Patterned Armor Performance Evaluation for 

Multiple Impacts 

by William S. de Rosset 

ARL-TR-3038 August 2003 

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Army Research Laboratory 

Aberdeen Proving Ground, MD 21005-5069 

ARL-TR-3038 August 2003 

Patterned Armor Performance Evaluation for 

Multiple Impacts 

William S. de Rosset 

Weapons and Materials Research Directorate, ARL 

Approved for public release; distribution is unlimited. 
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Patterned Armor Performance Evaluation for Multiple Impacts 

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William S. de Rosset 

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13. SUPPLEMENTARY NOTES 

14. ABSTRACT 

Patterned armor is characterized by an array of repeating cells, such as tiled ceramic armor or reactive armor boxes. 

Performance characteristics of an ideal patterned armor with respect to multiple hits are discussed, and the types of single-shot 

ballistic data needed to quantify that performance are presented. An approach to use these data is developed to provide a 

quantitative measure (probability of nonperforation after a given number of impacts) of the patterned armor performance against

multiple impacts. This performance measure can then be compared to a well-posed multiple-hit criterion to assess whether the 

patterned armor meets the criterion. 

15. SUBJECT TERMS

patterned armor, armor performance evaluation, limit velocity, ceramic armor, multiple-hit criterion, lightweight armor, 

multiple-hit capability 

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William S. de Rosset 

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iii

Contents 

List of Figures iv

Acknowledgments v

1. Introduction 1

2. Patterned Armor Performance Characteristics 1

3. Methodology for Assessing Patterned Armor Performance Against Multiple Hits 4

3.1 Basic Methodology..........................................................................................................4 

3.2 Patterned Armor Performance Comparison for Similar Designs....................................5 

3.3 Patterned Armor Performance Comparison for Different Armor Technologies.............6 

3.4 Velocity Effects...............................................................................................................7 

4. Summary 15

5. References 16

Appendix: General Formulation for the Probability of Nonperforation of a Patterned 

Armor With Cells Capable of Withstanding More Than One Impact 17
iv

List of Figures 

Figure 1. Patterned armor cell array showing D and A. .................................................................5 

Figure 2. Probability of nonperforation P(n) given n impacts for two cell types. ..........................6 

Figure 3. Performance comparison between tiled ceramic and B4600 Armor...............................8 

Figure 4. Impact velocity vs. range for the 12.7-mm AP-M2 machine gun round.........................9 

Figure 5. Possible outcomes from the first impact for N = 3........................................................10 

Figure 6. Possible outcomes from the second impact for N = 3...................................................11 

Figure 7. One-third of the possible outcomes from the third impact for N = 3 (not all 

configurations are represented in the figure) ...........................................................................12 

Figure 8. P(7) as a function of range, indicating velocity effects.................................................14 

Figure 9. Performance comparison between a one-impact cell and a two-impact cell armor. .....14 
v

Acknowledgments 

The author is indebted to Dr. Steven Segletes for his careful review of the draft manuscript and 

his general formulation of the probability of nonperforation of patterned armor containing cells 

capable of withstanding more than one impact, which is used as the Appendix to this report. 

Dr. Segletes provided the generalization to the combinatorial problem used concerning multiple 

hits on a given cell. 

Dr. Segletes would like to thank Dr. Paul Tanenbaum of the U.S. Army Research Laboratory’s 

Survivability/Lethality Analysis Directorate for discussing this problem at length and making 

very helpful suggestions, including that an inclusion/exclusion approach was applicable to a 

simpler, though similar, problem and should apply here as well. Dr. Tanenbaum’s review of the 

Appendix further corrected some deficient definitions and notations.
vi

INTENTIONALLY LEFT BLANK.
1

1. Introduction 

At the 14th Annual Ground Vehicle Survivability Symposium in Monterey, CA, on April 2003, 

LTG John S. Caldwell called for armors to be designed that would be perforated 0% of the time. 

This is certainly the goal of all armor designers, but, in practice, certain compromises must be 

made between protection levels, cost, and vehicle weight such that this goal is rarely attained. 

This is especially true for lightweight, air-transportable fighting vehicles that the Army is 

currently developing as part of the Future Combat System of Systems. 

The protection problem for lightweight armored vehicles is compounded by threats that feature 

multiple impacts, such as machine guns and fragmenting warheads. Armors designed for these 

types of vehicles must possess the capability to defeat a large number of closely spaced impacts 

in a given burst or explosion. Some of the armors being considered for lightweight vehicles are 

modular in nature, have a repeating geometric pattern, or both. These will be referred to as 

patterned armor. The module or repeating pattern element will be referred to as a cell. 

While specific cases of patterned armor performance have been examined in the past, there has 

been no general approach developed to evaluate patterned armor for lightweight armored 

vehicles attacked by automatic weapons or bursting munitions. Whatever level of performance 

is set as the multiple-hit criterion, there has to be an accepted means or methodology by which 

the results of single-shot ballistic data can be used to estimate the performance of a given 

patterned armor design against multiple impacts. 

Section 2 will contain a discussion of a possible testing strategy and the considerations that must 

be made in conducting the ballistic tests. Characteristics of effective patterned armor will be 

included in this section. 

Section 3 will provide a proposed methodology that shows how the individual ballistic data 

might be used to establish the performance level of the patterned armor against multiple impacts. 

It is based on a model recently published by de Rosset and Wald (2002). Several examples will 

be presented that exercise this methodology and show its usefulness in evaluating trade-offs in a 

given armor technology, trade-offs between competing armor technologies, and effects of 

velocity on armor performance. The final section will summarize the main ideas presented in the 

report. 

