Dynamic Reaction and Damage Evaluation of Reactive Powder Concrete Strengthened Reinforced Concrete Columns Subjected to Explosive Load

Abstract

China has an existing building area of 80 billion square meters, where reinforced concrete structures have a large quantity and a wide surface area. The risk of structures being subjected to blast loading is relatively high. Reactive powder concrete has the specialties of ultra-high toughness, super strength, and a high strength to ponderance ratio. Reinforced concrete (RC) structures strengthened by RPC are called RPC-RC structures, which can easily elevate the explosive load resistance of building structures while also strengthening the building. It is a significant method used in avoiding the collapse of structures under explosive loads. The dynamic reaction and damage evaluation approaches of RPC-RC columns under explosive load have not been deeply studied. For addressing this issue, numerical simulation of RPC strengthened RC columns under explosive load was carried out by LS-DYNA (R10), and the correctness of the numerical simulation was verified by comparing it with relevant experimental results. In this paper, a finite element model of an RPC-RC column was established, and the main factors affecting the anti-explosion performance of an RPC-RC column were studied. The influence of the RPC reinforcement layer parameters (RPC thickness, RPC strength, longitudinal reinforcement ratio, and stirrup ratio) on the dynamic reaction and damage degree of RPC-RC columns was examined. The consequences indicated that the failure mode of the columns after RPC reinforcement can alter from bending shear damage to bending damage. As the thickness and strength of the RPC increases, the longitudinal reinforcement ratio increases, the stirrup ratio increases, and the maximum horizontal deformation of the center point of the RPC reinforced RC columns decreases. For RPC-RC columns with a height of 3–4 m and a width of 300–400 mm under blast loading, columns with an axial compression ratio greater than 0.3 will collapse, while columns with an axial compression ratio less than 0.3 are less likely to collapse. In the light of the calculation outcomes, a formula for reckoning the damage index of RPC-RC columns was proposed, taking into account factors such as proportional distance, axial compression ratio, RPC thickness, longitudinal reinforcement ratio, and stirrup ratio.

1. Introduction

According to data from the National Bureau of Statistics and calculations from the China Academy of Building Research, the existing building area in China is approximately 80 billion square meters, with concrete structures accounting for over 80% of the total. The security of buildings decreases with age, and the demand for reinforcement and renovation of building structures is increasing day by day. RPC has high strength, durability, and toughness and is increasingly being used in the reinforcement of existing buildings. In production and life, safety accidents such as gas leakage, fuel leakage, and chemical explosion occur frequently, which is likely to cause the structures to be subjected to explosive loads and lead to significant economic losses and serious social impacts. For example, the unforeseen explosion event of the hazardous article storehouse in Tianjin had a huge impact and heavy losses.

The dynamic reaction and damage evaluation method of RPC reinforced RC columns are still unclear. Scholars all over the world have carried out research on the dynamic reaction and damage type of structures strengthened with CFRP, foam ceramics, and foam aluminum, such as reinforced concrete slabs, beams and columns, under explosive loading. Crawford et al. [1] conducted dynamic response tests on reinforced concrete columns reinforced with CFRP under close range explosion loads and found that CFRP underwent bending failure with minimal residual deformation at the midpoint under equal explosion load conditions. Hu et al. [2] studied the anti-explosion effect of CFRP on reinforced concrete columns after reinforcement and later established a numerical model based on their experiments. Through parametric analysis of the numerical model, the results showed that when the axial compression ratio was 0.2–0.6, increasing the axial compression could improve the residual axial compression load-bearing ability of CFRP reinforced RC columns. The maximum translation and residual translation of the column decrease with the rise of CFRP thickness, longitudinal reinforcement ratio, and stirrup ratio. For enhancing the anti-explosion ability of a concrete slab, the research group in [3,4] studied the dynamic reaction of a steel foam ceramic reinforced concrete slab (Steel FC-RC) under explosive load and found that increasing the thickness of the steel plate, foam ceramic, and RC slab can reduce the dynamic deformation of the slab. The calculation model of deformation and energy characteristics of a multi-layer foam aluminum protected reinforced concrete slab (MF-RC) is established, and the design method of multi-layer foam aluminum for improving the anti-explosion ability of an RC slab is proposed. Lan [5] introduced the original appliance of CA-RPC in bridge engineering. Through the experimental study on the bending resistance of an equal proportion precast CA-RPC slab, the sustained performance after adding coarse aggregate in RPC was verified, and the outcomes indicated that the slab has high bearing ability and good ductility.

