Damage Assessment of Laboratory-Scale Reinforced Concrete Columns Under Localized Blast Loading

Abstract

Reinforced concrete (RC) columns are structural components that carry loads and are vulnerable to damage and possible failure under blast loads. Understanding how damage accumulates and cracks propagate in these structural members is essential for improving their resilience and designing blast-resistant buildings. This study introduces an experimental approach to mitigate the fireball and fumes generated by an explosion, allowing for a more precise structural response assessment. With the help of high-speed cameras, this study experimentally investigates the real-time damage progression and crack formation in RC columns. To explore these failure mechanisms, laboratory-scale RC columns with a low reinforcement ratio are intentionally designed to experience significant damage, providing deeper insights into concrete-specific failure patterns. The tested columns are 1800 mm long and have a 100 mm diameter. Each specimen is reinforced with 3 mm longitudinal reinforcement bars and 2 mm transverse bars. An explosive driven shock tube (EDST) is used to apply blast loads, targeting the mid-height of the columns. High-speed digital image correlation (DIC) tracks the overall structural response. A numerical simulation is developed in LS-DYNA and compared with experimental data for validation. The findings demonstrate that the proposed FE model accurately simulates both the applied blast load and the resulting failure patterns. The difference between the mid-span lateral displacement predicted by the numerical simulation and the average experimental measurements remains within 15%.

1. Introduction

Recent reports show that the global number of casualties caused by explosive weapons has surpassed 30,000 globally [1]. This highlights the importance of studying the vulnerability of reinforced concrete (RC) structures, which are often damaged during such events, as seen in the Alfred P. Murrah Federal Facility [2] and the accidental explosion in China [3]. These incidents caused varying degrees of damage and failure to several structural elements, including RC columns. As a result, a large portion of the facility collapsed due to the inability to redistribute loads efficiently. Therefore, understanding how RC components respond to blast loading is crucial for accurately assessing the impact on RC structures. Evaluating the blast-induced damage in structural members is essential and can be approached in three ways: analytical, numerical, and experimental. The analytical methods use simplified models included in UFC-3-340-02 [4] or CSA S850 blast standards [5]. The numerical analysis involves finite element simulations using blast parameters from guidelines or explicit explosion modeling [6,7,8,9]. The experimental approach relies on full-scale blast tests.

The analytical methods are effective in predicting the behavior of RC structural members under blast loading when the response is primarily governed by flexure. However, their reliability decreases for more complex failure modes, such as a combined shear and flexural failure [10,11]. Using numerical approaches to model the response of a structural member to blast loading is challenging, as it demands substantial computational resources and highly advanced material models and numerical techniques. Large-scale experiments are costly, time-consuming, and often impacted by external factors, like weather, requiring sophisticated instrumentation [12]. Compared to large-scale experiments, reduced-scale tests are cost-effective and allow for testing under controlled conditions [13,14,15]. These experiments serve as a valuable tool for analyzing the effects of blasts on structures and materials.

Typically, structural members are subjected to blast loading generated from unconfined detonation. However, close-in detonations can have variable effects due to changes in the shape of the explosive charge, position, and quantity [16,17]. Researchers have noted issues with test repeatability and visibility problems caused by fireballs and dust clouds [17]. Hence, shock tubes offer a controlled alternative to unconfined detonations [18,19].

One approach for producing blast loading involves using an EDST. Placing it between the explosive material and the specimen effectively generates high reflected pressure and impulse values at a laboratory scale [12].

Numerous researchers have studied the response of RC columns under blast conditions, both from far-field and near-field explosion events. Woodson et al. [20] performed tests on reduced-scale RC columns. The test columns had a square profile and a height of approximately 900 mm. The longitudinal reinforcement bars had a 3.2 mm diameter, whereas the transverse reinforcement bars were 1.6 mm in diameter. Additionally, the concrete cover was 8.5 mm thick. The columns were tested under blast loading generated by 7.1 kg C4 explosives at a stand-off distance of approximately 1 m and 1.52 m. Moderate damage was observed in the blast-loaded columns. Xu et al. [21] tested five columns with a cross-section of 240 × 240 mm² and a height of 1600 mm. Each column contained longitudinal bars of 4 mm in diameter and ties of 2 mm in diameter. A TNT explosive charge of 40 g was placed at different distances between 200, 300, 400, and 500 mm. After observing only minor damage during the initial tests, the specimens were exposed to a contact explosion using the same explosive mass to intensify the damage. Zhang et al. [22] conducted an experimental campaign on small-scale RC beams. The primary goal was to examine how scaling the dimensions of RC beams affected the response when exposed to near-field blast loading. The RC specimens had three square cross-sections: 75 mm, 100 mm, and 125 mm on each side, with heights varying from 850 mm to 1100 mm and 1350 mm. The findings indicated that scaling the RC beams did not influence the damage modes and patterns. However, the extent of damage was significantly more severe in larger specimens.

