Monte Carlo simulation of steel spaced armor ballistic performance against fragment-simulating projectiles

Abstract

Terminal ballistics models are typically calibrated with data from projectile impacts into single-material-single-layer (monolithic) targets. Such models fail to consider the complexities of layered and spaced armor configurations, which are commonplace in force-protection applications. Approaches leveraging existing monolithic terminal ballistic models have been developed to approximate such configurations, estimating overall response by superimposing the individual layer responses. A Monte Carlo (MC) superposition method is proposed for modeling the ( $V_{50}$ ) of A36 steel layered and spaced armor configurations against fragment-simulating projectiles (FSPs). Its performance was examined relative to four existing superposition methods from literature, using experimental $V_{50}$ data. Each superposition methodology was applied on two underlying terminal ballistics models for the layers, demonstrating their ability to leverage existing monolithic terminal ballistic models. The proposed MC superposition methodology consistently yielded the lowest error regardless of the underlying terminal ballistics model.

Keywords

protection ballistic limit V50 terminal ballistics weapon effects impact fragment-simulating projectiles (FSP)A36 steel spaced armor layered armor fast-running model Monte Carlo simulation

Introduction

Terminal ballistics research is concerned with interactions between projectiles and targets, which are inherently complex, multivariate, and stochastic in nature since these interactions depend on projectile and target geometries, material properties, and impact conditions (Aptukov, 1990; Backman and Goldsmith, 1978; Ben-Dor et al., 2005; Ben-Dor et al., 2019; Wilkins, 1978). Target and projectile interactions with stop/perforation or pass/fail results are a quantal response problem (ARL, 2012; ARL, 2014; DARCOM, 1983), where the transition between projectile partial penetration (PP) to complete penetration (CP) does not occur at a discrete striking velocity. Instead, a zone of mixed results (ZMR) is frequently observed, where repeated tests at constant velocity levels result in a ratio of CP to PP outcomes, with the likelihood of CPs increasing with the velocity level. Because of this, ballistic perforation characterization is often based on response percentiles, with 50% being the most common.

The $V_{50}$ velocity is a stochastic parameter in terminal ballistics representing the speed in the ZMR at which the probability of perforation is 50% for a specific projectile-target pair. The $V_{50}$ enables quantitative benchmarking and comparison of protection performances between different armors against a specific projectile, or vice versa (BRL, 1992; Cunniff, 2014). From a kinetic energy balance standpoint, the $V_{50}$ is an indicator of energy threshold for target defeat. The experimental procedure for obtaining the $V_{50}$ is specified by the Department of Defense (DoD) MIL-STD-662F testing standard (DoD, 1997). MIL-STD-622F is the result of data reduction techniques and many decades of ballistic research since WWII and assumes a normal distribution of critical velocities. Other methodologies for $V_{50}$ calculation exist and require varying numbers of shots (D&PS, 1954; D&PS, 1957). Among them is the method of building probability curves based on the frequency of CPs at various velocity intervals (Delsasso et al., 1941; White, 1946) using 150 or more shots (D&PS, 1954), which makes it experimentally expensive and uncommon.

The main projectiles of interest in $V_{50}$ characterization are typically small arms ammunition and fragment-simulating projectiles (FSPs). Primary fragments emanating from rockets, artillery, and mortar (RAM) rounds pose a significant threat in military and force protection applications. These fragmenting munitions use high explosive detonations to break up and accelerate warhead casing ejecta or pre-formed fragments. RAM warheads produce fragments with varying materials, geometries, masses, and velocities, all of which affect their penetration ability (DoD, 1990). Full-scale arena tests for warhead fragmentation are costly, and acquiring warheads or manufacturing surrogates can be prohibitive. To address this, FSPs were designed to replicate the terminal ballistic effects of primary fragments with consistency in a ballistic laboratory setting (ATEC, 1983). FSPs come in standard discrete masses and diameters, manufactured in accordance with MIL-DTL-46593B (DoD, 2006).

Structural steels are a common material used in protective structures and retrofits (DoD, 2008a; DoD, 2008b; DoD, 2008c; FEMA, 2006; FEMA, 2011). Several models have been developed over time to predict terminal ballistic effects of primary fragments on mild steel targets (Ben-Dor et al., 2019). Common model types used in weapon effects research include empirical, analytical, and high-fidelity numerical models (Reed et al., 2002). Development of terminal ballistic models commonly relies exclusively on terminal ballistic data for single-material-single-layer (monolithic) targets, where both the projectile and the target are in pristine, undamaged conditions. This approach captures the baseline response of projectile-target pairs, isolates the effects of the main driving variables, and diminishes error sources. However, due to the weight of steel panels, they are often stacked to achieve a layered armor with the desired thickness while maintaining practicality during transport and installation. In protective applications, layered and spaced armor configurations are commonplace and add complexities that are not captured in monolithic target tests and models. Layered armors consist of one or more layers of materials in contact with each other. Spaced armors are essentially layered armors that contain a separation, i.e., an air gap, between two or more layers. These types of armor configurations can occur by design when the goal is to leverage and combine different material properties and induce behaviors that disrupt projectile penetration effectiveness. They can also occur by coincidence due to the projectile’s trajectory and obstacles in its path.

