Quantifying blast damage and failure mode transition in reinforced concrete columns with a dimensionless model

Abstract

Various accidental explosions and terrorist attacks have occurred frequently, posing a huge threat to the safety of building structures. To enhance the safety of building structures, the finite element model of a reinforced concrete column was established by ANSYS / LS-DYNA finite element analysis software. By comparing with the experimental results, the accuracy of the finite element model was verified, and the failure mode of reinforced concrete columns under different scaled distances was analyzed. The effects of four damage factors: charge weight, axial load ratio(ALR), concrete compressive strength, and stirrup ratio on the dynamic response of column damage were analyzed. Based on the dimensional analysis Π theory, the dimensionless relationship between the damage coefficient of the column based on the residual axial bearing capacity and the explosion scaled distance is obtained. Based on a large number of numerical simulation results, a damage assessment method for reinforced concrete columns subjected to explosion loads is proposed, utilizing dimensional analysis theory. The results show that with the increase of scaled distance, the failure mode of the column will gradually change from brittle shear failure to plastic bending failure. Increasing the axial load ratio, concrete compressive strength and, stirrup ratio, and reducing the charge weight can reduce the damage to the column and improve the blast-resistant performance of the column. It is verified that the proposed dimensionless prediction relationship can better predict the damage degree of reinforced concrete columns at different scaled distances, to provide some reference for the field of structural blast-resistant.

Introduction

With the development of human society, the turbulence of the international situation, and the intensification of political, economic, religious, and cultural conflicts, an increasing number of accidental explosions and terrorist attacks have occurred. Explosive attacks are often the main form of terrorist attacks¹. These events put forward new and urgent requirements and challenges for the blast-resistant design and damage assessment of building structures. It also renders research on blast-resistant reinforced concrete (RC) structures an urgent task^2,3,4.

In recent years, many scholars have carried out a lot of work on the failure mode and dynamic response of reinforced concrete members under blast loading^5,6,7. Kyei et al.⁸ carried out the explosion test and numerical simulation analysis of full-scale reinforced concrete column and studied the influence of different structural measures on the blast-resistant performance of reinforced concrete columns under explosion load. Wang Wei⁹ conducted a series of explosion tests of reinforced concrete beams and slabs, and analyzed the influence of scaled distance, size effect and reinforcement form on the dynamic response of reinforced concrete members. On this basis, the damage degree of components is evaluated, and the P-I curves of different components under different parameters are obtained. Wu Wenyan et al.¹⁰ analyzed the damage factors of reinforced concrete columns under explosive load. The effects of five damage factors, such as explosive equivalent, stirrup reinforcement ratio, concrete strength, standoff distance and longitudinal reinforcement ratio, on the dynamic response of column damage were studied by numerical simulation. Xuekang Guo et al.¹¹established a validated LS-DYNA model for reinforced concrete shear wall substructures under internal explosions, explored effects of TNT locations and masses on blast waves and structural responses, found perpendicular TNT shift amplified loads on farther walls, and proposed an efficient simplified uniformly distributed overpressure model with small prediction errors. Yang Yunkai et al.¹² studied the influence of different charge parameters on the damage law of reinforced concrete columns under explosion load through the combination of experiment and simulation. Li Chen et al.¹³ studied the effects of axial load ratio, scaled distance, charge diameter-length ratio, column length ratio, longitudinal reinforcement ratio and transverse reinforcement ratio on the failure characteristics and residual axial bearing capacity of columns. Under this premise, an empirical formula for predicting the damage degree of reinforced concrete columns under double-ended initiation considering both charge shape and column axial load ratio was obtained. Jun Li et al.¹⁴ studied the spall failure phenomenon of reinforced concrete columns under blast load through the three-dimensional finite element model of reinforced concrete columns. According to the data obtained from numerical simulation, the empirical formula for predicting the spall failure of columns by column size and steel mesh was obtained. Shi et al.¹⁵ established a numerical analysis model of reinforced concrete columns considering bond-slip between steel bars and concrete, and verified the validity of the model based on the classical scale frame explosion test. On this basis, the failure mode of reinforced concrete columns under explosion load was studied, and the influence of different influence parameters on its dynamic response was analyzed. Yan Qiushi et al.¹⁶ established a coupled analysis model of explosive-air-concrete columns, and the applicability of the model was verified by existing experiments. Through this model, the failure mode of subway station columns under explosive load is studied, and the damage assessment method of subway station columns based on bearing capacity is proposed. The damage of subway station columns is divided into different damage levels, and the damage of typical subway station columns under different explosives is evaluated. In addition to the air explosion scenario, Zhou et al.¹⁷used the ALE method to study the dynamic response and damage of pier columns under underwater explosion, and verified the model. A four-level damage assessment method based on residual bearing capacity was proposed, and a pier collapse risk vulnerability model containing 350 kg explosive was established.

