Mitigating casualty risks from primary fragmentation hazards

Abstract

Primary fragmentation from detonation of high-explosive metal-cased munitions imposes significant risks to the safety of related personnel and the public. Barricades or other protective structures are commonly used to stop fragments and reduce casualty risks caused by detonated munitions when a sufficient safety distance cannot be guaranteed. This study aims to provide decision support for the positioning of barricades that can reasonably mitigate primary fragmentation hazards from the detonation of large calibre munitions using a probabilistic risk assessment approach. This approach enables a stochastic characterization of fragment ejections, stacking effects, fragment trajectories, human vulnerability and fragment hazard reduction by barricade. In a case study, the assessments of casualty risks and effectiveness of barricades were conducted for a single and a pallet of 155 mm projectiles. It was found that barricades with heights exceeding the height of munitions can significantly reduce the hazardous fragment densities and casualty risks beyond the barricade. The benefit of increasing the barricade height becomes marginal when it exceeds the height of munitions.

Keywords

Primary fragmentation hazard metal-cased munition stacking effects stochastic analysis casualty risk mitigation barricades

Introduction

Accidental detonation of high-explosive munitions may occur during an activity of operation and deployment such as storage, transport and removal (Crull et al. 1996; DDESB 2009), which poses significant safety risks to personnel. Casualties are mainly caused by primary fragments when the detonation occurs in an open space. The probabilistic modelling of primary fragmentation hazards and the consequent casualty risks accounts for uncertainties involved in fragment ejections, trajectories and human vulnerability, which is essential for the determination of quantity distance (the minimum distance from the explosive to where people are at an acceptable safety risk) and risk reduction measures to mitigate fragment-induced safety risks (e.g. Van Der Voort and Weerheijm 2013; Stewart and Netherton 2019).

Several studies have adopted probabilistic and statistical approaches to assess safety risks from fragmentation hazards. Häring et al. (2009) proposed a quantitative assessment method for casualty risks from missiles exploding in the air. Crull and Hamilton (2012) and the SAFER model (DDESB 2009) used empirical equations to calculate quantities of interest such as fragment final velocities and density probabilities from high-explosive munitions that implicitly capture the uncertainty and stochasticity associated with fragment ejections and trajectories. Sielicki et al. (2021) conducted a probabilistic analysis considering the variability of launch angle, initial velocity and drag coefficient based on explosive field trial data of person-borne improvised explosive devices to predict casualty risks. Qin and Stewart (2021) proposed a probabilistic approach to assess casualty risks induced by the detonation of a single metal-cased cylindrical munition in an open space. The statistical fragment densities and casualty risks obtained from this study can be used to determine the quantity distance to prevent and mitigate safety risks induced by fragmentation hazards. While the quantity distance often cannot be satisfied to ensure safety during operation and deployment, barricades can be used to mitigate fragmentation hazards and reduce casualty risks (IATG 2015), and therefore, the positioning and sizing of the barricade need to be determined. In addition, rather than a single munition, munitions are often grouped in pallets during operation and deployment. The stacking effects due to sympathetic detonation and interactions between adjacent munitions result in different fragment distributions and initial launch conditions from those ejected by a single explosive munition (Powell et al. 1981; Crull and Hamilton 2012). Therefore, the probabilistic fragmentation and casualty risk assessment models need to consider stacking effects.

This study assesses the effectiveness of the positioning and sizing of barricades in mitigating fragmentation hazards from detonation of a single or a pallet of 155 mm M107 munitions using a probabilistic risk assessment approach. This type of munition is similar to that widely used in NATO countries. This risk assessment method enables a stochastic characterization of fragment ejections, stacking effects, fragment trajectories, human vulnerability and fragment hazard reduction by barricades. The barricade effectiveness is evaluated based on the hazardous fragment densities and casualty risks behind the barricade considering various heights and intercept angles of the barricade. This study provides decision support for mitigating safety risks induced by fragmentation hazards from high-explosive munitions by using barricade.

Fragmentation modelling

Fragment mass and shape

The fragment mass distribution (FMD) has been commonly modelled by the Mott distribution (Mott 1943). Another widely used model is the Held FMD (Held 1990a). Both agree well with the fragmentation experimental data and can reasonably model fragment mass distributions with acceptable accuracy (Held 1990b; Crull and Hamilton 2012). For the convenience of mathematical manipulation, Held FMD is used in this study. Other FMDs (e.g. Grady & Kipp 1985) may also be used and the differences of accuracy among these commonly used FMDs are deemed marginal with limited effects on the fragmentation hazard modelling.

