- Email: [email protected]
Simulation of standoff vertical-mortar fragment impact momentum on round targets
International Journal of Impact Engineering 65 (2014) 102e109
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Simulation of standoff vertical-mortar fragment impact momentum on round targets Luis A. de Béjar U.S. Army Engineer Research and Development Center, Geotechnical and Structures Laboratory, 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 October 2013 Accepted 14 November 2013 Available online 22 November 2013
The probability distribution of the fragment strike location on round targets within the scope of the expanding cylindrical metal case of a mortar at the center of a vertical arena test is formulated. Recognizing the axi-symmetric configuration, the description of the strike location in Cartesian coordinates is transformed into local polar coordinates. The theoretical distribution is experimentally verified approximately and the intervening parameters are identified. The number of fragments ejected upon case disintegration is estimated using the well-known Mott’s model. The distribution of the fragment weight was identified in a previous investigation as Weibull with specific parameters according to the fragment category: either finger or chunk. The impact momentum for the fragment rain upon striking the target is then simulated. An example of application illustrates the practical formulation of probabilistic statements concerning the vulnerability assessment of a set of round targets surrounding a central cylindrical mortar in a vertical arena test arrangement. Published by Elsevier Ltd.
Keywords: Fragment impact momentum Mortar fragmentation Mortar fragment simulation Target vulnerability under mortars Vertical arena test
1. Introduction The characterization of the strike location and the impact momentum of fragments colliding with a target after the natural disintegration of the metal case of cylindrical mortars upon detonation is fundamental for the evaluation of the fragment rain forces on transverse obstacles and the assessment of their vulnerability [1]. Some of the physical quantities contributing to that characterization exhibit a substantial amount of stochastic variation (e.g., the fragment weight and the strike location) while others (e.g., the impact velocity for moderate standoff distances) typically do not deviate significantly from their mean values and are postulated herein as deterministic. The number of significant fragments ejected upon cylindrical case disintegration is also taken as deterministic in this study. Mott’s model to predict the number of significant fragments has been experimentally verified extensively as fitting acceptably well the results for a broad range of the pair (type of explosive, case metal) in cylindrical munitions [2,3]. Although Mott’s prediction is not strictly deterministic, its dispersion about a mean value is relatively small by comparison with the scatter in the observations of those physical quantities considered of random variation in this model.
E-mail addresses: [email protected], [email protected]. 0734-743X/$ e see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ijimpeng.2013.11.006
Subsequently, the stochastic characterization of the strike location and the impact momentum of the fragments in a set allow the straightforward simulation of mortar experiments and the formulation of probabilistic statements concerning the vulnerability of targets in a vertical configuration around a cylindrical mortar. This paper introduces a model to describe the fragment impacts on round targets in standoff vertical mortar experiments. The model also shows the characterization of the physical quantities involved in the required formulations as supported by experimental evidence. A simple example illustrates the application of the model to evaluate the vulnerability of round targets in selected configurations. 2. Modeling the strike location Let a segment of length 2a of a standard cylindrical mortar of nominal diameter Dnom face a (quasi-) square target with center at C and side 2a on the lateral surface of the concentric cylinder of radius d, as in Fig. 1, and consider the round target with center also at C and radius a inscribed in the square. The target under consideration is contained within a circular band determined by the intersection of the cylinder generated by the uniform expansion of the 2a-segment of the mortar case and the cylindrical surface of radius d. The geometrical description of this arrangement may be referred to the set of orthogonal axes a, b, and g, at the center of
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
γ
2a
β X
B
2a
Y
O
ϕ
• C
α
A
Fig. 1. Schematic view of the idealized cylindrical expansion of fragments from a segment of length 2a of a vertical mortar case facing a vertical target within scope at a standoff distance d.
gravity of the mortar case O. The vertical axis g coincides with the mortar axis, whereas the horizontal axes a and b lie on a plane transverse to the mortar axis. For a fixed value of the standoff distance d, the location of the target is determined by the meridian angle 4. Given the axi-symmetrical configuration of this setup, the mathematical description of the fragment effect on the round target is independent of its location around the mortar (4). Any point P on the circular band may be projected on the mortar axis by drawing the internal normal to the band at the point. This normal will intersect first the mortar case and then the mortar axis at a point within the segment of length 2a. Likewise, any curve (set of points) on the circular band may be projected on the mortar axis, marking in the process a print on the mortar case. For example, the square of side 2a in Fig. 1 will mark on the case a vertical and curved strip, and the inscribed circle with center at C will mark on the case a vertical and curved ellipse with major semi-axis a, as shown schematically in Fig. 2. The first fundamental assumption in the model herein is that those fragments impacting the area within a closed curve on the circular band were originated from within the corresponding projected print on the mortar case. This assumption implies limitations in the model results.because real fragments have finite dimensions,
103
and fragments in the vicinity of the printed boundary of the source area on the case surface (Fig. 2) may actually impact the circular band slightly outside the corresponding target (This deviation from our idealization was actually observed for a few fragments in the experiments described below). The second fundamental assumption in this model is that the distribution of the location of individual elements in a set of fragments impacting the target is the same as that for the geometric centroid of the ideal point fragments in the set. This assumption is an approximation reflecting the case in which the set of fragments constitutes a thermo-dynamically isolated system of particles of equal mass. Obviously, this assumption also introduces inaccuracies in the results. Proceeding on the basis of these approximations, Fig. 3 shows an expanded view of a given round target and the location P of the center of gravity of the print caused by the impact of an arbitrary fragment on the target. Point P has locally either Cartesian coordinates (X,Y) or polar coordinates (r, q). Considering only the X-axis, because of axi-symmetry and neglecting the material imperfections within the metal case, the marginal probability density function for the abscissa X is expected to be Uniform, that is:
fX ðxÞ ¼
1 $½Hðx þ aÞ Hðx aÞ; 2a
(1)
where H(∙) ¼ the Heaviside’s unit-step function [4]. Considering next the Y axis, given a realization of the abscissa X ¼ x, because the mortar case is assumed to be relatively long and neglecting end effects, the conditional probability density function of the ordinate Y is expected to be also Uniform, that is:
fYjX ðyjxÞ ¼
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi$ H y þ a2 x2 H y a2 x2 : 2 2 2 a x (2)
Therefore, the joint probability density function for the pair (X,Y) is given by the product:
Y +a
FRAGMENT
dr P(r,θ) X
+a
• r x
θ
dθ -a C
2a
-a Fig. 2. Schematic view of the projection of the round target in Fig. 1 on the cylindrical case surface.
Fig. 3. Enlarged view of the round target with center C and radius a in Fig. 1.