The failure mode and dynamic response of reinforced concrete columns under blast load have attracted wide attention. However, due to the complexity of the problem, the dynamic response of reinforced concrete columns under blast load based on dimensional analysis theory is still rarely studied. In order to make up for the lack of research on the damage effect of structural members under explosion based on dimensional analysis theory, the finite element model of reinforced concrete columns is established by ANSYS/LS-DYNA software. The accuracy of the model is verified by comparing with the experimental results, and the failure modes of reinforced concrete columns under different scaled distances are analyzed. Taking the damage index based on the residual axial bearing capacity of reinforced concrete columns as the damage index, the effects of four damage factors: charge weight, axial load ratio, concrete compressive strength and stirrup ratio, on the dynamic response of column damage were studied. The dimensionless relationship between the damage index of the column based on the residual axial bearing capacity and the scaled distance is obtained by the Π theory of dimensional analysis. Based on the numerical simulation results, the dimensionless relationship formula between the dynamic response of the column damage and the scaled distance is summarized, in order to provide some reference for the field of structural blast-resistan.

Numerical calculation of reinforced concrete column under explosion load

Modelling

According to the Chinese Code for Design of Concrete Structures (GB 50010 − 2010)¹⁸, the size of the reinforced concrete column is 0.5 m × 0.5 m × 3.3 m, the concrete strength is C40, the reinforcement is HRB400, the longitudinal reinforcement is 6 φ 24, the stirrup is φ 10 @ 200, the thickness of the protective layer is 15 mm, the longitudinal reinforcement ratio is 1%, and the stirrup ratio is 0.3%. The finite element model and section reinforcement details of reinforced concrete columns are shown in Fig. 1. The finite element model adopts separate modeling. In the model, the concrete is modeled by SOLID164 unit, the steel bar is modeled by BEAM161 unit, and the bond relationship between the steel bar and the concrete is realized by the keyword * CONSTRAINED_BEAM_IN_SOLID. The constraint mode of reinforced concrete column is fixed at both ends, and the model adopts cm-g-us unit system. The explosion point is set directly above the middle of the column span, and the explosion load is applied by the S-ALE algorithm. When the reinforced concrete column is subjected to explosion load, the mesh will be deformed, resulting in inaccurate calculation. In order to solve the problem of grid distortion, on the basis of the previous trial calculation and reference to the research of Cao Yuhang et al.¹⁹, the erosion algorithm * MAT_ADD_EROSION is used to delete the failure unit of concrete in the model, and the threshold of deleting the failure unit is set to 0.1 maximum principal strain.

Fig. 1

Geometric model and finite element model of reinforced concrete slab. (a) is the Finite element model. (b) is the geometric model.

Damage evaluation criteria of RC columns

Many scholars have studied the damage assessment method of reinforced concrete columns under blast loading, which can be summarized as displacement-based evaluation criteria and force-based evaluation criteria. However, under the blast load, the reinforced concrete column may have a large damage, but it will not produce a large displacement. Therefore, the damage evaluation criterion based on displacement has inherent limitations. Therefore, this paper proposes to evaluate the damage of RC columns based on the residual axial bearing capacity of RC columns under explosive load, and defines the damage index D of RC columns :

$$\:D=1-\frac{{P}_{residual}}{{P}_{N}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(1)

where, $\:{P}_{residual}$ is the residual bearing capacity of the column, and $\:{P}_{N}$ is the axial bearing capacity of the column. In order to evaluate the damage of RC columns under blast load, according to the size of different damage index D values, the damage of RC columns is divided into four cases, as shown in Table 1.

Table 1 Evaluation standard of column damage degree.

Because the calculation process involves the application of the initial stress of the column and the application of the explosion load, it belongs to the combined action of static load and dynamic load. Therefore, it needs to be calculated in a certain order. As shown in the figure, the calculation steps are as follows:

• (1) The initial axial pressure is applied to the column by applying displacement, and the initial axial force is linearly increased to the design value.

• (2) When the stress initialization of the column is completed, the explosive impact load is applied to the column through the explosive-air-structure coupling effect.

• (3) After the explosion load is applied, the air domain is removed by the restart technique, and the load is continuously applied linearly until the column collapses to obtain the residual axial bearing capacity of the column. The schematic diagram of the load application process is shown in Fig. 2.