The number and mass of primary fragments naturally generated by the detonation of a single cylindrical metal-cased munition, modelled by the Held FMD, is given by

M (n) = M_{o b} (1 - e^{- B n^{λ}})

(1)

where M(n) is the cumulative mass or overall mass of all fragments from the munition, starting with the heaviest, M_ob is a parameter close to the total mass of fragments, n is the number of the n^th heaviest fragment and B and λ are distribution parameters to be determined based on fragmentation data. For the 155 mm M107 projectile considered in this study, M_ob = 28,318g, B = 0.0089 and λ = 0.6531 (Held 1990b). Taking the derivative of equation (1), the mass of the n^th heaviest fragment, M_n, is calculated by

M_{n} = \frac{d M (n)}{d n} = M_{o b} B λ n^{λ - 1} e^{- B n^{λ}}

(2)

Fragments from the detonation of cylindrical metal-cased munitions generally have irregular shapes. The mean presented area, A (m²), of the fragment is a key parameter for the characterization of air drag and fragment trajectory, which is calculated by (Zaker 1975)

A = {(m / k)}^{2 / 3}

(3)

where m is the fragment mass (kg) that can be obtained from the fragment mass distribution described above, and k is the shape factor (kg/m³). The shape factor is assumed to follow a lognormal distribution with a mean of 2600 kg/m³ for a single munition or 4740 kg/m³ for a stack of munitions (Zaker 1975; Crull and Hamilton 2012) and a coefficient of variation (COV) of 0.20 (Qin and Stewart 2021).

Initial velocity

The initial velocity of fragments calculated by the Gurney equation (Gurney 1943) is

V_{0 G} = \sqrt{2 E} {[\frac{C / M_{0}}{1 + 0.5 C / M_{0}}]}^{1 / 2}

(4)

where V_0G is the initial velocity from the Gurney equation,

\sqrt{2 E}

is the Gurney velocity that varies for different types of explosives (UFC 2014), C is the charge weight (kg) and M₀ is the weight of the cylindrical casing (kg). For the 155 mm M107 projectile (Composition B) considered in this study, M₀ = 32.151 kg, C = 7.01 kg and V_0G is calculated as 1230 m/s. The Gurney equation generally overpredicts the initial velocities according to experimental data (Anderson et al. 1985). To account for the model uncertainty and variations of initial velocity along the longitudinal axis of the metal casing, it is assumed that the initial velocity V₀ follows a normal distribution with a mean of 0.9V_0G and a coefficient of variation (COV) of 0.15 that are estimated from the test data given by Karpp and Predebon (1975) and Pearson (1990). Other test data and empirical equations (e.g. Huang et al. 2015; Grisaro and Dancygier 2015) may also be used for the modelling of fragment initial velocity, the accuracy of which is deemed comparable to the current approach.

Launch angle

The Taylor angle (Taylor 1963) is widely used to model the polar launch angle α (e.g. elevation angle relative to the horizontal axis for a standing cylindrical munition, see Figure 1) of a fragment. However, the Taylor angle has some discrepancies with the observations in experiments (e.g. Anderson et al. 1985; König 1987). The actual launch angle is a complex function of charge-to-case weight ratio, type of explosive and length-to-diameter ratio of the cylindrical casing (König 1987). Based on the experimental data, analytical and numerical solutions from Giroux (1971), Anderson et al. (1985) and König (1987), and cognizant of keeping the model easy to use, the polar launch angle α of fragments for a single or a pallet of upright cylindrical shells is assumed to be uniformly distributed (i) from −20^◦ to 0^◦ for 20% of fragments, (ii) from 0^◦ to +10^◦ for 60% of fragments and (iii) from +10^◦ to +80^◦ for 20% of fragments. This also considers the influence of the curved geometry of the munition as those experimental data, analytical and numerical solutions are for perfect cylindrical cases.

Figure 1.

Polar (α) and azimuthal (φ) launch angles.

The azimuthal launch angle φ (see Figure 1) for a single upright cylindrical shell is assumed to be uniformly distributed over all azimuthal directions (i.e. 0^◦ to 360^◦). As inferred from the test data by Powell et al. (1981), most fragments from the detonation of a standard pallet of eight upright 155 mm projectiles are concentrated within narrow azimuthal ranges over the long sides of the rectangular stack (see Figure 2), and hence the casualty risks are the highest for personnel in the directions facing the long side of the stack. The quantity (or safety) distance for a pallet of munitions is also governed by the hazard fragment density in these directions. The azimuthal launch angle for fragments ejected from the top long side of the pallet in Figure 2 is approximately modelled as uniformly distributed (i) from 0^◦ to 20^◦ and 340^◦ to 360^◦for 80% of the fragments, and (ii) from 20^◦ to 45^◦ and 315^◦ to 340^◦ for 20% of the fragments for a pallet of 155 mm projectiles based on the test data from Powell et al. (1981).