The research on RPC reinforced RC components focuses on static performance. Professor Zheng [6] used RPC as a template to constitute a composite structure of RPC and RC, conducted research on bending, shear performance, and design mean, and used the composite to precast buildings. Professor Liu [7] systematically discussed the influence of steel fiber on the mechanical performance of RPC materials. Through the test, the compressive strength, bending strength, modulus of elasticity, and other mechanical properties of RPC specimens with different steel fiber were tested. Deng [8], Po et al. [9], and Talayeh [10], respectively, studied RPC reinforcement of RC bending, compression, and shear members. The above research demonstrates the merits and foreground of RPC in the reinforcement and improvement of buildings. In the study of dynamic performance of reinforced concrete columns reinforced with RPC, Fan et al. [11] proceeded numerical simulation analysis of RPC reinforced columns subjected to vehicle impact. The study showed that the resistance of columns to vehicle impact was significantly improved after RPC reinforcement compared to unreinforced columns. When a medium-sized truck (total weight 8 t) was impacted at a speed of 60 km/h, the reinforced columns only suffered minor damage, while the unreinforced columns were severely damaged. Mobaraki [12] studied that installing explosion-proof walls can significantly reduce overpressure inside tunnels. Vaghefi [13] carried out numerical simulation with LS-DYNA software to study the dynamic response of a concrete bridge under explosion load. Fu et al. [14] studied the P-I curve method for damage assessment of reinforced columns. Xu [15] conducted explosive resistance tests on RPC targets and high-strength concrete (HSC) targets and carried out numerical simulation analysis. The experimental data show that the explosive resistance of RPC is superior to that of HSC. AkbarzadehBengar [16] investigated the blast resistance performance analysis of an RPC slab and NSC (normal strength concrete) slab. The results obtained show that the blast resistance of RPC is superior to that of normal strength reinforced concrete.

In terms of structural evaluation, Mostafaei [17] use modal analysis as a ponderable instrument for acquiring a structure’s modal parameters. The offered modal identification algorithm is tremendously precise and robust and generates highly reproducible results.

In previous simulation studies, there was a lack of material models for RPC under blast loading. This paper adopts an effective material model for RPC under blast, which can be used to study the influence of blast loading on RPC-RC columns. The influence of RPC reinforcement layer parameters (RPC reinforcement layer thickness, RPC strength, longitudinal reinforcement ratio, and stirrup ratio), column section form, and axial compression ratio on the dynamic reaction and damage index of RPC-RC columns under explosive loads is still unclear. In response to the above issues, this paper studies the dynamic reaction and breakage mode numerical analysis method of RPC-RC columns under explosive loads, revealing the influence of RPC reinforcement layer parameters, column cross-sectional form, and axial compression ratio on the dynamic response of RPC-RC columns. The damage index is used to calculate the degree of reduction in the axial compression bearing capacity of RPC-RC columns after explosion, and a calculation method for the damage index of RPC-RC columns is proposed, which can provide reference for the design of the blast resistance performance of RPC reinforced RC columns. This paper only studies the blast resistance of concrete columns with some regular sizes, and the blast resistance of RPC reinforced plates, beams, and frame structures may be studied in the future.

2. Finite Element Model of an RPC-RC Column Under Blasting Load

LS-DYNA was used to analyze the dynamic response of RPC-RC columns under blasting loads.

2.1. Finite Element Model for Blast Resistance Analysis of RC and RPC Columns

2.1.1. RC Column and RPC Column Explosion Resistance Test

Baylot and Bevins [18] conducted blast tests on a 1/4 scale two span double-layer reinforced concrete frame to study the dynamic reaction and break of the structure. This article aims to validate the correctness of the established RC column model by numerically simulating the explosion response of columns in the reinforced concrete at the bottom layer of the frame. In the experiment, 7.1 kg of C-4 high-energy explosive (equivalent to 8.0 kg of TNT [19]) was used, with a height of 0.229 m from the ground and a horizontal distance of 1.07 m from the frame column. The form of the frame column is shown in Figure 1.

Reference [20] conducted explosion resistance tests on four RPC columns with fixed supports at both ends, with a column height of 2.5 m, a cross-sectional length of b × h = 200 mm × 200 mm, a compressive strength of 145 Mpa for RPC cubes, a longitudinal reinforcement of 16mm, and a yield strength of 1450 MPa; the diameter of the hoop reinforcement is 8mm, and the yield strength is 300 MPa. The geometric dimensions, longitudinal reinforcement, and hoop reinforcement arrangement of RPC columns are shown in Figure 2. In the experiment, 0.71 kg emulsion explosive (equivalent to 0.5 kg TNT [19]) was used, and the detonation position was located 1.5m directly above the column span.

