The Effect of High-Speed Fragment Impact on the Overall Strength of Concrete Columns Under Pressure Load

Abstract

As a common engineering building material, concrete material is widely used in buildings, bridges, and protective structures. Concrete load-bearing columns are one of the main load-bearing components in buildings. In order to analyze the change rule of strength of plain concrete column under small size impact damage, the impact concrete test of 11 mm prefabricated tungsten alloy spherical fragment at different speeds was carried out, and the damage parameters of concrete were obtained. The numerical simulation was carried out with the concrete material model under the experimental strength. Based on the obtained material parameters, five initial variables of load (10–30 MPa), column side length (0.1–0.3 m), fragment velocity (500–1500 m/s), impact angle (0–45°), and position height (200–400 mm) were numerically simulated. Based on the action law of each variable on the concrete column, a comprehensive numerical calculation of orthogonal optimization with five variables and five levels was carried out. The calculation results show that the structural strength of concrete is mainly affected by the side length of the column, and the initial velocity of the fragment determines the size of the loss mass. The greater the load on the concrete column, the greater the height of the position, and the more easily the column collapses; when the side length of the concrete column reaches more than 250 mm, the fragment has little effect on the overall strength of the concrete column. Through the results obtained in this paper, it can be further extended to the evaluation of damage of building components under different loads, so as to obtain whether the bearing level of damaged concrete components can meet the requirements.

1. Introduction

Concrete is an indispensable material in modern construction engineering. Because of its rich raw materials, low cost, strong plasticity, and high compressive strength, it occupies a dominant position in civil and protective engineering. Compared with materials, such as metal, composite materials, or thick glass used in specific protective structures, concrete has become the first choice for constructing large-area load-bearing components due to its excellent economy and overall structure. Under impact load, concrete dissipates a lot of energy through the generation and expansion of internal micro-cracks, which is a controllable and positive failure mode, rather than a simple elastic deformation. The high density and high compressive strength of concrete make it have huge mass inertia, which can effectively resist and buffer the momentum of the impact, which is crucial to resist the explosion shock wave and large projectiles. Concrete not only provides physical barriers but also has good fire, explosion, and radiation shielding properties, which are unmatched by a single functional metal or glass material. The most important thing is that concrete raw materials are widely available, low cost, and can be poured into any complex shape, which is very suitable for large-scale defense fortifications, infrastructure protection, and other projects [1,2].

In the field of civil engineering, load-bearing columns are the core supporting elements of building structures, and their failure may lead to catastrophic progressive collapse. The current design specifications mainly focus on static and conventional dynamic loads. However, in scenarios such as conflicts or accidental explosions, the structure is more vulnerable to impact threats from small-sized, high-speed flying fragments such as fragmented shrapnel in addition to static loads. Such threats are essentially different from the penetration of large-caliber projectiles: the latter aims to completely penetrate and destroy the target, while small-sized fragments may cause widespread local damage on the load-bearing components. Although it does not necessarily lead to immediate penetration, it will seriously weaken its residual bearing capacity and pose a potential risk to the overall stability of the structure.

Based on the consideration of concrete bearing safety, 60–70% of its maximum bearing capacity is usually used as the basis for determination. When the concrete column is damaged by external action, it will damage the concrete structure formed on its surface or inside. Therefore, it is very important to study the residual structural strength of concrete under specific external conditions. Hansapinyo, C. et al. [3] analyzed the influence of continuous small impact load on reinforced concrete beams through ultimate static load test and obtained the residual stress of the members. Zhao [4] studied the relationship between the compressive strength of concrete and the load path under the constraint of steel tube and provided a reference for the calculation method of concrete strength under external conditions. In view of the deformation effect of different concrete members under high loading rate, scholars have studied the effective compressive strength of concrete members under different loading paths, load angles, and sustained high loads [5,6,7]. It is found that the load conditions and methods have a significant effect on the failure mode of concrete. On the tensile strength and damage of concrete, Shen, Q.C. [8] analyzed the influence of aggregate strength on the fracture surface of concrete with different strength. In the process of different loads, the mechanical properties of concrete columns will continue to be affected, and the compressive strength of concrete under dynamic load will gradually decrease with the increase in initial static load [9].

For large-caliber projectiles, their attack targets are usually clear and are mainly used to destroy core facilities. The research on the penetration damage of medium- and large-caliber projectiles to concrete with different strength has made important progress. Early scholars used energy conservation theory and other methods to verify the impact test results of concrete [10,11,12,13,14]. In the study of the impact of small-caliber projectiles on concrete, Beppu, M. et al. [15] analyzed the local damage of concrete target plates with a thickness of 30–130 mm under high-speed impact. The drop hammer test can be used to study the impact response of large concrete members in low-speed range, including the maximum mid-span displacement, residual displacement response of concrete shear wall under impact load, and back-crack damage of concrete with different thickness [16,17,18]. In addition, a large number of studies have been carried out on target damage and ballistic analysis of projectiles non-normally penetrating concrete [19,20,21]. In the numerical simulation, it is very important to adopt the appropriate concrete material model, state equation, contact algorithm, and yield surface definition for the accurate simulation of concrete structural members [22,23,24,25]. Although the research on projectile impact concrete has achieved fruitful results, there are obvious limitations in the existing work. First of all, a large number of studies have focused on the penetration effect of medium- and large-caliber projectiles on concrete slabs or on the response of components under low-speed impact (such as the drop hammer test). Secondly, there are few studies on the impact of small-sized fragments (such as centimeters) in the ultra-high-speed range above 1000 m/s, and it is even rarer to link these studies with the damage behavior of building load-bearing columns under sustained compression, which constitutes a key knowledge gap. In the actual scene, the load-bearing column is always under continuous compressive load, and its stress state significantly affects the damage evolution and failure mode of the material. Ignoring this key factor will make a huge gap between the impact damage assessment under laboratory conditions and the actual structural response.