Figure 2.

One standard pallet of eight cylindrical munitions (top view).

Stacking effects

For a pallet of munitions (see Figure 2), only the outside surface of the rectangular stack of cylindrical shells substantially contributes to the density of hazardous fragments, while fragments generated within the shaded area are contained within the stack and not ejected to the field (DDESB 2009; Crull and Hamilton 2012). The initiation of a donor munition, the interactions between munitions in a stack and the effects of sympathetic detonations result in different mass distributions and initial velocities from those corresponding to a single munition. Compared to fragments from a single munition, the average fragment mass resulted from the detonation of a stack of munitions may increase by about 50% with reduced number of fragments, while the initial velocity of fragments may increase up to two folds (Zaker 1975; Powell et al. 1981; Crull and Hamilton 2012). Two multipliers are assumed to model the stacking effects on fragment mass distribution and initial velocity. The first multiplier M_stack is applied to the fragment masses from a single munition, while the number of fragments is divided by M_stack. The second multiplier V_stack is applied to the initial velocities for fragments generated from a single munition. Based on the knowledge and test data from Powell et al. (1981) and Crull and Hamilton (2012), M_stack is assumed to follow a truncated normal distribution with a mean of 1.50 and a COV of 0.20 (truncated to ensure positive values), while V_stack follows a truncated normal distribution with a mean of 1.60 and a COV of 0.10 (truncated between 0 and 2.0).

The space between two adjacent shells on the side of a stack is defined as the interaction area (Powell et al. 1981) where the stacking effects will apply. The fragment concentration within a certain azimuthal range described in Section 2.3 results from fragments ejected from the three interaction areas on the long side. For one long side, the number of fragments from these interaction areas in the outer surface of an 8-projectile pallet in Figure 2 is assumed to be equivalent to the number of fragments from 1.5 projectiles (i.e. 0.5 projectile for each interaction area) in a stack. Fragments from 0.5 projectiles in the two non-interaction areas also contribute to the fragment densities on the long side of the pallet. The stacking effects on fragment shape factors and azimuthal launch angles are described in Sections 2.1 and 2.3.

Trajectory modelling

For a given azimuthal launch angle, the two-dimensional trajectory of a fragment after ejection is modelled by the equation of motion considering air drag force and gravity. The effects of lift, spin and wind-induced side drift of fragments are not considered. The equation of motion is then expressed as

m \frac{d \vec{v}}{d t} = {\vec{F}}_{D} + {\vec{F}}_{g}

(5)

where v is the velocity of fragment,

{\vec{F}}_{D}

is the drag force and

{\vec{F}}_{g}

is gravity, given by

{\vec{F}}_{D} = - \frac{1}{2} ρ_{a} C_{D} A | \vec{v} | \vec{v}

(6)

{\vec{F}}_{g} = m \vec{g}

(7)

where a is the air density, A is the mean presented area given by equation (3), C_D is the drag coefficient and g is gravitational acceleration. For a given azimuthal angle, equation (5) can be rewritten as the following differential equations in terms of fragment velocity and location in the polar coordinate system shown in Figure 1 (x as the horizontal axis and y as the vertical axis)

{\begin{cases} \frac{d x}{d t} = v_{x} \\ \frac{d y}{d t} = v_{y} \\ \frac{d v_{x}}{d t} = - γ v_{x} \sqrt{{v_{x}}^{2} + {v_{y}}^{2}} \\ \frac{d v_{y}}{d t} = - γ v_{y} \sqrt{{v_{x}}^{2} + {v_{y}}^{2}} - g \end{cases}

(8)

where x and v_x are the fragment coordinate and velocity in the horizontal direction, respectively, and y and v_y are the fragment coordinate and velocity in the vertical direction, respectively. The ballistic coefficient γ is

γ = \frac{ρ_{a} A C_{D}}{2 m}

(9)

The drag coefficient in equations (5) and (9) is modelled as a function of Mach number (i.e. the ratio of fragment velocity to the speed of sound) during the flight path to account for the variation of drag force in supersonic and subsonic velocity ranges. For fragments with irregular shapes from the detonation of cylindrical metal-cased munition, the relation between drag coefficient and Mach number is shown in Figure 3 (McCleskey 1988a), where C1 is the drag coefficient when Mach number is 0.10. To account for inherent and model uncertainties, C1 is assumed to follow a lognormal distribution with a mean of 0.95 and a COV of 0.20, which is estimated based on the drag coefficient data for irregular fragments obtained from wind tunnel testing (McCleskey 1988b). In this study, the correlation between drag coefficient and the shape characteristics of fragments (e.g. mean presented area) is neglected due to a lack of empirical information. The variability of drag coefficient with fragment shape is implicitly taken into account by the assumed probability distribution of drag coefficient.