2. Patterned Armor Performance Characteristics 

For the sake of clarity and purposes of discussion in this report, it is necessary to distinguish 

between the terms “ballistic performance” and “multiple-hit criterion.” Ballistic performance of 
2

a patterned armor vs. multiple impacts can be determined from an analysis of individual ballistic 

results and is the subject of this report. Multiple-hit criterion is a specified level of performance 

on a given armor cell or armor cell array. This level of performance should be set high enough 

to allow the armored vehicle to survive most realistic battlefield encounters. For instance, the 

multiple-hit criterion may state that the armor array must withstand perforation of a 10-round 

burst from a 12.7-mm machine gun at a distance of 500 m, with a probability of 90%. The 

weapon-to-target distance (range) and the weapon characteristics give the dispersion of the 

impacts on the target as well as the striking velocity. 

The procedures used to evaluate patterned armor that are presented in this section are 

straightforward and based on the nature of the armor being tested. Most, if not all, of these 

procedures have been and are being used by the U.S. Army Research Laboratory in testing 

ceramic tiled armor arrays. These procedures are to be distinguished from such tests as depth-of 

penetration (DOP) tests or other types of screening tests that are used in the early stages of 

evaluating armor concepts and materials. The procedures presented here are to be used for armor 

arrays that have already undergone some initial tests and have high potential for actual fielding 

on an armored vehicle. 

The main characteristic of patterned armor is that the armor is made up of repeating cells. There 

is generally a border or line of demarcation between the cells that is a potential weak point in the 

armor array. For instance, the seam between ceramic tiles generally has lower ballistic 

performance than the center of the tile. It may also be that performance of an individual cell 

depends on the hit location within the cell. Reactive armor is a good example of this (de Rosset, 

1998). To ensure overall armor array performance, an impact in one cell should leave adjoining 

cells intact and capable of defeating a subsequent impact (if it had not been previously struck). 

The properties of an ideal patterned armor can thus be summarized as follows: 

• ability of an undamaged cell to defeat a single threat round at a velocity corresponding to 

the specified range (based on limit velocity as indicated next); 

• uniform ballistic performance over the entire area of the cell; 

• no loss of performance in the area between cells (seam effect); and 

• no effect on adjoining cells from impact on a given cell. 

The first step in the evaluation procedure is to determine the limit velocity of the threat against 

an individual cell. This has usually been accomplished though an initial screening process that 

has determined suitable materials and design. By suitable, we mean that the armor material and 

design produces a limit velocity that is high enough to meet the multiple-hit criterion for a single 

shot (rather than the entire burst). The limit velocity can be determined in either of two ways. 

First, a plot of the residual velocity vs. striking velocity will yield a limit velocity VL through 
3

a Lambert-Jonas (Lambert and Jonas, 1976) fit to the data. Second, a V50 can be determined 

through standard test procedures. Where there is little or no zone of mixed results, the two 

values are approximately equal. For purposes of discussion, this report will assume that there is 

a single value for the limit velocity. Above this velocity, the armor is perforated by the threat, 

and below this value, the threat is defeated. This simplistic approach can be modified to take 

into account a more sophisticated concept of probability of perforation. However, the essence of 

the methodology presented in this report would not be affected with this more involved 

approach. 

The reduction in striking velocity as a function of range (weapon to target distance) is rather 

steep for small arms ammunition. It is conceivable that there is a range at which a given tile can 

withstand two impacts due to the low-striking velocity. The range where two impacts can be 

sustained by a single tile might vary slightly as a function of the distance between impacts on 

that tile. It is expected that with new ceramic tile encapsulation technology, the probability of a 

single tile sustaining multiple hits will increase. The next section indicates how the evaluation 

methodology can be adapted to take into account this increase in performance. 

The next step is to determine the uniformity (or lack thereof) of the ballistic performance over 

the entire area of the cell. Ideally, the limit velocity is the same at all impact points within the 

cell. However, if there is some falloff in performance in some particular area of the cell, a 

performance map of the cell needs to be generated as input data for the evaluation methodology. 

Clearly, it is more advantageous to apply efforts to making the performance homogeneous over 

the entire cell than to expend resources to show where the cell fails to meet the requirements. 

However, there may simply be physical reasons why the performance cannot be uniform. 

The third step is to determine the performance of the patterned armor in the area between cells. 

For these tests, multiple cells must be employed in an array that simulates to the greatest extent 

practical the actual arrangement of the cells in the armor design. Limit velocities should be 

determined not only for straight-edge seams but also for individual points where three cells 

might meet (triple point). 

Finally, ballistic tests must demonstrate that an impact on one cell does not affect the 

performance of an adjacent cell upon subsequent impact. This is done by shooting at a target 

with an array of cells put together similar to what is envisioned for the fielded armor design. 

Impacts on adjacent cells should be defeated consistently at the velocity specified in the 

multiple-hit criterion. 

Clearly, it may not be possible to design a patterned armor with all the ideal characteristics 

noted, especially when there are limits to the allowable areal density and cost. However, a 

sufficient amount of ballistic data must be gathered on the armor designs under consideration so 

that the trade-offs that may be required can be quantified to the greatest extent possible. The 

next section presents a methodology that can be used in this type of study. 
4

3. Methodology for Assessing Patterned Armor Performance Against 

Multiple Hits 

3.1 Basic Methodology 

The basic methodology for assessing patterned armor performance has already been presented in 

a previous report (de Rosset and Wald, 2002). The basic concepts for ceramic armor are readily 

transferable to patterned armor in general. The methodology is based on the calculation of the 

probability of perforating a given patterned armor array with one or more impacts. The 

dispersion of the impacts on the armor array is assumed to be in a bivariate normal distribution 

and is related to the range-dispersion characteristics of the weapon firing the ammunition. The 

report also shows that using a random distribution of impacts on the armor array gives similar 

results, with the largest differences in the calculations occurring for a small number of impacts 

(less than five) at short ranges (less than 250 m). For simplicity, this report will use a random 

distribution of impacts on the armor array. 