In the field of engineering practice, the design and evaluation of ballistic impact resistance of concrete members have formed a series of mature technical standards and guidelines. Among them, the most influential is the U.S. Department of Defense’s ‘Unified Facility Guidelines’ UFC 3-340-01 [26], which provides the most systematic design method, covering the complete process from load calculation to component reinforcement. UFC 4-023-03 [27] focuses on the design of the building as a whole and components, including how to select the protection level, assess the vulnerability of existing buildings, and upgrade measures. In addition, EN 13124 [28]’s test methods for impact, fragmentation, and pressure loads on building envelopes have important reference value for concrete component testing. They define performance levels and rigorous laboratory testing procedures. However, the existing standards and most studies often focus on the anti-penetration performance of the material itself or the component response under ideal boundary conditions. There is still a lack of in-depth research on the mechanical behavior and failure mechanism under the coupling of axial load and high-speed penetration in the actual structure. The purpose of this study is to fill this gap. Through the experiment and numerical simulation of coupled static and dynamic loads, the influence of preloading stress on the anti-penetration performance of concrete members is revealed, which provides a basis for improving the relevant design theory.

In this paper, the damage behavior of concrete columns subjected to the impact of centimeter-level tungsten alloy fragments, with a speed of up to 1500 m/s under continuous axial pressure load, is systematically studied. This working condition more realistically simulates the actual working state of the building load-bearing components under the attack of explosive fragments. By combining experimental observation and refined numerical simulation, the influence of multiple parameters, such as initial compressive stress, fragment velocity, and component size on the damage mode and residual bearing capacity of concrete columns, is revealed. The research results aim to provide direct data support and theoretical basis for the protection design of important buildings and critical infrastructure, especially for evaluating the progressive collapse resistance of building structures after local impact, which has clear engineering guidance value. Based on this, this paper first carried out the penetration experiment of centimeter-level penetrator on concrete and obtained the dynamic response and failure mode of the specimen. Furthermore, based on the material parameters calibrated by the experiments, a numerical model that can reflect the compressive state is established. The damage evolution process of concrete columns under different initial conditions (applied load, component size, initial velocity of fragments) is systematically analyzed, and the influence mechanism of each parameter on the damage degree is discussed in depth.

2. Fragment High-Speed Penetration Concrete Experiment

The experimental platform is a comprehensive system integrating emission, constraint, and measurement. It aims to study the high-speed penetration effect of tungsten alloy spherical fragments on concrete targets. The whole system adopts a modular steel structure, and the stability and safety of the experimental process are ensured by rigid connection. Its core consists of the following three subsystems:

Launch system. The system takes a ballistic launcher with a diameter of 12.7 mm as the core. The device is rigidly fixed on the steel launch pad by bolts to ensure the stability of the launch. Its power source is propellant. By accurately changing the mass of propellant in the shell, the continuous and accurate adjustment of the initial velocity of the fragment in the range of hundreds of meters per second to supersonic speed can be realized. In order to effectively launch spherical fragments that are difficult to fix, the system uses a special polymer (plastic) sabot. The sabot has two functions of positioning package and airtightness: on the one hand, it stabilizes the fragment at the head of the shell; on the other hand, it ensures that the gunpowder gas is sealed efficiently, thus fully promoting the fragment acceleration.

The specimen constraint and recovery system. The system is responsible for the fixation of concrete specimens and the recovery of fragments after the experiment, mainly including the target box and the recovery box. The target box is a welded square steel box, which is used to install standard concrete specimens, with a size of 300 mm × 300 mm × 100 mm and a compressive strength of 35 MPa. The design takes into account the convenience of operation and the safety of the experiment. The top is an open structure, which is convenient for the installation and replacement of the specimen. A circular hole with a diameter of 150 mm is symmetrically opened on both sides of the front and back as the incident and outgoing channels of the fragment. The bottom is rigidly connected to the experimental platform through anchor bolts. In the experiment, the top is covered with a steel plate and tightened with bolts to effectively prevent the displacement of the specimen or box during the impact process and ensure the stability of the boundary conditions. Recycling box: It is a fixed steel box placed behind the target box to collect fragments after penetrating the concrete specimen.

Data acquisition system. The system is based on the on–off velocity measurement method, which is used to accurately measure the velocity before and after the fragment hits the target plate. The core components of the system include speed measuring target paper, rigid target frame, six-channel digital chronometer, and wire. Arrangement: Before and after the concrete target box, a speed measuring unit is arranged. Each unit is composed of a rigid target frame and two sensitive to broken target papers pasted in parallel on its front and rear surfaces. Working principle: Before the experiment, the reference distance between the two speed measuring units is accurately input into the time measuring instrument in advance. In the experiment, the speed measuring circuit is in the conduction state. When the fragment penetrates the two pairs of target papers of the front and rear units in turn, it will cause an instantaneous short circuit of the circuit, and the timer records the time difference in penetrating each pair of target papers accordingly. Data output: The system automatically calculates the impact velocity and residual velocity of the fragment according to the known target paper spacing and the recorded time difference.

The whole experimental platform integrates the launch system, the specimen constraint system, and the data acquisition system through modular design, and realizes the unity of launch stability, boundary constraint reliability, and data acquisition accuracy. This configuration not only fully meets the safety requirements of high-speed impact experiments but also ensures that the experimental data obtained have good repeatability and comparability. The complete configuration of the system is shown in Figure 1.