Fig. 2

Schematic diagram of the calculation process. (a) is the schematic diagram of load application process (b) is the finite element calculation model.

Grid convergence verification

In order to improve the calculation efficiency while taking into account the calculation accuracy, the convergence of the mesh of the finite element model is verified. The residual axial bearing capacity of the column under explosive load is used as the convergence analysis index. The mesh sizes are 15 mm, 20 mm, 25 mm and 30 mm, TNT is 27 kg, and the standoff distance is 1.2 m. As shown in Fig. 3, the damage of concrete under different grid sizes is basically the same. When the grid size is 20 mm, the residual axial bearing capacity obtained has converged. Therefore, in the subsequent research and analysis, the grid size is 20 mm.

Fig. 3

Grid convergence verification. (a) is the damage cloud diagram of concrete columns with different mesh sizes.(b) is the residual axial bearing capacity and calculation time of concrete columns with different grid sizes.

Materials parameters

The concrete in this paper uses the * MAT_RHT material model. The model is more suitable for simulating the dynamic characteristics of concrete under explosion load. The elastic limit surface, failure surface and residual strength surface are used to describe the variation of initial yield strength, failure strength and residual strength of concrete under explosion load²⁰. The * MAT_RHT material model is widely used to simulate the dynamic behavior of concrete. In their research, Wang and Zhou et al.^21,22,23 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 explosion load. 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($\:{}_{\upepsilon}^{.}\:$) :

$$\:{Y}_{fail}={Y}_{TXC\left(P\right)}\cdot{R}_{3\left(\theta\:\right)}\cdot{F}_{RATE\left({}_{\upepsilon}^{.}\:\right)}$$

(2)

$\:{Y}_{TXC}$=$\:{f}_{c}$|$\:{({P}^{\text{*}}-{P}_{spall}^{\text{*}})}^{N}$|, $\:{f}_{c}$ is uniaxial compressive strength, A is the failure surface constant, N is the failure surface exponent, $\:{P}^{\text{*}}$ is the pressure normalized by $\:{f}_{c}$, and $\:{P}_{spall}^{\text{*}}$ is defined as $\:{P}^{\text{*}}$($\:{f}_{t}/{f}_{c}$),

$$\:{F}_{RATE}=\left\{\begin{array}{c}{\left(\frac{{}_{{\upepsilon}}^{.}\:}{{{}_{{\upepsilon}}^{.}\:}_{0}}\right)}^{D\:\:}\:\:\:P>{f}_{c}/3\\\:{\left(\frac{{}_{{\upepsilon}}^{.}\:}{{{}_{{\upepsilon}}^{.}\:}_{0}}\right)}^{\alpha\:}\:\:\:\:P < {f}_{t}/3\end{array}\right.$$

(3)

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:

$$\:{R}_{3\left(\theta\:\right)}=\frac{2(1-{Q}_{2}^{2})\text{cos}\theta\:+(2{Q}_{2}-1)\sqrt{4(1-{Q}_{2}^{2}){\text{cos}\theta\:}^{2}-4{Q}_{2}+5{Q}_{2}^{2}}}{4(1-{Q}_{2}^{2}){\text{cos}\theta\:}^{2}+{(1-2{Q}_{2})}^{2}}$$

(4)

$$\:\text{cos}3\theta\:=\frac{3\sqrt{3}{J}_{3}}{{2}^{\raisebox{1ex}{$3$}\!\left/\:\!\raisebox{-1ex}{$2$}\right.}\sqrt{{J}_{2}}}$$

(5)

$$\:{Q}_{2}={Q}_{\text{2,0}}+{BQP}^{*},\:\:\:\:\:\:0.5\le\:\:{Q}_{2}\le\:1,\:\:\:\:\:\:{BQ}^{\:}=0.0105$$

(6)

where, $\:{Q}_{\text{2,0}}$ 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:

$$\:{Y}_{elastic}={Y}_{fail}\cdot{F}_{elastic}\cdot{F}_{CAP\left(P\right)}$$

(7)

$\:{F}_{elastic}$ 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, used to limit elastic stress under static hydrostatic pressure:

$$\:{F}_{CAP\left(P\right)}=\left\{\begin{array}{c}1\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:, p\le\:{p}_{u}\\\:\sqrt{1-{(\frac{p-{p}_{u}}{{p}_{0}-{p}_{u}})}^{2}\:\:\:\:\:\:\:}\\\:0\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:, p\ge\:{p}_{0}\end{array}\right.,{p}_{u}<p<{p}_{0}$$

(8)

• (3) Strain hardening.