Several assumptions go into this methodology. First, it is assumed that all the relevant ballistic 

data have been obtained. In particular, any nonuniformity in ballistic performance over the cell 

face has been determined. Second, it is assumed that the armor has been designed such that 

impact on one cell will not affect the performance of adjacent cells. This would appear to be a 

reasonable requirement on a viable patterned armor. Third, it is assumed that when a cell is 

impacted, it is no longer capable of defeating a subsequent impact by the threat round. (This 

assumption will be modified in a variation of this methodology later in the report.) We also take 

a square with side length D as the cell geometry, although other shapes can be used in the 

analysis (hexagons, triangles, and rectangles, for instance). The area A (assumed to be a circle) 

over which the impacts fall on the target can be related to the range-dispersion characteristics of 

the attacking weapon. 

Consider the arrangement of square cells as shown in Figure 1. Each of the cells has uniform 

ballistic performance and can defeat the threat round at the specified velocity (i.e., the limit 

velocity is above the threat velocity). For computational simplicity, the bullet diameter is taken 

as 0. Thus, it cannot impact two cells at one time. The first round that impacts the target has a 

100% probability of being defeated (i.e., no perforation). The second round will have a finite 

probability of perforating the target since it may land on the first tile struck. If N tiles cover the 

area A (A = N*D2, neglecting corner effects), the probability P2 that the second round is defeated 

is given by 

 P2 = P1*(A – D2)/A = P1*(1 – D2/A), (1) 
5

D 

Area A 

Figure 1. Patterned armor cell array showing D and A. 

where P1 = 1. As subsequent cells are struck and become ineffective in stopping the threat 

rounds, the probability P(n) of the target not being perforated after n rounds have impacted it is 

given by 

 P(n) = (1 – D2

/A)*(1 – 2*D2/A)*(1 – 3*D2/A)*...*(1 – (n – 1)*D2/A). (2) 

De Rosset and Wald (2002) also showed that if the cell contained a zone at the edge of the cell 

that could not stop the threat round, the probability P(n) of stopping n shots was given by 

 P(n) = (1 – D2/A)*(1 – 2*D2/A)*...*(1 – (n – 1)*D2/A)*(1 – 2*δ/D)2n, (3) 

where δ is the width of the area of low performance. Note that P(n) is not defined as the 

probability that the nth shot is defeated. 

3.2 Patterned Armor Performance Comparison for Similar Designs 

Equations 2 and 3 can be used to determine which of two similar patterned armor designs is 

better. Suppose, for instance, that the armor designer is able to produce a patterned armor with a 

cell of side length D that has constant ballistic performance across its face and stops the threat at 

the specified velocity. It was shown previously (de Rosset and Wald, 2002) that the performance 

of a ceramic tile array could be increased with a decreasing cell size. What if another design is 

produced that has a smaller cell side length but has a small zone of low performance near the 

edges? Which design is better will obviously depend on the actual ballistic data. For 

demonstration purposes, several input parameters will be chosen arbitrarily and fixed to 

demonstrate how the methodology can be applied. 

Take A = 2m2

 (corresponding to a dispersion of about 500 mm), the large cell edge length as 

100 mm, and the small cell edge length as 90 mm. Figure 2 shows the results for a value of 

1 mm for δ, the width of the low-performance zone. 
6

n, Number of Impacts

0 5 10 15 20 25

P

0.0

0.2

0.4

0.6

0.8

1.0

1.2

D= 100 mm, delta = 0

D = 90 mm, delta = 1 mm

Figure 2. Probability of nonperforation P(n) given n impacts for two cell types. 

The methodology shows the value of having no zone of weak performance. The calculations 

continue with smaller cell sizes but the same value of delta (δ). The difference in armor 

performance grows due to the fact that as the number of cells increases in the impact zone (area 

A), the negative effect of increasing vulnerable area overwhelms the positive effect of having a 

smaller cell size. The methodology can easily be applied to more complicated situations where 

delta is nonzero and varies with cell size. 

3.3 Patterned Armor Performance Comparison for Different Armor Technologies 

Consider now a patterned armor with the following properties. First, it has a very small cell 

size, ~3 actual bullet diameters on a side. (Of course, the calculations still use a bullet with 

0 diameter.) Second, ballistic tests show this particular armor can defeat the threat bullet 19 out 

of 20 times. That is, the threat is able to perforate the target 1 out of 20 times, even when 

impacting an undamaged cell. Third, there is no zone of low performance (δ = 0). These 

characteristics are purely hypothetical, and the example is offered to show how the proposed 

methodology might be used to evaluate a patterned armor where performance across the cell face 

may vary in a way that is different from that generally associated with ceramic tiles. The armor 

concept will be referred to as B4600 Armor. 

The methodology is modified in the following way. The cell performance is modeled by 

assuming that there is a 5% probability that any impact on an intact (i.e., undamaged) cell will 
7

perforate the cell. We use a square unit cell and proceed as before. If the impact area 

corresponding to a given dispersion is A, then the probability P1 that the armor stops the first shot 

is 

 P1 = 0.95*N*D2

/(N*D2) = 0.95, (4) 

where N (an integer) is the number of cells contained in A and D is the length of the edge of the 

unit cell. As before, N*D2

 is taken as equal to A (a good approximation for large A and small 

D). 