A substantial body of research has been published on the layering and spacing of armor materials. Ben-Dor et al. (2019) provide a comprehensive review of this work and the current state of the art. Due to contradictory results, a consensus on the effects on armor ballistic properties has not been established (Ben-Dor et al., 2006). The effects of spacing, layering, and the order of layers are ultimately highly dependent on the target material, projectile material and geometry, and impact conditions (Ben-Dor et al., 2019). Understanding the contributions of these variables yields improved terminal ballistic models. Since typical monolithic terminal ballistic models cannot account for the effects of layered and spaced armor configurations on their own, a common practice for estimating terminal ballistic effects on multi-material layered and spaced armors is to superpose the individual layer effects calculated by the monolithic models. Such approaches usually assume (1) the projectile interacts with one layer at a time, (2) target layers do not interact with one another, and (3) pristine projectile impacts at each layer, meaning there is no projectile mass loss nor projectile deformation when the projectile impacts the next layer. Several methods for performing this analytical superposition are available in the literature and rely on the law of conservation of energy. Ben-Dor et al. (2019), Kasano and Abe (1997), Recht and Ipson (1963), and Rosenberg and Dekel (2016) describe a root-sum-squared (RSS) layer $V_{50}$ superposition methods for non-plugging, spherical-nosed, and blunt-nosed projectiles. The method recommended in DoD design guidance in both the UFC 3-340-02 (DoD, 2008a) and UFC 4-023-07 (DoD, 2008c) is a successive calculation of residual velocities for each layer in the configuration, using the previous layer’s exit conditions as the impact conditions for the next, and solving for the highest first-layer striking velocity that yields a last-layer residual velocity of zero.

The Monte Carlo (MC) method is a computational technique that is often a cost-effective alternative to expensive or impractical experimental procedures. Since the 1940s, MC simulations have been used to analyze the probabilistic response of complex systems (Metropolis and Ulam, 1949). An advantage of the MC method is its ability to represent these systems using deterministic models, introducing variance by means of random sampling of input variables through numerous trial repetitions to simulate the effects on overall system behavior. Johnson et al. (2014) leveraged the MC method in the field of ballistics to assess the uncertainty in $V_{50}$ resulting from different sequential sensitivity testing methods, estimators, and stopping criteria, and determine their efficiency and accuracy. Riley et al. (2012) used MC simulations to assess the impact of starting velocity, total number of shots, and armor performance on the uncertainty of $V_{50}$ performance estimates. Davis et al. (2021) used MC simulations to evaluate the perforation probability of a single impact by a small-caliber ammunition against live-fire shoot house containment systems. The use of MC simulations in these studies demonstrates the applicability of the method to explore and characterize complex terminal ballistic probabilistic behaviors.

This paper presents an enhanced layer superposition method to estimate the $V_{50}$ of spaced steel armor configurations against FSPs. The method leverages MC simulations, using deterministic monolithic terminal ballistic models and probability curve estimation from CP frequencies in the ZMR. The objectives of this research were to (1) evaluate the performance of the MC layer superposition methodology relative to other fast-running layer superposition methods from literature and (2) prove accurate solutions are determined for spaced armor configurations using existing monolithic terminal ballistic models.

Experimental setup and results

In this investigation, ASTM A36 mild steel is the target material. A36 is a low-carbon steel specified by the American Society for Testing and Materials (ASTM) Standard Specification A36/A36M standard carbon structural steel (ASTM, 2019). The United States Army Corps of Engineers Engineering Research Development Center (ERDC) Ballistics Facility conducted several test series investigating the $V_{50}$ for several spaced armor configurations during the 2012 to 2021 period. These experiments were conducted in accordance with MIL-STD-662F, and the $V_{50}$ values were calculated using the four-shot ballistic limit (2 PPs and 2 CPs within an 18 m/s span constraint). A 0.51 mm thick 2024 T3 aluminum witness panel was placed 152 mm behind and parallel to the target. The determination of PP or CP was based on perforation of the witness panel. This CP criterion provides a $V_{50}$ protection ballistic limit (V₅₀PBL); however, the term $V_{50}$ will be used going forward. Residual mass data was not collected in any of the tests. The A36 layer plates were dimensioned 305 mm square. The steel plates were securely clamped to a metal frame capable of adjusting the spacing distance in between, as presented in Figure 1. The average hardness for the A36 plates was 83.1 on the Rockwell B hardness scale, corresponding to a Brinell hardness (BHN) of 159.

Figure 1.

Target frame for spaced armor configurations. The plate on the left is held at a fixed location, while the location of the plate to the right is adjustable to create the desired air gap. The target frame is clamped to the table to prevent it from sliding upon ballistic impacts.

The FSP projectiles were manufactured in accordance with MIL-DTL-46593B. The FSPs were launched from rifled Mann barrels using a small-caliber gun system. The experiments were all performed at an impact obliquity angle of 0 deg. Four Oehler Model 57 infrared photoelectric velocity screens connected to two chronographs were placed downrange before the target to measure projectile striking velocity. The distance $L_{1}$ between screens was 762 mm. Two primary velocities were calculated, $V_{1}$ using the first and third screens and $V_{2}$ using the second and fourth screens. The average deceleration was calculated from $V_{1}$ and $V_{2}$ over the distance $L_{1}$ . The distance $L_{2}$ from the third screen to the target face varied with the configuration being tested, ranging from 1397 to 1954 mm. Jordan and Naito (2014) present a general schematic of this ballistic laboratory setup. The projectile striking velocity $V_{S}$ was calculated by linearly extrapolating the deceleration over $L_{2}$ using the following equation:

V_{S} = V_{2} + (\frac{L_{2}}{L_{1}}) (V_{1} - V_{2})

(1)

Two Phantom v710 high-speed cameras operating at a sample rate of 8300 fps were placed perpendicular to the projectile trajectory, one in front and the other behind the target. High-speed digital video cameras served as backup for striking velocity measurements and to determine striking yaw angle. Shots with a striking yaw angle exceeding 5 deg were discarded per MIL-STD-662F.