The 11 mm diameter fragment ensures that the research results have a direct reference value for assessing the vulnerability of building components in real-world conflict or accidental explosion scenarios. Tungsten alloy is selected because of its high density, high hardness, and excellent penetration performance. This makes the fragment produce extremely high impact pressure and clear penetration trajectory under high-speed impact, which is helpful to observe the typical concrete penetration and drop damage phenomenon in the experiment. The setting of the velocity range is designed to systematically cover the complete damage spectrum from critical penetration to hypervelocity impact to reveal different failure mechanisms. The lower limit is set near the estimated ballistic limit. The purpose is to study the critical state of the fragment from ‘embedment’ to ‘penetration’, which is crucial for assessing whether the component has penetrating damage, loss of tightness, or load-bearing function. The upper limit corresponds to the upper limit of the velocity of a typical explosion-driven fragment. At this velocity, the shock wave effect is more and more significant, which can study the transition from quasi-static penetration to hydrodynamic response, and observe more serious back shock and internal spallation damage.

A total of eight tests were carried out on the experiment of high-speed impact of fragments on concrete, and the experimental parameters are shown in

Table 1. Initial velocity and residual velocity (m/s) measured in 8 experiments.

Initial Velocity	608	743	924	987	1096	1121	1305	1476
Residual Velocity	0	0	22	57	165	172	291	384

. In view of the cost and complexity of conducting large-scale penetration experiments, a total of eight sets of tests were designed for the experimental scheme of this study. These test conditions are designed to systematically study the effect of penetration velocity on structural response and failure mode. The design principle is to cover the main physical process from local penetration to overall structural instability, so as to provide high-quality and representative benchmark data for the calibration of subsequent numerical models. The experimental results show that under different parameter combinations, the damage morphology of concrete (such as the size of the front aperture and the range of the back shock zone) shows a consistent and explainable change rule. This systematic trend is a strong basis for calibrating and validating numerical models. The uncertainty in the experiment mainly comes from the inherent heterogeneity and batch dispersion of concrete materials, the slight deviation of the impact attitude and position of the projectile, and the systematic error of high-speed dynamic measurement.

It can be seen from Table 1 that the fragment can pass through the concrete specimen near 900 m/s, which can be considered as the ultimate penetration velocity of the fragment. With the increase in the initial velocity of the fragment, the residual velocity of the fragment increases continuously. When the initial velocity of the fragment is 1400 m/s, the residual velocity of the fragment is still 26%. After the experiment, the concrete specimens were photographed after the middle impact. The failure mode consists of a cone hole on the impact surface, a circular hole with a slightly larger size than the fragment in the middle, and a cone hole on the back of the impact, which is the same as the typical damage form of concrete members, as shown in Figure 2.

3. Numerical Simulation of Small-Size Fragments Impact on Concrete Columns Based on Experimental Materials

In order to study the impact damage of small-sized fragments on concrete columns under certain loads at higher speeds, it is necessary to combine the experimental results of the upper part to obtain the material parameters of tungsten alloy materials and concrete materials under high-speed impact for further research.

The ANSYS LS-DYNA18.2 engineering calculation software is used to simulate the impact process of prefabricated tungsten alloy spherical fragments on concrete specimens. In the numerical calculation, the concrete material is selected as the RHT material model, and the tungsten alloy is selected as the JK material model. The ERODDING failure model is added to the concrete material, so that the failure unit can be deleted in time, thus presenting a damage state similar to the experimental results. The same as the experimental conditions, the numerical simulation was performed for a total of eight times. The comparison between the calculation results and the experimental results is shown in

Table 2. Experimental results and numerical results and error (m/s).

Initial velocity	608	743	924	987	1096	1121	1305	1476
Experimental residual velocity	0	0	22	57	165	172	291	384
Calculate the residual velocity	0	0	10	32	138	143	272	374
Error (%)	0	0	54.5	43.9	16.4	16.9	6.5	2.6

.

It can be seen from Table 2 that the maximum difference between the numerical simulation results and the experiment is 54.5%, and the average value is 17.6%, which has good consistency. It can also be seen from the concrete damage diagram in Figure 2 and Figure 3 that the numerical simulation results show the same failure characteristics as the experiment.

In order to study the damage characteristics of concrete columns subjected to high-speed impact of small-size fragments in building structures, it is necessary to consider the initial parameters of fragments and the structural parameters of concrete columns in many aspects. For high-speed moving fragments, the initial velocity of the fragment and the position of the impact cylinder will have different effects on the damage of the cylinder. The initial velocity of the fragment includes the size and direction, that is, |v| and the angle of the three directions α, β, and δ; the location of the landing point also includes two main factors, the height h from the fixed end and the distance d from the side center point. For concrete columns, the side length l of the column is the main factor affecting the structural strength. In addition, the strength grade C of the concrete material itself is also one of the main influencing factors. The compressive strength of the concrete directly determines its ability to withstand the load. Finally, the load p borne by the concrete column determines its stability; the greater the load, the more likely the concrete column is to lose stability. Through the analysis of the above influencing factors, we mainly analyze the control variables of several main factors. Finally, the orthogonal optimization method is used to analyze the results of the concrete column with 200 mm side length. The load applied to the column is to simulate the axial pressure of the column in practical engineering, aiming to study the influence of stress state on the anti-penetration ability of the material. This range covers the complete spectrum from the normal working state to the critical state before failure and can clearly reveal the nonlinear effect of the load level on penetration damage. Among them, 10 MPa represents a lower working stress level, and 20 MPa is a typical or higher design working stress. Close to or reaching the ultimate strength of ordinary concrete is 30 MPa, representing the ultimate or overload condition. The height of the cylinder mainly considers the size effect and the boundary effect of stress wave propagation. By changing the height, researchers can distinguish whether the anti-penetration performance of the material itself is working, or the dynamic response of the overall structure is working. The 100 mm is a small-sized specimen, and the time for the stress wave to reflect back from the impact point to the bottom fixed end is very short. The 300 mm is medium size, and the stress wave has enough space to develop and interact. The 500 mm is closer to the scale of the actual engineering component. At this scale, the influence of the end boundary condition on the local area of the penetration is reduced, which can better simulate the structural response in the real situation. Penetration velocity is the core input condition in the penetration problem, which directly determines the dynamic mechanism and load rate effect of penetration. This velocity range covers the main penetration scenarios from typical to extreme, and the influence of velocity on the damage mode can be studied comprehensively. The speed of 500 m/s belongs to medium-speed penetration. It is common in the impact velocity of shells and rockets. The speed of 1000 m/s belongs to high-speed penetration. The speed of 1500 m/s belongs to ultra-high-speed penetration. The penetration position height studies the influence of the position of the impact point on the cylinder on the failure mode, especially the interaction with the boundary conditions. For non ‘semi-infinite’ structures, the location of the impact point is crucial. It affects the propagation path of the stress wave, the reflection superposition, and the final structural failure mode. The 200 mm impact point is located in the middle and lower part of the cylinder. The impact point of 300 mm is located in the middle and upper part of the cylinder. The 250 mm for the middle position, used for comparative analysis, confirms the trend. The horizontal penetration angle simulates oblique penetration or non-normal impact, in which the trajectory is not perpendicular to the structural surface. The influence of the penetration angle on the critical condition of bouncing projectile, the stability of penetration trajectory, and the change in damage mode is studied. A small angle oblique penetration is 15°. The angle between the projectile trajectory and the normal line of the cylinder surface is small, and its behavior is close to normal penetration. A medium angle oblique penetration is 30°. The bounce effect and deflection effect are obvious, and the penetration depth will be significantly smaller than the positive penetration. A large angle oblique penetration is 45°.