2.1.2. Material Model

The K&C model is used for ordinary concrete, and the uniaxial compressive strength of the concrete is the input. The rest of arguments are automatically calculated by the program. The material model of the steel bars adopts the MAT-PIECEWISELINEAR-PLASTICITY model, which can simulate the isotropic hardening, dynamic hardening, strain rate effect, and other characteristics of steel bars. It can also simulate the failure of steel bar elements by specifying effective plastic strain and can better reflect the true mechanical properties of steel bars. We added the keyword MAT-ADD-EROSION to the K file to specify the concrete element failure criterion, with the maximum principal strain reaching 0.1 as the element failure criterion. When taking this value, the simulation results and the experimental results match well [21].

RPC adopts a modified K&C model, and the rest of arguments automatically generated by the K&C model are only used for ordinary concrete. On account of the high strength and dense structure of RPC, direct use will result in inaccurate simulation results. On the basis of the dynamic tensile and compressive properties and constitutive model of RPC, the research group [22] considered the strain rate effect, damage effect, strain strengthening, and softening effect of RPC material, and modified the overall failure surface parameters, damage function, and state equation of the K&C model. The parameters of the RPC material model and the state equation are shown in

Table 1. 72-R3 material model parameters.

FT	A0	A1	A1F	B1	B2	B3	UCF
6.946	32.81	0.4463	0.4417	0.5	1.35	1.15	145

and

Table 2. Tabulated-compaction state of equation parameters.

E0	V0	EV1	EV2	EV3	C1	C2	C3
0	1	0	−0.0015	−0.0043	0	31.35	90.6

.

2.1.3. Element Types and Boundary Conditions

In the finite element model of reinforced concrete columns, SOLID164 elements are used for concrete with a size of 5 mm and BEAM161 elements are used for steel reinforcement with a size of 10 mm. In the RPC finite element model, SOLID164 elements with a size of 10 mm are used for concrete and BEAM161 elements with a size of 10 mm are used for steel reinforcement. The grid size in the model needs to correspond to the size of the test specimen, and the concrete column is a scaled-down column, so the selected size is relatively small. Without considering the bond slip between the steel bars, concrete, and RPC, the keyword CONSTINED_LAGRANGE-IN_SOLID is added to the K file to fully bond the two [22]. The steel bars are modeled separately, and the mesh division is also independent of the concrete elements. According to the compression situation of the structural column in the experiment, a vertical load equivalent to 5% of the vertical axial compressive bearing capacity needs to be applied at the top of the column to simulate the upper load that the reinforced concrete column actually bears. The preloading is applied through the keyword LOAD SEGMENT SET. The RC column finite element model and the RPC column finite element model are shown in Figure 3.

2.1.4. Explosion Load Effect

When using LS-DYNA to simulate RC columns under explosive loads, the explosive load is simplified as a linear descending triangular load, and the overpressure data collected from the experiment is directly input into the load curve. When simulating the explosion of RPC columns, the CONWEP method is used, using the keyword LOAD_BLAST_ENHANCED. In the simulation of the blast response of columns with infill walls on both sides, this method has a better simulation effect. The equivalent M of the explosive, coordinates (x, y, z), detonation time t, and the setting of the explosion overpressure segment are inputted. It has good applicability in simulating both close range and long-distance explosions. The proportion distance used in the CONWEP algorithm is between 0.15–40 m/kg^1/3, and the proportion distance verified in this article is between 0.29–0.54 m/kg^1/3, which meets the requirements.

2.1.5. Contrast Between Experimental Results and Numerical Simulation Results

The simulation results of the midpoint deformation of the RC column in the framework over time are contrasted with the experimental results, as shown in Figure 4. The reinforcement of the column is shown in

Table 3. Reinforcement arrangement values.

Longitudinal Reinforcement Ratio (%)	Stirrup Ratio (%)	Ultimate Strength of Reinforcement (MPa)	Yield Strength of Reinforcement (MPa)
1.4	0.12	600	450

. The maximum horizontal deformation at the midpoint of this simulated column is 10.5 mm, and the residual horizontal deformation is 8.9 mm. The maximum deformation in the experiment is 12.4 mm, and the residual deformation is 7.8 mm. The simulated values differ from the experimental values by 15.3% and 12.4%, respectively. The frame column has undergone bending shear joint failure, which is consistent with the experimental results. There are multiple obvious cracks in the tensile area of the concrete at the mid span, and the distribution of effective stress and plastic strain along the column is similar to the experimental results in reference [18]. The correctness of the RC column established in this simulation has been verified.

The experiment in reference [20] measured the deformation time history curve of RPC columns at mid span. Figure 5 shows the time history curve of the mid span deformation of the simulated RPC column, with a maximum deformation of 2.26 mm. The maximum deformation measured in the experiment is 2.01 mm, with a difference of 11.06% compared to the simulated results. It can be concluded that the simulated values are in good agreement with the experimental values. The column did not show significant damage, and the plastic effective deformation generated was relatively small, which is similar to the experimental results, verifying the correctness of the finite element model.