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

$$\:{{Y}^{*}=Y}_{elastic}+\:\frac{{\upepsilon}_{p1}}{{\upepsilon}_{p1}(pre-softening)}\:({Y}_{fail}-{Y}_{elastic})$$

(9)

where, $\:{{\upepsilon}}_{p1}(pre-softening)$=($\:{Y}_{fail}$-$\:{Y}_{elastic}$)/3$\:{G}_{\:}$·[$\:{G}_{elastic}$-$\:{G}_{plastic}$].

• (4) Residual failure surface.

Definition of residual failure surface:

$$\:{Y}_{resid}^{*}={BP}^{*M}$$

(10)

$\:B$ 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:

$$\:{D}^{\:}=\sum\:\frac{{\varDelta\:{\upepsilon}_{p1}}_{\:}}{{{\upepsilon}_{p}}_{\:}^{failure}}$$

(11)

$$\:{{\upepsilon}_{p}}_{\:}^{failure}={D}_{1}{({P}^{*}-{P}_{spall}^{*})}^{{D}_{2}}\ge\:{\upepsilon}_{f}^{min}$$

(12)

$\:{D}_{1}$ and $\:{D}_{2}$ are damage constants, $\:{{\upepsilon}}_{f}^{min}$ is the minimum failure strain, and the failure surface after damage is:

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

(13)

Shear modulus after damage:

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

(14)

$\:{G}_{residual}$ is the residual shear modulus.

The RHT material parameters of concrete taken in this paper⁹ are shown in Table 2:

Table 2 Concrete material parameters.

The steel bar adopts * MAT_PLASTIC_KINEMATIC material constitutive model, which is an elastic-plastic material model related to strain rate and easy to fail. As shown in Fig. 4, the model considers the strain rate effect of material strength under dynamic loading, and selects isotropic strengthening or kinematic strengthening by adjusting the β value [0–1]. The Cowper-Simmonds model is used to describe the strain rate effect, as shown in Formula 15.

$$\:{f}_{y}=1+{\left(\frac{{\dot{\upepsilon\:}}_{d}}{C}\right)}^{\frac{1}{P}}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(15)

where, C and P are strain rate parameters. The material parameters of the steel bars used in this paper²⁴ are shown in Table 3.

Table 3 Rebar material parameters.

Fig. 4

PLASTIC_KINEMATIC model.

The material of TNT explosive is described by * MAT _ HIGH _ EXPLOSIVE _ BURN material model. When the material model is used, the * EOS _ JWL state equation must be used to describe the explosive material²⁵. The expression of JWL explosion pressure is shown in Eq. 16.

$$\:P=A{\left(1-\frac{\omega\:}{{R}_{1}V}\right){e}^{-{R}_{1}V}+B\left(1-\frac{\omega\:}{{R}_{2}V}\right){e}^{-{R}_{2}V}+\frac{\omega\:{E}_{0,TNT}}{V}}^{\:}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(16)

where, V represents the relative volume of detonation products; A, B, R₁, R₂ and ω are parameters related to the type of explosive. $\:{E}_{0,TNT}$ denotes the internal energy per unit volume of the explosive. The material parameters and equation of state parameters of explosives used in this paper²⁶ are shown in Table 4.

Table 4 TNT material model and state equation parameters.

Air is regarded as an ideal gas, and * MAT _ NULL material model and * EOS _ LINEAR _ POLYNOMIAL state equation are used to describe it. The pressure is determined by Eq. 17:

$$\:P={C}_{0}+{C}_{1}\mu\:+{C}_{2}{\mu\:}^{2}+{C}_{3}{\mu\:}^{3}+\left({C}_{4}+{C}_{5}\mu\:+{C}_{6}{\mu\:}^{2}\right){E}_{0}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(17)

where, C₀, C₁, C₂, C₃, C₄, C₅ and C₆ are coefficients of linear polynomial equations; $\:{E}_{0}$ is the internal energy per unit volume. The material parameters and state equation parameters of the air used in this paper²⁴ are shown in Table 5.

Table 5 Air material model and state equation parameters.