In most previous experimental campaigns [23,24,25,26,27,28,29,30], blast loading was uncontrolled, often engulfing columns in fireballs and fumes. This lack of control creates a gap between the moment the explosion impacts the RC component and its subsequent residual response. Consequently, researchers have focused only on the post-mortem analysis of blast-induced damage, limiting their ability to fully assess the structural behavior during and immediately after the blast event. Recently, Mohamed et al. [31] proposed an experimental approach to control blast loading while studying the response of small-scale columns under shock-tube-produced blast loading. Their test columns exhibited only superficial damage due to their relatively high stiffness, which prevented a detailed investigation of damage accumulation over time. In contrast, the present study addresses this gap using laboratory-scale RC columns with a low reinforcement ratio to sustain significant damage. The choice to vary the reinforcement ratio was motivated by previous studies, which have proven that column detailing significantly influences the extent of blast-induced damage [32,33,34,35,36,37]. Circular RC columns were chosen for this study due to the limited experimental data available on their behavior under blast loading. Additionally, their unique shape resulted in lower blast loads than other cross-sectional shapes, making them an important subject for further investigation.

This paper presents a novel approach for analyzing the behavior of RC columns with a circular section, focusing on crack evolution and damage accumulation. This study examines specimens subjected to controlled blast loading through the DIC technique. The blast loading is produced using an EDST.

The goals of this paper are as follows:

(1)

To investigate damage mechanisms in blast-loaded circular RC columns.

(2)

To analyze crack evolution and damage accumulation over time.

(3)

To explore the impact of column detailing on damage severity.

(4)

To validate the numerical model by comparing predicted fracture patterns, displacements, and failure modes with experimental results.

This paper is organized into four key sections. Firstly, an overview of the experimental setup is provided. Secondly, the numerical model implemented with LS-DYNA R10 is outlined. Thirdly, a comparative analysis of the numerical results with the experimental findings is presented. Lastly, a summary of the key results is offered.

2. Overview of the Test Configuration

2.1. Introduction to the Study

Eight RC columns experience blast loading produced by an EDST, as shown in Figure 1. A spherical C4 charge mass of 30 g is used for the different configurations, as indicated in Figure 1. The charge mass is positioned at the shock tube’s entrance. An electric detonator (1 g of TNT) initiates the detonation. Figure 1 presents an in-depth illustration of the explosive charge, the detonator, and its position. The steel structure adjusts the height of the EDST. To determine the reflected pressure and impulse, a 5 mm aluminum column is selected. This thickness is adopted to avoid deformation during the blast loading. The EDST adopted in this investigation features specific dimensions and characteristics detailed in [31].

A metal frame ensures the test specimen’s proper alignment. High-speed cameras capture lateral displacements, as illustrated in Figure 1. To establish a simply supported configuration, 4 mm thick steel tubes measuring 40 mm in outer diameter and 300 mm in length are adopted at both extremities of the column.

A total of eight columns, representing four distinct configurations, are subjected to testing, as detailed in

Table 1. Main test cases.

Test n°	Charge Mass [g]	Configuration n°	Longitudinal Reinforcement Type	Longitudinal Reinforcement Ratio [%]
1–2	30	1	A	0.56
3–4		2	A	0.28
5–6		3	A	0.14
7–8		4	B	0.28

. Two tests are performed for each configuration to verify the reproducibility of the results.

