NISTIR 7962
 

Recommendations for Specifying Idealized

Bust Surrogates for the Testing of Female

Stab Resistant Armor

Amanda L. Forster, Kirk D. Rice, Michael A. Riley, and

Aaron M. Forster

NISTIR 7962
 

Recommendations for Specifying Idealized

Bust Surrogates for the Testing of Female

Stab Resistant Armor

Amanda L. Forster, Michael A. Riley, Kirk D. Rice

Materials Measurement Science Division

Material Measurement Laboratory

Aaron M. Forster

Materials and Structures Division

Engineering Laboratory

September 2013

U.S. Department of Commerce

Penny Pritzker, Secretary

National Institute of Standards and Technology

Patrick D. Gallagher, Under Secretary of Commerce for Standards and Technology

1 INTRODUCTION
 

Abstract

This document describes recommendations provided to the National Institute of Justice’s (NIJ) Special Technical Committee (STC) for the revision

of the Stab Resistant Body Armor Standard related to the specification of

idealized bust surrogates for armor testing. This document summarizes the

literature reviewed in making recommendations, relevant data collected, and

the rationale behind the recommendations. The STC and NIJ are under no

obligation to use these recommendations, they are provided for informational

and documentation purposes only.

1 Introduction

Female stab resistant armor recently became the subject of interest during

the Special Technical Committee (STC) Meeting, convened by the National

Institute of Justice (NIJ), to develop a revised standard for the performance

and testing of stab resistant body armor. The current NIJ Standard-0115.00,

published in September 2000, describes performance requirements and protection classes for stab-resistant body armor. Standardized stab implements

are specified for the protection classes defined by the standard. Armor may

be engineered to meet the requirements for the Spike protection class, the

Edged Blade protection class, or both. The test methods described in the

standard were intended to apply to all conventional armor designs, and options were included to accommodate shaped or structured armor designs,

including female armor designs. The guidelines for supporting shaped armor

for testing are described in the standard, part of which is excerpted below [1].

For all flat and flexible armor samples, a slab of composite

backing material shall be used. For shaped female armors or

where armors have a preformed curve, it may not be possible for

the rear face of the armor sample to lie in close contact with the

surface of the composite backing material. If this occurs, then

the following should be attempted in the order described:

1. If the armor design permits, the armor shall be flattened so

that the curved portions of the armor make good contact

with the backing material.

2. If this cannot be done, then the backing material shall be

laid over a curved wooden form to ensure that the armor lies

in contact with the surface of the backing material.

3. If it is still not possible to achieve good contact between the

armor and the backing material, then smaller pieces of the

neoprene backing material shall be cut, and then stacked

together to make good contact with the tight curvature of

the armor.

1
1 INTRODUCTION
 

Currently, 287 models of body armor are listed on the NIJ stab-resistant

body armor Compliant Products List (CPL) [2]. Of that total, 229 are

Spike protection class only, 18 are Edged Blade protection class only, and

the remaining 40 are some combination of Spike and Edged Blade protection

classes [2]. Only 13 female stab-resistant body armor models are listed on

the CPL, and all 13 belong to the Spike-only protection class [2]. On a

percentage basis, female-specific stab-resistant body armor models represent

less than 5 % of the stab-resistant models listed on the CPL. Since spikeresistant armor materials are typically soft and flexible, many female armor

models rely entirely on a darted and seamed armor carrier with traditional

flat armor panels inserted in the carrier. These panels conform to the shape of

the carrier in an effort to provide comfortable armor to female users. During

discussions with the STC, none of the laboratory representatives could recall

seeing a female vest that included cups, darts, or stitching as part of the

armor panel design. The implication of this observation is that none of the

female armor models are believed to be non-planar or shaped, but instead

would have been tested the same as any other flat and flexible armor.

Another source of information with ties to the market demand for body

armor can be found in the public records of the Bulletproof Vest Partnership (BVP) Grant Program [3]. The BVP records were examined for the

period FY2007 through FY2012, inclusive, to determine the quantities of

stab-resistant armor requested on grant applications. Of the 13 currently

listed female stab-resistant models, only three of them were covered by grant

request applications, for a total of 540 units of body armor, over the six

year period. During that same period, applications covered requests for

1,079,987 units of body armor (either ballistic-resistant, stab-resistant, or

both). Ballistic-resistant body armor accounted for 962,476 units. Stabresistant body armor accounted for 117,511 units. The 540 units of female

body armor represents only 0.46 % of the total number of units of stabresistant body armor units on BVP applications during this six-year period.