2.2. Contrast of Dynamic Deformation Time History Curves

A finite model of the RC column reinforced with RPC is established in Section 2.1. The thickness of the RPC reinforcement layer is 10 mm. When dividing the grids, the element size is taken as 10 mm, the strength grade of the RPC reinforcement layer is 145 MPa, the thickness of the protective layer is 3 mm, and the steel bar configuration and strength are shown in Table 1. Pre-pressure is applied to both the ordinary concrete and RPC parts of the column end to increase the axial compression ratio of the reinforced column to 0.05. The bonding slip effect between RPC and ordinary concrete is not considered at this time, and RPC and ordinary concrete are completely bonded through common nodes.

To contrast the blast resistance representation of RC columns and RPC-RC columns, the model of RPC-RC columns is subjected to the same explosive load as in Section 2.1 (proportional distance Z = 0.535 kg/m^1/3). The construction method of RC columns reinforced with RPC is to chisel and tie steel bars on the surface of the original column, and plant reinforcement on the concrete surface to increase the shear resistance between interfaces. Formwork is erected around the column to pour the RPC. Figure 6 and Figure 7 show the effective plastic strain diagrams of RC columns and RPC-RC columns under explosive load damage. It can be seen from the figures that unreinforced columns undergo shear failure, while reinforced columns do not undergo shear failure and have a smaller degree of bending deformation. Figure 8 shows the deformation time history curves of the RC column and RPC-RC column at the mid span of the column. It can be seen that the maximum deformation of the RPC-RC column is 4.6 mm, and the maximum deformation of the RC column is 10.4 mm. The residual deformation of the RPC-RC column is 3.4 mm, and the residual deformation of the RC column is 8.9 mm. After reinforcement, the maximum deformation and residual deformation at the mid span are reduced by 55.8% and 61.7%, respectively. By comparison, it can be seen that the anti-explosion performance of RC columns reinforced with RPC has been improved compared to unreinforced columns.

2.3. Comparison of Damage Index

To compare the degree of damage between RC columns and RPC-RC columns, the damage index D proposed in reference [23] is used to represent the degree of reduction in the compressive bearing capacity of the column after explosion. The calculation formula is as follows:

$$D=1-{\displaystyle \frac{P_{res}}{P_{ini}}}$$

(1)

where P_ini is the axial compressive bearing ability under RC column explosion load and P_res is the residual axial compressive bearing ability after column explosion load. The residual bearing capacity can be simulated using post-processing using the LS-DYNA software. After the explosion is over, vertical deformation is added to the top of the column until the column collapses. The deformation is applied through the keyword BOUNDARY-PRESRIBED-MOTON-SET, and the peak vertical force applied is the residual bearing capacity. The compressive bearing capacity P_ini of RC columns under explosive loads is calculated using the formula in the ACI-318 standard. The initial bearing ability of RPC-RC columns is directly simulated by applying vertical deformation at the top of the column, and the peak force at the top is the initial bearing ability.

$$P_{ini}=0.85f_c(A_c-A_s)+f_y{A}_s$$

(2)

where f_c is the compressive strength of concrete; f_y is the yield strength of longitudinal reinforcement; A_c represents the cross-sectional area of the column; and A_s is the area of steel reinforcement.

As shown in

Table 4. Residual axial bearing capacity and damage index of the columns.

	Section Size Height Width (mm)	Pini (kN)	Pres (kN)	Damage Index D
RC column	89 × 89	3.32 × 102	1.55 × 101	0.953
RPC-RC column	99 × 99	8.3 × 102	2.63 × 102	0.683

, the residual vertical bearing capacity of the RC column after the explosion load is 15.5 kN, and the damage index D is 0.953. After the explosion load, the vertical bearing capacity of the RPC-RC column is 263 kN, and the damage index D is 0.683. By comparison, it can be seen that using RPC reinforcement can reduce the degree of column damage.

3. RPC Reinforcement Layer Parameter Analysis for an RC Column

The geometric parameters of the RC column section and the reinforcement configuration are shown in

Table 5. Geometric parameters and reinforcement dimension parameters of the RC columns.