Model verification

Verification of bearing capacity of RC column

Shi et al.²⁷ proposed a formula for calculating the axial bearing capacity of reinforced concrete columns :

$$\:{P}_{N}=0.85{f}_{c}\left({A}_{G}-{A}_{S}\right)+{f}_{y}{A}_{S}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(18)

where, $\:{A}_{S}$ is the cross-sectional area of longitudinal reinforcement; $\:{A}_{G}$ is the cross-sectional area of reinforced concrete column; $\:{f}_{y}$ is the yield strength of longitudinal reinforcement; $\:{f}_{c}$ is the axial compressive strength of concrete. In this paper, $\:{A}_{S}$ = 0.002714 m², $\:{A}_{G}$ = 0.25 m², $\:{f}_{y}$ = 400 MPa, $\:{f}_{c}$ = 40 MPa, the calculated $\:{P}_{N}$ = 9.493 × 10⁶ N. As shown in Figure.5, the axial bearing capacity of the column obtained by simulation is 1.02 × 10⁷ N, and the error of the axial bearing capacity of the RC column obtained by simulation is 6.9%. It shows that the established finite element model of RC column can better characterize the axial bearing capacity of RC column.

Fig. 5

The axial force displacement variation diagram of reinforced concrete column.

Verification of residual axial bearing capacity of RC columns under blast load

Shi et al.²⁸ carried out an experimental study on the residual axial bearing capacity of reinforced concrete columns under blast loading. In the experiment, TNT charges of different masses were placed in front of the middle span of the RC column, ranging from 18 kg to 24 kg. The column size was 250 mm × 250 mm × 3000 mm, the thickness of the protective layer was 25 mm, the longitudinal reinforcement was four HRB400 steel bars with a diameter of 22 mm, and the stirrup was HRB400 steel bars with a diameter of 8 mm. In order to verify the correctness of the modeling method and material constitutive model adopted in this paper, by establishing the same finite element model as the test column shown in Fig. 6, the residual axial bearing capacity of the four groups of columns under simulation is obtained. Compared with the test results shown in Table 6, the average relative error between the simulation and the test results is 3.9%, which proves that the established model can well predict the residual axial bearing capacity of reinforced concrete columns under explosive load. The numerical calculation model of the experiment is established by the same modeling method and material constitutive model as in this paper.

Fig. 6

Field blasting test. (a) is the test setting. (b) is the FEM of the reinforced concrete column.

Table 6 Comparison of test²⁸ and simulation results.

Verification of damage characteristics of RC columns under blast loading

In order to verify the damage characteristics of RC columns under blast loading, the test results of Ke-Chiang Wu et al.²⁹ were compared and verified. Figure.7 shows the geometric model and cross-section shape of the reinforced concrete column test carried out by Ke-Chiang Wu et al. The axial compressive strength of concrete is 40 MPa, and the yield strength of longitudinal reinforcement and stirrup is 420 MPa and 280 MPa, respectively. The reinforced concrete column is placed horizontally on the ground without applying axial load. The explosive is placed above the column, and the equivalent is 25 kg of TNT explosive. TNT is placed at the axial distance of 900 mm from the base of the column, and the distance from the column surface is 200 mm.

Fig. 7

Geometric model and section shape of reinforced concrete column.

In order to accurately simulate the dynamic response of RC columns under blast loading, a finite element model consistent with the experimental RC columns was established under the same blast loading. Figure 8 is the comparison diagram of test and simulation. It can be seen from the picture that the concrete under the explosive was completely destroyed. The deformation of longitudinal and transverse steel bars is large. The reinforced concrete column can be regarded as completely destroyed without residual axial bearing capacity. The length of the left erosion zone of the concrete column is 236 mm, and the error is 18% compared with the experimental value of 200 mm. The length of the right erosion zone is 1078 mm, and the error is 7.8% compared with the experimental value of 1000 mm.

Fig. 8

Comparison diagram of test and simulation.

By comparing with the experimental results^28,29, it is found that the calculation model established in this paper can be well characterized whether it is the dynamic response or the damage characteristics of the structure. The finite element model established in this paper can accurately predict the damage of reinforced concrete columns under explosive load.

Result analysis

Analysis of failure mode

In general, there are three possible failure modes of reinforced concrete columns under load : bending failure, bending-shear coupling failure and shear failure³⁰. The specific reasons for the three failure modes are as follows : (1) Shear failure : In the case of a small scaled distance, the peak pressure of the explosion load is large and the duration of the positive pressure is short. Therefore, while the mid-span bending deformation of the reinforced concrete column has not yet developed, the shear force at the support exceeds its ultimate shear stress and failure occurs. (2) Bending-shear coupling failure : With the increase of the scaled distance, the pressure generated by the explosion can make the shear stress of the bearing reach the ultimate shear stress and cause shear failure. At the same time, there is enough duration to make the component produce bending deformation and produce bending failure. (3) Bending failure : The pressure generated by the explosion is small but the duration is long, and the mid-span bending deformation of the column is fully developed.