104
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
3.5
fXY ðx; yÞ ¼ fYjX ðyjxÞ$fX ðxÞ
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2ffi$½Hðx þ aÞ Hðx aÞ$ H y þ a x 4a2 1 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii H y a2 x2 : (3)
Now transforming this joint density into the polar coordinate system, one obtains [5e7]:
fRQ ðr; qÞ ¼ fXY ½xðr; qÞ; yðr; qÞ$J$½HðrÞ Hðr aÞ$½HðqÞ Hðq 2pÞ; (4) where
J ¼
cos q vðx; yÞ ¼ sin q vðr; qÞ
r$sin q ¼ r r$cos q
3
Probability density function
¼
2.5
2
1.5
1
0.5
0
is the Jacobian of the transformation leading to:
fRQ ðr; qÞ ¼
(6)
0.5
0.6
0.7
0.8
0.9
1
Z1 k3 $FðkÞdk ¼ 0:556;
(9b)
leading to a value of the coefficient of variation (C.O.V.) of n ¼ 1/3 . Curiously, the cumulative distribution function:
Zk FK ðkÞ ¼
x$FðxÞ$dx;
(10)
0
where x is a dummy variable of integration, may be closely approximated by an associated truncated Normal distribution:
Zp=2 fRF ðr; 4Þd4 3p=2
1
FK ðkÞ ¼ pffiffiffiffiffiffi $ 2pðn$m1 Þ
Zp=2
d4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi $½HðrÞ Hðr aÞ r 2 1 a $sin2 4 0 r r $½HðrÞ Hðr aÞ; ¼ 2 $F a a
fK ðkÞ ¼ k$FðkÞ$½HðkÞ Hðk 1Þ;
(8)
Z1 0
and the mean square:
1=2$
e
xm1 n$m1
2 dx:
(11)
0
(7)
which appears plotted in Fig. 4. Notice that the area under this density is unity, as required, and that the first two moments of the distribution are given by the mean:
k2 $FðkÞdk ¼ 0:708;
Zk
Fig. 5 shows a plot of the distribution functions (10) and (11) for a qualitative comparison.
where F(∙) ¼ the complete elliptic integral of the first kind [8]. Defining the non-dimensional variable k ¼ r/a, one obtains, upon transformation, the corresponding density function as:
m1 ¼
0.4
0
The marginal density for the magnitude of the radius vector to the fragment impact location is obtained after integrating over the domain of ø as:
r ¼ 2$ a
0.3
(5)
m2 ¼
fR ðrÞ ¼
0.2
Fig. 4. Probability density function for the normalized radial distance from the center of the round target C to the fragment point of impact P.
Defining the change of variables ø ¼ p/2 q, one gets the joint density:
r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi $½HðrÞ Hðr aÞ fRF ðr; 4Þ ¼ r 2 4a2 1 a $sin2 4
p 3p H 4 : $ H 4þ 2 2
0.1
k
r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi $½HðrÞ Hðr aÞ$½HðqÞ r 2 2 4a 1 a $cos2 q Hðq 2pÞ:
0
(9a)
3. Experimental verification of probabilistic models A vertical arena test centered at the standard cylindrical mortar was conducted on three recovery CELOTEXÒ panels (side ¼ 2a) equally spaced along the circumference at a standoff distance d, as shown in Fig. 6 [9,10]. After detonation, the weight, shape, and impact location of the fragments retained in the panels were recorded for later analysis and experimental verification of the underlying probabilistic models. The data synthesis for a typical recovery panel (CEL1 with center at C) is shown in Figs. 7 and 8. The upper graph in Fig. 7 shows the Uniform quantile plot for the normalized abscissa X/a of the points of impact of fragments retained in this panel (Note that one recovered fragment actually impacted the slightly oversized panel just outside the range [a,þa]). This is the type of quantile plot that best fits the recorded data; consequently, the Uniform probability
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
105
1
0.8
0.5
X/a
0.7
0.0
0.6
NORMAL DISTRIBUTION (k) -0.5
0.5
CDF (k)
0.4
-1.0
0.3
-1.0
-0.5
0.0
0.5
1.0
Uniform Distribution (Min= -0.97,Max=1.08) 1.0
0.2 0.1 0
0.8
Cumulative probability distribution function
1.0
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 0.6
k
0.0
density function (pdf) U[1,þ1] is approximately verified. As a qualitative corroboration of this.hypothesis, the lower graph in the figure shows a satisfactory comparison between the cumulative distribution functions (CDF) for simulated data using the postulated
0.2
0.4
Fig. 5. Comparison between the cumulative probability distribution function (CDF) for the normalized radial distance k and a corresponding approximate Normal model.
CEL3
-1.0
-0.5
0.0
0.5
1.0
Dotted Line is the Empirical CDF of X/a
Fig. 7. (Upper Graph) Uniform quantile plot for the normalized abscissa X/a of a fragment point of impact P. (Lower Graph) Corresponding CDF comparison between the postulated and the empirical models.
A
1.0 m
mortar
C A
2a
CEL1
d
Y CEL2
X C
2a
2a VIEW A-A Fig. 6. Plan view of vertical arena test in which three recovery CELOTEXÒ panels face a central mortar at a standoff distance d for the experimental verification of postulated models and the identification of the constitutive parameters.
Uniform model and the corresponding empirical counterpart [11,12]. On the other hand, the results of a quantitative test on the model goodness-of-fit are listed in Table 1 [5,11]. Again, a simulated data sample using the postulated Uniform model for the normalized abscissa X/a and the empirical data sample are subjected to a two-sample KolmogoroveSmirnov test giving (1) a relatively small value of the KeS statistic at all practical levels of confidence, and (2) a relatively high p-value suggesting the clear acceptance of the null hypothesis that the variate is Uniformly distributed. Similarly, the upper graph in Fig. 8 shows the Normal quantile plot for the normalized magnitude of the radius vector R/a of the point of impact of the fragments retained in this panel. This is also the type of quantile plot that best fits the recorded data. Therefore, the empirical Gaussian pdf N(mean ¼ 0.69, C.O.V. ¼ 0.40) is identified and should be compared with the theoretical prediction of a Gaussian pdf N(mean ¼ 0.708, C.O.V. ¼ 1/3 ). The postulated model is approximately verified. As a qualitative corroboration of this hypothesis, the lower graph in the figure shows a satisfactory comparison between the CDF for simulated data using the postulated Gaussian model and the corresponding empirical counterpart [11,12]. On the other hand, the results of two separate quantitative tests on the model goodness-of-fit are listed in Table 1 [5,11]. Again, a simulated data sample using the postulated Gaussian model for the normalized radial distance R/a and the empirical data sample are subjected to separate two-sample tests: a Kolmogorove
106
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
1=2 M0 mMk NðmÞ ¼ $e ; 2 2Mk
1.2
(12a)
where N(m) ¼ number of fragments whose mass exceeds m; M0 ¼ mass of the metal cylinder, and: 0.8
1=3
R/a
Mk ¼ B$t 5=6 $Di
0.0
0.4 0.6 0.8 1.0 1.2 Normal Distribution (Mean= 0.69, C.O.V.= 0.40)
1.4
0.6
0.8
1.0
0.2
(12b)
is a dimensionless distribution factor. In Eq. (12b), t ¼ wall thickness of the metal cylinder; Di ¼ inside diameter of the cylinder; and B ¼ an experimental constant dependent on the specific pair (explosive material, case metal) being implemented, given in consistent units. Factor B is readily available for commonly used explosives cased by mild steel cylinders (as in mortars) [2]. Considering the axi-symmetry of a vertical arena test configuration, the total number of significant fragments ejected N(m) may be assumed as uniformly spread over the lateral surface of the expanding mortar cylindrical case. Therefore, the number of significant fragments hitting a vertical obstacle may be taken as proportional to its presented area. For example, the number of significant fragments to strike the round target of radius a in Fig. 1 is estimated as:
0.4
0.0
t $ 1þ Di
p$a2 a $NðmÞ; ¼ 4d 2pd$ð2aÞ
(13)
0.4
N ¼ NðmÞ$
0.0
0.2
where M0 in Eq. (12a) corresponds to the cylindrical segment of length 2a. For the square CELOTEXÒ target of the experimental verification in this investigation, the expected number of striking fragments whose mass exceeds 1 grain-mass (0.065 g-mass) is estimated as: 0.2
0.4
0.6
0.8
1.0
1.2
1.4
Dotted Line is the Empirical CDFof R/a
Fig. 8. Upper Graph) Normal quantile plot for the normalized magnitude R/a of the radius vector of a fragment point of impact P. (Lower Graph) Corresponding CDF comparison between the postulated and the empirical models.