2.2. Details of the Test Specimens

Figure 2 shows the specific configurations of the columns. The specimens are 1800 mm in length and have a cross-sectional width of 100 mm. Each specimen is reinforced with 3 mm longitudinal bars and 2 mm transverse bars, following the approach outlined in the literature [23,28,33,38,39,40,41]. The column size is determined based on a parametric study conducted in LS-DYNA using a calibrated simulation that analyzes the influence of column parameters on mid-span displacement and blast-induced damage. Modifications, including increased column height and adjusted reinforcement ratios, ensure a higher damage level for studying crack evolution and damage accumulation under blast loading. The tie spacing is shown in Figure 2. Additionally, a 10 mm thick layer of concrete cover is applied. The process of casting the columns is indicated in Figure 3. The columns are cast using Ordinary Portland Cement (OPC) with a grade of 32.5, fine sand (0 to 5 mm), and aggregates ranging in size from 2 mm to 8 mm.

Due to the size of the laboratory-scale columns, significant consideration is given to the concrete mix design. To meet the concrete cover requirements for the laboratory-scale models, the typical aggregate size (ranging from 4.75 mm to 19 mm) is replaced with roll gravel of 2–8 mm. A water-to-cement ratio of 0.55 is adopted for pouring the specimens. In the concrete composition for 1 m³, roll gravel represents 39.1%, while fine sand accounts for 32.7%. The air content of the mix is around 1.8%.

The new mix is flowable, prepared using a concrete mixer, and discharged directly into the model. Only slight vibration is required to settle the mix properly.

2.3. Material Properties of Concrete and Reinforcement Bars

Tensile tests are conducted on the reinforcement bars to determine their mechanical properties following standard testing procedures [42]. Three experiments were conducted to confirm the consistency of the results. The rebars for the longitudinal direction, Ø 3 and M 3, possess yield stresses of 941 ± 4.69 MPa and 430 ± 8.39 MPa, respectively, and are referred to as types A and B. Additionally, the transverse ties exhibit a yield stress of 443 ± 16.68 MPa. Figure 4 indicates the tensile results for the rebars.

Table 2. Mechanical characteristics of the steel rebars.

Steel Rebar	Diameter (mm)	Elastic Modulus (GPa)	Elastic Limit		Maximum
			Strain (-)	Stress (MPa)	Strain (-)	Stress (MPa)
Longitudinal Ø 3	3	209	0.0037	941	0.15	961
Longitudinal M 3	3	209	0.0026	474	0.038	491
Transverse	2	199	0.0021	443	0.113	513

presents the mechanical properties of the reinforcement bars. The average compressive strength of the concrete after 28 days of curing is 22.4 ± 1.6 MPa, determined according to NBN EN 12390-3 [43]. Figure 5 shows the concrete response curves.

2.4. Deflection Analysis Using DIC

DIC is a non-contact imaging technique that uses a series of photographs taken by a pair of fast-frame cameras. To achieve the necessary high-speed imaging, two Photron Fastcam SA5 cameras, synchronized for this purpose, are arranged stereoscopically, facing the backside of the columns (Figure 1). The two cameras are positioned 1700 mm away from the RC column. A pattern of high-contrast speckles is created in the region of interest visible to both cameras on the specimen. The cameras are positioned 900 mm above the ground, focusing on the central section of the test samples. A frame rate of 20,000 fps, with a shutter time of 25 microseconds, is selected to capture the dynamic behavior of the specimens. The aperture of the cameras is reduced since significant out-of-plane deflections (OPDs) are expected. This reduction influences the amount of light coming through lenses. The columns are lit with a single high-intensity LED. The targeted area measures approximately 157 mm by 300 mm and is captured with a 512 × 512 pixel resolution. For processing with VIC3D 2007 software, a subset size of 21 by 21 pixels and a 3-pixel spacing between subsets are selected [31].

3. Test Results

3.1. Characterization of the Blast Loading

In the initial phase, an aluminum column is adopted, featuring a 5 mm thickness, a 100 mm inner width, and a total length of 1800 mm. The distance between the exit of the EDST and the column is equal to 4 mm (Figure 6a). An explosive charge in a sphere weighing 10 g is used. High-frequency pressure sensors, precisely the PCB QUARTZ ICP 113 B22 type [44], are adopted with a recording rate of 1 MHz [44]. The pressure transducers are mounted on M10 special adaptors [12] to minimize vibrations of the EDST wall caused by the explosion. The pressure sensors mounted on the EDST record both the incident and reflected pressures, as shown in Figure 6b.

The first test campaign measures the blast loads exerted on the test specimens. The cylindrical column has a pressure sensor at the mid-point, as illustrated in Figure 6b.