Table 1 presents the experimental matrix and specifies the FSP projectile diameter, the armor configuration tested, and the experimental V₅₀. The notation used for armor configurations in this study denotes thicknesses of solid armor layers as numbers outside of parentheses, while numbers inside parentheses indicate the corresponding air gap distances separating those layers. The notation is arranged so it’s read in the same direction the projectile traveled. Note that cases 1, 10, 11, and 15 did not generate a

V_{50}

value due to either system limitations or specific project goals. The inequality stated in these cases is the highest velocity shot without achieving CP, and the

V_{50}

is expected to be equal or higher.

Table 1.

Experimental matrix.

Case	FSP diameter^a [mm]	FSP Mass^a [g]	Armor Configuration [mm]	V₅₀ [m/s]
1	7.62	2.85	6.35 (1016) 6.35	≥1910
2	12.7	13.41	6.35 (0) 6.35	1153
3	12.7	13.41	6.35 (13) 6.35	1143
4	12.7	13.41	6.35 (25) 6.35	1201
5	12.7	13.41	6.35 (51) 6.35	1155
6	12.7	13.41	6.35 (76) 6.35	1125
7	12.7	13.41	6.35 (89) 6.35	1083
8	12.7	13.41	6.35 (140) 6.35	1084
9	12.7	13.41	6.35 (1016) 6.35	1109
10	12.7	13.41	9.53 (89) 6.35	≥1636
11	12.7	13.41	6.35 (89) 9.53	≥1645
12	12.7	13.41	7.94 (76) 6.35	1379
13	12.7	13.41	7.94 (89) 6.35	1288
14	12.7	13.41	6.35 (89) 7.94	1415
15	12.7	13.41	15.9 (76) 6.35	≥2197
16	20	53.78	6.35 (1016) 6.35	651

^aNominal.

Terminal ballistics models

Two underlying monolithic terminal ballistic models, referred to here as the THOR and ERDC models, were used to evaluate the performances of the layer superposition methods discussed below. Two additional models from the DoD United Facilities Criteria (UFC) documents UFC 3-340-02 (DoD, 2008a) and UFC 4-023-07 (DoD, 2008c) were also studied but are not presented in this document due to their low accuracy in modeling experimental data for baseline A36 single-layer targets (Rios, 2021).

The THOR empirical equations developed by the Ballistics Research Laboratory (BRL) in the 1960s (Dusenberry, 2010) are perhaps the most popular empirical models for fragment terminal ballistics. These equations can model the resistance to perforation by mild steel fragments of several non-metallic (BRL, 1963) and metallic (BRL, 1961) materials of military relevance. Each material available in the THOR models is characterized by ten constants,

C_{1}

through

C_{10}

. Constants

C_{1}

through

C_{5}

are used to calculate residual fragment velocity, while

C_{6}

through

C_{10}

are for residual fragment mass. The current MIL-DTL-46593B requires FSPs to be manufactured from 4337H or 4340H steel with a Rockwell C scale hardness of 30 (DoD, 2006), making them less deformable and less prone to mass loss during penetration relative to the SAE 1020 mild steel used in the fragment simulators for the THOR experiments. For this reason, it was assumed that no significant mass loss was incurred by the FSPs for the configurations examined. This assumption is defense-conservative and is closer to what has been observed in previous A36 plate versus FSP experiments at ERDC. Table 2 presents the THOR coefficients for calculating fragment residual velocity for mild steel targets.

Table 2.

THOR equation coefficients for mild steel targets (Dusenberry 2010).

Material	$C_{1}$	$C_{2}$	$C_{3}$	$C_{4}$	$C_{5}$
Steel, mild homogeneous	2.801	0.889	−0.945	1.262	0.019

THOR’s fragment residual velocity equation is as follows:

V_{R} = V_{S} - 10^{C_{1} + 7 C_{2} + 3 C_{3}} {(T A)}^{C_{2}} m_{f}^{C_{3}} {\sec (θ)}^{C_{4}} V_{S}^{C_{5}}

(2)

where

V_{R}

is the residual velocity in m/s,

V_{S}

is the striking velocity in m/s,

T

is the plate thickness in m,

A

is the fragment presented area in m²,

m_{f}

is the fragment mass in kg, and

θ

is the impact obliquity angle in deg.

Solving equation (2) for the striking velocity with the residual velocity equal to zero transforms the relationship to:

V_{L} = {(10^{C_{1} + 7 C_{2} + 3 C_{3}} {(T A)}^{C_{2}} m_{f}^{C_{3}} {\sec (θ)}^{C_{4}})}^{\frac{1}{1 - C_{5}}}

(3)

where

V_{L}

is the deterministic limit velocity for perforation in m/s. In this study, this limit velocity is assumed to approximate the

V_{50}

The U.S. Army ERDC developed empirical terminal ballistic models for $V_{50}$ and residual velocity from historical data of FSPs against A36 plates. These models use similarity methods (Baker et al., 1991; White, 1946) for scaling ballistic data of different FSP calibers, impact obliquity angles, and target thicknesses into a single curve for regression analysis.

The ERDC model is as follows:

V_{50} = {(6.92 \times 10^{9} \frac{D^{3}}{m_{f}} {[\frac{T \sec (θ)}{D}]}^{2.03})}^{0.5}

(4)

where

V_{50}

is the

V_{50}

PBL in m/s,

D

is the fragment diameter in m,

T

is the plate thickness in m,

m_{f}

is the fragment mass in kg, and

θ

is the projectile obliquity in deg.

