Risk evaluation of ballistic penetration by small caliber ammunition of live-fire shoot house facilities with comparison to numerical and experimental results

Abstract

The development of advanced small caliber weapon systems has resulted in rounds with more material penetration capabilities. The increased capabilities may mean that existing live-fire facilities will no longer be adequate for the training and certification of military and law enforcement personnel. Constraints on training in many live-fire shoot house facilities are already in place, with some allowing only single round impact during training. With little understanding of the probability of perforation, or failure, of existing containment systems, this study evaluates risk by studying the single round impact of small caliber ammunition against live-fire shoot house containment systems constructed from AR500 steel panels with two-inch ballistic rubber covering. An analytical and numerical study was conducted using an existing model for steel penetration developed by Alekseevskii-Tate and the EPIC finite element code. A modified form of the advancing cavity model for the ballistic resistance of the target material was used to account for the relatively unconfined material resulting from the studied impacts. These results are then compared to experimental tests conducted by Goodman for rounds of various small calibers impacting live-fire facility containment systems. Projectile and target characteristics were then modeled as continuous random variables, and Monte Carlo simulations were conducted using the validated analytical model to estimate the probability of a single round impact perforating the live-fire facility containment system. An importance sampling scheme was used to reduce the variance of the solution and provide a more accurate estimate of the probability of failure. The Alekseevskii-Tate model was found to provide accurate estimates of the depth of penetration when compared to experimental and numerical results at ordnance velocities and an estimate of the probability of failure is on the order of 1x10^-5. This study provides useful tools for the analysis of existing live-fire facilities against future and existing ammunition, and for the design of new facilities. When coupled with Monte Carlo simulation techniques, a risk-based approach to certify live-fire facilities for use with any variety of small arms ammunition can be applied.

Keywords

Steel penetration risk analysis finite element analysis Alekseevskii-Tate model target resistance live-fire facilities Monte Carlo simulation

Introduction

As weapon systems evolve and munitions with more material penetration capabilities are fielded, existing live-fire facilities may no longer be adequate for the training and certification of military and law enforcement personnel. The previous generation of ammunition was developed using traditional lead core material, leading to lower velocities, more energy dissipation from plastic deformation of the projectile, and less penetration ability against hard targets. In the pursuit of lead-free ammunition to minimize environmental impact on range facilities, projectiles using steel cores are becoming commonplace, increasing muzzle velocities and penetration capability, a characteristic that is often preferred in military and law enforcement applications, but can increase damage and risk of failure in training facilities. Constraints on training in many live-fire facilities are already in place when using lead-free ammunition, with some very conservatively allowing only single round impacts during training (US Army, 2017). Previous experimental studies by Kasilingam et al. (2019), Stewart (2019), and Abdel-Wahed (2010) of similar projectiles striking mild steel targets exist, where some attempt of probabilistic assessment is made by Stewart, but no studies are known by the authors to exist of small caliber projectiles impacting the abrasion resistant materials examined in this report. Numerical and theoretical studies of the analytical model proposed by Alekseevskii-Tate were conducted by He (2019), Lan and Wen (2010), Rosenberg (1990), Wen (2010), and Wen and Lan (2010), building on and confirming the original authors’ work conducted over a series of experimental and analytical studies (Alekseevskii, 1966; Tate, 1967, 1969, 1986a, 1986b).

This study focuses on the risk of single round perforation of AR500 steel targets with two inches of ballistic rubber, a commonly used system to resist ballistic impacts and protect facility users, shown in Figure 1. A nominally 0.30 caliber ogive projectile consisting of a copper jacket and lead core is studied and depicted in Figure 2. Experimental data and finite element analysis are used to validate the accuracy of an energy based analytical model, the Alekseevskii-Tate Model, which is then used to conduct Monte Carlo simulations with an importance sampling variance reduction scheme to estimate the probability of perforation of a live-fire facility. This estimate is intended for use when assessing the risk to users and for establishing restrictions on facility use when necessary. This study demonstrates a risk-based design and analysis technique that is widely applicable to other small caliber projectiles for abrasion resistant steel panel-based facilities or facilities constructed from other materials where accurate analytical models exist, such as cellular concrete or granular material-based containment systems. To the author’s knowledge, the work presented herein is the first study focused specifically on the safety and associated risk of live-fire facilities through the application of analytical models and Monte Carlo simulation techniques.

Figure 1.

AR500 steel panel facility and close up of protective system.

Figure 2.

Sketch of analyzed projectile.

Target and projectile characteristics

There exist three prevailing containment systems in live-fire facilities: ballistic cellular concrete, gravel or sand-filled walls, and AR500 steel with two inches of ballistic rubber. This study focuses on the performance of AR500 steel panel facilities. AR500 steel is an abrasion resistant steel with a mean yield strength of approximately 200 KSI, modulus of elasticity of 31,900 KSI, and a Brinell Hardness of nominally 500 HB (Jamil et al., 2016). Ballistic rubber panels are constructed from recycled tire rubber that is shredded, compressed, and fused using a binding material. Pistols and carbine style rifles are commonly fired in shoot house facilities, with the highest energy projectiles being nominally 0.30 caliber and ogival nosed. The projectile studied in this report is a conventional copper jacketed round with a lead core. This study neglects the effects of ballistic rubber panels on the depth of penetration of the projectile into the target.