To guarantee reproducibility, three experiments are performed for each configuration. A Butterworth low-pass filter of second order is used in MATLAB R2021a to mitigate high-frequency measurement noise [12,31].

Figure 7 displays the time histories of reflected pressures and impulses from a 10 g C4 spherical explosive mass.

The reflected over-pressure as a function of time is limited to 1.6 ms, as no significant changes are detected past this moment. The recorded peak reflected pressure is 15.1 ± 0.9 MPa, while the average reflected impulse is 2.2 ± 0.1 MPa·ms.

3.2. Dynamic Behavior of Test Specimens

3.2.1. Deflection Versus Time Data

Two consecutive experiments are performed for each configuration to assess the reliability of the measurements. The DIC method enables the measurement of the distribution of out-of-plane deflection in a defined focus region. As an example, the deflection field of specimen T 3 (configuration n°2) is extracted from the VIC 3D V8 software for various time frames, as shown in Figure 8. Data loss is observed around the center section of the column, starting at 1.4 ms. This loss is attributed to the initiation and propagation of the mid-span crack. To facilitate comparison, particular attention is given to the time histories of the out-of-plane deflections at positions P1, P2, and P3.

The out-of-plane deflections as functions of time for test columns under blast loading are indicated in Figure 9. Figure 9a–d represents configurations 1, 2, 3, and 4, respectively. In test n°5, one of the high-speed cameras failed to record the images of the deflection of the specimen. Hence, only the results of test n°6 are adopted.

For configuration n°1, average maximum deflections of 18.8 mm, 17.3 mm, and 16.8 mm are noted for P1, P2, and P3, respectively. Minor variations in maximum deflection between locations P1 and P3 can be attributed to the asymmetry in the support conditions, as indicated in [31].

Based on Figure 10, when the longitudinal reinforcement ratio is reduced from 0.56% to 0.28% and from 0.56% to 0.14%, respectively, a 42% and 110% increase in mid-span deflection (corresponding to position P2) is observed. Furthermore, a 60% and 300% increase in residual deflection is found when comparing the results of configuration 1 to those of configuration 2 and configuration 3, respectively.

Even though configuration 2 and configuration 4 columns possess the same longitudinal reinforcement ratio, an important difference in the dynamic behavior is observed due to the use of different steel bars. When comparing the results of the two configurations, maximum mid-span deflection increases by 102%. Under the same blast loading, columns T7 and T8 do not exhibit ductile behavior.

3.2.2. Damage Pattern

Cracks with varying opening widths highlighted in red are present in each of the columns, as indicated in Figure 11. When examining the group of columns for configurations 1, 2, and 3, it is observed that reducing the longitudinal reinforcement ratio results in greater damage sustained by the test specimens. On the non-loaded side, an average of five flexural cracks is observed under a 30 g explosive charge. The arrangement of flexural cracks extends throughout the full height of the column.

The DIC technique effectively captures the evolution of damage in concrete. For example, specimen T7 is chosen for detailed analysis. The damage process for this column is selected in Figure 12. The first and second cracks emerge at the center section of the specimen at 1.6 ms. By 16.2 ms, the first and second cracks propagate, and a third crack initiates between them. In the meantime, the longitudinal bars are assumed to yield. At 38.1 ms, the deflection at the column’s center reaches 45.2 mm, accompanied by the ejection of a concrete block on the tension side of the column at the mid-span, leaving a crater. This crater has a width of 54 mm and a height of 21 mm. A post-test examination reveals a complete failure in the longitudinal bar of the non-loaded side.

The DIC technique enables the quantification of damage patterns in columns subjected to blast loads from the shock tube [31]. Test No. 2, which concerns the RC column, is selected as an illustrative example. Figure 13 illustrates the progression of cracks in the chosen specimen. The initial crack forms in the center section of the column at 0.95 ms, corresponding to a maximum out-of-plane deflection of 2.8 mm. By 18.1 ms, during the inbound phase, the RC column reaches its maximum deflection, accompanied by the propagation of an additional crack (crack n°2). Subsequently, the specimen returns to its initial state, and the cracks slowly close by 38.2 ms, as illustrated in Figure 13.