The second round will see a target with one unit cell missing, as well as a 5% probability of 

perforating each of the remaining N − 1 unit cells. Thus, the probability of stopping the second 

impact will be given by 

 P2 = (0.95*(N – 1)*D2

)/(N*D2). (5) 

For n shots, the probability P of defeating them all is then 

 P(n) = P1*P2*P3*...*Pn 

 = 0.95*N*0.95*(N – 1)*0.95*(N – 2)*...0.95*(N – n + 1)*D2n/(N*D2)

n

 , (6) 

or 

 P(n) = 0.95n*N*(N – 1)*(N – 2)*...*(N – n + 1)/Nn . (7) 

The B4600 Armor can be compared to a traditional ceramic tile array using equations 3 and 7. 

As in the previous example, choice of input parameters is arbitrary, and the comparison is made 

for demonstration purposes only. For the B4600 Armor, a cell edge length of 25 mm is selected. 

Using an area of 2 m2

 gives N = 3200. Figure 3 compares the performance of B4600 Armor to 

that of a patterned armor, with a square cell 90 mm on a side and a delta of 1 mm. 

For these particular input parameters, the B4600 Armor has a slightly lower performance for 

n < 5 but then outperforms the ceramic tile armor for n > 5. Different results would be obtained 

with different input parameters. 

3.4 Velocity Effects 

Based on the dispersion characteristics of the threat weapon, the methodology presented to this 

point takes into account the effect of weapon-to-target distance (range) through an increase in 

impacted area. That is, as the range is increased, A increases and the probability of no 
8

n, Number of Impacts

0 5 10 15 20 25

P

0.0

0.2

0.4

0.6

0.8

1.0

D = 90 mm, delta = 1 mm

B4600 Armor

Figure 3. Performance comparison between tiled ceramic and B4600 Armor. 

perforations for a given number of impacts goes up. However, an increase in range also means 

that the impact velocity of a given round decreases. For small- and medium-caliber weapons, 

this decrease can be substantial. For instance, the impact velocity as a function of range is 

shown in Figure 4 for the 12.7-mm AP-M2 machine gun round (Mascianica, 1976). The large 

reduction in velocity has not been taken into account up to now. However, the examples given 

show that it is quite easy to incorporate velocity effects into the methodology, provided that the 

ballistic data exists. 

A trivially obvious consideration is the range at which the impact velocity equals the limit 

velocity of a given cell. For ranges less than this value, perforations will occur with all impacts. 

For ranges greater than this value, the methodology presented here can be employed. Clearly, a 

goal of the armor designer is to produce a cell that has a limit velocity above the muzzle velocity 

of the threat weapon. 

Next, consider the case where the limit velocity of a given cell against a specified threat is higher 

than the weapon’s muzzle velocity. Suppose, however, there is a zone of low performance that 

is dependent on velocity. At some impact velocity, the round can be defeated in the zone of low 

performance. The armor performance thus increases with increasing range for two reasons— 
9

Range, m

0 200 400 600 800 1000 1200

Impact Velocity, m/s

400

500

600

700

800

900

Figure 4. Impact velocity vs. range for the 12.7-mm AP-M2 machine gun round. 

increase in A and an increase in cell area capable of defeating the round. The second reason will 

result in an incremental increase in P at the range corresponding to the velocity where the zone 

of weak performance is eliminated. 

Finally, it is a reasonable assumption that at some impact velocity, a given cell will be able to 

defeat more than one impact. It may not happen in all cases, especially if the hit locations on a 

given cell are very close. In addition, the assumption might not hold for reactive armor. For the 

sake of argument, however, assume that at some velocity, a cell can defeat two impacts. At the 

range corresponding to this velocity, there will be another incremental increase in P, the 

probability that the armor array is not perforated after some number of impacts. 

To calculate this probability, consider an array of N cells representing a patterned armor. Let 

P(m) be the probability that all shots up to and including the mth shot are stopped by the array. If 

PA(m) is the probability of the mth shot hitting a cell that has not been struck before in an array 

that has survived without perforation and PB(m) is the probability of the mth shot hitting a cell 

that has been hit just once in an array that has not been perforated, then 

 P(m) = PA(m) + PB(m). (8) 

Now let QA(m – 1) equal the sum of all undamaged tiles for all possible variations of target 

configurations that have survived up to (but not including) shot number m. In a similar manner, 

QB(m – 1) is the sum of all single-impacted tiles in arrays that have survived up to (but not 

including) shot number m. The number of cells in all combinations of outcomes through the mth 

shot is Nm. Thus, 
10

 PA(m) = QA(m – 1)/Nm , and PB(m) = QB(m – 1)/Nm . (9) 

To delineate the possible combinations of outcomes from the shots on the patterned array, the 

following notation will be employed. Let (x0, x1, x2, x3,...) represent an array that has x0 cells 

with no impacts, x1 cells with one impact, x2 cells with two impacts, and so forth. Clearly, 

Σ ixi = m, 

and (10) 

Σ xi = N, 

where it goes from 0 to the number equal to the largest number of impacts in a cell in the array. 

There will be many combinations for a given set of xi, and the frequency of each combination 

may also be different. The primary interest is in the number of cells with no impacts (x = x0) or 

just a single impact (x = x1). Note also that xi is either 0 or a positive integer and cannot exceed 

m. 