Analytical model

Selecting an appropriate analytical model to describe the behavior observed during this study presented challenges due to the deforming projectile and target material, and the desire for a model that is not purely empirical to allow for meaningful simulation results and a wider application for analysis and design. Existing non-hydrodynamic models generally assume that either the target or the projectile is perfectly rigid. If the target is assumed to be rigid, as described by Taylor (1948) and Whiffin (1948), Lee and Tupper (1958), Wilkins and Guinan (1973), Recht, (1978), and Tate, (1986a), then the penetration of the plate cannot be estimated. Models that assume a perfectly rigid penetrator, such as those proposed by Poncelot and Helie (Goldsmith, 1960), Nishiwaki, (1951), Burkhardt (Recht, 1986), Recht (1990), and Zukas (1982), are capable of estimating the depth of penetration but greatly overpredict for a projectile that is eroding and deforming and are of limited use when studying conventional small arms projectiles. Hydrodynamic models account for the erosion of the projectile and target materials but are generally used at velocities just outside of the ordnance range of approximately 3000 $\frac{ft}{s}$ .

Christman and Gehring (1966) summarize hydrodynamic penetration into four phases, with the majority of penetration occurring during the first two: The Transient, and Primary phases. The transient phase begins upon contact with the target and is a very short period of high pressure lasting until the shock wave detaches from the interface of the deforming projectile and the target material. The transient material response is dependent on the nose geometry of the projectile, but the penetration during this phase is negligible compared with the overall penetration of a high velocity projectile (Tate, 1986a). At the end of the transient phase, the ogive of the projectile is severely deformed as it enters the primary phase. As a result, the transient phase and ogive are considered to have negligible impact on the penetration depth of the projectile and are thus not considered in this analysis. In the primary phase the projectile and target material deform in a fluid manner and achieve steady state penetration. Allen and Rogers (1961) proposed a simple hydrodynamic model of the primary phase that accounted for the relative strengths of the projectile and target materials as a function of their densities and an empirically defined constant, R’, and assumed that the projectile had no strength. Their model was shown to effectively predict penetration of ductile plates by deformable projectiles at intermediate velocities but underpredict the penetration at higher velocities, where the strength of the projectile significantly influences the solution. Due to these challenges, modeling the penetration depth was accomplished using a hydrodynamic model developed independently by V. Alekseevskii and A. Tate, improving on the work of Allen and Rogers by introducing terms that account for the strength of the target and the projectile (Alekseevskii, 1966; Tate, 1967).

The Alekseevskii-Tate model is a hydrodynamic analogy, idealizing the target and projectile materials as fluids due to the extreme pressures at the base of the impact crater governed by a modified Bernoulli Equation

\frac{1}{2} ρ_{p} {(V_{i} - u)}^{2} + Y_{p} = \frac{1}{2} ρ_{t} u^{2} + R_{t}

(1)

where the variable $ρ$ is the density of the material, V_i is the impact velocity, and u is the instantaneous velocity at the base of the projectile (Carlucci and Jacobson, 2014). The subscripts p and t denote the projectile and the target, respectively. The complex transition of the target and projectile material from acting as rigid bodies to behaving as fluids is represented by $R_{t}$ and $Y_{p}$ —the ballistic resistance of the target and projectile, respectively—the pressure above which it is assumed that materials behave hydrodynamically. Ballistic resistance of the projectile material can be determined experimentally by testing very low velocity projectiles and measuring the initial length ( $L_{o}$ ) and the length of the remaining material ( $L_{r}$ ) through the relationship (Shin et al., 2018; Zook et al., 1992)

Y_{p} = \frac{\frac{1}{2} ρ_{p} {Vi}^{2}}{\ln \frac{L_{o}}{L_{r}}}

(2)

When very low velocity test results where some length of projectile remains after impact are not available, previous studies indicate that the ballistic resistance of a material is proportional to its dynamic yield stress, $σ_{dy}$ . Recht (1978) developed the relationship

σ_{d y} = 4.2 x 10^{6} B H N (P a)

(3)

for dynamic yield stress based on the Brinell Hardness (BHN) of a material, and Tate (1986b) proposed a relationship of

Y_{p} = (1 + λ) σ_{dyp}

(4)

with an estimate of $λ = . 7$ based on his experimental results for the ballistic resistance of the projectile material and is used for this analysis. To account for the composite construction of the projectile, the weighted average of the Brinell Hardness of the copper jacket and lead core was used to estimate the dynamic yield stress of the projectile.