The width of crack n°1 for test n°2 is plotted as a function of time and compared to the deflection at points P1, P2, and P3 in Figure 14. The crack width evolution can be divided into four phases. In the initial phase, the blast load transfers momentum to the column within a brief duration. This momentum, of sufficient magnitude, induces acceleration in the column even after the shock wave ceases. As the column initiates deflection and the tensile stress surpasses the dynamic tensile strength of the concrete, it leads to the onset of the flexural crack n°1, as seen in Figure 14. Once the crack is formed, the load is redistributed. In this phase, the concrete and reinforcement work together. The law of mixtures gives the equivalent moment of inertia. In the second phase, the propagation of crack n°1 persists, and its width reaches a peak corresponding to 0.93 mm at 3.5 ms. This propagation induces a decrease in the equivalent moment of inertia, diminishing the flexural stiffness of the column. As a result, a second crack (crack n°2) appears. Hence, a reduction of crack opening is seen from 3.5 to 4.45 ms. Meanwhile, a variation in the deflection pattern occurs in the deflection-time data for points P1, P2, and P3. Concurrently, owing to the opening of the cracks, the column sustains continued displacement, achieving a peak deflection of 18.4 mm at 18.1 ms. Following this event, the column returns to its initial position, with a gradual reduction in the extent of crack n°1 opening. Subsequently, starting from 38.2 ms, the column rebounds, leading to a progressive increase in the extent of crack opening.

Figure 15a illustrates the position of crack n°1 within the area of interest, as identified through the post-test examination of the test specimen. This crack is highlighted in red for clarity. Figure 15b presents the plot of the vertical deflection’s spatial derivative (Vd) as a function of the Z coordinate. The crack is located at a distance of 17 mm along the vertical axis. Analyzing the gradient of vertical deflection is crucial for crack detection because it allows for a more precise identification of localized changes in the deflection pattern that correspond to crack formation. Using the DIC technique, full-field data on the column’s lateral and vertical deflections are obtained within a defined area of interest. By calculating the spatial derivative (Vd) with respect to the Z coordinate (height), the location of cracks can be accurately determined, as discontinuities in the deflection gradient curve along the column’s height indicate crack positions.

Following UFC-3-340-02 [4], the damage intensity classification for the RC specimens is categorized into three levels based on the peak midspan deflection ${\mathsf{\delta }}_{\mathrm{m}}$ and the rotation at the fixity (${\mathsf{\theta }}_{\mathrm{m}}$). The classification is divided into three levels: superficial (${\mathsf{\theta }}_{\mathrm{m}}$ < 1°), moderate (1° ≤ ${\mathsf{\theta }}_{\mathrm{m}}$ < 2°), and heavy (2° ≤ ${\mathsf{\theta }}_{\mathrm{m}}$).

The observed responses and the classification based on the UFC-3-340-02 response limit agree well. Details about this comparison are included in

Table 3. Results of blast-induced response in RC columns.

Test n°	Explosive Mass [g]	${\mathsf{\delta }}_{\mathbf{m}}$ [mm]	${\mathsf{\delta }}_{\mathbf{m}\mathbf{e}\mathbf{a}\mathbf{n}}$ [mm]	Fixity Rotation Ɵm [°]	UFC-3-340-02	Failure Mode
1	30	19.9	18.9	1.35	Moderate damage	Multiple fine cracks on the unloaded side extend along the entire length of the column.
2		17.8
3		26.5	26.1	1.9	Moderate damage	Multiple fine cracks on the unloaded side extend along the entire length of the column.
4		25.6
5		--	39.4	2.8	Heavy damage	Significant cracks on the unloaded side extend along the entire length of the column.
6		39.4
7		52.9	54.7	3.9	Heavy damage	Plastic deformation of the first steel rebar on the loading side. Failure of the second longitudinal bar of the non-loaded side. Deep flexural crack in the central region of the column. Significant cracks on the unloaded side extend along the entire length of the column.
8		56.4

-- No data is obtained.

. Compared to the previous work [31], the column characteristics are modified in the present study by increasing the column height, adjusting the reinforcement ratios, and modifying the spacing of transverse reinforcement. According to UFC-3-340-02, these changes result in a shift from superficial to moderate damage, with the maximum mid-span displacement increasing by 1.5 times under the same blast loading conditions.