As an example, take N = 3. Before any impacts occur, QA(0) = 3 and QB(0) = 0. In general, 

before there are any impacts, the only possible arrangement is (N), and 

 QA (0) = N, and QB (0) = 0. (11) 

The probability of nonperforation resulting from the first shot for our specific example is given 

by 

 P(1) = 3/31 + 0/31 = 1. (12) 

In general, 

 P(1) = N/N1 + 0/N1 = 1. (13) 

The possible target arrangements for the N = 3 array after the first impact are shown in Figure 5. 

A cell that is impacted is indicated by O. 

Figure 5. Possible outcomes from the first impact for N = 3. 

This shot results in an array designated by (2, 1). That is, there will be two undamaged cells and 

one damaged cell for each possible array. In general, the resulting configuration will be 

designated as (N – 1, 1). For the specific case, the total number possible of undamaged cells is 

six, and there are three total possible damaged cells. In general, 

 QA(1) = (N – 1)N and QB(1) = N. (14) 
11

For the next shot (m = 2), the probability of nonperforation for the specific case of N = 3 is given 

by 

 P(2) = 3*2/32

 + 3/32 = 1. (15) 

In general, 

 P(2) = (N – 1)N/N2 + N/N2 = 1. (16) 

The second shot will produce the possibilities shown in Figure 6. 

Figure 6. Possible outcomes from the second impact for N = 3. 

The number of undamaged cells shown in Figure 6 is 12, and the number that have been 

impacted once is also 12. Thus, QA(2) = 12, QB(2) = 12, and 

 P(3) = 12/33 + 12/ 33 = 8/9 (17) 

for N = 3. 

In general, the third shot will impact a target represented by either (N – 1, 0, 1) or (N – 2, 2, 0). 

There will be N combinations where a single cell has been hit twice. All other combinations will 

have two cells with one impact and N – 2 undamaged cells. There are N(N – 1) of these 

combinations (N > 1). Thus, 

 QA(2) = (N – 1)N + (N – 2)N(N – 1), and QB(2) = 2N(N – 1). (18) 

This gives 

 P(3) = ((N – 1)N + (N – 2)N(N – 1))/N3

 + ((2N(N – 1))/N3

 = ((N)3 – N)/(N)3 = 1 – 1/N2 . (19) 

To see the result for N = 3, note that there are six different arrangements resulting from the 

second impact. Symmetry arguments can be used to reduce this number to three. Figure 7 

shows, in effect, one-third of the outcomes for the third shot. All configurations are not 

represented in this figure. For instance, the case where the array has three impacts in the center 

cell is not shown. However, the notation for this outcome (N – 1, 0, 0, 1) is the same as that of 
12

Figure 7. One-third of the possible outcomes from the third impact for N = 3 (not all 

configurations are represented in the figure). 

the array in the first column and first row in Figure 7. One cell in the top left array in Figure 7 

has been perforated, so that P(3) = 8/9 for N = 3. 

The same line of reasoning can be applied for the fourth shot. After the third shot, there will be 

N combinations (N – 1, 0, 0, 1), where a single cell has been hit three times. This set of 

combinations will no longer be considered in calculating QA and QB, since it has been perforated. 

A second combination (N – 2, 1, 1, 0) will be one that has a single hit and a double hit. This can 

happen if a shot impacts an undamaged cell in an array that has two hits in one box or the shot 

impacts an array in a cell that has already been hit once. The number of ways where this can 

happen for these two cases is N(N – 1) and 2N(N – 1), respectively. The third arrangement will 

be (N – 3, 3, 0, 0) for arrays with three or more cells. Its frequency is N(N – 1)(N – 2). 

Multiplying the number of cells with no impacts in all unperforated arrays times the frequency of 

occurrence gives 

 QA(3) = (N(N − 1) + 2N(N − 1))(N − 2) + N(N – 1)(N – 2)(N – 3). (20) 

Similarly for one-impact cell arrays, 

 QB(3) = (N(N – 1) +2N(N – 1)) + 3N(N – 1)(N – 2). (21) 

Grouping terms and simplifying, 

 P(4) = 1 – 4/N2

 + 3/N3 . (22) 

By similar reasoning, it can be shown that 

 P(5) = 1 – 10/N2 + 15/N3 – 6/N4 , (23) 

 P(6) = 1 – 20/N2 + 45/N3 – 26/N4 , (24) 
13

and 

 P(7) = 1 – 35/N2 + 105/N3

 – 56/N4 – 105/N5 + 90/N6 . (25) 

Note that P(3) = P(4) = P(5) = P(6) = P(7) = 0 for N = 1, P(5) = P(6) = P(7) = 0 for N = 2, and 

P(7) = 0 for N = 3. 

A general solution with a somewhat different approach and using a combinatorial notation has 

been developed and is presented in an appendix to this report (Segletes, 2003). Using this 

solution, the probabilities of nonperforation for larger values of impacts can be calculated, 

although the algebra begins to be quite lengthy as the number of impacts increases. There is no 

good compact solution for the general case (Tanenbaum, 2003). 

It is straightforward to show how these velocity effects would affect the value of P(m), the 

probability of nonperforation through the mth shot. Take a patterned armor with a square cell 

edge length of 100 mm. It has a zone of low performance at the cell edge of 2 mm. Assume the 

limit velocity of the cell is 800 m/s against the 12.7-mm AP-M8 machine gun round. Assume 

the zone of low performance goes away at a striking velocity of 700 m/s and the cell can defeat 

two impacts at a velocity of 600 m/s. Let the dispersion of the machine gun be 1 mil and use the 

relation between A and the standard deviation of impacts from the aim point as derived in 

de Rosset and Wald (2002). 

Figure 8 shows how P(7) varies as a function of range. There are steep increases in 

performance as a function of range. This steepness results from the simplification that the 

performance changes at a specific velocity. In reality, the transition would be smoother. 