The numerical simulation conditions and orthogonal optimization conditions of the control variables are shown in

Table 3. The working conditions of each variable on the strength of concrete columns.

Variable	Load (MPa)	Height (mm)	Speed (m/s)	Side Length (mm)	Angle (°)
working condition 1	10	100	500	200	15
working condition 2	20	300	1000	250	30
working condition 3	30	500	1500	300	45

and

Table 4. 5 variables 5 levels orthogonal optimization conditions.

	Working Condition	Level 1	Level 2	Level 3	Level 4	Level 5
Variable
load p (MPa)		10	15	20	25	30
height h (mm)		100	150	200	250	300
angleα (°)		0	11.25	22.5	33.75	45
speed v (m/s)		300	600	900	1200	1500
side length l (mm)		100	150	200	250	300

. The central impact force of the fragment coincides with the axis of the cylinder, which mainly causes local penetration damage and axial stress wave propagation of the component. The overall response is mainly axial compression. The impact force of eccentric impact will produce a significant bending moment on the square column. This makes one part of the cross section of the component bear higher compressive stress, while the other part may bear tensile stress. This stress redistribution will lead to asymmetric damage morphology, and the damage on the impact side will be more serious, while the shock zone on the back may show an asymmetric shape. Therefore, if a column that may only lose part of its cross section but still stand under the central impact is subjected to eccentric impact under the same initial conditions, the initial bending may be continuously amplified due to the bending effect, which eventually leads to the instability of the component and even the progressive collapse of the structure. The eccentric impact will excite more complex overall vibration modes of the component, while the central impact mainly excites the axial vibration. Energy will be more allocated to the overall dynamic response of the component, not just local material breakage, which may lead to a decrease in local penetration depth, but an increase in overall structural damage.

Figure 4 is a schematic diagram of the relevant variables of the high-speed penetration of fragments into concrete under pressure load.

3.1. The Influence of Each Variable on the Concrete Column

Based on the numerical simulation results of Table 3, the following will analyze the damage degree of concrete columns by five variables of load, height, speed, side length, and angle in turn, and briefly analyze the action law of each variable.

3.1.1. Uniform Load p on the Cylinder

The uniform load above the concrete column represents the load on the column. Considering the strength of the concrete itself, if the load exceeds the maximum strength grade of the concrete, the concrete column will directly fail and collapse. When the fragment hits the concrete column with a high degree, it will cause permanent failure damage to the concrete near the impact surface, causing the concrete above the impact position to lose support and move downward under the action of load, thus reducing the bearing area above and increasing the load. In the numerical simulation, the initial velocity of the fragment is set to be 1000 m/s, the height is 100 mm, the side length of the concrete column is 300 mm, the impact angle is 0°, and the loads applied above the column are 10 MPa, 20 MPa, and 30 MPa, respectively. Figure 5 is the damage state of the concrete column by the fragment at 3 ms. Figure 6 is the top-view position of Figure 5.

It can be seen from Figure 6 that as the initial load increases, the collapse area of the concrete column gradually increases, and the collapse expansion to both sides of the column occurs at 30 MPa. The area ratios of the non-downward displacement above the concrete column under the three load conditions are 100%, 91.96%, and 54.35%, respectively. However, there is no significant difference in the direct damage caused by the high-speed impact of the fragments, and there is no significant difference in the trajectory depth and opening diameter of the fragments in the concrete column.

3.1.2. Fragment Penetration Position Height h

The height h of the fragment penetration is the same as the penetration angle α, which indicates the relative position relationship between the fragment and the concrete cylinder when the fragment begins to impact. The height h represents the relative position relationship with the lower fixed constraint end and the upper free end. When the fragment impacts the concrete column at high speed, the closer to the free end above, the more likely it is to cause damage. The closer to the fixed end below, the more difficult it is to cause damage. In the numerical simulation, the side length of the concrete column is set to 200 mm, the pressure load is 30 MPa, the initial speed is 1500 m/s, the penetration angle is 0°, and the distance from the lower fixed end is 200 mm, 300 mm, and 400 mm. Figure 7 is the collapse state of concrete columns corresponding to different heights h at 3.0 ms.

It can be seen from Figure 7 that the larger the penetration height h, the faster the response speed of the upper pressure load and the faster the concrete collapse process. At 3.0 ms, the displacements of the top center of the concrete column at three heights are 28.96 mm, 32.56 mm, and 86.87 mm, respectively. The higher the penetration position of the fragment, the faster the damage and collapse of the cylinder.