The residual velocity model is:

V_{R} = V_{50} (0.5209 {\{{[{(\frac{V_{S}}{V_{50}})}^{2} - 1]}^{0.5}\}}^{1.253})

(5)

where

V_{R}

is the residual velocity in m/s,

V_{50}

is the velocity in m/s, and

V_{S}

is the striking velocity in m/s.

Modeling methods for layered and spaced armors

Analytical methods from literature for layered and spaced armors are based on the principle of conservation of energy and momentum for the armor configuration. Residual velocity models for a specific layer all share the same general form (Ben-Dor et al., 2019; Kasano and Abe, 1997; Recht and Ipson, 1963; Rosenberg and Dekel, 2016):

V_{R_{i}} = α_{i} \sqrt{V_{S_{i}}^{2} - V_{50_{i}}^{2}}

(6)

where for the i^th configuration layer,

V_{R_{i}}

is the residual velocity in m/s,

V_{S_{i}}

is the striking velocity in m/s, and

V_{50_{i}}

is the layer’s

V_{50}

in m/s, and where the value of term

α_{i}

depends upon the respective underlying assumptions discussed next.

Rigid non-plugging projectiles: Root-sum-squared superposition (NP-RSS)

This superposition method describes perforation by rigid, non-plugging projectiles, for which $α_{i} = 1$ . The $α_{i} = 1$ assumption means that perforation imposes a fixed kinetic-energy loss of $\frac{1}{2} m_{f} V_{50}^{2}$ , and for impact velocities above the $V_{50}$ threshold the projectile exits without additional mass loading, momentum sharing, or velocity attenuation by the target, which means no target material plug is produced and there is no projectile mass loss. Solving the energy balance for the overall armor $V_{50}$ yields the following expression:

V_{50}^{N P - R S S} = \sqrt{\sum_{i = 1}^{n} V_{50_{i}}^{2}}

(7)

where

V_{50}^{N P - R S S}

is the configuration

V_{50}

in m/s,

n

is the number of layers in the configuration, and

V_{50_{i}}

is the

V_{50}

for the i^th configuration layer m/s.

Spherical-nosed projectile: Root-sum-squared superposition (SN-RSS)

This method considers a spherical-nosed projectile that causes the formation of a target material plug upon layer perforation, where $α_{i} = {(\frac{m_{f} + M_{i - 1}}{m_{f} + M_{i}})}^{1 / 2}$ , where $M$ is the plug mass and $i$ is the layer index. The $0.5$ exponent results from the conservation of energy, where the plug mass is accelerated by the projectile and there is no additional dissipation beyond the energy loss associated with the $V_{50}$ threshold. The plug is assumed to have the same diameter as the projectile and to fuse with the projectile before impacting the next layer, and for $T \leq D$ the plug thickness is assumed to be the same as the plate thickness (Rosenberg and Dekel, 2016). The overall armor $V_{50}$ yields the following expression:

V_{50}^{S N - R S S} = \sqrt{V_{50_{1}}^{2} + \sum_{i = 2}^{n} (V_{50_{i}}^{2} (1 + \frac{ρ_{T}}{ρ_{P} L_{P}} \sum_{j = 1}^{i - 1} T_{j}))}

(8)

where

V_{50}^{S N - R S S} is the configuration V_{50} in m / s

V_{50_{i}}

is the

V_{50}

for the i^th layer in m/s,

n

is the total number of layers,

ρ_{T}

and

ρ_{P}

are the target and projectile densities, respectively, assumed to be 7850 kg/m³ for both the A36 and 4340 steels of interest in this work,

L_{P}

is the projectile length in m, and

T_{j}

is the thickness of the j^th configuration layer (previous layer) in m.

Blunt-nosed projectiles: Root-sum-squared superposition (BN-RSS)

This method considers a cylindrical blunt projectile that causes the formation of a target material plug upon layer perforation and energy loss due to plastic impact, where $α_{i} = \frac{m_{f} + M_{i - 1}}{m_{f} + M_{i}}$ . The exponent change from $0.5$ to 1 results from a conservation of momentum instead of energy. The same plug diameter and fusing assumptions as the previous method hold. The plug thickness is assumed to be the same as the plate thickness. The overall armor $V_{50}$ yields the following expression:

V_{50}^{B N - R S S} = \sqrt{V_{50_{1}}^{2} + \overset{n}{\sum_{i = 2}} (V_{50_{i}}^{2} {(1 + \frac{ρ_{T}}{ρ_{P} L_{P}} \sum_{j = 1}^{i - 1} T_{j})}^{2})}

(9)

where

V_{50}^{B N - R S S}

is the configuration

V_{50}

in m/s,

V_{50_{i}}

is the

V_{50}

for the i^th layer in m/s,

n

is the total number of layers,

ρ_{T}

and

ρ_{P}

are the target and projectile densities previously detailed,

L_{P}

is the projectile length in m, and

T_{j}

is the thickness of the j^th configuration layer (previous layer) in m.

Successive striking and residual velocity superposition (V_S−V_R)

The successive application of the residual velocity terminal ballistic equations is the recommended method in UFC 3-340-02 Section 5-49.3 (DoD, 2008a) and UFC 4-023-07 Section 5-3.4.2.1.2 (DoD, 2008c) for dealing with multiple plate penetration problems. In this method, the fragment striking velocity upon each intermediate layer is the residual velocity after perforation of the previous layer. This model makes two defense-conservative assumptions: (1) the fragment remains intact during the penetration with no mass loss, and (2) the fragment does not deviate from a straight-line path as it crosses layer interfaces. The maximum striking velocity resulting in zero residual velocity at the last layer yields perforation limit velocity. This method’s general steps are summarized by the following:

V_{R_{N}} = 0

V_{S_{N}} = f_{N}^{- 1} (V_{R_{N}}, \dots)

V_{R_{i}} = V_{S_{i + 1}}

V_{S_{i}} = f_{i}^{- 1} (V_{R_{i}}, \dots); i = N - 1, N - 2, \dots, 1

V_{50}^{V_{S} - V_{R}} = V_{S_{1}}

(10)

where

V_{R}

is the residual velocity,

f

represents the layer residual velocity terminal ballistics model,

V_{S}

is the striking velocity,

i

is the configuration layer number, and

N

is the total configuration layer number.