From 0 to 223 m, P(7) is 0 because the striking velocity is above the limit velocity of the armor 

for this particular threat (12.7-mm machine gun). From 223 to 476 m, the armor is able to stop 

single impacts on a cell but cannot stop them if they hit the zone of poor performance. From 

476 to 754 m, the zone of poor performance goes away. Above 754 m, a single cell is able to 

defeat two impacts. Note that there is not much change in P(7) in going from a single-impact 

cell capability to a double-impact cell capability. It may be that, in many cases, the capability of 

the armor to stop multiple impacts at long ranges is so high that there is little room for 

improvement. 

Consider now the trade-off between an armor cell that has a high limit velocity but single-impact 

capability vs. an armor cell that has a lower limit velocity but double-impact capability. Let the 

high limit velocity be 900 m/s vs. the 12.7-mm machine gun and let the lower impact velocity 

cell be 800 m/s against the same threat. Both armors are 100 mm on a side, and neither has a 

zone of low performance near the seam. Figure 9 shows P(7) as a function of range for each of 

these patterned armors. 
14

Range, m

0 200 400 600 800 1000 1200

P(7)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

Figure 8. P(7) as a function of range, indicating velocity effects. 

Range, m

0 200 400 600 800 1000 1200 1400

P(7)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

One-Impact Cell Armor

Two-Impact Cell Armor

Figure 9. Performance comparison between a one-impact cell and a two-impact 

cell armor. 
15

At short ranges, the one-impact cell outperforms the two-impact cell; but above the 223-m range, 

the two-impact cell has a higher value of P(7). The value of having a high-performance (i.e., 

high limit velocity) armor is negated (to some extent) at very close range due to the multiple 

impacts hitting a small area. 

It is clear that there are many different types of comparisons that can be made between patterned 

armors with various performance characteristics. However, unless there is a well-posed 

performance criterion, the methodology presented here is of little value. Being well-posed 

means that the performance criterion recognizes the nature of automatic weapons fire. For 

instance, a well-posed criterion might be stated as the requirement of the armor to defeat a 

seven-round burst of a 12.7-mm machine gun at a 500-m range, with a probability of more than 

95%. The methodology presented here could then be used to ascertain whether the armor met 

the criterion. In the example presented in Figure 9, the two-impact cell armor meets the 

requirement while the one-impact cell armor does not. 

As an added note, the methodology presented here might also be used for monolithic armor if it 

is found that an impact affects an area adjacent to it where a second impact is not defeated. The 

cell size would be associated with the dimensions of the affected area, but there would be no 

zone of low performance. 

4. Summary 

The goal of the armor designer has always been to defeat any and all threats. While this is a 

worthy goal, it must be recognized that light armor attacked by automatic weapons fire will 

always be defeated under some circumstances. Consequently, realistic performance criteria need 

to be decided upon. 

Characteristics of an ideal patterned armor were presented. The types of data needed to 

determine performance were discussed. These included such items as the ballistic limit for a 

given cell, the variation in performance as a function of hit location on the cell, and the effect of 

an impact on a neighboring cell. These data could then be used in conjunction with the presented 

methodology to determine if the patterned armor met a well-posed multiple-hit criterion. The 

methodology presented here could also be used to conduct trade-off studies between competing 

armor technologies or examine the effect of parameter variations (cell size, limit velocity of a 

cell, individual cell capability to withstand multiple impacts, etc.). 
16

5. References 

de Rosset, W. S.; Wald, J. K. Analysis of Multiple-Hit Criterion for Ceramic Armor; ARL-TR2861; U.S. Army Research Laboratory: Aberdeen Proving Ground, MD, September 2002. 

de Rosset, W. S. Reactive Armor Model Sensitivity Studies; ARL-TR-1849; U.S. Army 

Research Laboratory: Aberdeen Proving Ground, MD, November 1998. 

Lambert, J. P.; Jonas, G. H. Towards Standardization in Terminal Ballistics Testing: Velocity 

Representation; BRL Report 1852, U.S. Army Ballistic Research Laboratory: Aberdeen 

Proving Ground, MD, January 1976. 

Mascianica, F. S. Ballistic Technology of Lightweight Armors; AMMRC TR 76-15; U.S. Army 

Materials and Mechanical Research Center: Watertown, MA, May 1976. 

Segletes, S. B. U.S. Army Research Laboratory, Aberdeen Proving Ground, MD. Private 

communication, May 2003. 

Tanenbaum, P. J. U.S. Army Research Laboratory, Aberdeen Proving Ground, MD. Private 

communication, May 2003.
17

Appendix: General Formulation for the Probability of Nonperforation of a 

Patterned Armor With Cells Capable of Withstanding More Than One 

Impact 

Combinatorial Problem for m Hits on an N-tile Target Array 

Before presenting the general form of the method, let us first consider a specific problem that 

will help identify the basic strategy for solution. Consider the problem of seven random impacts 

on a four-tile1 target array, where three hits upon any given tile are required in order to defeat the 

tile and thus the array. 