The damage nephogram of three different failure modes of the column under different explosion scaled distances is shown in Fig. 9. When the scaled distance is 0.17, due to the small standoff distance, the explosion shock wave directly acts on the mid-span of the column. The pressure generated by the explosion directly causes serious collapse damage at the mid-span of the column. When the scaled distance is 0.2, the pressure generated by the explosion is large and the action time is short. The overall displacement of the column is too late to respond, and the shear damage is directly generated at the support, and the shear failure of the column occurs. At the same time, due to the small standoff distance, the middle part of the column has a slight collapse damage. When the scaled distance is 0.31, the bearing of the column reaches the shear failure limit under the explosive load and produces shear failure. At the same time, with the increase of the duration, the displacement response of the column develops to a certain extent, and the bending failure cracks appear in the mid-span of the column, and the bending-shear coupling failure of the column occurs. When the scaled distance increases to 0.5, the peak overpressure generated by the explosion load is small and the duration is large. The displacement response of the column is fully developed, and the column is subjected to bending failure.

Fig. 9

The failure mode of the column under different scaled distances. (a) is Z = 0.17, (spallation failure) (b) is Z = 0.2, (shear failure) (c) is Z = 0.31, (Bending shear coupling failure) (d) is Z = 0.5, (bending failure).

Analysis of damage factors

The damage of reinforced concrete columns under blast load is affected by many damage factors, which together determine the damage of reinforced concrete columns under blast load. In order to study the influence of different damage factors on the failure of the plate, four damage factors were carried out : charge weight, axial load ratio, concrete compressive strength and stirrup reinforcement ratio.

Charge weight

In order to ensure the rationality and applicability of the damage analysis results of reinforced concrete columns under explosive load, the TNT values of various suitcases and luggage bombs are referred to, and the charge weights of 17, 27, 45 and 90 kg are taken for research^31,32,33,34. The change curve of the damage index of the column with the charge weight is shown in Figure.10.As the charge weight increases, the damage index of the column under the explosion load gradually increases, and the damage degree of the column gradually increases. However, the growth rate of the damage index gradually decreases. When TNT was 17, 27, 45 and 90 kg, the damage index of the column was 0.2167, 0.4382, 0.6529 and 0.851, respectively, and the damage growth rate was 102.2%, 48.99% and 30.34%.

Fig. 10

The influence of charge weight on column damage.

Axial load ratio

The column member bears the effect of bearing axial pressure in the building structure, which has an impact on the damage response of the column under explosion load. Therefore, according to GB50010, four research conditions of column axial load ratio are designed, and the axial load ratio is 0.1, 0.2, 0.3 and 0.4 respectively. As shown in Fig. 11, when the scaled distance is 0.1, due to the large explosion load, the columns under different axial load ratios have large damage. When the scaled distance is greater than 0.2, the damage index of the column increases gradually with the increase of the axial load ratio. This is because with the increase of the axial load ratio, the column has been subjected to high pressure stress in the static load stage, which weakens the confinement effect and dynamic fracture energy of the concrete and reduces the plastic strain reserve of the steel bar. Under the action of explosion load, the superposition of dynamic stress and high static load stress makes the concrete and steel bars reach the strength limit earlier, which leads to the damage mode dominated by brittle failure. Therefore, with the increase of the axial load ratio of the column, the damage index of the column increases, and the column is more likely to be damaged under the explosion load.

Fig. 11

Effect of ALR on column damage.

Concrete compressive strength

In order to study the influence of concrete strength on column damage, the concrete strength was set to 30, 40, 50 and 60 MPa respectively, and the damage degree of four kinds of concrete strength columns at different scaled distances was analyzed, as shown in Fig. 12. With the increase of concrete strength, the column damage index decreases significantly. When the scaled distance is 0.4, the damage indexes corresponding to the four concrete strengths are 0.5191, 0.3814, 0.1667 and 0.0671 respectively, and the damage reduction rates of the columns are 26.53%, 59.29% and 59.75% respectively.

Fig. 12

The influence of concrete compressive strength on column damage.

Stirrup reinforcement ratio

The actual stirrup diameter of the reinforced concrete column is 10 mm, and the reinforcement ratio is 0.3%. In order to study the influence of stirrup ratio on the damage of reinforced concrete columns, the diameters of steel bars are set to 13.35 mm, 18.89 mm and 23.13 mm respectively, and the corresponding reinforcement ratios are 0.5%, 1.0% and 1.5%. The damage of four stirrup ratios at different scaled distances is shown in Figure.13. High stirrup ratio increases the compressive strength and ultimate compressive strain of concrete by enhancing the three-way constraint effect of core concrete. At the same time, stirrups enhance the shear capacity of reinforced concrete columns by directly resisting shear stress and delaying the buckling of longitudinal reinforcement. Therefore, when the stirrup ratio of the column increases from 0.3 to 1.5%, the damage index of the column decreases by 22.91% on average.