3.1.3. The Initial Impact Energy of the Fragment(v)

The initial impact energy of the fragment represents the severity of the impact. From the perspective of momentum, it is not difficult to conclude that the greater the initial impact energy of the fragment, the greater the degree of concrete damage caused by the impact, specifically that the penetration depth becomes larger and the diameter of the hole becomes larger. The increase in the energy of the fragment directly affects the residual support area of the concrete column on the penetration plane, resulting in the stress concentration of the residual concrete part under the initial load at the top. In the numerical simulation, the side length of the concrete cylinder is 300 mm, the penetration height is 100 mm, the initial pressure load is 20 MPa, the penetration angle is 0°, and the initial impact energy of the fragment is 1525 J, 6100 J, and 13,725 J, respectively. Figure 8 shows the damage of concrete columns at three initial impact energy at 3.0 ms. Figure 9 is the pressure cloud diagram of the middle section of concrete at different impact energy.

It can be seen from Figure 9 that with the increase in the initial impact energy of the fragment, the damage area above the concrete column increases continuously, and the area ratios that are not affected by the fragment are 100%, 92.94%, and 52.46%, respectively. As mentioned above, the reason for this situation is very direct, that is, the higher the fragment speed, the greater the penetration depth and the more concentrated the stress on the penetration position. The figure is the concrete cross-section pressure cloud diagram at three speeds at 3.0 ms. The stress on the right side of the penetration path increases significantly compared with the upper and lower sides. The path lengths on the plane where the penetration path is located are 28.11 mm, 56.83 mm, and 79.71 mm, respectively.

3.1.4. The Side Length of Concrete Column l

As one of the important parameters of the maximum load of the concrete column, the side length l of the concrete column directly determines the compressive area of the concrete column. Under the condition of constant concrete strength, the greater the l, the greater the total load it can bear. For the high-speed impact of fragments, if the side length l is larger, the remaining part of the concrete is retained more than the damage area of the fragment, which is equivalent to the reduction in the overall damage degree. In the numerical simulation, the initial pressure load is set to 20 MPa, the initial velocity of the fragment is 1500 m/s, the height of the fragment is 100 mm, the penetration angle is 0°, and the side lengths of the concrete column are 200 mm, 250 mm, and 300 mm, respectively. Figure 10 shows the damage state of concrete columns with three side lengths when 3 ms.

From Figure 10, it can be seen that with the increase in the side length of the concrete column, the damage degree of the concrete column decreases continuously. The undamaged area of the upper surface of the concrete column with three side lengths is 9.83%, 39.15, and 53.52%, respectively. From the structure of the concrete column, it can be seen that the damage of the cross section at the penetration position decreases with the increase in the side length is the main reason. In addition, the damage state diagram of the concrete can be seen. The reason for the gradual increase in the strength of the concrete is also the distance between the damage area on the surface of the impact column and the free surface around the column, as shown in the following diagram. With the increase in the side length of the column, the distance between the impact position of the concrete and the free surface continues to increase, so that the concrete damage caused by the impact cannot be effectively expanded, thereby reducing the overall damage of the concrete column.

3.1.5. Penetration Angle α

The penetration angle α is, like the penetration height h, the relative position parameter of the impact position compared to the concrete column. The angle α controls the impact trajectory direction of the fragment, resulting in different damage forms of the fragment to the concrete column. In the numerical simulation, the side length of the concrete cylinder is controlled to be 200 mm, the penetration height is 300 mm, the fragment velocity is 1000 m/s, the initial pressure load is 20 MPa, and the penetration angles are 15°, 30°, and 45°. Figure 11 is a schematic diagram of the damage state of concrete columns at different angles when 3 ms.

It can be seen from Figure 11 that when the penetration angle of the fragment changes, the contact position on the surface of the concrete cylinder changes. With the oblique penetration state of the fragment, the concrete at the position where the fragment hits the cylinder is damaged in different sizes. The failure mass of the concrete in the three cases is 0.379 kg, 0.391 kg, and 0.492 kg, respectively. Observing the damage area at the cross section of the penetration position, it can be seen that when the penetration angle increases, the impact position of the fragment is closer to the edge angle of the cylinder, and the concrete structure closer to the edge angle is more likely to be destroyed. But at the same time, it is worth noting that with the increase in the angle, the damage mass increases, but the crack propagation at the section decreases a lot. From the damage diagram of the three at 3.0 ms, it can be seen that with the increase in the angle, the damage of the column section is suppressed. Therefore, it can be seen that the change in the impact angle will affect the structural failure form of the column, resulting in different failure characteristics of the concrete.

Based on the above variable conditions, it can be seen that although the five variables have an impact on the strength of concrete columns, the magnitude of the impact is different and the mode of action is different. Among them, the parameters of the fragment itself (velocity, angle) mainly affect the damage size of the transient impact process, while the parameters of the cylinder (side length, pressure, position height) mainly affect the uniform load process after the impact, that is, the size of the collapse, the response speed, and the overall strength of the cylinder.

3.2. Orthogonal Optimization

In order to study the influence of each variable on the overall structural damage of the column when the prefabricated fragments impact the concrete column at high speed, 25 sets of orthogonal optimization numerical simulation calculations were carried out for 5 variables and 5 levels. The purpose is to analyze the relationship between variables from complex conditions, so as to provide reference for practical engineering applications. In order to describe the damage of concrete columns under the high-speed impact of fragments from multiple angles, the following indicators will be analyzed in detail: the proportion of collapse of concrete columns, the mass loss ratio of concrete column, and the final displacement of the upper surface of the concrete column. The final statistical results of the three parameters are shown in

Table 5. Statistics of three failure parameters of concrete columns.

