the reinforced concrete frame structure is as follows: explosion floor, charge weight, beam section reinforcement ratio, and explosion horizontal position.

1. Introduction

The conflict of the political situations around the world, the intensification of economic, religious, and other social contradictions, and the outbreak of local wars, have led to more and more terrorist attacks. Terrorist attacks usually target some important buildings as the main target and carry out terrorist activities by placing or dropping explosives [1]. According to the research results of the National Institute of Standards and Technology (NSIT), when the building structure is subjected to a bomb terrorist attack, 85% of the casualties are not caused by the bomb itself, but by the chain reaction of the collapse of the building structure. Therefore, it has become an urgent and important topic to study the damage factors affecting the RC frame structure under an internal explosion load.

Recently, many scholars have carried out a lot of work on the damage mechanism and damage influencing factors of RC frames under an explosion load. Xia et al. [2] established a FEM considering the fluid–solid coupling effect on the explosive–air–structure relationship and studied the damage mechanism of the RC structure under an explosion load. Alina Fatima et al. [3] simulated different types of blast loads using the finite element method and studied the damage effects of various types of blast loads on reinforced concrete frames. J.X. Lu et al. [4] studied the damage to a reinforced concrete frame filled with masonry wall under a blast load using numerical simulation. The results showed that the existence of a masonry wall can reduce the damage to a reinforced concrete frame. Yanchao Shi et al. [5] proposed a damage assessment method for a reinforced concrete frame under a blast load based on the substructure method. The results showed that this method can effectively predict the damage to a reinforced concrete frame under a near explosion. Chen Ou et al. [6] studied the response of the remaining structure of the reinforced concrete frame after the column was subjected to an impact load. The results showed that the damage degree of the remaining structure may be related to the order of the column removal. Similarly, Fan Yi et al. [7] also studied the failure response of reinforced concrete frames after the removal of impact loads on columns. The results showed that compared with direct removal, the removal of impact loads made the reinforced concrete frame more vulnerable to damage. Using numerical simulation, Zhou Jikai et al. [8] studied the influence of parameters such as the location of the failed component, the size of the component, the reinforcement ratio, and the standoff distance on the overall progressive collapse behavior of the frame structure under an explosive load. Zhang [9] studied the influence of different joint forms on the progressive collapse resistance of assembled reinforced concrete frame structures under a local explosion. The results showed that the progressive collapse resistance of assembled monolithic frame structures is better than that of fully assembled frame structures. Through numerical simulation, Yue [10] studied the anti-explosion performance of reinforced concrete structures under a contact explosion. Kuang et al. [11] used LS-DYNA to study the dynamic response of a three-story RC frame under blast loading, and analyzed the effects of the charge weight, section form, reinforcement arrangement, and reinforcement ratio on the structural response. M.Vaghefi et al. [12] used LS-DYNA to study the damage to reinforced concrete bridges under blast loading. The results showed that when the TNT charge weight increased by 39%, the damage effect of the TNT explosion on the bridge deck increased by 100%. B. Mobaraki et al. [13] also used LS-DYNA as a research tool to study the influence of soil type on the dynamic response of a tunnel under an explosion load. The results showed that when the soil type of the buried tunnel is silty clay sand, the damage to the tunnel under an explosion load is the least. Zhu Shengbo [14] studied the failure mode of a reinforced concrete frame under a near explosion using Autodyn. The results showed that the damage to a reinforced concrete structure under an explosion mainly depends on the explosives. Kim et al. [15] also studied the damage response of a reinforced concrete frame under an explosion load using Autodyn. The results showed that different reinforcement methods and stirrup spacing have different effects on the anti-explosion performance of a reinforced concrete frame. Ghada Mousa Hekal [16] studied the influence of the lack of middle columns caused by an explosion on the progressive collapse of the reinforced concrete frame using Abaqus. The results showed that compared with the direct removal of components, the removal of components by an explosion load can predict the collapse of a frame structure more realistically. Baylot et al. [17] carried out an experimental study on the internal and external explosion loads of a 1/4 scaled reinforced concrete frame structure and studied the influence of the external wall on the explosion load and response of the outer column of the structure. Gao et al. [18] designed and fabricated a 1/8 scaled three-story RC frame structure model and studied the influence of different charge weights and detonation positions on the dynamic response and failure mode of the frame structure. Gong [19] carried out an explosion test of a 1/2 scaled two-story RC frame structure and studied the influence of different charge weights and standoff distances on the dynamic response of the frame structure.