Fig. 13

Effect of stirrup reinforcement ratio on column damage.

Dynamic response prediction of plate based on Π theorem of dimensional analysis

Theoretical analysis

The dynamic response of reinforced concrete columns under blast loading is affected by many physical factors. Therefore, the influence of various physical factors on the dynamic response of reinforced concrete columns can be obtained by dimensional analysis³⁵. For reinforced concrete columns, the damage index under explosion load is the main factor to characterize the damage degree of columns. Therefore, the damage index of reinforced concrete columns under explosion load is mainly studied by dimensional analysis. Through analysis, the main physical parameters affecting the damage index of reinforced concrete columns are:

• (1) Charge parameters : charge weight $\:M$, standoff distance $\:S$, charge density $\:{\rho\:}_{e}$, detonation velocity $\:{V}_{d}$, unit mass energy density $\:\text{e}$, detonation product expansion coefficient $\:{\upgamma\:}$.

• (2) Reinforcement parameters : elastic modulus $\:{E}_{c}$, Poisson ‘s ratio $\:{\mu\:}_{c}$, reinforcement density $\:{\rho\:}_{c}$, reinforcement strength $\:{{\upsigma\:}}_{c}$.

• (3) Concrete parameters : elastic modulus $\:{E}_{0}$, Poisson ‘s ratio $\:{\mu\:}_{0}$, initial density $\:{\rho\:}_{0}$, uniaxial tensile strength $\:{{\upsigma\:}}_{t}$, uniaxial compressive strength $\:{{\upsigma\:}}_{p}$, reinforcement ratio $\:{\eta\:}_{c}$, column depth $\:d$.

Based on the above physical parameters:

$$\:D=f\left(\text{M},{S,\rho\:}_{e},{V}_{d},\text{e},{\upgamma\:},{E}_{c},{\mu\:}_{c},{\rho\:}_{c},{{\upsigma\:}}_{c},{E}_{0},{\mu\:}_{0},{\rho\:}_{0},{{\upsigma\:}}_{t},{{\upsigma\:}}_{p},{\eta\:}_{c},d\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(19)

where, D is the damage index of reinforced concrete columns under explosion load.

In order to comprehensively consider the influence of the charge weight and the standoff distance on the column damage index, the charge weight M and the standoff distance S can be combined into a scaled distance Z to represent:

$$\:Z=S/{M}^{1/3}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(20)

Based on the above analysis, the formula 19 can be rewritten as:

$$\:D=f\left(Z{,\rho\:}_{e},{V}_{d},\text{e},{\upgamma\:},{E}_{c},{\mu\:}_{c},{\rho\:}_{c},{{\upsigma\:}}_{c},{E}_{0},{\mu\:}_{0},{\rho\:}_{0},{{\upsigma\:}}_{t},{{\upsigma\:}}_{p},{\eta\:}_{c},d\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(21)

For the damage of reinforced concrete columns under blast loading, the basic physical quantities involved are mass, length and time, and M-L-T can be used as the basic dimension. As shown in Table 7, the dimension matrix can be written as:

Table 7 M-L-T dimensional matrix.

The basic dimensions are selected as $\:d$, $\:{\rho\:}_{e}$ and $\:{V}_{d}$. As shown in Table 8, the M-L-T dimensional matrix can be rewritten as:

Table 8 The transformed dimensional matrix.

The dimensionless similarity criterion obtained is:

$$\begin{aligned} & \:\varPi\:=D,{\varPi\:}_{1}=Z\sqrt[3]{{\rho\:}_{e}},{\varPi\:}_{2}=\frac{e}{{{V}_{d}}^{2}},{\varPi\:}_{3}=\gamma\:,{\varPi\:}_{4}=\frac{{E}_{c}}{{{\rho\:}_{e}{V}_{d}}^{2}},{\varPi\:}_{5}={\mu\:}_{c},{\varPi\:}_{6}=\frac{{\rho\:}_{c}}{{\rho\:}_{e}},{\varPi\:}_{7}=\frac{{\sigma\:}_{c}}{{{\rho\:}_{e}{V}_{d}}^{2}},\\ & \:{\varPi\:}_{8}=\frac{{E}_{0}}{{{\rho\:}_{e}{V}_{d}}^{2}},{\varPi\:}_{9}={\mu\:}_{0},{\varPi\:}_{10}=\frac{{\rho\:}_{0}}{{\rho\:}_{e}},{\varPi\:}_{11}=\frac{{\sigma\:}_{t}}{{{\rho\:}_{e}{V}_{d}}^{2}},{\varPi\:}_{12}=\frac{{\sigma\:}_{p}}{{{\rho\:}_{e}{V}_{d}}^{2}},{\varPi\:}_{13}={\eta\:}_{c}\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\end{aligned}$$