Previous authors Goodier (1965), Hanagud and Ross (1971), Hill (1950), Hopkins (1960), and Tate (1986b) have presented relationships treating the ballistic resistance of the target material as a constant based upon an advancing cavity model at velocities well above the ranges of small arms with penetrations that are highly confined and deep within the target material. These relationships are not suitable for this analysis and will yield inaccurate results. There is evidence that the ballistic resistance of the target material is not a fixed property and may be a function of impact velocity and confinement. Work by Kozhushko et al. (1991), Orphal et al. (1996, 1997), and Orphal and Franzen (1997) suggests that the magnitude of the ballistic resistance is sensitive to impact velocity and correlated with the dynamic yield strength of the material. Subramanian and Bless (1995) show that with aluminum, the ballistic resistance varies with impact velocity and Anderson and Royal-Timmons (1997) show that the ballistic resistance varies with the confinement of the target material (Hazell, 2015). Zook et al. (1992) writes that when the penetrator-target interface is close to the surface of the target that the ballistic resistance should be less than the resistance found using an advancing cavity model and proposes the relationship

R_{t} = R_{t, B H N} + (R_{t, A C} - R_{t, B H N}) (1 - e^{- c P})

(5)

where the subscript BHN describes an estimate of the ballistic resistance at the surface of the target based on Brinell Hardness and the subscript AC describes an estimate of the maximum ballistic resistance from the advancing cavity models. P is the depth of penetration into the target, the confinement, and c is an empirical constant. Zook’s approach directly accounts for the variation caused by confinement and indirectly accounts for the variation due to velocity, but as a function of penetration depth, limiting its application to posterior analysis where the depth of penetration is already known. To improve Zook’s approach, the following relationship was developed for this analysis

R_{t} = σ_{dyt} + (R_{t, A C} - σ_{dyt}) (1 - B e^{- c V_{i}})

(6)

This model accounts for the correlation with the dynamic yield strength of the target, target confinement, and the impact velocity. A second empirical constant, $B$ , is introduced to allow for greater flexibility in application of the model. The experimental data and the results of numerical simulations were used to estimate these variables at a wide range of velocities.

Zook compared five proposed relationships for ballistic resistance based on an advancing cavity model with experimental results and determined that the model presented by Tate (1986b) best described the target material behavior (Zook et al., 1992), where the ballistic resistance of the target is defined as

R_{t, A C} = σ_{d y t} (\frac{2}{3} + \frac{2 E_{t}}{3 σ_{d y t}})

(7)

and is adopted for this analysis.

Three cases of hydrodynamic models exist: where the ballistic resistance of the target is greater than the resistance of the projectile, where the ballistic resistance of the projectile is greater than the resistance of the target, and where the ballistic resistances of the materials are equal. This study discusses only the first case in which the ballistic resistance of the target is greater than the resistance of the projectile ( $R_{t} > Y_{P}$ ).

Equation (1), the modified Bernoulli Equation, is integrated numerically until the velocity ( $v$ ) of the projectile is equal to zero, or the projectile perforates the target, giving

u (v) = (\frac{1}{1 - γ^{2}}) (v - γ \sqrt{v^{2} + A})

(8)

γ = \sqrt{\frac{ρ_{t}}{ρ_{p}}}

(9)

A = \frac{2 (R_{t} - Y_{p}) (1 - γ^{2})}{ρ_{t}}

(10)

Should the projectile perforate the target, $u$ is the residual velocity of the projectile. When the projectile is at rest, a second numerical integration is done to calculate the depth of penetration of the projectile giving

P = \frac{ρ_{p}}{Y_{p}} \int_{V_{c}}^{V_{i}} u (v) l (v) dv

(11)

where $V_{c}$ is the velocity below which penetration into the target material ceases and is estimated through the equation

V_{c} = \sqrt{2 \frac{R_{t} - Y_{p}}{ρ_{p}}}

(12)

and the change in length of the projectile, $l (v)$ , due to deformation and erosion is described by the equation

\frac{l (v)}{L} = {(\frac{v + \sqrt{v^{2} + A}}{V_{i} + \sqrt{V_{i}^{2} + A}})}^{\frac{R_{t} - Y_{p}}{γ Y_{p}}} \exp {\frac{γ ρ_{p}}{2 (1 - γ^{2}) Y_{p}} [(v \sqrt{v^{2} + A} - γ v^{2}) - (V_{i} \sqrt{V_{i}^{2} + A} - γ V_{i}^{2})]}

(13)

The analytical model does not account for the ballistic rubber panel’s effect on the velocity of the projectile as it strikes the AR500 steel plate. The primary purpose of the rubber is to prevent injury to facility users from fragmentation of the projectile, not minimize damage to the facility. This study does not account for effects of the rubber on the projectile and provides a conservative estimate of the penetration depth. Estimates for the material properties used in this analysis are provided in Table 1.

Table 1.

Material properties used in analytical analysis.