4. Numerical Modeling

4.1. Finite Element Model

The specific setup of the numerical simulation is indicated in Figure 16 [31]. The RC column is modeled based on a Lagrangian formulation. The LS-DYNA R10 explicit solver is used to conduct the analysis. The concrete is represented using eight-node solid elements. Beam elements are used for the steel reinforcements. Similarly, the same type of elements used for the concrete are adopted to model the wooden plates, steel supports, and the EDST. In this study, translational movements of the wooden plates, steel supports, and EDST are restricted in all three spatial directions. Additionally, the lower surface of the column is restrained from vertical displacement along the z-axis. The Multi-Material Arbitrary Lagrange–Eulerian (MM-ALE) approach has been chosen to represent the air and explosive domains. For a complete description of the finite element approach, please refer to [31].

4.2. Modeling of Air and Detonation

The simulation consists of 2D axially symmetric and 3D ALE models. Using an LS-DYNA 2D-to-3D mapping technique, this interconnected simulation aims to precisely predict the detonation, propagation, and interaction of the blast wave with the column. For a complete description of the finite element approach, please refer to [31].

4.3. RC Column Modeling

4.3.1. Concrete Modeling

The Karagozian and Case material model is chosen for the concrete component. This model has been widely used to demonstrate the behavior of concrete when exposed to blast loading [11,27,45,46,47,48]. It adopts a plasticity-based approach, such as non-linear hardening and softening [49]. The model cannot simulate cracking formation [50]. Consequently, erosion is applied to represent the failure [40]. The parameters required for the concrete representation are specified in

Table 4. Karagozian and Case material model characteristics.

Material Characteristics	Parameter
Mass density, (kg/m3)	2255
Uniaxial compressive strength, (MPa)	22.4
Poisson’s Ratio	0.3

.

In the Karagozian and Case concrete model, the analytical formulas from the CEB [51] are used to consider the loading rate in compression:

$$\mathrm{D}\mathrm{I}{\mathrm{F}}_{\mathrm{c}}={\displaystyle \frac{{\mathrm{f}}_{\mathrm{c}}}{{\mathrm{f}}_{\mathrm{c}\mathrm{s}}}}=\left\{\begin{array}{c}{\left({\displaystyle \frac{\dot{\mathsf{\epsilon }}}{{\dot{\mathsf{\epsilon }}}_{\mathrm{s}}}}\right)}^{1.026\mathsf{\beta }},\dot{\mathsf{\epsilon }}\le 30 {\mathrm{s}}^{-1}\\ \gamma {\left({\displaystyle \frac{\dot{\mathsf{\epsilon }}}{{\dot{\mathsf{\epsilon }}}_{\mathrm{s}}}}\right)}^{1/3},\hspace{1em}\dot{\mathsf{\epsilon }}>30 {\mathrm{s}}^{-1}\end{array}\right.$$

(1)

where ${\dot{\mathsf{\epsilon }}}_{\mathrm{s}}=30\ast {10}^{-6}$ s⁻¹ represents the static rate of strain, and $\log \mathsf{\gamma }=6.156\mathsf{\beta }-2$, $\mathsf{\beta }={\displaystyle \frac{1}{5+9(\mathrm{f}\mathrm{c}\mathrm{s}/10)}}$. ${\mathrm{f}}_{\mathrm{c}}$ represents the compressive strength at a specific strain rate, whereas ${\mathrm{f}}_{\mathrm{c}\mathrm{s}}$ indicates compressive strength under static conditions. For concrete in tension, the formulas established by Malvar and Crawford [52] are used:

$$\mathrm{D}\mathrm{I}{\mathrm{F}}_{\mathrm{t}}={\displaystyle \frac{{\mathrm{f}}_{\mathrm{t}}}{{\mathrm{f}}_{\mathrm{t}\mathrm{s}}}}=\left\{\begin{array}{c}{\left({\displaystyle \frac{\dot{\mathsf{\epsilon }}}{{\dot{\mathsf{\epsilon }}}_{\mathrm{s}}}}\right)}^{\mathsf{\alpha }},\hspace{1em}\dot{\mathsf{\epsilon }}\le 1 {\mathrm{s}}^{-1}\\ \mathsf{\beta }{\left({\displaystyle \frac{\dot{\mathsf{\epsilon }}}{{\dot{\mathsf{\epsilon }}}_{\mathrm{s}}}}\right)}^{1/3},\hspace{1em}\dot{\mathsf{\epsilon }}>1 {\mathrm{s}}^{-1}\end{array}\right.$$