(22)

Taking into account the actual situation, the material density and other physical parameters of concrete and steel bar are constant, then the dimensionless variables $\:{\varPi\:}_{2}$, $\:{\varPi\:}_{3}$,…, $\:{\varPi\:}_{13}$ are constants:

$$\:\varPi\: = \alpha\:\cdot F\left({\varPi\:}_{1}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(23)

Therefore

$$\:D=\alpha\:\cdot\left(Z\sqrt[3]{{\rho\:}_{e}}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(24)

In the formula : $\:\alpha\:$ is a constant; $\:F\left(\cdot\right)$ is the function between damage index and charge parameters.

Prediction of dimensionless damage index

Taking the scaled distance as 0.1 $\:m/{kg}^{1/3}$, 0.15 $\:m/{kg}^{1/3}$, 0.2 $\:m/{kg}^{1/3}$ · ·, 0.25 $\:m/{kg}^{1/3}$, 0.5 $\:m/{kg}^{1/3}$, the variation curve of D with the dimensionless scaled distance $\:Z\sqrt[3]{{\rho\:}_{e}}$ is shown in Figure.14.As the dimensionless scaled distance increases, the damage index D of the column gradually decreases, the damage degree of the column gradually decreases, and the decrease rate of the damage index D gradually increases. When the dimensionless scaled distance increases from 1.1769 to 1.7653, the damage index D only decreases by 7.249%, but when the dimensionless scaled distance increases from 5.2959 to 5.8844, the damage index D decreases by 96.38%. This is because with the increase of the scaled distance, the explosion load acting on the reinforced concrete column decays rapidly, and the column can not cause too much damage.

Fig. 14

The damage index of the column under dimensionless scaled distance.

The damage index of the column under explosion load can characterize the strength of the blast-resistant ability of the column. By summarizing and analyzing the damage index of the column under different dimensionless scaled distances, the prediction formula can be developed to quickly predict the blast-resistant ability of the column and protect people ‘s lives and property in terrorist attacks and accidental explosion accidents. Table 8 shows the column damage index under different dimensionless scaled distances. Taking the dimensionless scaled distance as the independent variable and the damage index as the dependent variable, the fitting is obtained

$$\:D=\text{ln}\left(3.15083-0.35777Z\sqrt[3]{{\rho\:}_{e}}\right)\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:$$

(25)

Taking B1, B2 and B3 as the verification group, the D of Eq. 25 is 0.9568, 0.6790 and 0.2610. The damage index D obtained by simulation is 0.9641,0.7056 and 0.2814, and the error is 0.76%, 3.77% and 7.25%. Therefore, the proposed dimensionless relationship can better predict the damage degree of reinforced concrete columns under different scaled distances(0.1< Z ≤ 0.5).

Table 8 The damage index under different dimensionless scaled distances.

Conclusions

• (1) When the scaled distance is small, columns are prone to brittle shear failure. As the scaled distance increases, the failure mode of the column will gradually change from brittle shear failure to plastic bending failure.

• (2) The damage of reinforced concrete columns under blast load is positively correlated with the charge weight and the axial load ratio, and negatively correlated with the stirrup ratio and the compressive strength of concrete. Reducing the initial axial load of the column, increasing the stirrup ratio and the compressive strength of the concrete can reduce the damage index of the column and improve the blast-resistant performance of the column.

• (3) The damage index of reinforced concrete column under blast load can reflect the strength of the blast-resistant ability of the column. With the increase of the scaled distance, the damage index decreases and the blast-resistant ability becomes stronger. The dimensionless prediction formula of the column damage index is proposed. After testing, it can better predict the damage response of the column under different dimensionless explosion scaled distances, which provides an important basis for quickly predicting the damage of the column under explosion load.

Finally, due to the randomness of the explosion, there are still some limitations in this study. Although detailed simulations have been carried out in this study, the results are only applicable to the models used. At the same time, this paper only focuses on the damage law of square columns under explosion load, but does not conduct a comprehensive study on the damage law of other cross-section forms such as circular columns and different explosion constraints such as lateral constraints under explosion load. This may limit its practical applicability in a wide range of reinforced concrete structures, especially in the actual construction environment, structural layout and materials may change. In the future, we will carry out further research on this issue.