Smirnov test and a c2 test giving (1) a relatively small value of the KeS statistic at all practical levels of confidence and a corresponding p-value of relatively high magnitude, and (2) a relatively high p-value (0.31) in the c2 test. Both tests strongly suggest the acceptance of the null hypothesis that the variate is Normally distributed.
N ¼ NðmÞ$
ð2aÞ2 a ¼ $Nð1Þ: pd 2pd$ð2aÞ
(14)
Equation (14) yields a value N ¼ 79, after inserting the standardmortar information into Eqs. (12a) and (12b), which compares well with the empirical value Ne ¼ 72 (Table 1). Also, as compared to the fragment size and to the striking location, the random variable impact velocity exhibits relatively small dispersion, for a moderate standoff distance. Consequently, the fragment impact velocity is taken here as deterministic and given by its mean value, as estimated by simple particle mechanics of rectilinear motion through a viscous fluid [2,9]:
V ¼ V0 $el$s ;
4. Simulation of fragment effect on a round target Aiming now at a model for the standoff-mortar fragment momentum upon impact on round targets, the number of significant fragments ejected upon detonation is assumed as deterministic and well predicted by Mott’s model [2]:
(15)
where
VG ffi V0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi WT 1 WC 2
(16)
Table 1 Model goodness-of-fit tests for impact location variates. Two-sample KolmogoroveSmirnov and c2 tests Number of fragments sampled
72
Normalized variate
X/a R/a
Observed probabilistic model
Uniform [0.97,1.08] Normal
Empirical distribution second-order moments
Critical value of KolmogoroveSmirnov statistic (Ccrit)
KeS statistic
p-value in KeS test
c2 statistic (8 degrees of freedom)
p-value in c2 test
0.09
0.95
e
e
Accept
0.12
0.73
9.34
0.31
Accept
Confidence level
Mean
SDev
0.10
0.05
0.01
0.05
0.62
0.15
0.17
0.20
0.69
0.276
Analyst’s Attitude towards model
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
is the initial velocity of fragment flight upon detonation; VG ¼ the Gurney velocity, an experimental characteristic of the type of explosive encased; and WT/WC ¼ the ratio of the total munition weight to the weight of explosive, for a given munition segment; and:
l ¼
CD $Af $ga ; 2W
(17)
in which CD ¼ a drag coefficient; Af ¼ fragment presented area at the time of impact; ga ¼ unit weight of air at the physical conditions of the experiment environment; and W ¼ fragment weight; and s ¼ distance traveled by the fragment along its trajectory [2,3,13,14]. Equation (17) can be simplified if it is assumed that one of the fragment dimensions is the metal case thickness t. In this case the fragment weight may be estimated as:
WyAf $t$gs ;
(18)
where gs ¼ unit weight of the mortar case metal. Inserting Eq. (18) into Eq. (17), and putting the result into Eq. (15) produces: ga s
V ¼ V0 $e1=2$CD $ gs $t :
(19)
Now, the fragment population can be categorized into two broad classes: fingers (relatively elongated fragments) and chunks (fragments with no outstanding dimension). Experimental evidence with mortars suggests that, generally, in a given detonation, the number of fingers generated is about twice as many as the number of chunks [1,9]. Also, on the average, the value of the drag coefficient is CD ¼ 1.24, for fingers, and CD ¼ 1.00, for chunks [3]. As an example, by taking a typical value for the ratio ga/gs ¼ 0.165 103 and assuming a mortar metal case thickness of t ¼ 1 cm, Eq. (19) reduces to:
( V ¼
s $e100 ;
for fingers
s $e120 ;
for chunks
V0 V0
;
(20)
in which s is inserted in meters. A previous study [1] on the characterization of the random variable mortar fragment weight led to the conclusion that the weight of fingers is distributed according to a Weibull model [15] with a shape parameter of 0.825, and a characteristic value of 38.4 grains, whereas the weight of chunks also follows a Weibull model, but with a shape parameter of 1.33, and a characteristic value of 12.0 grains. Consequently, under the assumptions stated above, the simulation of the impact momentum from a standoffmortar fragment rain on a round target of radius a reduces to the following simple steps: (a) Estimate the number of fragments of significant size to strike the target, using Eq. (13); (b) Estimate the fragment impact velocity for the standoff distance s, using Eq. (19), or its simplified version Eq. (20); (c) Simulate realizations of the random properties for each individual fragment, i.e., (c1) Fragment category: generate a random digit u ¼ U[0,1] and compute the variate g ¼ u 1/3 . If g is negative, the fragment is a chunk; otherwise, the fragment is a finger; (c2) Fragment weight: If the fragment is a chunk, simulate its weight by the inverse transform method [17] on the model Weibull(1.33; char. value ¼ 12.0), but if the fragment is a finger, simulate its weight by the inverse transform method on the model. Weibull(0.825; char. value ¼ 38.4); and (c3) Fragment impact location: generate a random digit u ¼ U[0,1] and compute the abscissa of the impact location X ¼ a∙(2u 1); simulate the magnitude of the radius vector to the impact location by the inverse transform method on the Gaussian model
107
R ¼ Normal(mean ¼ a/O2; C.O.V. ¼ 1/3 ); and compute the ordinate of the impact location: Y ¼ (r2 x2)½ ∙ sgn(2u 1), where sgn is the signum function [4]. Fig. 9 is an example of simulation on a round target of radius a at the standoff distance d in a vertical arena test centered at a standard mortar. Point sizes in this bubble plot are proportional to the fragment impact momentum. Blue dots represent fingers, while red dots represent chunks.
5. Example of application Given a single round target of radius a facing a vertical standard mortar at a standoff distance d under the assumptions outlined above, if the strike of an individual fragment on the target is defined as lethal when two conditions are met independently and simultaneously: (1) its impact momentum p exceeds a critical value p*, and (2) its radial location R lies within a critical R*-neighborhood of the target center, then the probability of target survival under the effect of a single fragment may be estimated by:
h n oi P1 fsurvivalg ¼ 1 FK k* $ 1 P W W * ;
(21)
where k* ¼ r*/a, and W* ¼ (p*/V)∙g, in which g ¼ the acceleration of gravity. Upon conditioning, the probability that the fragment weight W does not exceed the critical value W* is given by Refs. [6,7]:
o n 2Wf W * þ Wc W * P W W* ¼ ; 3
(22)
where Wf (W*) ¼ the CDF for the weight in grains of finger fragments, i.e., Weibull(0.825; char. val. ¼ 38.4) evaluated at W*, and Wc (W*) ¼ the CDF for the weight in grains of chunk fragments, i.e., Weibull(1.33; char. val. ¼ 12.0) evaluated at W*. Under the effect of the whole rain of fragments, the probability of single-target survival is given by:
Fig. 9. Simulation of prints by a rain of impacting fragments on a round target of radius a. The size of the dots is proportional to the fragment impact momentum. Blue dots represent fingers, while red dots represent chunks.