Figure 3.

Drag coefficient versus Mach number (McCleskey 1988a).

The ricochet after a fragment hits the ground is also considered for the fragment trajectory modelling. Depending on the angle of impact, ricochet may or may not occur when a fragment hits the ground (McCleskey 1988a). If ricochet occurs, the fragment velocity and launch angle after ricochet for a given surface ground type (e.g. clay soil, sand) can be assessed by the empirical approach given by McCleskey (1988a) and Qin and Stewart (2021).

Given the initial conditions (i.e. location coordinates at fragment ejection, fragment initial velocity and launch angle), the trajectory of each fragment, modelled by the system of ordinary differential equations (ODEs) given by equation (8), can then be numerically solved by the fourth order Runge-Kutta method (Butcher 2016). By using reasonably small time steps, the location coordinates and velocities of each fragment can be obtained at each discrete time step. If ricochet occurs when a fragment hits the ground, the trajectory of fragment ricochet can also be obtained using the ODEs and the Runge-Kutta method with updated initial conditions (i.e. location coordinates at the point of impact, and the ricochet angle and velocity).

Casulty risk and quantity distance

Human vulnerability

Consider an average standing person (i.e. body height H_T = 1.75 m, exposed body area A_M = 0.50 m²) facing the detonation point of a single munition or a pallet of munitions (the person standing within the fragment concentration azimuth and facing the long side of a pallet) as shown in Figure 4. The probabilities of fatality, major and minor injuries as a function of the kinetic energy (KE) of the fragment hitting an average standing person (i.e. human vulnerability) are modelled by a lognormal function

P (F a t a l i t y | H) = Φ [\frac{\ln (K E) - μ_{I n}}{σ_{I n}}]

(10)

where

Φ

is cumulative distribution function (CDF) of standard normal distribution, and μ_ln = 4.662 and σ_ln = 0.5536 are the lognormal parameters based on the weighted average (Qin and Stewart 2021) of vulnerabilities for different body parts (i.e. head, thorax, abdomen and limbs) from relevant physiological studies (RCC 2000). Similarly, the lognormal parameters for major and minor injuries can also be obtained. Figure 5 shows the vulnerability curves for fatality, major and minor injuries based on RCC (2000), DDESB (2009) and Qin and Stewart (2021). Note that the probability of major injury in Figure 5 is the likelihood of an individual subjected to at least a condition of major injury (i.e. major injury or fatality), and the probability of minor injury is the likelihood for an individual experiencing a minor injury or worse (i.e. minor injury, major injury or fatality).

Figure 4.

An average standing person facing the detonation point at a standoff distance of R.

Figure 5.

Vulnerability curves for an average standing person if hit by a fragment.

Risk estimation

At a standoff distance of R, the number of potential impacting fragments (N_R) with location coordinates satisfying x = R and 0 ≤ y ≤ H_T for the standing person can be obtained from the trajectory model described in Section 2.5. The corresponding kinetic energy (KE = mv²/2) can also be obtained for each fragment. The probability of fatality, P_j(Fatality), due to impacting fragment j (j = 1, 2, 3, …, N_R) each with a given KE_j is

P_{j} (F a t a l i t y) = P (F a t a l i t y | H) P_{hit, j} (H)

(11)

where P(Fatality|H) is calculated using the lognormal vulnerability function given by equation (10). Considering the uniform azimuthal launch angles for fragments ejected from either a single upright cylindrical munition or the long side of a pallet of upright cylindrical munitions, the probability of being hit by fragment j, P_hit,_j(H) is

P_{hit, j} (H) = {\begin{cases} \frac{A_{M}}{2 π R H_{T}} single standing munition \\ \frac{A_{M}}{0.5 π R H_{T}} within 90 ° azimuth of the long side of a pallet of standing munitions \end{cases}

(12)

The fatality risk for the person with a standoff distance of R, P_Fatality, is then given by

P_{F a t a l i t y} = 1 - \prod_{j = 1}^{N_{R}} [1 - P_{j} (F a t a l i t y)]

(13)

Similarly, the major injury risk (P_MajInjury) and the minor injury risk (P_MinInjury) for a standing person can be obtained using the same approach for fatality risks, just replacing the vulnerability function for fatality with those for major and minor injuries.