Monte Carlo methodology

This study introduces an MC methodology to estimate the $V_{50}$ of spaced armor configurations. The method is built upon the successive V_S−V_R superposition and is enhanced by adding considerations for capturing projectile trajectory and impact orientation alterations and velocity decay due to drag in air gaps. These considerations replicate the effects of physical variations that occur during ballistic testing. This enhanced superposition is executed in MC simulations, where variability is introduced into key input variables in the underlying deterministic terminal ballistics models. The simulations generate a virtual quantal response from which the perforation likelihood can be estimated as a function of striking velocity.

Simulation setup

Because of the high dimensionality of the problem, subscript notation is used in this study for the simulation variables. The indices $x$ , $n$ , and $i$ denote velocity level, repetition number, and configuration layer number, respectively. Simulation setup requires defining the following values and conditions:

a. FSP: projectile mass $m_{f}$ and diameter $D$

b. Armor configuration: number of layers $N_{L}$ ; $N_{L} \geq 2$ , and their respective terminal ballistics models, thicknesses ${(T_{(i)})}_{i = 1}^{N_{L}}$ , and air gap distances ${(T_{(i)}^{A I R})}_{i = 1}^{N_{L}}$ .

c. Initial striking velocities: number of velocity levels $N_{V}$ , velocity levels to simulate $V = {v_{(x)}^{*}; x = 1, 2, \dots, N_{V}}$ , and number of repetitions per velocity $N_{R}$ .

d. Initial impact conditions: initial impact obliquity $θ_{0}$ and yaw $φ_{0}$ angles

e. MC details: MC variables with their respective probability distributions for sampling.

Scenarios in layered and spaced armor penetration

The simulation projectile striking and residual velocities and impact obliquity and yaw angles at the layer level are represented respectively by:

v_{(x, n, i)}^{S}, v_{(x, n, i)}^{R}, θ_{(x, n, i)}, φ_{(x, n, i)}; x = 1, 2, \dots, N_{V}; n = 1, 2, \dots, N_{R}; i = 1, 2, \dots, N_{L}

(11)

The projectile striking and residual velocities at the overall configuration level are represented by $v_{(x, n)}^{S C}$ and $v_{(x, n)}^{R C}$ , respectively. The selected FSP impacts the first layer at the overall configuration striking velocity $v_{(x, n)}^{S C} = v_{(x, n, 1)}^{S} = v_{(x)}^{*}$ . The overall configuration residual velocity $v_{(x, n)}^{R C}$ is the residual velocity at the last layer in the configuration $v_{(x, n, N_{L})}^{R}$ . If the projectile is stopped at any layer, then $v_{(x, n)}^{R C} = 0$ . Using this methodology, FSP penetration at any individual layer $i$ is assumed to fall within one of the following three scenarios. These scenarios are modular and can be successively superposed for each armor layer to approximate the overall terminal ballistic effects.

a. First layer impact ( $i = 1$ ):

The projectile impacts the first layer of thickness $T_{(1)}$ at an initial striking velocity $v_{(x, n, 1)}^{S}$ , initial projectile obliquity angle $θ_{(x, n, 1)}$ , and initial projectile yaw angle $φ_{(x, n, 1)}$ . The result is perforation at a residual velocity $v_{(x, n, 1)}^{R}$ . Figure 2 shows a visualization of the first-layer impact scenario.

b. Air-gapped layer impact ( $i > 1 a n d T_{(i - 1)}^{A I R} > 0$ ):

Figure 2.

First layer impact scenario for spaced and layered armor modeling. (1) first layer impact, (2) first layer perforation.

The projectile perforated the previous layer with a residual velocity $v_{(i - 1)}^{R}$ . Upon exit, its trajectory is altered to a new obliquity angle $θ_{(x, n, i)}$ and starts tumbling. The tumbling projectile travels across an effective air gap distance, experiencing velocity decay due to drag forces and decaying to the new striking velocity $v_{(x, n, i)}^{S}$ , which is lower than $v_{(x, n, i - 1)}^{R}$ , and is calculated as follows:

d_{(x, n, i - 1)}^{A I R} = \frac{T_{(i - 1)}^{A I R}}{\cos (θ_{(x, n, i)})}

(12)

v_{(x, n, i)}^{S} = v_{(x, n, i - 1)}^{R} e^{\frac{- C_{D} ρ_{a i r} A_{m} d_{(x, n, i - 1)}^{A I R}}{2 M}}

(13)

where

d_{(x, n, i - 1)}^{A I R}

is the effective air gap distance after the previous layer in m,

T_{(i - 1)}^{A I R}

is the actual air gap thickness after the previous layer in m,

θ_{(x, n, i)}

is the new obliquity angle in radians,

C_{D}

is the projectile drag coefficient,

ρ_{a i r}

is the air density (assumed to be 1.23 kg/m³), and

A_{m}

is the projectile mean presented area in m². A

C_{D}

value of 1.20 (U.S. Air Force, 1993) is used for calculations in this study.