NO.	Proportion of Collapses Occurring	Mass Loss (kg)	Upper Surface Displacement (mm)
1	0%	0.0375	0.1264 mm
2	100%	3.2872	115.993
3	100%	1.3826	46.5784
4	100%	2.9579	189.5456
5	100%	1.6231	51.3625
6	0%	2.0437	0.2878
7	0%	0.1425	0.2208
8	83.01%	0.567	0.1540
9	100%	4.4162	75.9834
10	100%	2.6815	37.8779
11	73.23%	0.2429	0.5615
12	41.12%	0.5216	0.6121
13	74.01%	0.2441	0.3394
14	0%	0.6676	0.2158
15	68.19%	0.749	0.2411
16	0%	0.0323	0.1990
17	0%	0.1985	0.1225
18	66.45%	0.6289	1.3284
19	25.83%	2.7923	1.7776
20	65.65%	0.4162	0.4725
21	0%	0.0293	0.443645
22	0%	0.1303	0.3099
23	0%	0.2958	0.2185
24	0%	0.3687	0.1280
25	14.87%	4.4602	6.9924

.

When the velocity of the fragment is large enough and the side length of the column is small, the concrete column is prone to local structural damage, and the collapse of the overall structure occurs under the action of pressure load. In the numerical simulation of orthogonal optimization, although the range of variables has been simplified as much as possible, the default trajectory of the fragment in the orthogonal design hits the center of the cross section of the column and does not consider the velocity component of the fragment in the z-direction, but still produces a variety of damage types of concrete columns, as shown in Figure 12.

3.2.1. The Proportion of Collapse of Concrete Columns

The proportion of the area of collapse above the concrete column to the overall area is defined as the proportion of collapse. It can be seen from Figure 12a that when the impact velocity is small, the damage part of the concrete column is very small, and the overall structure is basically intact. It can be considered that there is no effect on the overall structure of the column at this time, so the proportion of collapse is 0%. It can be seen from Figure 12d that when the fragment velocity is general, although it cannot affect the overall structure, local collapse still occurs under large pressure load, but the remaining structure is still intact and is not affected by the impact damage part, so the proportion of collapse is between 1% and 99%. It can be seen from Figure 12c that when the concrete column is subjected to a large impact and the pressure load is also large, the overall collapse phenomenon is prone to occur. At this time, the entire column will move down, and the proportion of collapse at this time is 100%.

According to the results of Table 5, the range analysis is carried out. Taking the collapse ratio of concrete column as the analysis object of orthogonal optimization, the following analysis results can be obtained as shown in Figure 13:

It can be seen from Figure 13 that under the combined influence of the five variables, the collapse ratio of the concrete column has a significant negative proportional linear relationship with the side length of the column, and the range is the largest. The range of fragment velocity and load is 0.55 and 0.51, respectively, while the range of impact angle and position height is only 0.33 and 0.32. From the parameters of obtaining the minimum collapse ratio, the side length and impact angle of the cylinder should be maximized, while the fragment velocity, position height, and uniform load should be minimized.

3.2.2. The Mass Loss Ratio of Concrete Column

With the high-speed impact of the fragment, the concrete column is inevitably subjected to the direct penetration of the fragment and the failure of the concrete material caused by the damage expansion under the pressure load, which causes the concrete column to undergo structural damage at different positions, and ultimately affects the overall structural strength.

In the numerical simulation, the mass loss of the concrete column not only reflects the effect of the above two damage processes but also plays a role in the analysis of the overall strength failure of the column to a certain extent. Therefore, the final mass loss ratio of concrete columns in 25 sets of numerical simulations is statistically analyzed, as shown in Table 5. Figure 14 is the average value of each factor of the proportion of concrete mass loss.

It can be seen from Figure 14 that when the concrete mass loss ratio is used as the evaluation result, the side length of the concrete column still plays a major role, and the range is the largest and the mass loss ratio is almost unchanged when the side length of the column exceeds 0.2 m. The uniform load size, fragment velocity, and position height all have a strong linear relationship with the mass loss of the cylinder. In addition to the impact angle, when the side length of the column increases, the speed decreases, the position height decreases, and the load decreases, the mass loss ratio of the concrete column can be reduced.

3.2.3. Final Displacement of the Upper Surface of the Concrete Column

The basis for analyzing the final displacement of the upper surface of the concrete column is that it can be used as the criterion for judging the overall strength of the concrete column, and at the same time, the speed of the collapse process of the concrete column is taken into account. The collapse ratio of the concrete column described in Section 3.2.1 reflects the extent to which the entire column structure is affected by the impact of fragments under pressure loads, and takes into account the final structural failure results, but does not reflect the rate of change in the collapse process of the column. Therefore, the displacement variation in the center of the upper surface of the cylinder is taken as the result data, and the result shown in Figure 15 is obtained.

From Figure 15, it can be seen that the range of the side length of the cylinder is the largest, which is 80.33 mm; the range of fragment velocity is 53.21 mm. The linear trend of the position height and the load is more obvious. With the increase in the impact angle, the displacement of the upper surface of the cylinder gradually decreases, which is the same as the pre-analysis results of the angle variable.

3.3. PLS Regression Analysis and Multivariate Analysis of Variance of the Results Parameters

3.3.1. Multi-Factor Analysis

In order to further analyze the influence of five variables on the above three results, the multi-factor variance analysis is carried out on the three results of the collapse ratio of the column, the mass loss ratio of the concrete, and the displacement of the upper surface of the column. The results are shown in

Table 6. The results of multivariate analysis of variance based on the proportion of column collapses.

	Quadratic Sum	df	Mean Square	F	p
Intercept	4.999	1	4.999	59.122	0.002
Side length	1.449	4	0.362	4.284	0.094
Speed	1.108	4	0.277	3.276	0.139
Angle	0.397	4	0.099	1.175	0.440
Height	0.148	4	0.037	0.439	0.778
Load	0.743	4	0.186	2.196	0.232
Residuals	0.338	4	0.085

:

From Table 6, it can be seen that the multi-factor variance analysis is used to study the difference relationship between the five items of column side length, fragment velocity, impact angle, position height, and uniform load on the proportion of collapse of the column. The square value of the model R is 0.9235, which means that the column side length, fragment velocity, impact angle, position height, and uniform load can explain the 92.35% change in the proportion of collapse. Additionally, the analysis shows that all five variables will not have a different relationship with the proportion of collapse.