Hazardous fragment density

Fragments with kinetic energy over 79 J are deemed as hazardous by UN (2015) and NATO (2015). The quantity distance is typically defined as the distance that ensures there is no more than one hazardous fragment per 56 m², that is, 0.0179 fragment/m² (UFC 2014; NATO 2015; UN 2015). For a pallet of munitions shown in Figure 2, the quantity distance is governed by the hazardous fragment density in the directions within the fragment concentration azimuth and facing the long side of a pallet. The hazardous fragment density at a standoff distance of R, F_D(R), is calculated by

F_{D} (R) = \{\begin{cases} \frac{N_{H} (R)}{2 π R H_{T}} single standing munition \\ \frac{N_{H} (R)}{0.5 π R H_{T}} within 90^{\circ} azimuth of the long side of a pallet of standing munitions \end{cases}

(14)

where N_H (R) is the number of hazardous fragments with location coordinates satisfying γ = R (for a given azimuthal angle, γ is the distance to the origin in Figure 1) and 0 ≤ y ≤ H_T that potentially impact an average standing person at a standoff distance of R.

Barricades

Barricades reduce fragment hazards primarily by stopping high-velocity low-angle fragments ejected from accidentally detonated single munition or a stack of munitions. This study considers using a wall-type barricade standing vertically. It is assumed that the material and thickness of the barricade are designed to guarantee that any fragment hitting the face of barricade will be stopped, and the barricade will not be destroyed due to the explosion before the hitting by fragments. This assumption may be slightly nonconservative as there might exist some fragments with very high energy impacting the barricade and exiting with reduced kinetic energies and altered trajectories. Since this is beyond the current modelling capacity and has a low occurrence probability for a properly designed barricade, its effects on the quantity distance and barricade size are not considered here but left for future research. The barricade design in this study is mainly to determine the height and separation distance of the barricade or the barricade intercept angle as shown in Figure 6. It is expected that both the hazardous fragment density and casualty risks will be reasonably reduced at standoff distances beyond the appropriately positioned barricade.

Figure 6.

Barricade size and position.

Monte Carlo simulation

Monte Carlo simulation (MCS) is employed to probabilistically assess the hazardous fragment density and casualty risks for a single munition or a stack of munitions. Barricades with different heights are also considered in the simulations to evaluate their effectiveness in mitigating fragment hazards. The outputs from the simulation are samples and statistics of hazardous fragment densities, fatality and injury risks for an average person standing at various distances from the detonation point as shown in Figure 4. Figure 7 shows a flowchart for the simulation procedures.

Figure 7.

Flowchart of the Monte Carlo simulation.

Casualty risk assessment and mitigation

The casualty risk assessments were conducted considering the detonation of a single and a pallet of 155 mm M107 projectiles vertically placed as shown in Figure 1. It is assumed that an average person in a standing position facing the detonation point is subjected to primary fragmentation hazards and without taking any evasive action. For a pallet of 155 mm M107 projectiles, the standing person is considered within 90^◦ azimuth of the long side of the pallet.

The fragment mass distributions for a single and a pallet of 155 mm munitions were generated by the Held FMD considering stacking effects as described in Sections 2.1 and 2.4. The fragment mass was generated from the heaviest in a descending order by equation (2). The mass of each fragment and the cumulative fragment mass for a single and a pallet (for the azimuthal concentration area facing one long side of the pallet) of 155 mm munitions are shown in Figure 8, which reflects the modelled feature that the average fragment mass considering stacking effects is generally greater than that for a single munition (Section 2.4).

Figure 8.

Mass of each fragment and cumulative fragment mass generated by Held FMD.

The fragment initial velocities considering stacking effects, launch angles, fragment shapes and drag coefficients can be generated based on the fragmentation modelling described in Section 2 that enables the trajectory analysis. The stochastic analysis was then conducted using the Monte Carlo simulation approach as outlined in Figure 7. Figure 9 shows the probability densities of fragment initial velocity that illustrates the high velocities and their variabilities as well as the stacking effects on initial velocities described in Section 2.4. Figure 10 shows the average fragment kinetic energies impacting a standing person. It illustrates how the average fragment kinetic energy decreases with increasing standoff distance and the differences between those corresponding to a single and a pallet of munitions.