In calculating the mean presented area for an FSP, its geometry is simplified to that of a right-circular cylinder (RCC) and calculated as follows:

A (φ) = D^{2} (L o D | \sin (φ) | + \frac{π}{4} | \cos (φ) |)

(14)

where

A

is the presented area in m² as a function of yaw angle

φ

in radians,

D

is the projectile diameter in m, and

L o D

is the projectile length-to-diameter ratio equal to 1.17 for FSPs (ARL, 2008).

The mean presented area of a tumbling RCC is calculated from equation (14):

A_{m} = \frac{2}{π} \int_{0}^{\frac{π}{2}} A (φ) d φ = \frac{D^{2} (4 L o D + π)}{2 π}

(15)

where

A_{m}

is the mean presented area in m²,

A (φ)

is the presented area in m², and

φ

is projectile yaw in radians. Due to the symmetry of an RCC, the average presented area is obtained using yaw angles from 0° to 90°.

Finally, due to tumbling, the projectile impacts the current layer of thickness $T_{(i)}$ at a new yaw angle of $φ_{(x, n, i)}$ and perforates the plate with a residual velocity $v_{(x, n, i)}^{R}$ . Figure 3 shows a visualization of the air-gapped layer impact scenario.

c. Contact layer impact ( $i > 1 a n d T_{(i - 1)}^{A I R} = 0$ ):

Figure 3.

Air-gapped layer impact scenario for spaced and layered armor modeling. (1) exit from previous layer, (2) obliquity angle alteration, (3) velocity decay and yaw angle alteration, and (4) layer perforation.

In the case of layers in contact, the projectile impacts the layer of thickness $T_{(i)}$ at a striking velocity $v_{(x, n, i)}^{S} = v_{(x, n, i - 1)}^{R}$ , projectile obliquity angle $θ_{(x, n, i)} = θ_{(x, n, i - 1)}$ , and projectile yaw angle $φ_{(x, n, i)} = φ_{(x, n, i - 1)}$ , resulting in perforation with a residual velocity $v_{(x, n, i)}^{R}$ . Figure 4 shows a visualization of the contact layer impact scenario.

Figure 4.

Contact layer impact scenario for spaced and layered armor modeling. (1) exit from the previous layer and (2) contact layer perforation.

Terminal ballistic models with limited input variables

Not all terminal ballistic models directly consider all variables needed for the proposed methodology. To address this limitation, these variables are incorporated indirectly by transforming to other common model inputs, with the constraint that at a minimum the projectile diameter $D$ and the layer thickness $T$ must be among the original model input variables. The transformations are as follows:

a. Effective thickness ( $T_{(x, n, i)}^{E F F}$ ): The effective thickness can substitute the layer thickness variable to approximate the effect of the obliquity angle when not explicitly considered by the terminal ballistics model. This trigonometric transformation represents the increased layer material thickness penetrated by the projectile due to its obliquity and assumes no obliquity change during the layer penetration process:

T_{(x, n, i)}^{E F F} = \frac{T_{(i)}}{\cos (θ_{(x, n, i)})}

(16)

where

T_{(x, n, i)}^{E F F}

is the effective thickness in m,

T_{(i)}

is the layer thickness in m, and

θ_{(x, n, i)}

is the projectile obliquity angle in deg.

b. Equivalent diameter ( $D_{(x, n, i)}^{E Q}$ ): The equivalent diameter transformation approximates the effect of the projectile yaw angle when neither the yaw angle nor the presented area is explicitly considered by the terminal ballistics model. The RCC-presented area as a function of yaw is calculated and transformed to an equivalent circle of the same area to solve for its diameter:

D_{(x, n, i)}^{E Q} = \sqrt{\frac{4}{π} A (φ_{(x, n, i)})}

(17)

where

D_{(x, n, i)}^{E Q}

is the equivalent diameter in m, and

A (φ_{(x, n, i)})

is the presented area in m² calculated using equation (14).

Random variable sampling

This investigation focused on introducing variability to the obliquity and yaw angles. The angle of obliquity for

i = 1

is a parameter that is relatively easy to set in the ballistic laboratory and is assumed to be bounded by

θ_{0} \pm 1 °

. The angle of yaw for

i = 1

exhibits more variability; per MIL-STD-662F, the maximum allowable yaw angle is

\pm 5 °

; anything exceeding this is considered an unfair hit and discarded from the

V_{50}

calculation. In the “Air-gapped layer impact” scenario, the modified obliquity angle is assumed to be bounded by

θ_{(x, n, i - 1)} \pm 15 °

. In the “Air-gapped layer impact” scenario, the projectile is assumed to be tumbling at a random angular velocity. Due to symmetry, the final orientation upon impact with the successive layer can be anything from a normal impact (yaw angle of

0 °

) to a side-on impact (yaw angle of

90 °

). Since the exact probability distributions describing obliquities and yaws are unknown in this study, uniform distributions were used as uninformative priors. Table 3 presents the key variables that are randomly sampled in the MC simulations.

Table 3.

MC random variables and assumed sampling distributions.