Column Height/mm	Section Size Height Width/mm	Longitudinal Reinforcement Diameter/mm	Hoop Spacing/mm	Protective Layer Thickness/mm
3300	300 × 300	20	200	25

. The concrete strength and the reinforcement strength are the same as in Section 2.1. The concrete strength is the strength at the time of reinforcement, which can be determined by testing in actual engineering. The improvement effects of various reinforcement parameters on the blast resistance of RPC-RC columns were compared and analyzed. The thickness of the RPC layers was 20 mm, 30 mm, 40 mm, or 50 mm, the strength of the RPCs was 150 MPa or 170 MPa, the longitudinal reinforcement ratios were 0.882%, 1.764%, 2.645%, 3.527%, and the stirrup ratios were 0.709%, 1.418%, 2.836%. The axial pressure ratio applied at the end of the column was 0.05 when the parametrized analysis was carried out, with the CONWEP method applied to the blast load, with the corresponding TNT explosive yield W = 60 kg. The horizontal distance of the blast center from the column R = 1.5 m, and the height of the explosive from the ground h = 1.65 m (i.e., the midspan position of the column). From the simulated results, the blast overpressure acting on the structure under this blast condition was 50,400 kPa, and the blast impulse was 10,900 kPa·ms.

3.1. RPC Thickness

To study the effect of RPC thickness on the blast resistance performance of RPC-RC columns, the strength of RPC (150 MPa), longitudinal reinforcement ratio (0.882%), and stirrup ratio (0.709%) were held constant and RPC thicknesses of 20 mm, 30 mm, 40 mm, and 50 mm were tested.

Taking RC columns as the basic control, the time history curves of mid span deformation of columns with different RPC thicknesses are shown in Figure 9. The maximum horizontal deformation at the midpoint of the column span is the maximum deformation, and the horizontal deformation at the midpoint of the span when the column vibration tends to be stationary is the residual deformation. The maximum deformation and residual deformation of the column mid span with the variation of RPC thickness are shown in Figure 10. The results of the maximum deformation and residual deformation of the column mid span under different thicknesses of RPC reinforcement under the same working condition are summarized in

Table 6. Maximum deformation and residual deformation of the column span with different RPC thicknesses.

RPC Thickness t (mm)	Maximum Deformation Smax (mm)	Residual Deformation Sres (mm)
0	40.9	22.0
20	27.7	19.8
30	22.7	17.9
40	19.4	16.2
50	17.0	13.9

. From Figure 9 and Figure 10, it can be seen that the maximum deformation and residual deformation at the mid span of the column decrease with the increase in the thickness of the RPC reinforcement layer. The failure mode of the column after RPC reinforcement changed from bending shear failure to bending failure. Comparing the four thicknesses of RPC, the one with a thickness of 20 mm has the most severe damage, and the degree of damage decreases with increasing thickness.

During the development of an effective plastic strain of RC column over time, when t = 1.8 ms, the overall lateral deformation of the RC column is significant, and shear failure occurs first at the support. As the lateral deformation further increases at t = 3.5 ms, significant bending deformation occurs in the mid span and the concrete in the tensile zone cracks, indicating that the failure mode of the unreinforced RC column under this explosive condition is bending shear failure. For an RPC thickness of 20 mm, when t = 3.1 ms, the mid span of the RC column first undergoes bending deformation and the concrete in the tensile zone cracks. The effective plastic strain development trend of the column is similar for the other three thicknesses (30 mm, 40 mm, 50 mm). Therefore, it can be concluded that the failure mode of the column after RPC reinforcement changes from bending shear failure to bending failure. In addition, compared with the three thicknesses of RPC, the one with a thickness of 20 mm suffered the most severe damage, and the degree of damage decreased with increasing thickness.

We calculated the initial axial compressive bearing capacity P_ini, residual axial compressive bearing capacity P_res, and damage index D of the RC columns and columns reinforced with RPC of different thicknesses. The results are shown in

Table 7. Bearing capacity and damage index the strengthened columns with different RPC thicknesses.

RPC Thickness t (mm)	Pini/103 kN	Pres/103 kN	Damage Index D
0	4.91	0.50	0.898
20	8.23	1.57	0.809
30	10.07	3.78	0.624
40	12.00	4.58	0.618
50	14.03	6.34	0.452

. From the table, it can be seen that the increase in RPC thickness leads to an increase in both the initial axial compressive bearing capacity and residual axial compressive bearing capacity of the column, while the damage index D decreases with increasing thickness. This is because as the thickness of RPC increases, the bending stiffness, shear stiffness, and bearing capacity of the column increase. Under the same load, the maximum deformation and residual deformation decrease, and the degree of damage to the column decreases.

3.2. RPC Strength

This section studies the effect of RPC strength on the blast resistance of RPC-RC columns, maintaining the thickness of the RPC layer (40 mm), the longitudinal reinforcement ratio (0.882%), and the stirrup ratio (0.709%). The RPC strength is taken as 150 MPa or 170 MPa. From Figure 11 and Figure 12, the maximum horizontal deformation and residual deformation in the middle of the column span decrease with the increase in the strength of the reinforced layer of RPC, but the increase in strength has little effect on the anti-blast performance of the column.