Material	Length (in)	Diameter (in)	Mass (grain)	Density (lbm/in3)	Yield strength (KSI)	Modulus of elasticity (KSI)	Brinell hardness
0.30 caliber projectile	1.141	0.308	147	0.378	16.5	–	–
AR500 steel	–	–	–	0.284	200	31900	470

Numerical model

The Finite Element Analysis (FEA) was conducted using the Elastic Plastic Impact Computations (EPIC), a Lagrangian finite-element code developed to simulate the mechanical responses of solid materials during events of short duration, such as projectile impacts (Johnson et al., 2019). A nominally 150 grain, 0.30 caliber ogival projectile with a lead core and copper jacket was simulated impacting a ½ inch thick AR500 steel plate at the minimum, maximum, and median strike velocities measured during the experimental testing and at velocities up to 4000 $\frac{ft}{s}$ . The simulations were completed using three-dimensional finite elements where highly distorted elements are converted to smooth particle hydrodynamics as they exceed the rupture strains of their materials, allowing for the conservation of mass. Examples of the simulation set-up and results are shown in Figure 3. A mesh refinement study was conducted comparing four models of 5.1 million elements, 6.6 million elements, 8.5 million elements, and 12.3 million elements. The percent change in the depth of penetration was 2.29% from the model with 5.1 million elements to 6.6 million elements, 0.70% from the model with 6.6 million elements to 8.5 million elements, and 0.11% from the model with 8.5 million elements to 12.3 million elements. Based on a convergence criteria of nominally 10^–3, the mesh with 8,549,376 tetrahedral elements with an average element size of 0.022 inches was determined to be sufficiently precise and computationally efficient. The projectile was modeled using 91,776 tetrahedral elements, giving a matching average element size of 0.022 inches. Natural boundary conditions were used to model the target restraints.

Figure 3.

Numerical simulation of projectile impacting steel plate.

The Johnson-Cook constitutive model of material flow stress was used to model the strength of each material and is described by the empirical equation

σ = (A + B ε^{n}) (1 + C \ln {\dot{ε}}^{*}) (1 - T^{* m})

(14)

where ${\dot{ε}}^{*}$ is the dimensionless strain rate and $T^{*}$ is the homologous temperature. The material constants $A, B, n, C, and m$ are fit to material test data conducted a different strain rates and temperatures (Johnson and Cook, 1983). The fracture characteristics of each material were captured using the Johnson-Cook fracture model where the strain at fracture is given by the equation

ε_{f} = (D_{1} + D_{2} e^{D_{3} σ^{*}}) (1 + D_{4} \ln {\dot{ε}}^{*}) (1 + D_{5} T^{*})

(15)

where $σ^{*}$ is the dimensionless pressure-stress ratio and the constants $D_{1} - D_{5}$ are fit to tension test data of notched specimens are different temperatures and strain rates. The equation of state for each material was described using the cubic form of the Mie-Gruneisen equation

P = (K 1 μ + K 2 μ^{2} + K 3 μ^{3}) (1 - \frac{Γ μ}{2}) + Γ E_{s} (1 + μ)

(16)

where $μ = \frac{ρ}{ρ_{o}} - 1$ , $Γ$ is the Gruneisen Coefficient, $E_{s}$ is the internal energy, and $K 1 - K 3$ are constants fit to shock Hugoniot experiments (Johnson et al., 2019; Kohn, 1969; Meyers, 2014).

A steel with a Brinell Hardness of 470 and the same yield strength, modulus, and density was parameterized by Meyer et al., 2012 and used to simulate the AR500 steel target material in this study. Based on testing and analysis of high hardness steels conducted by Jamil et al. (2016) and Kasilingam et al. (2019), it was determined that the differences between the steel varieties would have negligible impact on the computational results. Projectiles, including the core and jacket, were modeled based on their measured geometry and materials reported by the manufacturers. The copper jacketing was characterized using the parameters listed in Table 2 with strength data compiled from Johnson and Cook (1983), fracture data compiled from Johnson and Cook (1985) and Shock Hugoniot equation of state data compiled from Kohn (1969). The lead core was characterized using the parameters in Table 2 with strength and fracture data compiled by Johnson et al. (2019) and Shock Hugoniot equation of state data compiled from Kohn (1969).

Table 2.

Material properties used in numerical analysis.

Copper material properties					Lead material properties
Johnson-Cook strength parameters					Johnson-Cook strength parameters
A [MPa]	B [MPa]	n	C	m	A [MPa]	B [MPa]	n	C	m
90	292	0.31	0.025	0.019	21	43	0.43	0.038	1
Johnson-Cook damage parameters					Johnson-Cook damage parameters
D₁	D₂	D₃	D₄	D₅	D₁	D₂	D₃	D₄	D₅
0.54	4.89	−3.03	0.014	1.12	0	4	−1.15	0	0
Mie-Gruneisen equation of state parameters					Mie-Gruneisen equation of state parameters
K1 [GPa]	K2 [GPa]	K3 [GPa]	Γ		K1 [GPa]	K2 [GPa]	K3 [GPa]	Γ
137.2	175.1	564.2	1.96		50.1	49.8	201.8	2.2