Random variables		Probability distribution
Obliquity	$θ_{(x, n, 1)}$	$θ_{(x, n, 1)} \sim U (θ_{0} - 1 °, θ_{0} + 1 °)$
Obliquity	$θ_{(x, n, i)}$	$θ_{(x, n, i)} \sim U (θ_{(x, n, i - 1)} - 15 °, θ_{(x, n, i - 1)} + 15 °)$
Yaw	$φ_{(x, n, 1)}$	$φ_{(x, n, 1)} \sim U (- 5 °, 5 °)$
Yaw	$φ_{(x, n, i)}$	$φ_{(x, n, i)} \sim U (0 °, 90 °)$

Estimating V₅₀

The target configuration is tested virtually for the $N_{V}$ velocity levels in $V^{*}$ . Randomness is introduced by running $N_{R}$ repetitions per velocity level, where the key random variables are sampled from their predefined distributions in each repetition. The MC quantal response is obtained after $N_{V} \times N_{R}$ simulations. A V_S-V_R analysis is performed for each respective layer $i$ , and layer results are superposed using the scenarios previously described. For every velocity level $x$ and repetition $n$ , the overall configuration’s striking and residual velocities, $v_{(x, n)}^{S C}$ and $v_{(x, n)}^{R C}$ , are recorded. Values in $v_{(x, n)}^{R C}$ are converted to Boolean residual velocities $v_{(x, n)}^{B R C}$ based on whether the shot resulted in CP or PP. Negative residual velocities are not physically possible, so this conversion is as follows:

v_{(x, n)}^{B R C} = {\begin{cases} 0 \\ 1 \end{cases} \begin{array}{l} if v_{(x, n)}^{R C} \leq 0, \\ otherwise \end{array} for every x, n

(18)

The variability introduced by the random sampling will produce a virtual ZMR. Like the experimental approach, the virtual ZMR bounds are determined from the minimum velocity that yielded a CP and the maximum velocity that yielded a PP:

V_{C P}^{MIN} = \min {v_{(x, n)}^{S C} | v_{(x, n)}^{B R C} = 1}

(19)

V_{P P}^{MAX} = \max {v_{(x, n)}^{S C} | v_{(x, n)}^{B R C} = 0}

(20)

All the striking velocities within the ZMR that resulted in CP are:

V_{C P} = {v_{(x, n)}^{S C} | V_{C P}^{MIN} \leq v_{(x, n)}^{S C} \leq V_{P P}^{MAX} and v_{(x, n)}^{B R C} = 1}

(21)

The median velocity in $V_{C P}$ is the MC $V_{50}$ for the overall configuration ( $V_{50}^{M C}$ ):

V_{50}^{M C} = med {V_{C P}}

(22)

Results

$N_{R}$ convergence study

This study used a $V^{*}$ ranging from 1 to 3000 m/s at 1 m/s level increases ( $N_{V} = 3000$ ). A convergence study was performed to determine an appropriate $N_{R}$ . The study was performed using the case of a 12.7 mm FSP against a 6.35 (1016) 6.35 mm configuration. The underlying terminal ballistics model used was the ERDC model. Nine values for $N_{R}$ from 5 to 50,000 were examined. The $V_{50}^{M C}$ stabilized around $N_{R} = 1000$ at a value of 1105 m/s. The value of $N_{R} = 1000$ was used for the rest of this study. Additionally, this $N_{R}$ value kept the runtime at less than 1 min per calculation, which is desirable for fast-running applications.

MC methodology performance

The four superposition methodologies were implemented with each of the two underlying terminal ballistics models to estimate the

V_{50}

for the 16 experimental cases shown in Table 1. While

V_{50}

values were estimated for cases 1, 10, 11, and 15, these values were not considered when quantifying the performance of the respective methodologies. Each superposition methodology’s performance was calculated for each underlying terminal ballistics model using as a metric the mean absolute percentage error (MAPE) of the estimated

V_{50}

relative to

V_{50}^{EXP}

. Tables 4 and 5 present the estimated

V_{50}

V_{50}^{EXP}

, and MAPE for each methodology using the THOR and ERDC models, respectively.

Table 4.

Results for methodologies applied with the THOR terminal ballistics model.

		Layers terminal ballistics model: THOR
Case	FSP diameter [mm]	FSP mass [g]	Armor configuration [mm]	$V_{50}^{EXP}$ [m/s]	Armor configuration V₅₀ [m/s] for respective layer superposition method
Case	FSP diameter [mm]	FSP mass [g]	Armor configuration [mm]	$V_{50}^{EXP}$ [m/s]	NP-RSS	SN-RSS	BN-RSS	V_S−V_R	MC
1	7.62	2.85	6.35 (1016) 6.35	≥1910	968	1127	1357	1378	1895
2	12.7	13.41	6.35 (0) 6.35	1153	550	606	677	783	846
3	12.7	13.41	6.35 (13) 6.35	1143	550	606	677	783	1065
4	12.7	13.41	6.35 (25) 6.35	1201	550	606	677	783	1066
5	12.7	13.41	6.35 (51) 6.35	1155	550	606	677	783	1066
6	12.7	13.41	6.35 (76) 6.35	1125	550	606	677	783	1066
7	12.7	13.41	6.35 (89) 6.35	1083	550	606	677	783	1066
8	12.7	13.41	6.35 (140) 6.35	1084	550	606	677	783	1067
9	12.7	13.41	6.35 (1016) 6.35	1109	550	606	677	783	1072
10	12.7	13.41	9.53 (89) 6.35	≥1636	683	751	850	956	1254
11	12.7	13.41	6.35 (89) 9.53	≥1645	683	775	891	957	1354
12	12.7	13.41	7.94 (76) 6.35	1379	615	677	763	870	1162
13	12.7	13.41	7.94 (89) 6.35	1288	615	677	763	870	1163
14	12.7	13.41	6.35 (89) 7.94	1415	615	689	783	871	1211
15	12.7	13.41	15.9 (76) 6.35	≥2197	974	1054	1202	1288	1616
16	20	53.78	6.35 (1016) 6.35	651	329	350	376	468	640
MAPE					52.06%	47.27%	41.08%	31.84%	8.81%

Table 5.

Results for methodologies applied with the ERDC terminal ballistics model.