The failure modes of the RC strengthened lower columns of both strengths are bending failure, which shows bending cracking in the middle of the column span and large rotational deformation at the end of the rear bearing. Maximum deformation and residual deformation of columns with different RPC strengths are shown in

Table 8. Maximum deformation and residual deformation of columns with different RPC strengths.

RPC Strength frpc (MPa)	Smax (mm)	Sres (mm)
0	40.9	22.0
150	18.9	14.6
170	18.3	14.2

.

Table 9. Bearing capacity and damage index.

RPC Strength frpc (MPa)	Pini/103kN	Pres/103kN	Damage Index D
0	4.91	0.50	0.898
150	12.00	4.58	0.618
170	12.92	4.94	0.617

gives the initial axial compressive capacity P_ini, the residual axial compressive capacity P_res, and the damage index D for columns strengthened in different strength classes. From Table 9, it can be seen that the initial axial compressive capacity and residual axial compressive capacity of columns with increasing strength of RPC increase, and the damage index D decreases with increasing strength. This is because as the strength of the RPC increases, the bending stiffness, shear stiffness, and bearing capacity of the column increase, the maximum deformation and residual deformation under the same load decrease, and the damage degree of the column decreases.

3.3. Longitudinal Reinforcement Ratio

To investigate the effect of the longitudinal reinforcement ratio of RPC reinforcement layer on the blast resistance performance of RPC-RC columns, the RPC thickness (40 mm), RPC strength (150 MPa), and stirrup ratio (0.709%) were kept constant. According to the relevant design specifications and the actual needs of the project, the reinforcement ratio of the longitudinal reinforcement of the reinforcement layer should not be too small. For the convenience of study, the reinforcement ratios were selected as 0.882%, 1.764%, 2.645, 3.527%. Figure 13 shows the time history curves of mid span deformation for RC column and RPC-RC columns. Figure 14 shows the variation of the maximum deformation and residual deformation of the RPC-RC column at mid span with the longitudinal reinforcement ratio. It can be seen from the figure that as the longitudinal reinforcement ratio of the RPC reinforcement layer increases, the maximum deformation and residual deformation of the RPC-RC column gradually decrease. The failure mode of RPC-RC columns is bending failure. Maximum deformation and residual deformation of columns with different RPC longitudinal reinforcement ratios are shown in

Table 10. Maximum deformation and residual deformation of columns with different reinforcement ratios.

Reinforcement Ratio (ρs0/%)	Smax (mm)	Sres (mm)
0	40.9	22.0
0.882	19.5	16.2
1.764	18.2	14.1
2.645	17.2	12.7
3.527	16.3	11.6

. Comparing the failure modes of RC columns under the four different reinforcement ratios, the longitudinal reinforcement ratio of 0.882% shows the most severe failure, with obvious cracks appearing at the mid span and support ends. As the longitudinal reinforcement ratio increases, the cracks at the mid span and support decrease. When the longitudinal reinforcement ratio reaches 1.764%, the support does not fail, only bending cracks occur at the mid span.

Table 11. Bearing capacity and damage index.

Reinforcement Ratio (ρs0/%)	Pini/103 kN	Pres/103 kN	Damage Index D
0	4.91	0.50	0.898
0.882%	10.40	4.58	0.618
1.764%	10.59	4.77	0.550
2.645%	10.80	4.91	0.545
3.527%	10.97	5.39	0.509

shows the initial axial compressive bearing capacity P_ini, residual axial compressive bearing capacity P_res, and damage index D of columns reinforced with different longitudinal reinforcement ratios. From the table, it can be seen that as the longitudinal reinforcement ratio of the RPC reinforcement layer increases, both the initial axial compressive bearing capacity and residual axial compressive bearing capacity of the column increase, and the damage index D decreases with the increase in the longitudinal reinforcement ratio. This is because when the RPC longitudinal reinforcement ratio increases, the bending stiffness and bearing capacity of the column increase, and the maximum deformation and residual deformation decrease under the same load, reducing the degree of damage to the column.

3.4. Stirrup Ratio

This section studies the effect of stirrup ratio on the blast resistance of RPC-RC columns, keeping the thickness of the reinforced layer (40 mm), the strength of the reinforced layer (150 MPa), and the longitudinal reinforcement ratio (0.882%) unchanged. The stirrup ratios tested were 0.709%, 1.418%, and 2.836%.

From Figure 15 and Figure 16, the maximum deformation and residual deformation of RPC-RC columns decrease with an increase in the stirrup ratio of the RPC reinforced layers. Maximum deformation and residual deformation of columns with different stirrup ratios are shown in

Table 12. Maximum deformation and residual deformation of the column span under different stirrup ratios.