(2)

where ${\dot{\mathsf{\epsilon }}}_{\mathrm{s}}$ = ${10}^{-6}$ s⁻¹ presents the static rate of strain and $\mathsf{\beta }={10}^{7.11\mathsf{\alpha }-2.33}$; $\mathsf{\alpha }$ = ${\displaystyle \frac{1}{10+8(\mathrm{f}\mathrm{t}\mathrm{s}/10)}}$. In this context, ${\mathrm{f}}_{\mathrm{t}}$ denotes the tensile strength for a given rate of strain, while ${\mathrm{f}}_{\mathrm{t}\mathrm{s}}$ represents the quasi-static tensile strength.

4.3.2. Steel Rebars Modeling

Steel reinforcements are modeled using MAT_024, which captures elastoplastic behavior, including plastic deformation, strain rate effects (via an arbitrary curve or Cowper–Symonds formulation) and failure. Input parameters are detailed in

Table 5. MAT_024 material model characteristics.

Material Characteristics	Longitudinal Rebar (Type A)	Longitudinal Rebar (Type B)	Transverse Rebar
Mass density, (kg/m3)	7800	7800	7800
Yield strength, (MPa)	941	474	443
Elastic modulus, (GPa)	209	209	199
Poisson’s ratio	0.33	0.33	0.33
Failure strain	15%	3.8%	11.3%

.

A DIF is applied to the steel reinforcement since the mechanical properties of the steel are influenced by the loading rate. In this research, the DIF for the reinforcement steel is evaluated according to [53,54]:

$${\mathrm{D}\mathrm{I}\mathrm{F}=\left({\displaystyle \frac{\dot{\mathsf{\epsilon }}}{{\dot{\mathsf{\epsilon }}}_{\mathrm{o}}}}\right)}^{\mathsf{\theta }};\mathsf{\theta }=0.074-0.04{\displaystyle \frac{{\mathrm{f}}_{\mathrm{y}}}{414}}$$

(3)

where ${\dot{\mathsf{\epsilon }}}_{\mathrm{o}}={10}^{-4}$ s⁻¹ is the strain rate under static conditions, $\dot{\mathsf{\epsilon }}$ represents the strain rate of rebars, and ${\mathrm{f}}_{\mathrm{y}}$ (yield strength) is in MPa. This expression applies to strain rates from 10⁻⁴ s⁻¹ to 225 s⁻¹ and for ${\mathrm{f}}_{\mathrm{y}}$ between 290 and 710 MPa. Hence, for steel rebar type A, the Cowper and Symonds model is used [55]:

$$\mathrm{D}\mathrm{I}\mathrm{F}=1+{\left({\displaystyle \frac{\dot{\mathsf{\epsilon }}}{\mathrm{C}}}\right)}^{{\displaystyle \frac{1}{\mathrm{p}}}}$$

(4)

C and P are the model’s constants taken from [56]. The other essential parameters are included in Table 5.

5. Numerical Results

5.1. Blast Load

Figure 17 compares the pressure and impulse over time from the experiments with the results of the finite element model. The numerical reflected peak pressure and impulse are around 15 MPa and 2 MPa·ms, respectively.

5.2. Blast Load Response of the Columns

5.2.1. Deflection over Time

DIC is adopted to measure the lateral deflection of the test specimens at P1, P2, and P3 during the explosion. Maximum deflections of 17 mm, 18.6 mm, and 16.5 mm occur at P1, P2, and P3, respectively. Relative variations of 1.7%, 1.1%, and 1.8% occur at P1, P2, and P3, respectively, as shown in Figure 18. For configuration n°2, maximum deflections of 22.4 mm, 24.7 mm, and 22.3 mm are recorded at P1, P2, and P3, respectively. Variations of 1.2%, 5.3%, and 6.3% are observed at P1, P2, and P3, respectively, as shown in Figure 19.

In configuration n°3, the computed free vibration period is slightly shorter, and the corresponding displacements are larger, as shown in Figure 20. This discrepancy is primarily due to the model’s limitations in accurately accounting for material damage accumulation. Chen et al. [34] reported similar findings when applying the same concrete model to analyze the highly damaged RC columns’ response to a close-in explosion.