108
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
" ps ¼
!#N 2Wf W * þ Wc W * * 1 FK k $ 1 ; 3
0.4
(23) 0.35
P
nm n
o nm n survived ¼ Cm $pm ; s $ð1 ps Þ
(24)
0.3
0.25 Probability
where N is the number of significant fragments striking the target, as estimated by Eq. (13). Now suppose that a set of n disjoint targets at a common but uncertain standoff distance d.surround the mortar at the time of detonation, as Fig. 10 shows schematically; then, Equation (23) applies for each separate target. The conditional probability that m-out-of-n targets survive, given the standoff distance d is [7,16]:
0.15
n ¼ the number of possible combinations with n objects, m where Cm of them taken at a time. If the standoff distance d is distributed in the domain (0, dL) according to the probability density function fD(d), the probability that m-out-of-n targets survive is:
P
nm n
survive
o
¼
0.1
0.05
ZdL n o m survivex $fD ðxÞdx: P n
(25) 0
0
For example, if d can be represented as a discrete random variable with nd possible realizations, each di value having a probability of occurrence pi, then:
fD ðdÞ ¼
nd X
pi $dðd di Þ;
(26)
i¼1
where d(∙) is the Dirac’s delta function [4,6]. Inserting Eq. (26) into Eq. (25) gives:
P
m n
ZdL
survive ¼ 0
¼
n
!
m
n! m!ðnmÞ!
nm $ $pm s $ð1 ps Þ
nd X
pi $dðx di Þdx
i¼1 nd P i¼1
0.2
8
12
16 d/a
20
24
Fig. 11. Probability mass distribution for the normalized distance of the set of n disjoint round targets from the central mortar in the vertical arena test of the example.
round targets is given by the discrete probability mass distribution shown in Fig. 11. The distribution given in this figure is for the nondimensional distance normalized by the radius a of the round targets. The discrete values vary from 8 to 24, with increments of 4. For a standard steel mortar [1,3,9,10] filled with the explosive composition C-4 [2] and surrounded by 12 disjoint round targets in a vertical arena test configuration, Equation (27) gives the probability of survival of m-out-of-n targets.
im h inm h pi $ ps jx¼di $ 1 ps jx¼di ;
0.7
(27) where ps jx¼di means the value of ps (from Equation (23)), with N evaluated at d ¼ di (from Equation (13)). To present a specific scenario, suppose that the random distance from the central vertical mortar to each element in a set of disjoint
0.6
m=1
Prob[m-out-of-n survival]
0.5
0.4 m=2
m=0 0.3
3
4
0.2
5 6
0.1
7 0 0
Fig. 10. Example of application: set of n disjoint round targets surrounding a central mortar at a common random distance d in a vertical arena test configuration.
1
2
3
4
5
6 n
7
8 8
9 9
10
10
11
11
12
Fig. 12. Curves for the probability of survival of m-out-of-n disjoint round targets at a common random distance from the central mortar in the vertical arena test of the example.
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
the distribution of the magnitude of the radius vector from the center of a round target of radius a to the point of impact of individual fragments may be assumed as Gaussian with mean value a/O2 and coefficient of variation 1/3 . 2. Elaborating on the results of previous research on the effects of vertical mortar fragmentation on panels, in which fragments are classified according to their shape into fingers and chunks, the random variable fragment weight is modeled as Weibull. The number of significant fragments generated by the detonation and the impact velocity are considered in the simulations of this study as deterministic, given their relatively small amount of dispersion about their respective means. These ingredients allow a realistic simulation of the impact location and momentum of individual fragments striking mortar-facing targets in vertical arena test configurations. 3. An example of application illustrates the formulation of practical probabilistic statements concerning a set of targets surrounding a vertical mortar at a common standoff distance of moderate magnitude.
0.18 0.16 0.14 Prob[m-out-of-12 survival]
109
0.12 0.1 0.08 0.06 0.04 0.02
Acknowledgments 0
0
1
2
3
4
5
6 7 m
8
9 10 11 12
Fig. 13. Probability-mass distribution for the survival of m-out-of-12 disjoint round targets at a common random distance from the central mortar in the vertical arena test of the example.
The critical values of
2Wf W * þ Wc W * 2 and FK k* ¼ 0:051 ¼ 3 3 were estimated from previous experiments [1,9], and the results are presented in Fig. 12. This figure gives iso-survivability curves. That is, the independent variable is the number of exposed targets n, and the curves give the probability of m survivals, as a fixed parameter for each curve. These curves are not probability densities or distributions. However, they show some interesting features. For the event of one or no surviving targets, the probability of occurrence consistently decreases as the number of exposed targets n increases. But for m 2, the same statement is valid only if the number of exposed targets exceeds a critical value. If m 2, for a given survival number m (a fixed curve), the probability of the event m-out-of-n survivals increases with thenumber of exposed targets n, when this number is smaller than the critical value. This behavior resembles the principle of “diminishing returns”. To present probability distributions, one must fix the number of exposed targets n and evaluate the probability of occurrence of the event mout-of-n survivals. Fig. 13 shows the resultant discrete probability mass distribution for the particular case of n ¼ 12. 6. Conclusions This investigation has led to the following conclusions: 1. Experiments with cylindrical standard mortars in a vertical arena test arrangement have verified approximately the expected theoretical distribution for the impact location of the associated rain of fragments from the disintegration of the mortar metal case. At moderate standoff distances, generally,
This investigation was sponsored by the U.S. Department of the Army under the joint support of the Army Technology Objective entitled Protection against Terrorist and Conventional Attacks in Contingency Environments and the Army Military Engineering Basic Research Program. The author gratefully acknowledges this support. Permission to publish was granted by the Director of the Geotechnical and Structures Laboratory of the U.S. Army Engineer Research and Development Center.
References [1] de Béjar LA, Davis JL, Simmons L. Standoff-mortar-induced fragment forces. Int J Impact Eng 2008;35:1053e62. [2] Cooper PW. Explosives engineering. New York, NY: Wiley; 1996. [3] The Departments of the Army, Air Force, and Navy, and the Defense Special Weapons Agency. Technical manual: design and analysis of hardened structures to conventional weapons effects. Washington, D.C: U.S. Army Corps of Engineers, Headquarters, Office of the Chief of Engineers; 1998. [4] Greenberg MD. Foundations of applied mathematics. Englewood Cliffs, NJ: Prentice-Hall; 1978. [5] Benjamin JR, Cornell CA. Probability, statistics, and decision for civil engineersIn Random noise, signals, and dynamic systems, vol. II. New York, NY: McGraw-Hill; 1970. [6] Larson HJ, Shubert BO. Probabilistic methods in engineering sciencesIn Random variables and stochastic processes, vol. I. New York, NY: Wiley; 1979. [7] Pfeiffer PE. Concepts of probability theory. NY: Dover, New York; 1978. [8] Andrews LC. Special functions of mathematics for engineers. New York, NY: McGraw-Hill; 1992. [9] de Béjar LA, Simmons L, Davis JL. Standoff-mortar fragment velocity characterization before and after perforating conventional building walls. Int J Impact Eng 2008;35:1043e52. [10] Simmons L, Woodson SC, Davis JL. Fragment residual velocity experiments using mortars. In: Proceedings of the 74th Shock and Vibration Symposium, SAVIAC, Virginia Beach, VA 2004. [11] MathSoft. Sþ: modern statistics and advanced graphics. Seattle, WA: MathSoft; 2000. [12] Mood AM, Graybill FA, Boes DC. Introduction to the theory of statistics. 3rd ed. New York, NY: McGraw-Hill; 1974. [13] Kennedy JE. Gurney energy of explosives: estimation of the velocity and impulse imparted to driven metal. Albuquerque, NM: Sandia National Laboratories; 1970. [14] Kinney GF, Graham KJ. Explosive shocks in air. 2nd ed. New York, NY: Springer-Verlag; 1985. [15] Tobias PA. Applied reliability. 2nd ed. New York, NY: Van Nostrand Reinhold; 1995. [16] Rozanov YA. Probability theory. NY: Dover, New York; 1969. [17] Ross SM. Simulation. 2nd ed. San Diego, CA: Academic Press; 1997.