Table 7. Results of multivariate analysis of variance based on the percentage of column mass loss.

	Quadratic Sum	df	Mean Square	F	p
Intercept	0.083	1	0.083	34.763	0.004
Side length	0.095	4	0.024	10.001	0.023
Speed	0.021	4	0.005	2.201	0.232
Angle	0.006	4	0.001	0.591	0.689
Height	0.015	4	0.004	1.583	0.334
Load	0.028	4	0.007	2.971	0.158
Residuals	0.010	4	0.002

is the result of multivariate analysis of variance based on the proportion of column mass loss.

From Table 7, it can be seen that the multi-factor variance analysis is used to study the difference relationship between the five items of column side length, fragment velocity, impact angle, position height, and uniform load on the mass loss ratio. The model R square value is 0.947, which means that the column side length, fragment velocity, impact angle, position height, and uniform load can explain the 94.67% change in the mass loss ratio. According to the analysis, the side length of the column will have a significant difference in the proportion of mass loss (p < 0.05); the fragment velocity, impact angle, position height, and uniform load will not have a different relationship with the mass loss ratio.

Table 8. The results of multi-factor analysis of variance based on the displacement of the upper surface of the cylinder.

	Quadratic Sum	df	Mean Square	F	p
Intercept	11,401.518	1	11,401.518	8.465	0.044
Side length	23,420.000	4	5855.000	4.347	0.092
Speed	8035.687	4	2008.922	1.492	0.354
Angle	3279.051	4	819.763	0.609	0.679
Height	3454.738	4	863.685	0.641	0.661
Load	5579.221	4	1394.805	1.036	0.487
Residuals	5387.386	4	1346.846

shows the results of multi-factor analysis of variance based on the displacement of the upper surface of the cylinder.

The multi-factor analysis of variance was used to study the difference relationship between the five items of column side length, fragment velocity, impact angle, position height, and uniform load on the displacement of the upper surface of the column. The model R square value is 0.893, which means that the column side length, fragment velocity, impact angle, position height, and uniform load can explain 89.25% of the change in the upper surface displacement. And analysis shows that side length, speed, angle, height, load, a total of five, will not have a difference on the upper surface displacement.

3.3.2. PLS Regression Analysis

Partial least squares regression (PLS regression) is used to study the influence of multiple independent variables on multiple dependent variables. It can combine the design of 25 groups of orthogonal optimization results with better canonical correlation, principal component analysis, and multiple linear regression.

Table 9. Analysis results of regression coefficients of the relationship between 4 dependent variables Y and 5 independent variables X.

	Proportion of Collapses Occurring	Quality Loss Ratio	Displacement of Upper Surface	Mass Loss	Proportion of Collapses (Standardized)	Quality Loss Ratio (Standardized)	Upper Surface Displacement (Standardized)	Quality Loss (Standardized)
Constant	0.300	0.024	18.049	−2.559	0.000	0.000	0.000	0.000
Length of side	−3.566	−0.756	−355.198	−5.288	−0.600	−0.632	−0.561	−0.273
Velocity	0.000	0.000	0.023	0.001	0.429	0.235	0.220	0.460
Angle	−0.002	0.000	−0.327	0.012	−0.092	0.024	−0.116	0.144
Height	0.374	0.260	136.547	5.715	0.065	0.224	0.222	0.303
Load	0.023	0.004	1.686	0.105	0.383	0.371	0.266	0.543

is the analysis results of the regression coefficients of the relationship between the four dependent variables Y (result parameters) and the independent variable X.

Figure 16 is a standardized regression histogram of the proportion of variables and the positive and negative relationships corresponding to the four results obtained by PLS regression analysis.

From the results of PLS analysis and the histogram of standardized regression coefficient, it can be seen that the influence coefficient of the side length of the column is negative. The increase in the side length hinders the increase in various damage degrees of the column and occupies the largest influence in the first three result coefficients. The average influence factor of the four results is 0.52. In addition to the side length of the column, the angle of the fragment impacting the column also showed a negative effect in the two results, which indicated that with the increase in the impact angle, the overall strength damage of the concrete structure decreased instead, and the influence coefficient was the smallest among the five variables, with an average impact factor of only 0.09. Fragment velocity, as the most direct process, has a major impact on the mass loss of concrete and subsequent collapse, but has a general impact on the occurrence rate of columns. The impact factors are 0.43 and 0.46, respectively, and the average impact factor is 0.34. The influence degree of the uniform load on each result parameter is equivalent to the fragment velocity, which reflects the pressure bearing capacity of the building column. Therefore, for the same concrete column, the greater the uniform load it bears, the more likely it is to be damaged and accelerate the collapse rate. The average influence factor of the uniform load is 0.39. The location and height of the fragment impact on the concrete column determines the direct damage location of the column, accelerates the subsequent collapse rate, and affects the mass loss during the collapse of the concrete column. The average impact factor is 0.20. The following is the calculation formula of each result parameter based on the results of PLS regression analysis:

r_s = −0.600 × l + 0.429 × v − 0.092 × α + 0.065 × h + 0.383 × p

(1)

r_m = −0.632 × l + 0.235 × v + 0.024 × α + 0.224 × h + 0.371 × p

(2)

x_z = −0.561 × l + 0.220 × v − 0.116 × α + 0.222 × h + 0.266 × p

(3)

m_l = −0.273 × l + 0.460 × v + 0.144 × α + 0.303 × h + 0.543 × p

(4)

In the above Formulas (1)–(4), r_s is the proportion of collapse; r_m is the proportion of mass loss; x_z is the displacement of the upper surface; m_l is the mass loss of concrete. l is the side length of the cylinder; v is the fragment velocity; α is the impact angle; h is the position height; p is uniform load.