The damage mechanism and damage factors of an RC frame structure under an explosion load have attracted wide attention from scholars. However, research on the influence of various damage factors on an RC frame structure is still rarely conducted. Therefore, in this paper, the FEM of the RC frame is established using ANSYS/LS-DYNA finite element analysis software, and the influence of the four damage factors of the beam section reinforcement ratio, charge weight, explosion horizontal position, and explosion floor on the damage degree of the RC frame is analyzed. The variance analysis of the simulation results obtained by the orthogonal optimization design is carried out. The order of the influence degree of the four factors, such as the reinforcement ratio, charge weight, explosion horizontal position, and explosion floor, on the RC frame structure is obtained. The aim is to provide some references for the anti-explosion design and safety protection of an RC frame structure.

2. Establishment of FEM of Reinforced Concrete Frame

2.1. Geometric Model

The reinforced concrete frame model is composed of reinforced concrete beams, slabs, and columns. The geometric model of the building and its reinforcement details are shown in Figure 1. Among them, points A, B, C, and D are set as internal explosion points. According to the Chinese Code for Design of concrete structures (GB 50010-2010) [20], the height of the reinforced concrete beam is 400 mm, the width is 200 mm, and the section size of the reinforced concrete column is 400 × 400 mm. The transverse column spacing is 2200 mm, and the longitudinal column spacing is 3300 mm. The height of each floor of the building is 3300 mm; it has a total of nine floors, and each floor has a total of 24 columns and 38 beams. Four steel bars with a diameter of 24 mm are used in the column of the frame structure. The section reinforcement ratio is 1.13%, the diameter of the stirrup is 8 mm, and the spacing of the stirrups in the column is 300 mm. The beam adopts upper and lower double-layer reinforcements, which are two 16 mm steel bars, respectively; the section reinforcement ratio is 1%, and the stirrup spacing is 250 mm. The thickness of the floor is 150 mm, the floor adopts double-layer reinforcement, the diameter of the reinforcement is 8 mm, the transverse and longitudinal spacing is 500 mm, and the spacing of each layer of reinforcement is 100 mm.

2.2. FEM

In the modeling, the steel bar adopts the Beam161 element, because the Beam161 element is a three-dimensional beam element, which can simulate axial tension and compression, biaxial bending, and finite strain. It can be used to simulate the finite strain in many practical applications, such as steel bars. The schematic diagram is shown in Figure 2. In ANSYS, the section shape of the Beam161 element is selected as a circle to simulate the steel bar through the section type tab.

The Solid164 unit is used for concrete, air, and explosives. The Solid164 unit is a display three-dimensional solid unit, which is used for the unit division of the solid geometric model. Its geometric shape, coordinate system, and node position are shown in Figure 3. The Solid164 unit is defined by eight nodes, and each node has nine degrees of freedom, that is, translation, acceleration, and velocity in the X, Y, and Z directions.

To improve the calculation efficiency while taking into account the calculation accuracy, the convergence of the grid is verified. The maximum vertical displacement at the top of the RC frame is used as the convergence analysis index. The explosion condition is 300 kg. The third-layer C point is detonated, and the cross-section reinforcement ratio of the beam is 0.5%. The mesh sizes are set to 350 mm, 300 mm, 250 mm, 200 mm, and 150 mm, respectively. As shown in Figure 4, when the mesh size is 200 mm, the vertical displacement of the building has converged, and blindly reducing the grid greatly increases the calculation time. Therefore, the grid size of the beam and column is determined to be 200 × 200 × 200 mm, and the grid size of the floor is 200 × 200 × 75 mm.

For the interaction between the explosion load and frame structure, the S-ALE algorithm is used to simulate the process of the explosion acting on the frame structure. Keyword *CONSTRAINED_LAGRANGE_IN_SOLID is used for the fluid–solid coupling between the air, explosive, and frame structure. Compared with the ALE, the S-ALE is more convenient when generating and modifying the grid, and the calculation time and the required storage space are less [21]. The reinforcement and concrete are modeled separately, and the coupling of the steel bar and concrete is realized by *CONSTRAINED_BEAM_IN_SOLID. Through the *CONTACT_AUTOMATIC_SURFACE_TO_SURFACE keyword, contact is set between the longitudinal reinforcement and the stirrup, steel bar, and concrete. The friction coefficient is set to 0.3. Using *BOUNDARY_NON_REFLECTING, a non-reflective (outflow) boundary is set at the air boundary to simulate the infinite space domain. The bottom of the building is fixed by *BOUNDARY_SPC_SET, and the bottom of the building is set as a rigid surface. To simulate the collapse of the structure after the explosion, the gravity along the y-axis is applied to the structure by *LOAD_BODY_Y. To solve the problem of large mesh deformation, *MAT_ADD_EROSION is used to delete the failure elements. Because there is no uniform standard for the value of the keyword, after a large number of pre-trial calculations, and referring to the value of Ding Yang [22] and others, the failure strain of the concrete is set to a 0.15 maximum principal strain. Due to the huge scale of the model, to save the calculation time, the air domain only covers three layers of the structure where the explosive is located, allowing the observation of the impact of local damage on the overall damage effect. Additionally, because the explosion impact is instantaneous, the explosion duration is set to 100 ms. After the explosion loading is completed, the air and explosive parts are deleted using the restart technique, and the calculation time is set to 2 s to allow the frame structure to collapse under the force of gravity, which can significantly improve the simulation efficiency. The FEM is shown in Figure 5.