Stirrup Ratio (ρs1/%)	Smax (mm)	Sres (mm)
0	40.9	22.0
0.709	19.5	16.2
1.418	19.2	15.7
2.836	19	15.4

. In order to further compare the effect of improving the blast resistance of RPC-RC columns with the stirrup ratio, the initial axial compressive capacity P_ini, the residual axial compressive capacity P_res, and the damage index D were calculated for four cases (see

Table 13. Bearing capacity and damage index.

Stirrup Ratio (ρs0/%)	Pini/103kN	Pres/103kN	Damage Index D
0	4.91	0.50	0.898
0.709	10.40	4.58	0.618
1.418	10.40	5.08	0.512
2.836	10.40	5.54	0.467

). From the table, it can be seen that the initial axial compressive capacity of the column is unchanged but the residual axial compressive capacity is increased with the increase in the stirrup ratio of the RPC reinforced layer, and the damage index D decreases with the increase in the stirrup ratio. This is because the bending stiffness and load-carrying capacity of the columns remain unchanged when the ratio of the RPC hoopings is increased. The maximum deformation and residual deformation under the same load decrease, but the decrease is not obvious. The increase in stirrup ratio limits the deformation and crack development of RPC and concrete, and the damage degree of columns is reduced.

3.5. Effect of Section Form on the Dynamic Response of RPC-RC Columns

In view of the large quantity of rectangular columns and circular columns in engineering, the dynamic response law of RPC-RC columns with different section forms is not clear. Square RPC-RC columns and circular RPC-RC columns are established in this section. The square section internal RC columns are the same as the Section 2.1 RC columns. The parameters of the external RPC reinforcement layer are shown in

Table 14. Parameters of the reinforced concrete layer with RPC-RC column.

Sectional Form	RPC Thickness (mm)	RPC Strength (MPa)	Longitudinal Reinforcement Ratio (%)	Stirrup Ratio (%)
Circular (R = 170 mm)	20	150	0.882	0.709
Square (L = 300 mm)	20	150	0.999	0.709

. The circular section column has the same section area, number of columns, and stirrup ratio as the square section column; the external RPC reinforcement thickness is 20 mm; and the number of longitudinal reinforcement and hooping reinforcement is the same as the square section column.

The RPC-RC columns with square section and circular section were simulated under three different blast conditions, with the explosive horizontal distance R of 1.5 m from the column, explosive yields W of 40 kg TNT, 50 kg TNT, 60 kg TNT, and corresponding proportional distances Z of 0.439 m/kg^1/3, 0.407 m/kg^1/3, and 0.382 m/kg^1/3, respectively. The midspan deformation curve of the RPC-RC columns is shown in Figure 17. The maximum horizontal deformation of the square section columns is 14.1 mm when the scale distance is 0.439 m/kg^1/3, and the maximum horizontal deformation of the circular section columns is 10.1 mm. The latter is 28.4% lower than the former. When the scale distance is 0.407 m/kg ^1/3, the maximum horizontal deformation of the square section column is 20.5 mm, and that of the circular section column is 13.0 mm. The circular section column is reduced by 36.4% compared with square section column. When the scale distance is 0.382 m/kg^1/3, the maximum horizontal deformation of the square section column is 26.9 mm, and that of the circular section column is 16.3 mm. The circular section column is 39.4% lower than that of the square section column. From the mid span deformation of the two sections under the above three conditions, the circular section columns have better resistance to deformation than the square section columns under the same conditions. This is due to the transition of the section from square to circular, and the shock waves generated by the blast loads diffracted more visibly on either side of the column, reducing the impact force on the column to about half of its original at the same proportional distance, with less mid span deformation.

3.6. Effect of the Axial Pressure Ratio on the Dynamic Response of RPC-RC Columns

A column, unlike other members such as beams and plates, is a typical compression bending member in which vertical pressure has already been applied to the top of the column before blast loading. The effect of vertical axial pressure ratio and the ratio of vertical pressure to column bearing capacity (calculated from the measured value of material strength) on the blast resistance of the columns are not clear.

This section investigates the effect of axial pressure ratio on the blast resistance of RPC-RC columns, the parameters of which are the same as those of Section 1 RPC-RC columns. Vertical loads tested are 1040 kN, 3120 kN, 5200 kN, and 7280 kN, corresponding to axial pressure ratios of 0.1, 0.3, 0.5, and 0.7, respectively. A set of axial pressure ratios corresponding to six cases, keeping the horizontal distance of the explosive from the column constant at 1.5 m, adjusting the explosive equivalent explosive mass W to increase by 140 kg of TNT per 20 kg of TNT from 40 kg of TNT, for a total of 24 cases of blast load simulation, the outcomes of which are exhibited in Figure 18. It can be observed from the picture that columns with axial pressure ratios less than 0.3 do not collapse when explosive yield W is less than or equal to 80 kg.