Figure 9.

Probability densities of fragment initial velocity for a single and a pallet of 155 mm projectiles.

Figure 10.

Average kinetic energy of impacting fragments for a single and a pallet of 155 mm projectiles.

Fragment density

The hazardous fragment densities were obtained for a single or a pallet of 155mm projectiles. For a single munition, the hazardous fragment density is uniform over all azimuths, while for a pallet of munitions, the density is that in the direction within the fragment concentration azimuth and facing the long side of the pallet. Figure 11 shows the mean, 10^th and 90^th percentiles of the hazardous fragment density as a function of the standoff distance for a single or a pallet of 155 mm projectiles. The quantity distance criterion, that is, 0.0179 hazardous fragment/m², is also shown in this figure. It is suggested that there is a slightly higher variability for fragmentation hazards from a pallet of munitions. The quantity (or safety) distances determined by the mean, 10th and 90th percentiles of the hazardous fragment density are 116 m, 123 m and 130 m, respectively, for a single munition, and 268 m, 294 m and 320 m, respectively, for a pallet of munitions (see Table 1). The quantity distance for a single 155 mm projectile matches well with that given by the U.S. Department of Defense (DoD 2004) which is 126 m and less conservative than that given by Crull and Hamilton (2012) which is 137 m. The quantity distances for a pallet of 155 mm projectiles the Monte Carlo simulation (MCS) in this study which also aligns well with those given by DoD (2004) and Crull and Hamilton (2012) which are 291 m and 295 m, respectively. A major benefit of the current stochastic model is its ability to quantify the uncertainties involved in the assessment of the hazardous fragment density and quantity distance.

Figure 11.

Hazardous fragment densities for a single or a pallet of 155 mm projectiles.

Table 1.

Quantity distances from the MCS and the literature.

	MCS (m)			Literature (m)
	10th	Mean	90th	DoD (2010)	Crull and Hamilton (2012)
Single 155 mm projectile	116	123	130	126	137
A pallet of 155 projectiles	268	294	320	291	295

Casualty risks

Figure 12 shows the casualty risks where major injury risk is the likelihood of an individual subjected to at least a condition of major injury (i.e. major injury or fatality), and minor injury risk is the likelihood for an individual experiencing a minor injury or worse (i.e. minor injury, major injury or fatality). As expected, the individual fatality and injury risks decrease with an increasing standoff distance. The casualty risks from the detonation of a pallet of 155 mm projectiles are much higher than those from a single 155 mm projectile, which is consistent with the hazardous fragment density results shown in Figure 9. The curves for fatality, minor and major injuries corresponding to a pallet of projectiles are less smooth than those for a single munition because more uncertainties are involved to model fragments from a pallet of munitions while the number of simulation runs remains the same. According to Crull and Hamilton (2012), an alternate criterion for quantity distance is the distance to ensure a 1% probability of being struck by a hazardous fragment that results in incapacitation (i.e. major injury or worse). Based on this criterion, the quantity distance is 123 m for a single 155 mm projectile and 310 m for a pallet of 155 mm projectiles that are close to or consistent with those determined based on the hazardous fragment density in Table 1. It helps demonstrate the credibility of the stochastic analysis described herein.

Figure 12.

Casualty risks corresponding to the detonation of a single or a pallet of 155 mm projectiles.

Figure 13 shows the likelihood of the average standing person experiencing a fatality only, major injury only and minor injury only (‘only’ means the person exclusively experiences a fatality, a major or a minor injury). The figure suggests that the probability of major injury only is the lowest for both a single and a pallet of 155 mm projectiles. As fatality and injury are mutually exclusive, the probability of injury only is relatively low when the standoff distance is small with a high fatality risk. There is a threshold distance beyond which the person more likely suffers a minor injury rather than fatality. For a single 155 mm projectile, this distance is 76 m, while 179 m for a pallet of 155 mm projectiles.

Figure 13.

Probability of fatality only, major injury only and minor injury.