2.3. Material Model

The concrete uses the *MAT_RHT material model. This model is more suitable for simulating the dynamic characteristics of concrete under an explosion load. The elastic limit surface, failure surface, and residual strength surface are used to describe the variation in the initial yield strength, failure strength, and residual strength of concrete under an explosion load [23]. The RHT strength model can be divided into five basic parts: failure surface, elastic limit surface, strain hardening, residual failure surface, and damage model.

(1)

Failure surface is defined as a function of stress P, Lode angle θ, and strain rate (${}_{ℇ}{}^{.}$):

$$Y_{fail}=Y_{TXC\left(P\right)}·R_{3\left(\theta \right)}·F_{RATE\left({}_{ℇ}{}^{.}\right)}$$

(1)

$Y_{TXC}$ = $f_c$|$A{(P^{\mathrm{*}}-P_{spall}^{\mathrm{*}})}^N$|, $f_c$ is uniaxial compressive strength, A is the failure surface constant, N is the failure surface exponent, $P^{\mathrm{*}}$ is the pressure normalized by $f_c$, and $P_{spall}^{\mathrm{*}}$ is defined as $P^{\mathrm{*}}$($f_t/f_c$).

$$F_{RATE}=\left\{\begin{array}{c}{\left({\displaystyle \frac{{}_{\mathrm{ℇ}}{}^{.}}{{{}_{\mathrm{ℇ}}{}^{.}}_0}}\right)}^{D}P>f_c/3\\ {\left({\displaystyle \frac{{}_{\mathrm{ℇ}}{}^{.}}{{{}_{\mathrm{ℇ}}{}^{.}}_0}}\right)}^{\alpha }P<f_t/3\end{array}\right.$$

(2)

D is the compression strain rate index, and α is the tensile strain rate index.

Define $R_{3\left(\theta \right)}$ as the third invariant of the model:

$$R_{3\left(\theta \right)}={\displaystyle \frac{2(1-Q_2^2)\cos \theta +(2Q_2-1)\sqrt{4(1-Q_2^2){\cos \theta }^2-4Q_2+5Q_2^2}}{4(1-Q_2^2){\cos \theta }^2+{(1-2Q_2)}^2}}$$

(3)

$$\cos 3\theta ={\displaystyle \frac{3\sqrt{3}{J}_3}{2^{\raisebox{1ex}{$3$}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{J_2}}}$$

(4)

$$Q_2=Q_{\mathrm{2,0}}+{BQP}^{*},0.5\le Q_2\le 1,BQ=0.0105$$

(5)

where $Q_{\mathrm{2,0}}$ represents the ratio of stretching and compression of the meridians.

(2)

The elastic limit surface.

The elastic limit surface is determined by the failure surface:

$$Y_{elastic}=Y_{fail}\text{·}{F}_{elastic}\text{·}{F}_{CAP\left(P\right)}$$

(6)

$F_{elastic}$ is the ratio of elastic strength to failure surface strength, which can be determined based on the input parameters of tensile elastic strength $f_t$ and compressive elastic strength $f_c$. $F_{CAP\left(P\right)}$ is the hat function for the elastic limit surface and is used to limit elastic stress under static hydrostatic pressure:

$$F_{CAP\left(P\right)}=\left\{\begin{array}{c}1,p\le p_u\\ \sqrt{1-{\left({\displaystyle \frac{p-p_u}{p_0-p_u}}\right)}^2}\\ 0,p\ge p_0\end{array}\right.,p_u<p<p_0$$

(7)

(3)

Strain hardening.

Linear hardening is adopted before reaching the peak load, and during hardening, the current yield surface ($Y^{\mathrm{*}}$) is determined based on the elastic limit surface and failure surface:

$$Y^{*}=Y_{elastic}+{\displaystyle \frac{{ℇ}_{p1}}{{ℇ}_{p1}(pre-softening)}}(Y_{fail}-Y_{elastic})$$

(8)

where ${\mathrm{ℇ}}_{p1}\left(pre-softening\right)$ = ($Y_{fail}$ − $Y_{elastic}$)/3$G$ · [$G_{elastic}$ − $G_{plastic}$].

(4)

Residual failure surface.

Definition of residual failure surface:

$$Y_{resid}^{*}={BP}^{*M}$$

(9)

$B$ is the constant of the residual failure surface, and $M$ is the exponent of the residual failure surface.

(5)

Damage.