In summary, controlling the axial compression ratio of the RPC-RC column to within 0.3 can avoid collapse failure of the RPC-RC column under explosive load, and the axial compression ratio of the RPC-RC column can be reduced by adopting the RPC reinforcement, so the RPC reinforcement layer should be designed to reduce the axial compression ratio to within 0.3. This conclusion is suitable for the blast resistant reinforced design of RPC-RC columns at 3–4 m height and 300–400 mm width.

4. Damage Index Calculation of RPC-RC Columns Under Blast Loading

Based on the Section 2 simulation results, it is known that the damage index of RPC-RC columns is mainly influenced by two factors, the proportional distance and the axial compression ratio. The smaller the proportional distance, the larger the damage index. When the scale distance is determined, the damage index decreases first and then increases with the increase in axial compression ratio. The effects of longitudinal reinforcement ratio, stirrup ratio, and thickness of the reinforced layer on damage index are a linear negative correlation. Based on the damage indices obtained from all simulations, two parabolic curves were used for fitting, and the final fitting formula is as follows:

$$D1=\left\{\begin{array}{l}m{(a-i)}^2+D_0,\left(a\le i\right)\\ n{(a-i)}^2+D_0,\left(a>i\right)\end{array}\right\}$$

(3)

$$i=0.88-3.68e^{-{\displaystyle \frac{\mathrm{z}}{0.82{\mathrm{z}}_0}}}$$

(4)

$$D_0=0.26+37.63e^{-{\displaystyle \frac{\mathrm{z}}{0.073{\mathrm{z}}_0}}}(0.3\le z\le 0.5)$$

(5)

$$m=0.38 {{\displaystyle (\frac{\mathrm{z}}{{\mathrm{z}}_0})}}^{-1.02}$$

(6)

$$n=0.21 {{\displaystyle (\frac{\mathrm{z}}{{\mathrm{z}}_0})}}^{-4.02}$$

(7)

$$k_1=-0.0153{\displaystyle \frac{\mathrm{d}}{{\mathrm{d}}_0}}+1.6, (20 \mathrm{mm}\le \mathrm{d}\le 50 \mathrm{mm})$$

(8)

$$k_2=-3.18\mathrm{q}+1.04, (0.3\%\le q\le 5\%)$$

(9)

$$k_3=-7.5\mathrm{s}+1.04, (0.3\%\le \mathrm{s}\le 3\%)$$

(10)

$$D=D_1{k}_1{k}_2{k}_3, (0\le \mathrm{D}\le 1)$$

(11)

where D₀ is the minimum damage index, a is the axial pressure ratio, z is the proportional distance in m/kg^1/3, d is the thickness of the reinforced layer in mm, z₀ is the unit thickness of 1 mm, z₀ is the unit scale distance of 1 m/kg^1/3, q is the reinforcement ratio of the longitudinal reinforcement of the reinforced layer, s is the stirrup ratio of the reinforced layer, D₁ is the preliminary calculated damage index (considering only the scale distance and axial compression ratio), k₁ is the thickness influence coefficient of the reinforced layer, k₂ is the longitudinal reinforcement ratio influence coefficient of the reinforced layer, k₃ is the stirrup ratio influence coefficient of the reinforcement layer, and d is the final calculated damage index. The formula is suitable for the blast resistance of RPC-RC columns at 3–4 m height and 300–400 mm width. The contrast between the calculated values of the formula and the simulated results is shown in Figure 19; the R² value is 0.982, and the fitting effect is good.

5. Conclusions

(1)

A finite element model for dynamic reaction analysis of RPC-RC columns under blast loading is established, and the validity of the finite element model is verified. The dynamic response and damage degree of RC columns and RPC-RC columns under blast loading were compared. The maximum deformation and residual deformation of RPC-the RC columns in midpoint decreased by 55.8% and 61.7% compared with RC columns, and the damage index D decreased by 28.3%.

(2)

The midspan maximum deformation and residual deformation of RPC-RC columns decrease with the increase in the thickness of the RPC, the strength of the RPC, the longitudinal reinforcement ratio of the reinforced layer, and the hooping reinforcement ratio of the reinforced layer. Axial pressure ratio has great influence on dynamic response of RPC-RC columns. For RPC-RC columns at 3–4 m height and 300–400 mm width, the columns are likely to exhibit collapse failure under blast loading when the axial pressure ratio is greater than 0.3, and the columns are not likely to exhibit collapse failure when axial pressure ratio is less than 0.3.

(3)

Under the same blast loading, the circular section RPC-RC columns have better resistance to deformation than the square section with the same section area and reinforcement.

(4)

Proportional distance and axial pressure ratio have great influence on the damage index of RPC-RC columns. The calculation method of damage index of RPC-RC columns considering the arguments of reinforced layer and axial pressure ratio is proposed.