Risk mitigation by barricades

This study considers the barricade is placed at a separation distance of SB = 2 m and the height of standing munitions is HS = 0.76 m. The stochastic analysis is conducted considering various barricade heights (HB) or barricade intercept angles (β) as shown in Table 2. The mean hazardous fragment densities obtained for different barricade scenarios are shown in Figure 14. The figure suggests that barricades can significantly reduce hazardous fragment densities. For a slightly negative barricade intercept angle (i.e. Scenario 1 where the height of barricade is less than the height of munitions), the mean hazardous fragment densities behind the barricade are more than halved and reduces the quantity distance from 123 m to 59 m for a single 155 mm projectile, and from 294 m to 118 m for a pallet of 155 mm projectiles. For the non-negative intercept angles (i.e. Scenarios 2––6), the hazardous fragment densities beyond the barricade are much more significantly reduced compared to the negative intercept angle. Such reductions increase with the barricade height for standoff distances close to the barricade. However, the benefit of barricade becomes marginal as the barricade height increases. For a pallet of 155 mm projectiles, the quantity distance reduces from 17 m for an intercept angle of 0° (i.e. Scenario 2) to 6 m for an intercept angle of 45°. For a single 155 mm projectile, the hazardous fragment densities beyond the barricade become negligible when the barricade intercept angle is greater than 15°. Figure 15 shows the effects of barricades on the incapacitation risks (i.e., major injury risks) for a single and a pallet of 155 mm projectiles. Similarly, it suggests that barricades can effectively reduce the incapacitation risks beyond the barricade. The analysis also suggests barricades have similar mitigating effects on fatality and minor injury risks. A barricade with a non-negative intercept angle (i.e. Scenarios 2–6) would significantly reduce the major injury risk to less than 1% immediately or a very short distance beyond the barricade. As the benefit of a barricade becomes marginal with an increasing height for non-negative intercept angles, the choice of the barricade height depends on whether it is required to eliminate casualty risks immediately beyond the barricade or whether a very short distance with intolerable casualty risks is allowed beyond the barricade.

Table 2.

Barricade heights and intercept angles at a separation distance of 2 m.

Barricade scenario	Height (HB)	Intercept angle (β)
1	0.59 m	−5°
2	0.76 m (HS)	0°
3	0.93 m	5°
4	1.30 m	15°
5	1.91 m	30°
6	2.76 m	45°

Figure 14.

Mean hazardous fragment densities considering various barricade scenarios. (a) Single, (b) Pallet.

Figure 15.

Incapacitation risks (major injury risks) considering various barricade scenarios. (a) Single, (b) Pallet.

Note that a flat ground is considered for all above analyses in Sections 6.1–6.3. Fragment trajectories can be shortened up-hill and elongated down-hill if a constant slope is considered. In principle, more complex terrains (terrains with spatially varying elevations) may be considered in the model, however, the effects of terrain on assessing and mitigating casualty risks from fragmentation hazards are left for future study.

Conclusions

This study developed a probabilistic risk assessment (PRA) approach to provide decision support for the positioning of barricades that can reasonably mitigate primary fragmentation hazards from the detonation of large calibre munitions. This PRA approach enables a stochastic characterization of fragment generations, stacking effects, fragment trajectories, human vulnerability and fragment hazard reduction by barricade. The quantity distances obtained from the present PRA for unbarricaded single and a pallet of 155 mm projectiles align well with those quantity distances from existing standards and literature, which demonstrates the model capacity for predicting hazardous fragment densities and casualty risks. Barricades with various heights were then introduced. It was found that barricades can significantly reduce the hazardous fragment densities and casualty risks beyond the barricade for both a single and a pallet of 155 mm projectiles. The benefit of increasing the barricade height becomes marginal when it exceeds the height of munitions. The choice of the barricade height depends on whether it is required to eliminate casualty risks immediately behind the barricade or whether a very short distance with intolerable casualty risks is allowed beyond the barricade.

Footnotes

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: The support of the Australian Research Council grant DP210101487 is gratefully acknowledged.

ORCID iDs

Hao Qin

Mark G. Stewart

References

Anderson

JRCE

Predebon

Karpp

(1985) Computational modeling of explosive-filled cylinders. International Journal of Engineering Science 23(12): 1317–1330.

Butcher

(2016) Numerical Methods for Ordinary Differential Equations. UK: John Wiley and Sons.

Crull

Hamilton

(2012) Methodologies for Calculating Primary Fragment Characteristics. Alexandria, VA: Department of Defense Explosives Safety Board (DDESB). Technical Paper 16.

Crull

Watanabe

Manthey

, et al. (1996) OE Barricade Guide Development. Huntsville, AL: Army Engineering and Support Centre.

DDESB (2009) Approved Methods and Algorithms for DoD Risk-Based Explosives Siting. Alexandria, VA: Department of Defense Explosives Safety Board (DDESB).

DoD (2004) DoD Ammunition and Explosives Safety Standards 6055.09-STD. Washington DC: Department of Defense (DoD).