Contents lists available at ScienceDirect
International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng
Simulation of standoff vertical-mortar fragment impact momentum on round targets Luis A. de Béjar U.S. Army Engineer Research and Development Center, Geotechnical and Structures Laboratory, 3909 Halls Ferry Road, Vicksburg, MS 39180-6199, USA
a r t i c l e i n f o
a b s t r a c t
Article history: Received 29 October 2013 Accepted 14 November 2013 Available online 22 November 2013
The probability distribution of the fragment strike location on round targets within the scope of the expanding cylindrical metal case of a mortar at the center of a vertical arena test is formulated. Recognizing the axi-symmetric configuration, the description of the strike location in Cartesian coordinates is transformed into local polar coordinates. The theoretical distribution is experimentally verified approximately and the intervening parameters are identified. The number of fragments ejected upon case disintegration is estimated using the well-known Mott’s model. The distribution of the fragment weight was identified in a previous investigation as Weibull with specific parameters according to the fragment category: either finger or chunk. The impact momentum for the fragment rain upon striking the target is then simulated. An example of application illustrates the practical formulation of probabilistic statements concerning the vulnerability assessment of a set of round targets surrounding a central cylindrical mortar in a vertical arena test arrangement. Published by Elsevier Ltd.
Keywords: Fragment impact momentum Mortar fragmentation Mortar fragment simulation Target vulnerability under mortars Vertical arena test
1. Introduction The characterization of the strike location and the impact momentum of fragments colliding with a target after the natural disintegration of the metal case of cylindrical mortars upon detonation is fundamental for the evaluation of the fragment rain forces on transverse obstacles and the assessment of their vulnerability [1]. Some of the physical quantities contributing to that characterization exhibit a substantial amount of stochastic variation (e.g., the fragment weight and the strike location) while others (e.g., the impact velocity for moderate standoff distances) typically do not deviate significantly from their mean values and are postulated herein as deterministic. The number of significant fragments ejected upon cylindrical case disintegration is also taken as deterministic in this study. Mott’s model to predict the number of significant fragments has been experimentally verified extensively as fitting acceptably well the results for a broad range of the pair (type of explosive, case metal) in cylindrical munitions [2,3]. Although Mott’s prediction is not strictly deterministic, its dispersion about a mean value is relatively small by comparison with the scatter in the observations of those physical quantities considered of random variation in this model.
E-mail addresses: [email protected], [email protected]. 0734-743X/$ e see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ijimpeng.2013.11.006
Subsequently, the stochastic characterization of the strike location and the impact momentum of the fragments in a set allow the straightforward simulation of mortar experiments and the formulation of probabilistic statements concerning the vulnerability of targets in a vertical configuration around a cylindrical mortar. This paper introduces a model to describe the fragment impacts on round targets in standoff vertical mortar experiments. The model also shows the characterization of the physical quantities involved in the required formulations as supported by experimental evidence. A simple example illustrates the application of the model to evaluate the vulnerability of round targets in selected configurations. 2. Modeling the strike location Let a segment of length 2a of a standard cylindrical mortar of nominal diameter Dnom face a (quasi-) square target with center at C and side 2a on the lateral surface of the concentric cylinder of radius d, as in Fig. 1, and consider the round target with center also at C and radius a inscribed in the square. The target under consideration is contained within a circular band determined by the intersection of the cylinder generated by the uniform expansion of the 2a-segment of the mortar case and the cylindrical surface of radius d. The geometrical description of this arrangement may be referred to the set of orthogonal axes a, b, and g, at the center of
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
γ
2a
β X
B
2a
Y
O
ϕ
• C
α
A
Fig. 1. Schematic view of the idealized cylindrical expansion of fragments from a segment of length 2a of a vertical mortar case facing a vertical target within scope at a standoff distance d.
gravity of the mortar case O. The vertical axis g coincides with the mortar axis, whereas the horizontal axes a and b lie on a plane transverse to the mortar axis. For a fixed value of the standoff distance d, the location of the target is determined by the meridian angle 4. Given the axi-symmetrical configuration of this setup, the mathematical description of the fragment effect on the round target is independent of its location around the mortar (4). Any point P on the circular band may be projected on the mortar axis by drawing the internal normal to the band at the point. This normal will intersect first the mortar case and then the mortar axis at a point within the segment of length 2a. Likewise, any curve (set of points) on the circular band may be projected on the mortar axis, marking in the process a print on the mortar case. For example, the square of side 2a in Fig. 1 will mark on the case a vertical and curved strip, and the inscribed circle with center at C will mark on the case a vertical and curved ellipse with major semi-axis a, as shown schematically in Fig. 2. The first fundamental assumption in the model herein is that those fragments impacting the area within a closed curve on the circular band were originated from within the corresponding projected print on the mortar case. This assumption implies limitations in the model results.because real fragments have finite dimensions,
103
and fragments in the vicinity of the printed boundary of the source area on the case surface (Fig. 2) may actually impact the circular band slightly outside the corresponding target (This deviation from our idealization was actually observed for a few fragments in the experiments described below). The second fundamental assumption in this model is that the distribution of the location of individual elements in a set of fragments impacting the target is the same as that for the geometric centroid of the ideal point fragments in the set. This assumption is an approximation reflecting the case in which the set of fragments constitutes a thermo-dynamically isolated system of particles of equal mass. Obviously, this assumption also introduces inaccuracies in the results. Proceeding on the basis of these approximations, Fig. 3 shows an expanded view of a given round target and the location P of the center of gravity of the print caused by the impact of an arbitrary fragment on the target. Point P has locally either Cartesian coordinates (X,Y) or polar coordinates (r, q). Considering only the X-axis, because of axi-symmetry and neglecting the material imperfections within the metal case, the marginal probability density function for the abscissa X is expected to be Uniform, that is:
fX ðxÞ ¼
1 $½Hðx þ aÞ Hðx aÞ; 2a
(1)
where H(∙) ¼ the Heaviside’s unit-step function [4]. Considering next the Y axis, given a realization of the abscissa X ¼ x, because the mortar case is assumed to be relatively long and neglecting end effects, the conditional probability density function of the ordinate Y is expected to be also Uniform, that is:
fYjX ðyjxÞ ¼
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi$ H y þ a2 x2 H y a2 x2 : 2 2 2 a x (2)
Therefore, the joint probability density function for the pair (X,Y) is given by the product:
Y +a
FRAGMENT
dr P(r,θ) X
+a
• r x
θ
dθ -a C
2a
-a Fig. 2. Schematic view of the projection of the round target in Fig. 1 on the cylindrical case surface.
Fig. 3. Enlarged view of the round target with center C and radius a in Fig. 1.