Starting from the hardening stage, the additional plastic strain of the material leads to a decrease in damage and strength. Damage accumulates through the following equation:

$$D=\sum {\displaystyle \frac{∆{ℇ}_{p1}}{{ℇ}_p^{failure}}}$$

(10)

$${ℇ}_p^{failure}=D_1{(P^{*}-P_{spall}^{*})}^{D_2}\ge {ℇ}_f^{min}$$

(11)

$D_1$ and $D_2$ are damage constants, ${\mathrm{ℇ}}_f^{min}$ is the minimum failure strain, and the failure surface after damage is:

$$Y_{fractured}^{*}=(1-D)Y_{failure}^{*}+{DY}_{residual}^{*}$$

(12)

Shear modulus after damage:

$$G_{fractured}=(1-D){G+DG}_{residual}$$

(13)

$G_{residual}$ is the residual shear modulus.

The * MAT_RHT material model is widely used to simulate the dynamic behavior of concrete. Wang et al. and Zhou et al. [24,25,26] also used the RHT material model as the material model of concrete and achieved good results. This shows that the RHT material constitutive model can better simulate the response of concrete materials under an explosion load. The RHT material parameters of the concrete taken [27] are shown in

Table 1. Concrete material parameters.

Parameter	ρ/(g·cm−3)	${\mathit{f}}_{\mathit{c}}$/MPa	A	B	N	${\mathit{D}}_1$	${\mathit{D}}_2$
value	2.55	40	1.6	0.0105	0.61	0.04	1

.

The steel bar adopts the* MAT _ PLASTIC _ KINEMATIC material constitutive model, which is an elastic–plastic material model related to strain rate that easily fails. As shown in Figure 6, the model considers the strain rate effect of material strength under dynamic load and selects isotropic or isotropic strengthening by adjusting β [0–1]. The Cowper–Simmonds model is used to describe the strain rate effect, as shown in Equation (14).

$$f_y=1+{\left({\displaystyle \frac{{\dot{\epsilon }}_d}{C}}\right)}^{{\displaystyle \frac{1}{P}}}$$

(14)

where C and P are strain rate parameters; the main parameters [28] of the steel bars are shown in

Table 2. Rebar material parameters.

Parameter	ρ/(g·cm−3)	$\mathit{E}$/MPa	PR	SIGY/MPa	β	C	P
value	7.85	200	0.3	400	0	40	5

.

The material of the TNT is described by the * MAT_HIGH _ EXPLOSIVE _ BURN material model. When the material model is used, the * EOS_ JWL state equation must be used to describe the explosive material [29]. The explosion pressure of JWL is shown in Equation (15):

$$P=A\left(1-{\displaystyle \frac{\omega }{R_{1}V}}\right)e^{-R_{1}V}+B\left(1-{\displaystyle \frac{\omega }{R_{2}V}}\right)e^{-R_{2}V}+{\displaystyle \frac{\omega E_{0,TNT}}{V}}$$

(15)

where $V$ is the relative volume of the detonation product; $A$, $B$, $R_1$, $R_2$, $\omega$ are parameters related to the explosive type; $E_{0,TNT}$ is the internal energy per unit volume of the explosive. The material parameters and state equation parameters [30] of the explosives used in this paper are shown in

Table 3. TNT material model and state equation parameters.

Parameter	ρ/(g·cm−3)	$\mathit{A}$/MPa	$\mathit{B}$/GPa	R1	R2	ω
value	1.63	373.75	3.747	4.15	0.9	0.35

.

Air is regarded as an ideal gas, and the *MAT_NULL material model and * EOS_LINEAR_POLYNOMIAL state equation are used to describe it. The pressure is determined by Equation (16):

$$P=C_0+C_1\mu +C_2{\mu }^2+C_3{\mu }^3+\left(C_4+C_5\mu +C_6{\mu }^2\right)E_0$$

(16)

where $C_0$, $C_1$, $C_2$, $C_3$, $C_4$, $C_5$, and $C_6$ are coefficients of the linear polynomial equation; $E_0$ is the internal energy per unit volume. The material parameters and state equation parameters [28] of the explosives used in this paper are shown in

Table 4. Air material model and state equation parameters.

Parameter	ρ/(g·cm−3)	μ	C0, C1, C2, C3, C6	C4, C5	V0
value	0.00128	0	0	0.4	0.9

.