Giroux

(1971) HEMP User’s Manual. UCRL-51079. Livermore, CA: Lawrence Livermore Laboratory, University of California.

Grady

Kipp

(1985) Geometric statistics and dynamic fragmentation. Journal of Applied Physics 58(3): 1210–1222.

Grisaro

Dancygier

(2015) Numerical study of velocity distribution of fragments caused by explosion of a cylindrical cased charge. International Journal of Impact Engineering 86: 1–12.

10.

Gurney

(1943) Initial Velocities of Fragments from Bombs, Shell, Grenades. Report BRL 405, Army Ballistic Research Laboratories. MD: Aberdeen Proving Ground.

11.

Häring

Schönherr

Richter

(2009) Quantitative hazard and risk analysis for fragments of high-explosive shells in air. Reliability Engineering and System Safety 94(9): 1461–1470.

12.

Held

(1990a) Fragment mass distribution of HE projectiles. Propellants, Explosives, Pyrotechnics 15(6): 254–260.

13.

Held

(1990b) Fragment Mass Distribution of Debris. St. Louis, MO: Proceedings of the Twenty-Fourth DoD Explosives Safety Seminar.

14.

Huang

Feng

(2015) Axial distribution of fragment velocities from cylindrical casing under explosive loading. International Journal of Impact Engineering 76: 20–27.

15.

IATG (2015) Types of Buildings for Explosive Storage. New York: The UN Office for Disarmament Affairs (UNODA).

16.

Karpp

Predebon

(1975) Calculations of Fragment Velocities from Naturally Fragmenting Munitions. Report BRL-MR-2509, Army Ballistic Research Laboratories. MD: Aberdeen Proving Ground.

17.

König

(1987) A correction for ejection angles of fragments from cylindrical wareheads. Propellants, Explosives, Pyrotechnics 12(5): 154–157.

18.

McCleskey

(1988a) Quantity-Distance Fragment Hazard Computer Program (FRAGHAZ) (No. NSWC-TR-87-59). VA: Department of Defense Explosives Safety Board and Naval Surface Warfare Centre.

19.

McCleskey

(1988b) Drag Coefficients for Irregular Fragments (No. NSWC-TR-87-89). VA: Department of Defense Explosives Safety Board and Naval Surface Warfare Centre.

20.

Mott

(1943). A Theory of the Fragmentation of Shells and Bombs. Ministry of Supply Report No A.C. 4035.

21.

NATO (2015) AASTP-1 NATO Guidelines for the Storage of Military Ammunition and Explosives. Bruxelles, Belgium: The North Atlantic Treaty Organization (NATO).

22.

Pearson

(1990) Fragmentation model for cylindrical warheads. Report No. NWC TP 7124. CA: Naval Weapons Center.

23.

Powell

Smith

McCleskey

(1981) Fragment Hazard Investigation Program: Natural Communication Detonation of 155 Mm Projectiles. Dahlgren, VA: Naval Surface Weapons Centre.

24.

Qin

Stewart

(2021) Casualty risks induced by primary fragmentation hazards from high-explosive munitions. Reliability Engineering and System Safety 215: 107874.

25.

RCC (2000) Common Risk Criteria for National Test Ranges; Inert Debris, Supplement to Standard 321-00. NM: Range Commanders Council, U.S. Army White Sands Missile Range.

26.

Sielicki

Stewart

Gajewski

, et al. (2021) Field test and probabilistic analysis of irregular steel debris casualty risks from a person-borne improvised explosive device. Defence Technology 17(6): 1852–1863.

27.

Stewart

Netherton

(2019) A probabilistic risk-acceptance model for assessing blast and fragmentation safety hazards. Reliability Engineering and System Safety 191: 106492.

28.

Taylor

(1963). In: Batchelor

(ed), The Scientific Papers of Sir Geoffrey Ingram Taylor. Volume 3 - Aerodynamics and the Mechanics of Projectiles and Explosions. Cambridge University Press.

29.

UFC (2014) Structures to resist the effects of accidental explosions. Washington, DC: U.S Army Corps of Engineers, 3-340-02.

30.

UN (2015) International IATG Ammunition Technical 02.20 Guideline: Quantity and Separation Distances. New York: United Nations (UN).

31.

Van Der Voort

Weerheijm

(2013) A statistical description of explosion produced debris dispersion. International Journal of Impact Engineering 59: 29–37.

32.

Zaker

(1975) Fragment and Debris Hazards. Washington, DC: Department of Defense Explosives Safety Board (DDESB). Technical Paper 12.