104
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
3.5
fXY ðx; yÞ ¼ fYjX ðyjxÞ$fX ðxÞ
h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2ffi$½Hðx þ aÞ Hðx aÞ$ H y þ a x 4a2 1 a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffii H y a2 x2 : (3)
Now transforming this joint density into the polar coordinate system, one obtains [5e7]:
fRQ ðr; qÞ ¼ fXY ½xðr; qÞ; yðr; qÞ$J$½HðrÞ Hðr aÞ$½HðqÞ Hðq 2pÞ; (4) where
J ¼
cos q vðx; yÞ ¼ sin q vðr; qÞ
r$sin q ¼ r r$cos q
3
Probability density function
¼
2.5
2
1.5
1
0.5
0
is the Jacobian of the transformation leading to:
fRQ ðr; qÞ ¼
(6)
0.5
0.6
0.7
0.8
0.9
1
Z1 k3 $FðkÞdk ¼ 0:556;
(9b)
leading to a value of the coefficient of variation (C.O.V.) of n ¼ 1/3 . Curiously, the cumulative distribution function:
Zk FK ðkÞ ¼
x$FðxÞ$dx;
(10)
0
where x is a dummy variable of integration, may be closely approximated by an associated truncated Normal distribution:
Zp=2 fRF ðr; 4Þd4 3p=2
1
FK ðkÞ ¼ pffiffiffiffiffiffi $ 2pðn$m1 Þ
Zp=2
d4 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi $½HðrÞ Hðr aÞ r 2 1 a $sin2 4 0 r r $½HðrÞ Hðr aÞ; ¼ 2 $F a a
fK ðkÞ ¼ k$FðkÞ$½HðkÞ Hðk 1Þ;
(8)
Z1 0
and the mean square:
1=2$
e
xm1 n$m1
2 dx:
(11)
0
(7)
which appears plotted in Fig. 4. Notice that the area under this density is unity, as required, and that the first two moments of the distribution are given by the mean:
k2 $FðkÞdk ¼ 0:708;
Zk
Fig. 5 shows a plot of the distribution functions (10) and (11) for a qualitative comparison.
where F(∙) ¼ the complete elliptic integral of the first kind [8]. Defining the non-dimensional variable k ¼ r/a, one obtains, upon transformation, the corresponding density function as:
m1 ¼
0.4
0
The marginal density for the magnitude of the radius vector to the fragment impact location is obtained after integrating over the domain of ø as:
r ¼ 2$ a
0.3
(5)
m2 ¼
fR ðrÞ ¼
0.2
Fig. 4. Probability density function for the normalized radial distance from the center of the round target C to the fragment point of impact P.
Defining the change of variables ø ¼ p/2 q, one gets the joint density:
r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi $½HðrÞ Hðr aÞ fRF ðr; 4Þ ¼ r 2 4a2 1 a $sin2 4
p 3p H 4 : $ H 4þ 2 2
0.1
k
r qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi $½HðrÞ Hðr aÞ$½HðqÞ r 2 2 4a 1 a $cos2 q Hðq 2pÞ:
0
(9a)
3. Experimental verification of probabilistic models A vertical arena test centered at the standard cylindrical mortar was conducted on three recovery CELOTEXÒ panels (side ¼ 2a) equally spaced along the circumference at a standoff distance d, as shown in Fig. 6 [9,10]. After detonation, the weight, shape, and impact location of the fragments retained in the panels were recorded for later analysis and experimental verification of the underlying probabilistic models. The data synthesis for a typical recovery panel (CEL1 with center at C) is shown in Figs. 7 and 8. The upper graph in Fig. 7 shows the Uniform quantile plot for the normalized abscissa X/a of the points of impact of fragments retained in this panel (Note that one recovered fragment actually impacted the slightly oversized panel just outside the range [a,þa]). This is the type of quantile plot that best fits the recorded data; consequently, the Uniform probability
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
105
1
0.8
0.5
X/a
0.7
0.0
0.6
NORMAL DISTRIBUTION (k) -0.5
0.5
CDF (k)
0.4
-1.0
0.3
-1.0
-0.5
0.0
0.5
1.0
Uniform Distribution (Min= -0.97,Max=1.08) 1.0
0.2 0.1 0
0.8
Cumulative probability distribution function
1.0
0.9
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1 0.6
k
0.0
density function (pdf) U[1,þ1] is approximately verified. As a qualitative corroboration of this.hypothesis, the lower graph in the figure shows a satisfactory comparison between the cumulative distribution functions (CDF) for simulated data using the postulated
0.2
0.4
Fig. 5. Comparison between the cumulative probability distribution function (CDF) for the normalized radial distance k and a corresponding approximate Normal model.
CEL3
-1.0
-0.5
0.0
0.5
1.0
Dotted Line is the Empirical CDF of X/a
Fig. 7. (Upper Graph) Uniform quantile plot for the normalized abscissa X/a of a fragment point of impact P. (Lower Graph) Corresponding CDF comparison between the postulated and the empirical models.
A
1.0 m
mortar
C A
2a
CEL1
d
Y CEL2
X C
2a
2a VIEW A-A Fig. 6. Plan view of vertical arena test in which three recovery CELOTEXÒ panels face a central mortar at a standoff distance d for the experimental verification of postulated models and the identification of the constitutive parameters.
Uniform model and the corresponding empirical counterpart [11,12]. On the other hand, the results of a quantitative test on the model goodness-of-fit are listed in Table 1 [5,11]. Again, a simulated data sample using the postulated Uniform model for the normalized abscissa X/a and the empirical data sample are subjected to a two-sample KolmogoroveSmirnov test giving (1) a relatively small value of the KeS statistic at all practical levels of confidence, and (2) a relatively high p-value suggesting the clear acceptance of the null hypothesis that the variate is Uniformly distributed. Similarly, the upper graph in Fig. 8 shows the Normal quantile plot for the normalized magnitude of the radius vector R/a of the point of impact of the fragments retained in this panel. This is also the type of quantile plot that best fits the recorded data. Therefore, the empirical Gaussian pdf N(mean ¼ 0.69, C.O.V. ¼ 0.40) is identified and should be compared with the theoretical prediction of a Gaussian pdf N(mean ¼ 0.708, C.O.V. ¼ 1/3 ). The postulated model is approximately verified. As a qualitative corroboration of this hypothesis, the lower graph in the figure shows a satisfactory comparison between the CDF for simulated data using the postulated Gaussian model and the corresponding empirical counterpart [11,12]. On the other hand, the results of two separate quantitative tests on the model goodness-of-fit are listed in Table 1 [5,11]. Again, a simulated data sample using the postulated Gaussian model for the normalized radial distance R/a and the empirical data sample are subjected to separate two-sample tests: a Kolmogorove
106
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
1=2 M0 mMk NðmÞ ¼ $e ; 2 2Mk
1.2
(12a)
where N(m) ¼ number of fragments whose mass exceeds m; M0 ¼ mass of the metal cylinder, and: 0.8
1=3
R/a
Mk ¼ B$t 5=6 $Di
0.0
0.4 0.6 0.8 1.0 1.2 Normal Distribution (Mean= 0.69, C.O.V.= 0.40)
1.4
0.6
0.8
1.0
0.2
(12b)
is a dimensionless distribution factor. In Eq. (12b), t ¼ wall thickness of the metal cylinder; Di ¼ inside diameter of the cylinder; and B ¼ an experimental constant dependent on the specific pair (explosive material, case metal) being implemented, given in consistent units. Factor B is readily available for commonly used explosives cased by mild steel cylinders (as in mortars) [2]. Considering the axi-symmetry of a vertical arena test configuration, the total number of significant fragments ejected N(m) may be assumed as uniformly spread over the lateral surface of the expanding mortar cylindrical case. Therefore, the number of significant fragments hitting a vertical obstacle may be taken as proportional to its presented area. For example, the number of significant fragments to strike the round target of radius a in Fig. 1 is estimated as:
0.4
0.0
t $ 1þ Di
p$a2 a $NðmÞ; ¼ 4d 2pd$ð2aÞ
(13)
0.4
N ¼ NðmÞ$
0.0
0.2
where M0 in Eq. (12a) corresponds to the cylindrical segment of length 2a. For the square CELOTEXÒ target of the experimental verification in this investigation, the expected number of striking fragments whose mass exceeds 1 grain-mass (0.065 g-mass) is estimated as: 0.2
0.4
0.6
0.8
1.0
1.2
1.4
Dotted Line is the Empirical CDFof R/a
Fig. 8. Upper Graph) Normal quantile plot for the normalized magnitude R/a of the radius vector of a fragment point of impact P. (Lower Graph) Corresponding CDF comparison between the postulated and the empirical models.