2.4. Model Verification

Zhu et al. [31] carried out an experimental study on a 1/6 scale model of a reinforced concrete frame under an implosion load. The specific parameters of the test can be found by referring to [31]. To verify the accuracy of the calculation model adopted in this paper, the experimental results were compared with the numerical simulation results. In the experiment, the blasting point is located in the center of the frame structure, and the blasting point is 0.235 m away from the base. The explosive is a 50 g and 200 g emulsion explosive. Two acceleration measuring points, A2 and A3, are set at the mid-span of the left beam and the right beam to obtain the acceleration response of the structure under the explosion load. The longitudinal reinforcement of the beam and column is the HRB400 steel bar, and the measured yield strength is 450 MPa. The stirrups of the beam and column are galvanized iron wire, and the measured yield strength is 600 MPa. The concrete is C40 concrete, and the compressive strength is 45.5 MPa. The numerical calculation model of the experiment is established by the same modeling method and material constitutive model used in this paper. The test and FE models are shown in Figure 7. As shown in Figure 8, to improve the computational efficiency while taking into account the calculation accuracy, the mesh size of the finite element model is 10 mm, as determined through mesh convergence verification.

2.4.1. Comparison and Verification of Structural Dynamic Response

In the verification of the dynamic response of the frame structure, the charge weight is a 50 g emulsion explosive. The acceleration responses of A2 and A3 obtained by experiment and simulation are compared. At the same time, to more effectively evaluate and compare the dynamic response obtained by the test and simulation [32], the frequency–power spectral density (PSD) diagram of the structure is obtained using the fast Fourier transform (FFT) method, and the comparison result is shown in Figure 9. It can be seen from the figure that with the increase in explosion distance, the acceleration response of point A2 is smaller than that of point A3. There is no maximum acceleration response at the time point when the first peak arrives. Under the action of structural reflection pressure, the acceleration response of the frame structure gradually increases. With the dissipation of the explosion shock wave, the acceleration response decreases.

Table 5. The peak acceleration obtained by test and simulation.

Measuring Point	Acceleration/g			Power Spectral Density
	Test	Simulation	Error/%	Test	Simulation	Error/%
A2	8000	7000	12.5	80.63	68.55	14.9
A3	12,000	10,100	15.8	329.87	274.98	16.6

lists the peak acceleration measured by experiment and simulation and the maximum power spectral density obtained by FFT. The peak acceleration of A2 obtained by numerical simulation is 7000 g, and the error with the experimental results is 12.5%. The peak acceleration of A3 obtained by numerical simulation is 10,100 g, and the error with the experimental results is 15.8%. The errors of the maximum power spectral density of A2 and A3 obtained by simulation and experiment are 14.9% and 16.6%, respectively. The discrepancy in dynamic response between the experimental results and numerical simulations is minimal, indicating that the dynamic response of the structure obtained by the two is in good agreement and that the established FEM can better predict the dynamic response of the structure.

2.4.2. Comparison and Verification of Structural Damage Characteristics

The damage characteristics of the structure under the action of a 200 g emulsion explosive are shown in Figure 10. Through comparative analysis, it is found that the numerical simulation is in good agreement with the experimental structure. The crack at the junction of the beam and plate is characterized by punching failure. The roof is characterized by punching failure, and the crack distribution is consistent. The beam–column joints also show punching failure characteristics.

In summary, whether it is the dynamic response or damage characteristics of the structure, the calculation model established in this paper can be well characterized, and the established model can accurately predict the damage to the frame structure under the implosion load.

3. Analysis of RC Frame Damage Factors Under Implosion Load

The damage to the RC frame structure under an internal explosion load is affected by many damage factors, and the damage to the RC frame under an internal explosion is determined by these damage factors. To study the damage and failure of an RC frame under an implosion load, this paper studies the damage to an RC frame using four damage factors: beam section reinforcement ratio, charge weight, explosion horizontal position, and explosion floor.

3.1. Section Reinforcement Ratio of Beam

To quantitatively study the damage to the RC frame under the implosion load and to characterize the overall damage to the RC frame under the explosion load, this paper refers to the research of Ding Yang [22] and Tian Li [33] and takes the vertical displacement at the middle of the top span of the RC frame under the internal explosion load as the damage evaluation index of the RC frame.

The actual longitudinal reinforcement diameter of the reinforced concrete beam in the RC frame is 16 mm, and the section reinforcement ratio is 1%. To study the damaging effect of the section reinforcement ratio of the beam on the RC frame, the longitudinal reinforcement diameters of the beam are set to 11.28 mm, 19.54 mm, and 22.56 mm, respectively, and the corresponding section reinforcement ratios are 0.5%, 1.5%, and 2%. The curve of the vertical displacement at the top of the beam with the change in the section reinforcement ratio of the beam is shown in Figure 11. With the increase in the reinforcement ratio, the vertical displacement decreases gradually, and the damage degree of the RC frame decreases gradually. This outcome results from the increased reinforcement ratio of the beam section, and the bending capacity of the beam increases. When the beam is damaged by the explosion load, the loss of bending capacity decreases, and the damage degree of the RC frame decreases. When the reinforcement ratio is 1%, the vertical displacement of the RC frame is 7.09% lower than that when the reinforcement ratio is 0.5%. When the reinforcement ratio is 1.5%, the vertical displacement of the RC frame is 5.18% lower than that when the reinforcement ratio is 1%. When the reinforcement ratio is 2%, the vertical displacement of the RC frame is 8.15% lower than that when the reinforcement ratio is 1.5%.