Let us name the tiles “a,” “b,” “c,” and “d.” With a problem of this limited scale, one may 

deduce from inspection that the only combinations for surviving seven hits on four tiles are of 

the form “aabbccd,” where one of the tiles (in this case, “d”) is struck once and all the other tiles 

(“a,” “b,” and “c”) are each struck twice. The two hits on tile “a” may occur at the following 

possible positions among the seven-hit impact sequence: 

 

aa····· ·aa···· ··aa··· ···aa·· ····aa· ·····aa 

a·a···· ·a·a··· ··a·a·· ···a·a· ····a·a

a··a··· ·a··a·· ··a··a· ···a··a

a···a·· ·a···a· ··a···a

a····a· ·a····a

a·····a 

where the “·” denotes an as yet undesignated impact upon a tile other than “a.” This yields 21 

possibilities, which also equals C (7, 2) = 7!/(5! 2!) = 21, signifying the number of two-element 

subsets from a seven-element set. For each of these 21 possibilities, the question is then how 

could the two impacts upon tile “b” be arranged from the remaining five undesignated impacts. 

The two element subsets for tile “b” from the five remaining undesignated impacts are as 

follows: 

 

bb··· ·bb·· ··bb· ···bb 

b·b·· ·b·b· ··b·b

b··b· ·b··b

b···b 

 1

 In this appendix, a tile is equivalent to a cell. 
18

This yields 10 possibilities equaling C (5, 2), the number of two-element subsets from a fiveelement set. Because each of these impact sequences upon tile “b” applies to the undesignated 

shots for each of the two-element subsets for impacts on tile “a,” the total ways to order two 

impacts on “a” and two impacts on “b” among the seven-shot sequence produces 21 × 10 = 210 

possibilities. Of the remaining three undesignated impacts, the ways to choose two impacts on 

tile “c” are as follows: 

cc· ·cc 

c·c 

This yields three possibilities equaling C (3, 2). The number of ways, therefore, to order two 

impacts on “a” and two impacts on “b” and two impacts on “c” among the seven-shot sequence 

produces 210 × 3 = 630 possibilities. There is only one way to choose from among the lone 

remaining undesignated impact equaling C (1, 1) = 1, which must be upon tile “d.” And so there 

remain 630 × 1 = 630 possibilities to order two impacts on tile “a” and two impacts on “b” and

two impacts on “c” and one impact on “d.” 

Finally, the combinations examined were for the assumed case of tile “d” receiving only one 

impact. In fact, there are four possibilities to choose from as to which one tile gets struck only 

once. So, having solved the problem for one hit on tile “d,” multiplying the result by C(4, 1) = 4 

will cover the chances that it was, in fact, tiles “a,” “b,” or “c” being hit only once. So the 

possible orderings of survivable impacts will number 630 × 4 = 2520 possible ways of the seven 

impacts being spread 2 + 2 + 2 + 1 upon the four tiles. 

The number of ways that seven distinct shots can be distributed randomly among the four target 

tiles is simply 47 = 16384, since for each of the seven successive impacts, there are four possible 

outcomes as to which tile is struck. The survivability rate for this seven-shot sequence on the 

four-tile target array, assuming that all shots are independently and uniformly distributed, is 

therefore P = 2520/16384 = 0.1538. 

In this example, we explicitly counted the array-survival orderings because the combinations of 

ways to survive were so limited–namely limited to a spread of 2 + 2 + 2 + 1 impacts upon the 

elements of the four-tile array. It the number of tiles were to be increased, the orderings of ways 

that seven impacts could produce a surviving target array would grow quite rapidly, quickly 

outstripping the number of impact-sequence orderings resulting in a defeat of the array. And 

while it is largely a matter of preference, the general method will be developed to explicitly 

count not the number of surviving impact sequences, but rather the number of defeating impact 

sequences. The survival probability is, at that point, simply calculated by subtracting the 

probability of defeat from unity. 

The example shown, however, does not highlight the one problem that arises when tabulating 

impact sequences for cases when the array is randomly struck by a large number of impacts. For 

the situation of counting impact sequences that cause array defeat, the following applies: when 
19

there exists a sequence of m impacts in which two (or more) of the N tiles can be individually 

defeated, the method of totaling the orderings for the array, in which the total number of tiles 

multiplies the number of “defeat” orderings that a single tile contributes, must be subjected to a 

correction. A subtractive correction is necessary because a single impact sequence that results 

in, for example, two tiles being defeated is overcounted with the basic method, since the 

sequence shows up in the “defeated” tabulations for both tiles “a” and “b,” even though the 

twice-counted defeat arose from only a single impact sequence. Unfortunately, the correction 

itself requires a subtractive correction, when the possibility arises that there are three tiles that 

could have been simultaneously defeated by the impact sequence. Since two successive 

subtractive corrections make for an addition, successive corrections will alternate in sign. This 

alternating correction might need to be successively applied up to and until the limiting case of 

all N tiles being defeated by the m impacts, should m be large enough to do so. In the field of 

combinatorics, this approach to tabulating the impact sequences is known, as an 

inclusion/exclusion approach. 

Note that an analogous correction is required when counting impact sequences that do not result 

in array defeat, and so there is no trivial way around this problem. It, likewise, must rely on an 

inclusion/exclusion approach, after a similar fashion. 

We now present the general result to the problem of calculating the likelihood that an N-tile 

target array will survive an m-shot sequence if k shots are required upon any single tile to defeat 

it and thus the array. First, we lay out the following explicit definitions: 

• P − probability that a target array will survive a random volley of impacts (i.e., hits). 

• N – number of (identical) target tiles in the target array. 

• m – number of hits randomly spread upon the target array. 

• k – threshold number of hits on a given tile required to defeat that tile (and thus the array). 

• jn − index denoting number of hits upon the n’th defeated tile arising from the m-shot 

sequence. 

• 















k

n − the binomial coefficient that counts the number of ways one can assign each member 

of an n-set with either of two labels—k of them getting labeled chosen and the remaining 

n − k getting labeled not chosen. The binomial coefficient represents the number of kelement subsets from an n-element set, is equal to ( )! !