Smirnov test and a c2 test giving (1) a relatively small value of the KeS statistic at all practical levels of confidence and a corresponding p-value of relatively high magnitude, and (2) a relatively high p-value (0.31) in the c2 test. Both tests strongly suggest the acceptance of the null hypothesis that the variate is Normally distributed.
N ¼ NðmÞ$
ð2aÞ2 a ¼ $Nð1Þ: pd 2pd$ð2aÞ
(14)
Equation (14) yields a value N ¼ 79, after inserting the standardmortar information into Eqs. (12a) and (12b), which compares well with the empirical value Ne ¼ 72 (Table 1). Also, as compared to the fragment size and to the striking location, the random variable impact velocity exhibits relatively small dispersion, for a moderate standoff distance. Consequently, the fragment impact velocity is taken here as deterministic and given by its mean value, as estimated by simple particle mechanics of rectilinear motion through a viscous fluid [2,9]:
V ¼ V0 $el$s ;
4. Simulation of fragment effect on a round target Aiming now at a model for the standoff-mortar fragment momentum upon impact on round targets, the number of significant fragments ejected upon detonation is assumed as deterministic and well predicted by Mott’s model [2]:
(15)
where
VG ffi V0 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffi WT 1 WC 2
(16)
Table 1 Model goodness-of-fit tests for impact location variates. Two-sample KolmogoroveSmirnov and c2 tests Number of fragments sampled
72
Normalized variate
X/a R/a
Observed probabilistic model
Uniform [0.97,1.08] Normal
Empirical distribution second-order moments
Critical value of KolmogoroveSmirnov statistic (Ccrit)
KeS statistic
p-value in KeS test
c2 statistic (8 degrees of freedom)
p-value in c2 test
0.09
0.95
e
e
Accept
0.12
0.73
9.34
0.31
Accept
Confidence level
Mean
SDev
0.10
0.05
0.01
0.05
0.62
0.15
0.17
0.20
0.69
0.276
Analyst’s Attitude towards model
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
is the initial velocity of fragment flight upon detonation; VG ¼ the Gurney velocity, an experimental characteristic of the type of explosive encased; and WT/WC ¼ the ratio of the total munition weight to the weight of explosive, for a given munition segment; and:
l ¼
CD $Af $ga ; 2W
(17)
in which CD ¼ a drag coefficient; Af ¼ fragment presented area at the time of impact; ga ¼ unit weight of air at the physical conditions of the experiment environment; and W ¼ fragment weight; and s ¼ distance traveled by the fragment along its trajectory [2,3,13,14]. Equation (17) can be simplified if it is assumed that one of the fragment dimensions is the metal case thickness t. In this case the fragment weight may be estimated as:
WyAf $t$gs ;
(18)
where gs ¼ unit weight of the mortar case metal. Inserting Eq. (18) into Eq. (17), and putting the result into Eq. (15) produces: ga s
V ¼ V0 $e1=2$CD $ gs $t :
(19)
Now, the fragment population can be categorized into two broad classes: fingers (relatively elongated fragments) and chunks (fragments with no outstanding dimension). Experimental evidence with mortars suggests that, generally, in a given detonation, the number of fingers generated is about twice as many as the number of chunks [1,9]. Also, on the average, the value of the drag coefficient is CD ¼ 1.24, for fingers, and CD ¼ 1.00, for chunks [3]. As an example, by taking a typical value for the ratio ga/gs ¼ 0.165 103 and assuming a mortar metal case thickness of t ¼ 1 cm, Eq. (19) reduces to:
( V ¼
s $e100 ;
for fingers
s $e120 ;
for chunks
V0 V0
;
(20)
in which s is inserted in meters. A previous study [1] on the characterization of the random variable mortar fragment weight led to the conclusion that the weight of fingers is distributed according to a Weibull model [15] with a shape parameter of 0.825, and a characteristic value of 38.4 grains, whereas the weight of chunks also follows a Weibull model, but with a shape parameter of 1.33, and a characteristic value of 12.0 grains. Consequently, under the assumptions stated above, the simulation of the impact momentum from a standoffmortar fragment rain on a round target of radius a reduces to the following simple steps: (a) Estimate the number of fragments of significant size to strike the target, using Eq. (13); (b) Estimate the fragment impact velocity for the standoff distance s, using Eq. (19), or its simplified version Eq. (20); (c) Simulate realizations of the random properties for each individual fragment, i.e., (c1) Fragment category: generate a random digit u ¼ U[0,1] and compute the variate g ¼ u 1/3 . If g is negative, the fragment is a chunk; otherwise, the fragment is a finger; (c2) Fragment weight: If the fragment is a chunk, simulate its weight by the inverse transform method [17] on the model Weibull(1.33; char. value ¼ 12.0), but if the fragment is a finger, simulate its weight by the inverse transform method on the model. Weibull(0.825; char. value ¼ 38.4); and (c3) Fragment impact location: generate a random digit u ¼ U[0,1] and compute the abscissa of the impact location X ¼ a∙(2u 1); simulate the magnitude of the radius vector to the impact location by the inverse transform method on the Gaussian model
107
R ¼ Normal(mean ¼ a/O2; C.O.V. ¼ 1/3 ); and compute the ordinate of the impact location: Y ¼ (r2 x2)½ ∙ sgn(2u 1), where sgn is the signum function [4]. Fig. 9 is an example of simulation on a round target of radius a at the standoff distance d in a vertical arena test centered at a standard mortar. Point sizes in this bubble plot are proportional to the fragment impact momentum. Blue dots represent fingers, while red dots represent chunks.
5. Example of application Given a single round target of radius a facing a vertical standard mortar at a standoff distance d under the assumptions outlined above, if the strike of an individual fragment on the target is defined as lethal when two conditions are met independently and simultaneously: (1) its impact momentum p exceeds a critical value p*, and (2) its radial location R lies within a critical R*-neighborhood of the target center, then the probability of target survival under the effect of a single fragment may be estimated by:
h n oi P1 fsurvivalg ¼ 1 FK k* $ 1 P W W * ;
(21)
where k* ¼ r*/a, and W* ¼ (p*/V)∙g, in which g ¼ the acceleration of gravity. Upon conditioning, the probability that the fragment weight W does not exceed the critical value W* is given by Refs. [6,7]:
o n 2Wf W * þ Wc W * P W W* ¼ ; 3
(22)
where Wf (W*) ¼ the CDF for the weight in grains of finger fragments, i.e., Weibull(0.825; char. val. ¼ 38.4) evaluated at W*, and Wc (W*) ¼ the CDF for the weight in grains of chunk fragments, i.e., Weibull(1.33; char. val. ¼ 12.0) evaluated at W*. Under the effect of the whole rain of fragments, the probability of single-target survival is given by:
Fig. 9. Simulation of prints by a rain of impacting fragments on a round target of radius a. The size of the dots is proportional to the fragment impact momentum. Blue dots represent fingers, while red dots represent chunks.