Monte Carlo simulation

Monte Carlo (MC) simulation is a tool for conducting experiments numerically and can be used to obtain simulated data. For engineering purposes, MC can be utilized to study the predicted response of a system by sampling probability distributions of inputs to a governing equation of interest. For this reason, MC is an analytical sampling technique that can quantify predicted error in the outcome of a physical system (Ang and Wilson, 1984). To account for the uncertainty of the material properties and the stochastic nature of ballistic events, input parameters of the Alekseevskii-Tate model are described as continuous random variable distributions where $Y = g (\overset{⇀}{X})$ , $Y$ is the response of the target material, and $\overset{⇀}{X} = {X_{1,} X_{2} \dots X_{n}}$ is a vector of input variables with a joint probability function $f_{\overset{⇀}{X}} (\overset{⇀}{x})$ . $X$ represents a random variable of uncertain value and $x$ represents a realization of the variable or known value. The cumulative distribution function for $Y$ is given by

F_{Y} (y) = P (Y \leq y) = \int_{{g (\vec{x}) \leq y}} \dots \int^{} f_{X_{1}, \dots, X_{n}} (x_{1}, \dots, x_{n}) d x_{1} \dots d x_{n}

(17)

= \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} I [g (\overset{⇀}{x}) \leq y] f_{x_{1}, \dots x_{n}} (x_{1}, \dots x_{n}) d x_{1} \dots d x_{n}

(18)

where I [g (\overset{⇀}{x}) \leq y] = {\begin{matrix} 1, g (\overset{⇀}{x}) \leq y \\ 0, g (\overset{⇀}{x}) > y \end{matrix}

(19)

and F_{Y} (y) \approx \frac{1}{N} \sum_{k = 1}^{N} I [g (\overset{⇀}{x}) \leq y]

(20)

By simulating realizations of $\overset{⇀}{X}$ many times, one can calculate an estimate of $F_{Y} (Y)$ , the probability that the target material fails, as well as estimates of the statistical properties of the output parameter, or the depth of penetration (Gilbert, 2019).

The velocity of the projectile impact was modeled as a continuous random variable using the recorded experimental velocities. The dataset of velocities was placed into bins using the Freedman-Diaconis rule to reduce the error between the empirical probability mass function and the selected theoretical distribution, and selecting an appropriate random variable distribution to minimize the error of the distribution fit (Freedman and Diaconis, 1981). A T-Location Scale distribution was selected due to its continuous nature, allowing for its use during importance sampling, and its ability to minimize the standard error of the model and closely match the mean and standard deviation of the experimental results. A probability density function of

\frac{Γ (\frac{v + 1}{2})}{σ \sqrt{v π} Γ (\frac{v}{2})} {[\frac{v + {(\frac{x - μ}{σ})}^{2}}{v}]}^{- (\frac{v + 1}{2})} w h e r e {\begin{matrix} μ = 2776.15 \\ σ = 41.1954 \\ v = 2.84411 \end{matrix}

(21)

was fit to the results as depicted in Figure 4. Table 3 describes the specific distributions and parameters utilized for each input variable. Where sufficient data was not available, triangular distributions were assumed based on reported and measured average, minimum, and maximum values.

Figure 4.

Projectile velocity probability distribution function.

Table 3.

Probability density functions.

Probability density functions
Variable	Distribution	Parameters
Variable	Distribution	Minimum	Peak (Mean)	Maximum	µ	σ	ν
Target yield strength (KSI)	Triangular	195	200	215
Target density (lbm/in³)	Triangular	0.281	0.284	0.287
Target modulus of elasticity (KSI)	Triangular	31200	31900	32500
Target Brinell hardness (BHN)	Triangular	450	470	520
Projectile yield strength (KSI)	Triangular	14.85	16.5	18.15
Projectile density (lbm/in³)	Triangular	0.374	0.378	0.382
Projectile length (in)	Triangular	1.121	1.141	1.161
Projectile velocity (ft/s)	T-Location Scale				2776.15	41.1954	2.84411
Importance sampling probability density functions
Variable	Distribution	Parameters
Variable	Distribution	Minimum	Peak (Mean)	Maximum	α	β	γ
Target Brinell hardness (BHN)	Triangular	450	462.5	475
Projectile density (lbm/in³)	Triangular	0.378	0.38	0.382
Projectile velocity (ft/s)	T-Location Scale				3139	41.1954	2.84411

A weakness of MC is the computational effort required to accurately estimate the small probabilities generally allowable in structural engineering applications. Therefore, it is important to quantify the accuracy by expressing the estimate of the probability as a confidence interval in relation to the true probability and to estimate the required number of simulations prior to performing the analysis. For large numbers of realizations, the estimate of the parameters will approach a normal distribution by the Central Limit Theorem. This expectation allows for an estimate of the number of realizations required to define a confidence interval of the probability of an event through the relationship