!

n k k

n

− , and is alternately represented as 

C(n, k). These are the familiar elements of Pascal’s triangle. 
20

• 















t k k k

n

, , ..., 1 2

 − the multinomial coefficient, a generalization of the binomial coefficient, that 

counts the number of ways one can assign each member of an n-set one of t labels. The 

label indices k1, ..., kt are nonnegative integers, and k1 + ··· + kt = n. The multinomial 

coefficient is equal to ! ! !

( ) ( ) !

, , ..., 1 2

1 1

3

1 2

2

1

1 2 1 t t

t

t k k k

n

k

n k k

k

n k k

k

n k

k

n

k k k

n

⋅ ⋅ ⋅ =













 − + ⋅ ⋅ ⋅ + ⋅ ⋅ ⋅ 













 − +















 −















 =













 − . 

The form that follows arises from explicitly counting the m-shot sequences that defeat the array, 

dividing them by the total possible number of m-shot sequences on an array of N target tiles 

(N m

), and subtracting that probability from 1 to get the probability of surviving a sequence of m

shots at the array. The following formula would seem to apply (though it has not been rigorously 

proven): 

 

() ( )

( ) 3 ... . 3 , , ,

2

2 , , 11

1 1

2

1 2 3 1 2 3

1 1 2 1 2

1

1

2

1 2

3

1 2 3

1

1

2

1 2

1

1





 − − 















− − − 















+





 − 















− − 













 − − 





























 = −

∑∑ ∑

∑ ∑ ∑

−

=

− −

=

− −

=

− − −

−

=

−

=

− −

=

−

m k

j k

m j k

j k

m j j

j k

m j j j

m k

j k

m j

j k

m j j m

j k

m j

m

N

j j j m j j j

N m

N

j j m j j

N m

N

j

N m

N

P

(A-1) 

 

The first summation in the braces counts out the number of impact sequences for which a given 

tile is defeated (impacted k, k + 1 ..., or m times) and multiplies the result by the number of tiles, 

i.e., C(N, 1). Beyond the first summation term, the finite sequence of subsequent summations 

represent correction terms needed to subtract out duplicately counted failures and/or corrections 

when more than one tile has been subjected to the lethal threshold of hits. The second 

summation grouping represents the correction for those limited cases when exactly two tiles in 

the array are defeated from the m-shot sequence. The third summation sequence corrects the 

correction for those cases when exactly three tiles are defeated, etc. 

In the second term, for example, 















2

N denotes the number of possible pairs of (defeated) tiles from 

an array of N tiles, and ∑

−

=

m k

j k 1

denotes a summation over the possible values for the number of 

impacts on the first tile of the pair. It shows that the first defeated tile must have at least k

impacts. An upper limit is m − k impacts, since second defeated tile must have at least k impacts 

as well. ∑

−

=

1

2

m j

j k

 denotes a summation over the possible values for the number of impacts on the 

second tile of the pair. It shows that the second defeated tile must have at least k impacts but is 

bounded by m − j1 impacts since there are only m impacts in total j1, which are on the first 

defeated tile, 















1 2 − 1 − 2 j , j ,m j j

m is the multinomial coefficient representing 













 −

















2

1

1 j

m j

j

m . In this product, 

















1j

m denotes the number of ways that j1 impacts on the first defeated tile may be ordered among 

the m impacts, and 













 −

2

1

j

m j denotes the number of ways that j2 impacts on the second defeated tile 
21

may be arranged among the remaining m − j1 undesignated impacts. 1 2 ( 2)

m j j N − − − denotes the 

number of ways that the m − j1 − j2 undesignated nonlethal impacts may be ordered among the 

remaining N − 2 nondefeated tiles. 

There are exactly N summation-term groupings in the solution, with leading multipliers C(N, 1), 

C(N, 2)...C(N, N), since a maximum of N tiles may simultaneously fail under any condition. In 

practical application, however, the maximum number of summation-term groupings is m/k

(i.e., in this case, the “floor” operator indicates the largest integer not greater than m/k) since this 

is the actual maximum number of tiles that may simultaneously fail from firing m shots onto tiles 

with a lethal threshold of k shots. The solution accounts for this since terms beyond such a value 

of m/k will have the leading summation of the term performed zero times. Thus, the number of 

summation-term groupings actually required is min [N, m/k ], depending on whether the lethal 

impact sequences are limited by the overall number of tiles (N) in, or the number of actual shots 

(m) on, the target array. 

And while the summation-term groupings in the general solution may seem foreboding, the 

actual application of the equation to a quantifiable scenario is much less so. Take, for example, 

the case of an eight-hit distribution upon a 10-tile target, where three hits are required to defeat 

any given tile and thus the following array: 

 

{ } [ ] [ ( )]

70( ) 4 8 1 56 0.637.

10 56 9 70 9 56 9 28 9 8 9 1 45 56 10 8 5 8 1

10

1 1 5 4 3 2 2

8

+ ⋅ + + =

P = − ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + − ⋅ + ⋅ + (A-2) 

Similarly, if those same eight impacts had been distributed over three tiles, instead of ten, at least 

one of the tiles would necessarily have received three or more impacts, rendering that tile and 

thus the array defeated. Such a response is accurately captured by the following formula: 

 

{ } [ ] [ ( )]

70( ) 4 1 1 56 0.

3 56 2 70 2 56 2 28 2 8 2 1 3 56 10 1 5 1 1

3

1 1 5 4 3 2 2

8

+ ⋅ + + =

P = − ⋅ + ⋅ + ⋅ + ⋅ + ⋅ + − ⋅ + ⋅ + (A-3) 
22

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