108
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
" ps ¼
!#N 2Wf W * þ Wc W * * 1 FK k $ 1 ; 3
0.4
(23) 0.35
P
nm n
o nm n survived ¼ Cm $pm ; s $ð1 ps Þ
(24)
0.3
0.25 Probability
where N is the number of significant fragments striking the target, as estimated by Eq. (13). Now suppose that a set of n disjoint targets at a common but uncertain standoff distance d.surround the mortar at the time of detonation, as Fig. 10 shows schematically; then, Equation (23) applies for each separate target. The conditional probability that m-out-of-n targets survive, given the standoff distance d is [7,16]:
0.15
n ¼ the number of possible combinations with n objects, m where Cm of them taken at a time. If the standoff distance d is distributed in the domain (0, dL) according to the probability density function fD(d), the probability that m-out-of-n targets survive is:
P
nm n
survive
o
¼
0.1
0.05
ZdL n o m survivex $fD ðxÞdx: P n
(25) 0
0
For example, if d can be represented as a discrete random variable with nd possible realizations, each di value having a probability of occurrence pi, then:
fD ðdÞ ¼
nd X
pi $dðd di Þ;
(26)
i¼1
where d(∙) is the Dirac’s delta function [4,6]. Inserting Eq. (26) into Eq. (25) gives:
P
m n
ZdL
survive ¼ 0
¼
n
!
m
n! m!ðnmÞ!
nm $ $pm s $ð1 ps Þ
nd X
pi $dðx di Þdx
i¼1 nd P i¼1
0.2
8
12
16 d/a
20
24
Fig. 11. Probability mass distribution for the normalized distance of the set of n disjoint round targets from the central mortar in the vertical arena test of the example.
round targets is given by the discrete probability mass distribution shown in Fig. 11. The distribution given in this figure is for the nondimensional distance normalized by the radius a of the round targets. The discrete values vary from 8 to 24, with increments of 4. For a standard steel mortar [1,3,9,10] filled with the explosive composition C-4 [2] and surrounded by 12 disjoint round targets in a vertical arena test configuration, Equation (27) gives the probability of survival of m-out-of-n targets.
im h inm h pi $ ps jx¼di $ 1 ps jx¼di ;
0.7
(27) where ps jx¼di means the value of ps (from Equation (23)), with N evaluated at d ¼ di (from Equation (13)). To present a specific scenario, suppose that the random distance from the central vertical mortar to each element in a set of disjoint
0.6
m=1
Prob[m-out-of-n survival]
0.5
0.4 m=2
m=0 0.3
3
4
0.2
5 6
0.1
7 0 0
Fig. 10. Example of application: set of n disjoint round targets surrounding a central mortar at a common random distance d in a vertical arena test configuration.
1
2
3
4
5
6 n
7
8 8
9 9
10
10
11
11
12
Fig. 12. Curves for the probability of survival of m-out-of-n disjoint round targets at a common random distance from the central mortar in the vertical arena test of the example.
L.A. de Béjar / International Journal of Impact Engineering 65 (2014) 102e109
the distribution of the magnitude of the radius vector from the center of a round target of radius a to the point of impact of individual fragments may be assumed as Gaussian with mean value a/O2 and coefficient of variation 1/3 . 2. Elaborating on the results of previous research on the effects of vertical mortar fragmentation on panels, in which fragments are classified according to their shape into fingers and chunks, the random variable fragment weight is modeled as Weibull. The number of significant fragments generated by the detonation and the impact velocity are considered in the simulations of this study as deterministic, given their relatively small amount of dispersion about their respective means. These ingredients allow a realistic simulation of the impact location and momentum of individual fragments striking mortar-facing targets in vertical arena test configurations. 3. An example of application illustrates the formulation of practical probabilistic statements concerning a set of targets surrounding a vertical mortar at a common standoff distance of moderate magnitude.
0.18 0.16 0.14 Prob[m-out-of-12 survival]
109
0.12 0.1 0.08 0.06 0.04 0.02
Acknowledgments 0
0
1
2
3
4
5
6 7 m
8
9 10 11 12
Fig. 13. Probability-mass distribution for the survival of m-out-of-12 disjoint round targets at a common random distance from the central mortar in the vertical arena test of the example.
The critical values of
2Wf W * þ Wc W * 2 and FK k* ¼ 0:051 ¼ 3 3 were estimated from previous experiments [1,9], and the results are presented in Fig. 12. This figure gives iso-survivability curves. That is, the independent variable is the number of exposed targets n, and the curves give the probability of m survivals, as a fixed parameter for each curve. These curves are not probability densities or distributions. However, they show some interesting features. For the event of one or no surviving targets, the probability of occurrence consistently decreases as the number of exposed targets n increases. But for m 2, the same statement is valid only if the number of exposed targets exceeds a critical value. If m 2, for a given survival number m (a fixed curve), the probability of the event m-out-of-n survivals increases with thenumber of exposed targets n, when this number is smaller than the critical value. This behavior resembles the principle of “diminishing returns”. To present probability distributions, one must fix the number of exposed targets n and evaluate the probability of occurrence of the event mout-of-n survivals. Fig. 13 shows the resultant discrete probability mass distribution for the particular case of n ¼ 12. 6. Conclusions This investigation has led to the following conclusions: 1. Experiments with cylindrical standard mortars in a vertical arena test arrangement have verified approximately the expected theoretical distribution for the impact location of the associated rain of fragments from the disintegration of the mortar metal case. At moderate standoff distances, generally,
This investigation was sponsored by the U.S. Department of the Army under the joint support of the Army Technology Objective entitled Protection against Terrorist and Conventional Attacks in Contingency Environments and the Army Military Engineering Basic Research Program. The author gratefully acknowledges this support. Permission to publish was granted by the Director of the Geotechnical and Structures Laboratory of the U.S. Army Engineer Research and Development Center.
References [1] de Béjar LA, Davis JL, Simmons L. Standoff-mortar-induced fragment forces. Int J Impact Eng 2008;35:1053e62. [2] Cooper PW. Explosives engineering. New York, NY: Wiley; 1996. [3] The Departments of the Army, Air Force, and Navy, and the Defense Special Weapons Agency. Technical manual: design and analysis of hardened structures to conventional weapons effects. Washington, D.C: U.S. Army Corps of Engineers, Headquarters, Office of the Chief of Engineers; 1998. [4] Greenberg MD. Foundations of applied mathematics. Englewood Cliffs, NJ: Prentice-Hall; 1978. [5] Benjamin JR, Cornell CA. Probability, statistics, and decision for civil engineersIn Random noise, signals, and dynamic systems, vol. II. New York, NY: McGraw-Hill; 1970. [6] Larson HJ, Shubert BO. Probabilistic methods in engineering sciencesIn Random variables and stochastic processes, vol. I. New York, NY: Wiley; 1979. [7] Pfeiffer PE. Concepts of probability theory. NY: Dover, New York; 1978. [8] Andrews LC. Special functions of mathematics for engineers. New York, NY: McGraw-Hill; 1992. [9] de Béjar LA, Simmons L, Davis JL. Standoff-mortar fragment velocity characterization before and after perforating conventional building walls. Int J Impact Eng 2008;35:1043e52. [10] Simmons L, Woodson SC, Davis JL. Fragment residual velocity experiments using mortars. In: Proceedings of the 74th Shock and Vibration Symposium, SAVIAC, Virginia Beach, VA 2004. [11] MathSoft. Sþ: modern statistics and advanced graphics. Seattle, WA: MathSoft; 2000. [12] Mood AM, Graybill FA, Boes DC. Introduction to the theory of statistics. 3rd ed. New York, NY: McGraw-Hill; 1974. [13] Kennedy JE. Gurney energy of explosives: estimation of the velocity and impulse imparted to driven metal. Albuquerque, NM: Sandia National Laboratories; 1970. [14] Kinney GF, Graham KJ. Explosive shocks in air. 2nd ed. New York, NY: Springer-Verlag; 1985. [15] Tobias PA. Applied reliability. 2nd ed. New York, NY: Van Nostrand Reinhold; 1995. [16] Rozanov YA. Probability theory. NY: Dover, New York; 1969. [17] Ross SM. Simulation. 2nd ed. San Diego, CA: Academic Press; 1997.