P (| \hat{P} - p | \leq ε) = 95 % C o n f i d e n c e = 2 Φ (\frac{ε \sqrt{N}}{\sqrt{p (1 - p)}}) - 1

(22)

where the number of realizations required to provide a 95% confidence interval, given an estimate of the true probability (p) and the percent error of the result (% Error in P) is

N_{r e q d} \approx {(\frac{196}{% Error i n p})}^{2} (\frac{1 - p}{p})

(23)

To evaluate the precision of calculated moments of the penetration depth, the relationship

P (| \hat{M_{Y}} - μ_{Y} | \leq ε) = 95 % Confidence = 2 Φ (\frac{ε \sqrt{N}}{σ_{Y}}) - 1

(24)

can be evaluated, resulting in

N_{reqd} \approx 2 {(\frac{196}{% Error in σ_{Y}^{2}})}^{2} + 1

(25)

and N_{reqd} \approx {(\frac{196}{% Error in μ_{Y}})}^{2} {δ_{Y}}^{2}

(26)

where the number of realizations required to produce a desired level of precision with a 95% confidence interval is a function of the level of precision of the estimate of the mean or variance and the estimated coefficient of variation ( $δ_{y}$ ) of the simulation results (Gilbert, 2019; Sheldon, 2013).

Estimating the true variance of the penetration depth to within ± 1% with 95% confidence requires approximately 76,833 realizations from equation (25). An estimate the coefficient of variation of the mean was made using three sets of 100 realizations each, resulting in an estimated $δ_{y}$ of 0.771. Using this estimate of $δ_{y}$ and equation (26), approximately 22,836 realizations are required to estimate the true mean penetration depth to within ± 1% with 95% confidence. To estimate the mean and variance of the penetration to within ± 1% of the true value with 95% confidence, 10 sets of 8000 realizations was used in this analysis.

Assuming the probability of failure is approximately 1 × 10^–5 and at that probability of failure an estimate to within ± 25% of the true probability with 95% confidence is sufficient, then approximately 6 × 10⁶ simulations are required from equation (23). Ten sets of 6 × 10⁵ simulations were conducted to allow for comparison of the variability in the results to the predicted variability. The increased effort to reduce the variance in the result is due to the need for the event to occur during simulation to estimate its probability, leading to wasted effort generating realizations when the event does not occur. To further increase the accuracy and efficiency of calculations, importance sampling was utilized to cause more events to occur during sampling and reduce the variance in the estimate of the true probability. The probability of an event was obtained through the process described by equations (17)–(20). In importance sampling, these equations are restated as

F_{Y} (y) = P (Y \leq y) = \int_{- \infty}^{\infty} \dots \int_{- \infty}^{\infty} {I [g (\overset{⇀}{x}) \leq y] \frac{f_{\overset{⇀}{x}} (\overset{⇀}{x})}{h_{\overset{⇀}{x}} (\overset{⇀}{x})}} h_{\overset{⇀}{x}} (\overset{⇀}{x}) d (\overset{⇀}{x})

(27)

and F_{Y} (y) \approx \frac{1}{N} \sum_{k = 1}^{N} I [g ({\overset{⇀}{x}}_{k}^{'}) \leq y] \frac{f_{\overset{⇀}{x}} ({\overset{⇀}{x}}^{'})}{h_{\overset{⇀}{x}} ({\overset{⇀}{x}}^{'})}

(28)

where $h_{\overset{⇀}{x}} (\overset{⇀}{x})$ is an importance sampling joint probability density function and ${\overset{⇀}{x}}_{k}^{'}$ is a realization of $\overset{⇀}{x}$ from $h_{\overset{⇀}{x}} (\overset{⇀}{x})$ . $h_{\overset{⇀}{x}} (\overset{⇀}{x})$ is selected so that sampling from that density function results in more realizations where $g ({\overset{⇀}{x}}_{k}^{'}) \leq y$ occurs, increasing the probability of occurrence and decreasing the variance of the estimate. This larger probability is then scaled back to correspond to $f_{\overset{⇀}{x}} (\overset{⇀}{x})$ , the original joint probability density function (Sheldon, 2013), (Gilbert, 2019). To select $h_{\overset{⇀}{x}} (\overset{⇀}{x})$ , the distributions of the Brinell Hardness of the target, projectile density, and projectile velocity were shifted to focus on the failure region of the Alekseevskii-Tate model that was determined during a sensitivity analysis of the input variables. The importance sampling distributions can be referenced in Table 3. The method presented herein can be replicated and applied to other unique projectiles and targets.

Experimental results

Experimental testing was conducted by (Goodman, 2016) against ½ inch AR500 steel targets faced with two inches of ballistic rubber using nominally 0.30 caliber projectiles at distances commonly seen in live-fire facilities. The mean, median, standard deviation, and sample size for the experimental results are presented in Table 4.

Table 4.

Experimental results statistics.