3.2. Charge Weight

The vertical displacement of the building with the change in the charge weight is shown in Figure 12. With the increase in the charge weight, the vertical displacement of the RC frame increases, and the damage to the building increases. When the charge weight changes from 400 kg to 500 kg, the vertical displacement of the building increases sharply, and the damage degree of the building increases. The vertical displacements of the top span of the RC frame under 300, 400, 500, and 600 kg are 432, 831, 3080, and 4652 mm, respectively, and the growth rates of the vertical displacement are 92.36%, 270.64%, and 51.04%, respectively.

3.3. Explosion Horizontal Position

To study the influence of the different horizontal positions of the explosion on the damage to the RC frame, and taking into account the symmetry of the building, four different horizontal explosion points, A, B, C, and D, were set. The vertical displacement of the mid-span at the top of the RC frame under different explosive equivalents varies with the horizontal position of the explosion, as shown in Figure 13. In the case of the same charge weight, the vertical displacement of the initiation at point B is the largest, and the damage to the building is the most serious. When the explosion point is set at point A, due to the existence of the explosion vent, the shock wave generated by the explosion is rapidly attenuated, and the pressure acting on the surrounding components is reduced. Therefore, the initiation at point A leads to the minimum vertical displacement of the building and the minimum damage. Compared with point A, the vertical displacement of point B increased by 46.14% on average.

3.4. Explosion Floor

To study the damaging effect of the floor where the explosion point is located on the building when the RC frame explodes, the explosion points are set on the first, second, third, and fifth floors, respectively, and the horizontal position of the explosion at point A is kept unchanged. The influence of the explosion floor on the vertical displacement of the building is shown in Figure 14. With the increase in the floor where the explosion point is located, the vertical displacement of the building decreases, and the damage to the RC frame also decreases. This is because the axial force carried by the columns at the bottom floor of the building is the greatest, and damage to these columns can lead to significant redistribution of internal force within the RC members. Therefore, the lower the explosion floor, the greater the vertical displacement of the building and the greater the damage. When the charge weight is 500 kg, the average reduction rate of the vertical displacement of the building is the largest.

4. Analysis of Variance of RC Frame Damage Results

4.1. Orthogonal Optimization Simulation Scheme Design

When studying the test results under the interaction of multiple levels and multiple factors, if each factor and each level are tested with each other, the number of tests required is amazing. Therefore, there is an urgent need for a scientific design method to design the experiment. The orthogonal optimization experimental design has the advantages of orthogonality, uniformity, and representativeness; so, the orthogonal optimization experimental design method has become one of the most commonly used experimental design methods [34].

To study the influence of four damage factors on the reinforced concrete frame structure under the action of the implosion load, the vertical displacement of the top center of the frame structure is taken as the evaluation index, and the four damage factors—the beam section reinforcement ratio, charge weight, explosion horizontal position, and explosion floor—are taken as the relevant factors, and a column of empty columns is added as the error column. The orthogonal optimization design of five factors and four levels is carried out for the simulation test conditions. The general form of the orthogonal table head is: $L_n\left(r^m\right)$, where $L$ is the orthogonal table symbol, $n$ is the number of tests, indicating that $n$ tests are carried out in total, and $r$ is the number of levels and represents a total of $r$ levels under one factor. $m$ is the number of factors, indicating that there are $m$ factors in the experiment. By looking up the table [35], the orthogonal optimization table head of this paper is designed as $L_{16}\left(4^5\right)$. The factor level table of the simulation test in this paper is shown in

Table 6. Factor level table.

Level	Factor
	Charge Weight/kg	Explosion Horizontal Position	Explosion Floor	Reinforcement Ratio/%	Empty Column
1	500	B	2	1.5%	1
2	600	D	3	1%	2
3	400	C	5	2%	3
4	300	A	1	0.5%	4

.

Table 7. Numerical simulation test conditions.

Scheme	Factor and Level
	Charge Weight/kg	Horizontal Position	Explosion Floor	Reinforcement Ratio/%	Empty Column	Displacement/mm
1	500	B	2	1.5	1	995
2	500	D	3	1	2	674
3	500	C	5	2	3	52.1
4	500	A	1	0.5	4	3961
5	600	B	3	2	4	2620
6	600	D	2	0.5	3	4340
7	600	C	1	1.5	2	4800
8	600	A	5	1	1	61.1
9	400	B	5	0.5	2	45.4
10	400	D	1	2	1	2651
11	400	C	2	1	4	610
12	400	A	3	1.5	3	206
13	300	B	1	1	3	654
14	300	D	5	1.5	4	46.2
15	300	C	3	0.5	1	161
16	300	A	2	2	2	423

is the simulation test scheme designed by the orthogonal test and the vertical displacement of the mid-span at the top of the building of each scheme.