Finally, the dimensionless relationship of each variable to the damage degree of the cylinder is obtained:

$$\delta ={\displaystyle \frac{vhp}{l\alpha C}}$$

(5)

4. Conclusions

This study clearly distinguishes the instantaneous penetration damage under the impact of high-speed fragments from the subsequent collapse process under sustained load. Their interaction is the key to assessing the ultimate failure of the structure and directly points to the possibility of progressive collapse. Firstly, the high-speed impact of the fragment causes serious local damage to the load-bearing column in a very short time. This includes, but is not limited to, large-scale spalling of concrete sections, exposure or even fracture of internal steel mesh, and significant weakening of effective load-bearing sections of members. This stage is a dynamic and local process. After that, the original static balance of the building was broken. The sustained axial load on the column does not disappear but needs to be borne by the remaining section which has been weakened. If the residual bearing capacity of the remaining section is lower than the actual load, the column will undergo further crushing, buckling, or shear failure. The failure of this column causes the load it originally bears to be transferred to adjacent beams, plates, and other columns. Adjacent members may not be considered to bear this additional transferred load during design. If these components are also overloaded, the load will be transmitted downward again.

In this study, the damage mechanism of confined concrete columns under the impact of high-speed small-size fragments was systematically revealed through experiments and numerical simulations. Through active and systematic protection design, the structural vulnerability under such threats can be significantly reduced. In the design, the concept of ‘minimum impact resistance section’ should be introduced on the basis of meeting the requirements of static load. For key load-bearing columns, the section size should be evaluated to ensure that the expected damage zone (crater depth + back shock depth + safety margin) can be accommodated to avoid penetrating damage. Compared with square section, polygonal or circular section has better mechanical properties when subjected to eccentric impact. They can distribute the shock wave more evenly, reduce the serious spalling and cracking caused by stress concentration at the corners, and maintain the integrity of the core bearing area. Steel fibers or synthetic fibers are incorporated into concrete in key areas. The fiber can effectively bridge the micro-cracks generated by the impact load and inhibit the crack propagation, thereby significantly reducing the size and quantity of the shattering fragments and improving the toughness and anti-stripping ability of the material. A layer of protective net made of high-strength steel wire mesh or fine steel mesh is added to the surface of the column, which can effectively reduce the concrete debris generated by the drop of the back surface, maintain the integrity of the component, and prevent the sudden drop of bearing capacity.

In this paper, the failure behavior of concrete columns under the impact of high-speed fragments is quantitatively analyzed by orthogonal experimental design and system numerical simulation. Based on the experimental verification model, the range analysis, multivariate analysis of variance and partial least squares (PLS) regression were used to clarify the influence degree and interaction of each variable. The following conclusions were drawn based on the specific data results:

(1)

There is a key threshold for the size of the column: the orthogonal range analysis shows that the side length of the column is the most dominant factor affecting the residual bearing capacity. Specifically, when the side length is ≥ 200 mm, the residual bearing capacity of the column can be maintained above the static load design value within the parameter range set in this paper, indicating that the structure has an effective safety margin against substantial damage.

(2)

Quantitative correlation between variables and damage modes: Multivariate analysis of variance confirmed that the influence of each variable on the three types of result parameters (penetration depth, size of the shock zone, residual bearing capacity) was small, and the interaction was significant. The fragment velocity is the decisive variable to control the local penetration depth. The impact height and angle jointly determine whether the failure mode changes from local crushing to overall bending fracture. The axial pressure and side length jointly regulate the collapse risk. Under the combined conditions of high pressure and small size, the stability coefficient will plummet.

(3)

PLS regression model (accurately quantify the positive/negative effects of variables. The results show that increasing the side length is the most effective positive measure to improve the impact resistance, and its effect is far more than reducing the load. At the same time, the variable importance projection diagram shows that the top area of the cylinder is the weakest link in the overall protection system. This is the implementation of differentiated protection strategies, giving priority to strengthening the top area and systematically increasing the central protection.

As discussed, although laboratory-scale specimens can effectively reveal the local damage mechanism, they cannot fully reproduce the overall bending, shear, and more complex dynamic responses of full-size columns. When the conclusion of this study is extrapolated to the actual project, the scale amplification analysis needs to be carried out through the verified numerical model. Although the RHT concrete model can well reproduce the macroscopic damage mode, there is still a large error in predicting the residual velocity under the critical penetration state. This indicates that the model has inherent simplifications in simulating the meso-mechanical behavior of concrete at extremely high strain rates and during the complete failure stage, and the calibration process of material parameters still needs to be further optimized. This study focuses on the ideal condition of central impact to control variables. However, eccentric and oblique impacts, which are more common in actual threats, introduce bending moment and torque, which may significantly change the failure mode and more easily lead to overall instability. The applicability of the current conclusions under such complex conditions needs to be further verified.

Based on the results of this study and the above limitations, we propose the following key directions for future research: Extend the research scope from central positive impact to eccentric impact and oblique impact with large inclination angle, and quantitatively analyze the influence of impact position and angle on failure mode, residual bearing capacity, and structural stability. The response of concrete column under the combined action of explosion shock wave and high-speed fragment is studied, and the synergistic damage effect of the two threats is revealed, which provides a basis for protective design for more real scenes. The validated numerical model is used as a design tool to evaluate the protective effectiveness of strengthening measures, such as high-performance fiber-reinforced concrete and steel sleeve composite wrapping under high-speed impact, and to optimize the design. The residual bearing capacity of the damaged column obtained in this study is used as the input parameter to carry out the collapse simulation of the substructure or the whole structure, aiming to establish the prediction model and design method from ‘local impact damage’ to ‘overall system failure’. Beyond the single-fragment impact, the cumulative damage effect and overall performance degradation mechanism of concrete members under the distributed impact of small-sized fragment groups are studied.