Mean penetration (in)	0.021875
Median penetration (in)	0.0195
Standard deviation (in)	0.0105
Maximum velocity (ft/s)	2849
Minimum velocity (ft/s)	2751
Median velocity (ft/s)	2802
N	48

The results of that testing are presented in Figure 5. These results serve as a benchmark for comparison and validation of the analytical model for use in Monte Carlo Simulation and the numerical modeling.

Figure 5.

Experimental penetration depth into AR500 steel with rubber panel.

Numerical results

The minimum, maximum, and median velocities from the experimental results were used for the comparison of the numerical and experimental results and velocities up to 3900 $\frac{ft}{s}$ were modeled to confirm the analytical model at velocities that would likely cause failure of the plate. Figure 6 shows a comparison of the experimental, analytical, and numerical results within the region of the experimental results and Figure 7 compares the analytical model and numerical results up to 3900 $\frac{ft}{s}$ . The numerical model results correlate with the experimental and analytical results and produces reasonable estimates of the depth of penetration. Errors in the results could be caused by minor differences in the material properties used in the modeling from those used during the experimental testing. The simulated impacts also occur exactly perpendicular to the target and do not account for any yaw, pitch, or roll of the projectile resulting in increased penetration depth. Finally, the effects of the ballistic rubber are neglected, causing additional over-prediction of the penetration depth.

Figure 6.

Numerical and analytical penetration depth at ordnance velocities.

Figure 7.

Numerical and analytical penetration depth at all velocities.

Analytical results

Figure 6 and 7 shows a comparison of the experimental results with the analytical prediction of the penetration depth using the Alekseevskii-Tate model. The parameters of the target ballistic resistance model from equation (6) were calibrated by back-solving for the ballistic resistance that would result in the depth of penetration recorded from the numerical results and then compared to the experimental results for verification. A “B” parameter value of 1.251 and “c” parameter value of 0.000713 was used throughout this study.

Over the velocity range of 2750 $\frac{f t}{s}$ to 2780 $\frac{f t}{s}$ the average experimental depth of penetration is 0.016 inches with a maximum penetration of 0.035 inches and minimum penetration of 0.015 inches. The analytical model results in a depth of penetration between 0.017 and 0.025 inches over that range and the numerical model results in a depth of penetration of 0.019 inches at 2750 $\frac{f t}{s}$ . Over the velocity range of 2780 $\frac{f t}{s}$ to 2820 $\frac{f t}{s}$ the average experimental depth of penetration is 0.022 inches with a maximum penetration of 0.045 inches and minimum penetration of 0.013 inches. The analytical model results in a depth of penetration between 0.025 and 0.035 inches over that range and the numerical model results in a depth of penetration of 0.030 inches at 2800 $\frac{f t}{s}$ . Over the velocity range of 2820 $\frac{f t}{s}$ to 2850 $\frac{f t}{s}$ the average experimental depth of penetration is 0.027 inches with a maximum penetration of 0.061 inches and minimum penetration of 0.019 inches. The analytical model results in a depth of penetration between 0.035 and 0.042 inches over that range and the numerical model results in a depth of penetration of 0.042 inches at 2850 $\frac{f t}{s}$ .

The model produces results that are a reasonable approximation of the average penetration depths, given the stochastic nature of impact events. The model does not account for the yaw of the projectile at impact and assumes that the angle of obliquity of the impact is zero degrees. The model also neglects the potential effect of the rubber panel on the strike velocity. These factors result in the model overpredicting penetration compared with the experimental results. However, the overprediction is by design because the outcome provides a conservative assessment of penetration that is appropriate for the purposes of assessing risk and informing designers of protective structures. The model also makes significant simplifications in the transition behavior from elastic to plastic and from plastic to hydrodynamic, also contributing some uncertainty.

The Alekseevskii-Tate model was developed for long rod projectiles penetrating purely hydrodynamically, while the penetration behavior of the projectiles studied herein is a combination of multiple failure mechanisms at different ranges of its velocity. The approach above serves as an adequate model of the experimental results and provides a conservative estimate of the penetration depth. The behavior of deforming projectiles can be described at ordnance velocities with reasonable accuracy by correlating the ballistic resistance term with the dynamic yield strength of the material as a function of velocity. As the velocity increases and the projectile begins to penetrate in a hydrodynamic manner, and the ballistic resistance of the target approaches that defined by the advancing cavity models, maintaining accuracy at velocities beyond 3000 $\frac{ft}{s}$ (Tate, 1986a). The strength of this model is its reliance on hydrodynamic theory and limit of the use of empirical factors to estimate specific material properties that cannot be easily measured.