4.2. Damage Process of Reinforced Concrete Frame Under Internal Explosion Load

The damage process of the frame structure under the action of scheme 2 is shown in Figure 15. Under the action of the internal explosion, the floor slab is first subjected to explosion impact, and the floor slab at the location of point D is directly subjected to punching failure. The steel bar in the slab is radially broken, and the column near the explosion source produces brittle failure. With the increase in time, the external support column sustained plastic damage, the damage degree of the internal components intensified, and the upper and lower slabs of the third floor of the building underwent direct shear failure. The concrete at the beam–column joints was broken and peeled off, and the supporting beams were bent and sheared. Finally, under the force of gravity, the building above the three floors of the frame structure collapsed downward.

4.3. Analysis of Variance of Simulation Results Based on Damage Factor Analysis

4.3.1. Basic Principles of Variance Analysis

Variance analysis is a mathematical statistical method based on the variance of each factor to determine the degree of influence of each factor on the test results. In the process of experimental data processing, variance analysis is a very practical and effective method, which can be used to test the significance of the influence of relevant factors on the experimental results during the experiment [35]. There is no limit on the number of factors and the level of factors, and it can be used for the analysis of multi-factor and multi-level test results. The main principle is to first calculate the sum of squares of the deviation of each factor and error column and then calculate the degree of freedom, mean square, and F value. Higher F values indicate a stronger influence of the respective factor on the experimental outcomes, and vice versa.

4.3.2. Calculation Steps of Analysis of Variance

The relevant factors are $X_l\left(l=1,2,···,m\right)$, and the evaluation index is $y_i\left(i=1,2,···,n\right)$. In this paper, the reinforcement ratio, explosion equivalent, explosion horizontal position, explosion floor, and empty column are listed as relevant factors, which are expressed as $X_1$, $X_2$, $X_3$, $X_4$, and $X_5$, respectively. Taking the vertical displacement of the top of the building as the evaluation index, it can be expressed as $y_1$, $···$, $y_{16}$. The basic steps of variance analysis are:

(1)

Calculate the sum of the squared deviations

$$\overline{y}={\displaystyle \frac{1}{n}}{\sum }_{i=1}^n{y}_i$$

(17)

$$Q=\sum _{i=1}^n{y}_i^2$$

(18)

$$P={\displaystyle \frac{1}{n}}{\left(\sum _{i=1}^n{y}_i\right)}^2$$

(19)

Then, the total sum of the squared deviations is:

$${SS}_T=\sum _{i=1}^n{\left(y_i-\overline{y}\right)}^2=Q-P$$

(20)

The sum of the squared deviations of each factor is:

$${SS}_{X_l}={\displaystyle \frac{r}{n}}\left(\sum _{i=1}^n{K}_i^2\right)-P$$

(21)

In Equation (21), $K_i$ is the sum of the corresponding experimental results when the horizontal number is $i$ ($i$ = 1, 2, 3, 4 in this paper).

The sum of the squared deviations of the errors:

$${SS}_e=\sum {SS}_{emptycolumn}$$

(22)

(2)

Calculate degrees of freedom

Total degree of freedom:

$${df}_T=Totalnumberoftests-1=n-1$$

(23)

The degree of freedom of any column:

$${df}_j=Thelevelnumberofthiscolumn-1=r-1$$

(24)

Degree of freedom of error:

$${df}_e=\sum {df}_{emptycolumn}$$

(25)

(3)

Calculate the mean square

$${MS}_{X_l}={\displaystyle \frac{{SS}_{X_l}}{{df}_{X_l}}}$$

(26)

The mean square of the test error is:

$${MS}_e={\displaystyle \frac{{SS}_e}{{df}_e}}$$

(27)

After calculating the mean square, if the mean square of a factor column is less than the mean square of the error, it should be included in the error column to form a new error mean square.

(4)

Calculate the F value

The mean square of each factor is divided by the mean square of the error to obtain the F value:

$$F_{X_l}={\displaystyle \frac{{MS}_{X_l}}{{MS}_e}}$$

(28)

The larger the F value of a factor, the greater the influence of the factor on the experimental results, and vice versa.

4.3.3. Analysis of Variance of Simulation Results

The simulation results obtained by the orthogonal test design are analyzed by variance analysis (Equations (17)–(28)), and the influence degree of the four damage factors on the reinforced concrete frame structure under the implosion load is obtained. The results are shown in

Table 8. Ranking of F values and degree of influence.

Damage Factor	Charge Weight	Explosion Horizontal Position	Explosion Floor	Beam Section Reinforcement Ratio
$F$	10.4283	1.1847	12.5845	3.6578
Ranking of impact levels	2	4	1	3

.