For configurations n°3and 4, as indicated in Figure 20 and Figure 21, the experimental measurements capture only the inbound phase. However, the numerical results provide in-depth insights into the behavior of the specimens throughout the loading and unloading phases.

The comparisons of the maximum mid-span deflection obtained from the numerical model and test results are listed in

Table 6. Comparisons of the numerical and tested maximum average mid-span deflections.

Columns	Maximum Mid-Span Deflections
	Exp [mm]	Num [mm]	Ratio [Num/Exp]
C1	18.9	18.6	0.98
C2	26.1	24.7	0.95
C3	39.4	40.4	1.03
C4	54.7	59.9	1.1

.

The maximum deflection predicted by the finite element model closely aligns with the measured values, showing a discrepancy below 15%, as illustrated in Figure 22. This reasonable deviation demonstrates that the proposed finite element model can reliably calculate the maximum deflection of RC columns subjected to localized blast loading.

5.2.2. Numerical Analysis of the Damage Pattern

The scale damage (SD) is defined to quantify the damage in the (Karagozian and Case) concrete material model, as indicated in Equation (5):

$$\mathrm{S}\mathrm{D}={\displaystyle \frac{2\mathsf{\lambda }}{\mathsf{\lambda }+{\mathsf{\lambda }}_{\mathrm{m}}}}$$

(5)

where ${\mathsf{\lambda }}_{\mathrm{m}}$ represents $\mathsf{\lambda }$ at its peak failure surface. $\mathsf{\lambda }$ is a non-decreasing parameter.

It is important to note that an improper SD value may not accurately characterize severe damage or crack patterns. Figure 23 indicates the development of the SD under compression. When reaching 1, the material attains its peak strength. When ranging from 1 to 2, the material surpasses its maximum capacity and transitions into the softening phase [31,50]. Consequently, the minimum SD value is 1 to describe the onset of residual damage and crack patterns in RC specimens.

Figure 24 illustrates the damage to the RC columns based on effective plastic strain, represented on a scale ranging from 1 to 2. A comparison of the experimental and numerical findings indicates that the cracks, marked in red, are primarily located in the middle section of the columns. A good correlation is observed between the experimental and simulated concrete damage profiles.

6. Conclusions

This study experimentally investigated the progressive damage accumulation and crack propagation in RC columns subjected to localized blast loading. Unlike previous research, where damage assessment was limited to post-mortem analysis due to a lack of blast control, this study introduced an approach that mitigates fireball and fume interference. This allowed for real-time evaluation of the structural response. Laboratory-scale RC columns with a low reinforcement ratio were designed to sustain significant damage, providing better insight into concrete-specific failure mechanisms.

High-speed stereoscopic DIC captured the columns’ dynamic response, while numerical simulations using LS-DYNA showed good agreement with experimental results.

The conclusions derived from this analysis are as follows:

(1)

A reduction in the longitudinal reinforcement ratio is associated with increased damage severity in RC columns. The rotation-based damage assessment of columns subjected to a 30 g explosive charge (tests n°1 and 2) indicates moderate damage, with a fixity rotation (ϴ) of 1.35°. In comparison, specimens from tests n°5 and 6 show heavy damage, characterized by a considerably higher fixity rotation (ϴ) of 2.8°.

(2)

The DIC technique effectively identifies the crack distribution in the blast-loaded RC column. The locations of individual cracks along the columns are accurately determined by analyzing the gradient of the vertical deflection field.

(3)

The DIC technique investigates the time histories of crack propagation and failure progression.

(4)

The FE model can replicate the blast effects from an EDST, particularly concerning the maximum pressures and impulse. The variations observed in the maximum reflected pressures and impulse are 1.3% and 7.3%, respectively.

(5)

The FE model provides valuable insights into the damage mechanism of RC columns subjected to a local blast load produced by an EDST.

(6)

The FE model effectively replicates the blast response of RC columns. For instance, in config. n°1, the relative discrepancy between the model and the experimentally captured peak deflections at positions P1, P2, and P3 is 1.7%, 1.1%, and 1.8%, respectively.

(7)

The damage pattern simulated by the numerical model shows good agreement with the observed experimental data.

In conclusion, this work’s experimental and numerical analyses offer critical insights on the effects of blast load produced with an EDST on the behavior of circular reinforced concrete members. The approach used in this study can serve as a reliable basis for subsequent investigations.