Risk analysis from Monte Carlo simulation

Using the Alekseevskii-Tate model shown to provide reasonable estimates of penetration of 0.30 caliber projectiles into AR500 steel plates, Monte Carlo simulations were conducted to estimate the average penetration depth and probability of perforation. The results of both simulations are presented in Table 5. This study estimates that the mean probability that a fully copper jacketed, 0.30 caliber, lead core-ogive nose projectile, will perforate a ½ inch AR500 steel target lies within $1.3884 x 10^{- 5}$ and $1.3892 x 10^{- 5}$ with 95% confidence. This level of risk is equivalent to the level recommended for the ultimate capacity in other structural engineering applications and corresponds to a reliability index of 4.19 (Probabilistic Model Code: Basis of Design, 2001). The expected value for the mean depth of penetration from simulation is estimated to be within 0.021156 and 0.021158 inches with a mean standard deviation of between 0.01619 and 0.01620 with 95% confidence, compared to an experimental mean of 0.021875 with a standard deviation of 0.0105 from 48 trials.

Table 5.

Monte Carlo simulation results.

Set	E (Penetration) (in)	σ (Penetration) (in)	P (Penetration > 0.5 in)	Importance sampling
Set	E (Penetration) (in)	σ (Penetration) (in)	P (Penetration > 0.5 in)	Set	P (Penetration > 0.5 in)
1	0.02120	0.01564	0	1	9.5648E-06
2	0.02126	0.01586	0	2	1.6379E-05
3	0.02141	0.01628	0	3	9.2690E-06
4	0.02099	0.01632	0	4	2.3668E-05
5	0.02137	0.01584	0	5	9.7619E-06
6	0.02111	0.01826	1.25E-04	6	1.8909E-05
7	0.02100	0.01587	0	7	1.2719E-05
8	0.02098	0.01539	0	8	1.1017E-05
9	0.02110	0.01699	0	9	1.0662E-05
10	0.02114	0.01550	0	10	1.6927E-05
Sample mean	0.02116	0.01619	1.2500E-05	Sample mean	1.3888E-05
Sample Std. Dev.	0.00015	0.00086	3.9528E-05	Sample Std. Dev.	4.8691E-06
n	80000	80000	80000	n	6000000
Standard error	5.4470E-07	3.0510E-06	1.3975E-07	Standard error	1.9878E-09
2.5 percentile	0.021156	0.01619	1.2226E-05	2.5 percentile	1.3884E-05
97.5 percentile	0.021158	0.01620	1.2774E-05	97.5 percentile	1.3892E-05
				Beta	4.19

Conclusion

An Alekseevskii-Tate model for the hydrodynamic penetration of an AR500 steel plate was shown to have reasonable agreement with experimental test results compiled by Goodman (2016) when modifying existing cavity expansion models of the target resistance to account for the unconfined material based on the results of numerical simulations. Both the analytical model using the modified target resistance and numerical results showed reasonable agreement with the experimental testing. Using the analytical model, Monte Carlo simulation was used to estimate the mean depth of penetration and an importance sampling scheme was employed to estimate the probability of a single round perforating the AR500 steel containment system. The probability of failure was shown to be on the order of magnitude of the ultimate capacity accepted in other structural engineering applications. The methods presented herein can be used to evaluate the design of existing and future facilities to ensure safe, affordable, and reliable operation, while accurately describing appropriate restrictions on facility use when considering different classes of ammunition.

Future work

The techniques described in this study should be applied to cellular concrete and gravel and sand filled wall facilities using existing analytical models for those materials developed by Forrestal et al. (1994) and Forrestal and Luk (1992). To improve the accuracy of the numerical results and expand the modeling to other live-fire facility containment systems, high strain rate constitutive modeling of cellular concrete, ballistic rubber, plywood, dimensional lumber, field samples of AR500 steel, and projectile materials should be undertaken. These will provide tools for facility designers and operators to assess the performance of these facilities against a wide variety of weapon systems using numerical simulation tools, when available.

The aforementioned analysis focuses on the single round impact of projectiles against AR500 steel but does not address the effects from multiple impacts over the life cycle of the facility containment system. Some testing of the multiple impact durability of these materials exists, and a Bayesian framework with additional experimental testing could be used to forecast the containment system performance with additional simulation utilized to estimate the proximity impacts. An analytical model based on this framework could then be developed by normalizing the energy of the projectile and the relative ballistic resistances of the materials to develop a universal assessment tool and establish proof testing to certify facilities for different projectiles. This technique could provide a better metric for assessing facility reliability and establishing restrictions based on an acceptable level of risk to the user and could be validated through numerical simulation. An existing analytical model for multiple projectile impacts into concrete was developed by Gomez and Shukla (2001) and could be used in an approach similar to the one presented in this study.

Footnotes

Acknowledgements

The authors would like to thank Mr. Shawn Spickert-Fulton, Mr. Ray Chaplin, Mr. Vincent D’Anton, and Mr. Mark Minisi for their technical expertise regarding round development and the development of the models presented in this study. The views expressed herein are those of the authors and do not purport to reflect the position of the United States Military Academy, the Department of the Army, or the Department of Defense.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Combat Capabilities Development Command – Armaments Center at Picatinny Arsenal, NJ.

ORCID iD

Brad Gregory Davis

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