		Layers terminal ballistics model: ERDC
Case	FSP diameter [mm]	FSP mass [g]	Armor configuration [mm]	$V_{50}^{EXP}$ [m/s]	Armor configuration V₅₀ [m/s] for respective layer superposition method
Case	FSP diameter [mm]	FSP mass [g]	Armor configuration [mm]	$V_{50}^{EXP}$ [m/s]	NP-RSS	SN-RSS	BN-RSS	V_S−V_R	MC
1	7.62	2.85	6.35 (1016) 6.35	≥1910	1218	1418	1708	1686	1881
2	12.7	13.41	6.35 (0) 6.35	1153	719	792	886	995	1016
3	12.7	13.41	6.35 (13) 6.35	1143	719	792	886	995	1097
4	12.7	13.41	6.35 (25) 6.35	1201	719	792	886	995	1096
5	12.7	13.41	6.35 (51) 6.35	1155	719	792	886	995	1097
6	12.7	13.41	6.35 (76) 6.35	1125	719	792	886	995	1098
7	12.7	13.41	6.35 (89) 6.35	1083	719	792	886	995	1096
8	12.7	13.41	6.35 (140) 6.35	1084	719	792	886	995	1097
9	12.7	13.41	6.35 (1016) 6.35	1109	719	792	886	995	1105
10	12.7	13.41	9.53 (89) 6.35	≥1636	921	1007	1134	1206	1312
11	12.7	13.41	6.35 (89) 9.53	≥1645	921	1049	1208	1293	1440
12	12.7	13.41	7.94 (76) 6.35	1379	816	896	1007	1099	1202
13	12.7	13.41	7.94 (89) 6.35	1288	816	896	1007	1099	1205
14	12.7	13.41	6.35 (89) 7.94	1415	816	916	1043	1144	1269
15	12.7	13.41	15.9 (76) 6.35	≥2197	1388	1485	1666	1654	1762
16	20	53.78	6.35 (1016) 6.35	651	447	476	511	619	685
MAPE					36.88%	30.59%	22.47%	12.90%	5.80%

The performance calculations show the MC superposition methodology consistently brings $V_{50}$ estimation closest to the experimental value regardless of the underlying terminal ballistics model. This was also the case with the UFC 3-340-02 and UFC 4-023-07 underlying monolithic terminal ballistic models not presented in this study. In these two cases, the MC superposition also yielded the lowest error among the methods with MAPEs of 31.60% and 22.31%, respectively.

Conclusions

This investigation introduced an MC approach to modeling the $V_{50}$ of layered and spaced A36 armor configurations against FSPs. The performance of the MC method was evaluated relative to four layer superposition methodologies from the literature. The objective of the study was not to assess the intrinsic accuracy of any individual terminal ballistic model, but rather to examine the relative performance of the superposition frameworks when applied to multi-layer targets. Although the underlying terminal ballistic models employed are known to provide reasonable predictions for monolithic targets under appropriate impact conditions, conventional superposition methods rely on idealized, deterministic formulations that capture only a subset of the relevant penetration physics. These approaches neglect stochastic effects which are known to influence experimental outcomes in layered and spaced armor systems and which the proposed MC framework seeks to represent. If the underlying terminal ballistic model itself is not accurate for the monolithic material, predictive accuracy cannot be achieved regardless of the superposition method used. Two independent underlying terminal ballistics models were used to show that relative performance advantage was not model-specific. Simulation results converged at $N_{R} = 1000$ repetitions per velocity, using a velocity level resolution of 1 m/s. This number resulted in fast-running calculations of under 1 min. The performance for each method was evaluated relative to the experimental $V_{50}$ data. The MC method yielded the lowest error among the methodologies, with a MAPE of 8.81% for the THOR terminal ballistic model and 5.80% for the ERDC model. Overall, the next best superposition methodology was the V_S−V_R methodology, validating the DoD guidance.

Additional research is required to identify and incorporate additional random variables relevant to the problem. Characterizing the probability distributions that best model the random variables will boost the accuracy of the methodology. Additionally, validating the accuracy of the variance of the MC quantal response will be necessary to determine percentiles other than 50%. Future work could also include exploring the validity of the method for other projectile types and geometries, and for non-zero initial obliquity angles. Optimizing the algorithm to balance the striking velocity resolution and the number of simulations per velocity value could further increase performance. Finally, although this method shows excellent preliminary agreement when modeling multi-material layered and spaced armor configurations, further study is required to assess the applicability of the method for these configurations.

Footnotes

ORCID iDs

Daniel H. Rios-Estremera

Matthew W. Priddy

Jesse A. Sherburn

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research was funded by the U.S. Army Corps of Engineers ERDC. Any opinions, findings, and conclusions expressed in this presentation are those of the writers and do not necessarily reflect the view of ERDC. ERDC also conducted the ballistic testing presented herein.

Declaration of conflicting interests

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

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Monte Carlo simulation of steel spaced armor ballistic performance against fragment-simulating projectiles

Abstract

Keywords

Introduction

Experimental setup and results

Terminal ballistics models

Modeling methods for layered and spaced armors

Rigid non-plugging projectiles: Root-sum-squared superposition (NP-RSS)

Spherical-nosed projectile: Root-sum-squared superposition (SN-RSS)

Blunt-nosed projectiles: Root-sum-squared superposition (BN-RSS)

Successive striking and residual velocity superposition (VS−VR)

Monte Carlo methodology

Simulation setup

Scenarios in layered and spaced armor penetration

Terminal ballistic models with limited input variables

Random variable sampling

Estimating V50

Results

N R convergence study

MC methodology performance

Conclusions

Footnotes

ORCID iDs

Funding

Declaration of conflicting interests

References

Successive striking and residual velocity superposition (V_S−V_R)

Estimating V₅₀

$N_{R}$ convergence study