It can be seen from Table 8 that the influence degree of the four damage factors on the reinforced concrete frame structure under the action of the implosion load can be ranked from the greatest to the least as follows: explosion floor > charge weight > reinforcement ratio of beam section > explosion horizontal position.

To further confirm the relative influence of each factor on the vertical displacement of the building, the range analysis method is used to further analyze Table 7. The range calculation results are shown in

Table 9. Ranking of F values and degree of influence.

	Charge Weight	Explosion Horizontal Position	Explosion Floor	Beam Section Reinforcement Ratio
K1	1420.525	1078.6	1592	1151.8
K2	2955.275	1927.8	915.25	499.775
K3	878.1	1405.775	51.2	1436.525
K4	321.05	1162.775	3016.5	2126.85
K	2634.225	849.2	2965.3	1627.075

, where K1, K2, K3, and K4 represent the mean value, and K is the range.

In the study, the greater the range of each factor, the greater the influence of this factor on the selected index. It can be seen from Table 9 that the results are consistent with those obtained in Table 8. Among the four damage factors, the floor where the explosion is located has the greatest influence on the vertical displacement of the building, followed by the explosion equivalent, the reinforcement ratio of the beam section, and the horizontal position of the explosion.

Based on this finding, we propose a series of detailed and clear anti-explosion design suggestions. First of all, given the influence of the floor where the explosion point is located, it is suggested to strengthen the structural strength of the key floors, such as by increasing the cross-section size of the column, using high-strength concrete, and optimizing the connection of the joints. Secondly, given the influence of charge weight, targeted structural design and reinforcement should be carried out to ensure that the structure can withstand the explosion load that may be suffered. At the same time, appropriately increasing the reinforcement ratio of the beam section to enhance the bending and shear resistance of the beam is an effective means of reducing the impact of the blast load on building damage. Although the impact of the horizontal location of the explosion point is relatively small, it should still be considered in the design, by setting up protective barriers, such as explosion-proof doors, windows, and other measures, to block the explosion shock wave and debris. In addition, these anti-explosion design suggestions are not only applicable to the construction industry but can also be extended to high-risk areas such as transportation hubs and chemical industries. In practical applications, anti-explosion design should be taken into account as a whole, and strict anti-explosion standards and norms should be formulated to ensure that new and existing buildings and facilities have sufficient anti-explosion ability, to improve the safety level of the whole society.

5. Conclusions

In this paper, the reinforcement ratio of the beam section, charge weight, explosion horizontal position, explosion floor, and a column of empty columns are taken as relevant factors, and the vertical displacement of the mid-span at the top of the building is taken as the evaluation index. The simulation test scheme is designed using the orthogonal optimization design method, and the variance analysis of the test results is carried out. The influence degree of four damage factors on the RC frame under the implosion load is obtained. The results show that:

(1)

Increasing the reinforcement ratio of the beam section, reducing the charge weight, and setting the explosion point near the outer window of the high-rise building can reduce the vertical displacement of the building after the internal explosion, reduce the damage degree of the building, and improve the anti-explosion ability of the building.

(2)

Under the internal explosion load, the influence degree of various damage factors on the reinforced concrete frame can be ranked from the greatest to the least as follows: explosion floor > charge weight > beam section reinforcement ratio >explosion horizontal position.

(3)

When carrying out the anti-explosion design of RC structures, we should focus on strengthening the key structural parts of the reinforced concrete frame, including strengthening the structural strength of the floor where the explosion point is located (such as by increasing the column section, using high-strength concrete, and optimizing the joints), designing and strengthening according to the possible charge weight, improving the reinforcement ratio of the beam section to enhance the bending and shear capacity, and setting up protective barriers to reduce the impact of different locations where explosions may occur. These methods are not only applicable to the construction industry but also to high-risk areas such as transportation and chemical industries. It is necessary to formulate strict anti-explosion standards to ensure that buildings and facilities have sufficient anti-explosion ability, to improve the overall safety of society.

(4)

The simulation scheme obtained by orthogonal optimization design has good uniformity and representativeness and is a scientific experimental design method. The analysis of variance can simply and efficiently obtain the influence degree of each influencing factor on the experimental results. There is no limit on the number of factors and the level of factors, and it can be effectively applied to engineering practice.

Finally, considering the uncertainty of accidental explosion accidents, this study still has some limitations. Although this study conducted a detailed simulation, the results are for the model used. At the same time, this paper only focuses on the influence of the damage to the beam member on the structure under the explosion load but does not conduct a comprehensive study of the influence of the damage to the column member on the structural response under the explosion load. This may limit the practical applicability in a wide range of RC structures, especially in the actual construction environment, where structural configurations and materials may change. In the future, we will conduct further research on this topic.