The BVP does not report on combination armors, which tend to be grouped

with ballistic-resistant armor, so if female officers are wearing combination

stab and ballistic armor, it is difficult to tell from the BVP statistics.

Whether by model count or units requested on BVP applications, female

stab-resistant body armor appears to represent a small fraction of the market demand. The corrections community represents the largest user group of

stab-resistant body armor, and interestingly, surveys of corrections facilities

indicate that females represent a much larger proportion of the corrections

officer workforce than suggested by the body armor market demand indicators. For example, an article published by Cheeseman in 2012, states [4]

“According to the American Correctional Associations September 30, 2007

report on Adult Correctional Personnel by Gender and Race, women represented 37 percent of adult correctional personnel (144,274, excluding federal

prison personnel) and 51 percent of juvenile corrections personnel. The ratios of male to female personnel varied from state to state. This is in sharp

2
1 INTRODUCTION
 

contrast to 1969 when women made up 12 percent of the correctional work

force.” These incongruous findings suggest that female corrections officers

are either not wearing body armor, are wearing body armor obtained without

BVP funding, or are wearing body armor typically worn by males, without

structuring to conform to their bodies [5, 6]. The Committee’s renewed interest in female armor stems from several concerns and questions that arose

during discussions.

1. Are female officers being adequately protected from stab and spike

threats?

2. Do supported and unsupported regions of armor (such as the cleavage

area on a female armor) behave differently during testing?

3. How does one design idealized female bust surrogates for non-planar

areas of armor?

The first question touches on several aspects of protection that are beyond the scope of research that could be performed in the limited time the

committee has been engaged in a standard review and revision process. Key

to addressing the first question are answers to still more questions:

1. Is the female anatomy more susceptible to injury compared to the male

anatomy?

2. Are current	 armor designs worn by females exposing them, in some

manner, to more severe threats?

3. Are the current body armor test and evaluation methods appropriate

for both the female anatomy and contemporary body armor design

strategies?

While not definitive, the committee was not aware of data indicating that

current levels of protection were inadequate. Future research directed toward

these questions posed above would help to resolve the uncertainty, although

such research is not presently planned at NIST. Given the apparent limited

usage of female armor, addressing some of the questions will be difficult.

The second question concerning supported and unsupported regions of

armor will be the focus of continuing research at NIST. Recent research led

to the development of an instrumented drop mass for characterizing the stab

impact event and foam materials used in the composite backing material

for standardized testing. Research that utilizes the same instrumented drop

mass to study impacts on supported, unsupported, and non-planar regions

of armor is currently underway and will be the subject of a future report.

The third question concerning the design of an idealized bust surrogate

is the subject of this document, which describes NIST’s efforts to assist in

developing a suitable bust surrogate.

3
2 DEFINITION OF IDEALIZED SURROGATES FOR NON-PLANAR

ARMOR TESTING

The Committee’s sense is that improvements in test methods, including

the specification of an idealized bust simulant, will stimulate greater interest

in female body armor and lead to greater market demand.

2	 Definition of idealized surrogates for nonplanar armor testing

At the June 2012 meeting of the stab resistant armor STC, the Committee

established requirements to base the bust surrogates on common bra sizes.

The committee wanted two bra cup sizes defined, one smaller and one larger

to cover a large percentage of the female population. The committee also

established a requirement for the bust shapes to be simple, primarily to minimize variability and cost in forming the surrogates. A request was made

during the meeting for NIST to provide recommended dimensions for idealized surrogates for testing non-planar female stab resistant body armor.

NIST agreed to a limited amount of research to advise the committee. The

committee’s approach toward supporting shaped or non-planar armor is different in a significant manner from the approach in NIJ Standard–0115.00,

the salient portion of which was previously discussed [1]. The difference lies

in how non-planar armor should be filled, with the current standard including provisions for shaping the contour of the backing material to ensure that

the armor model lies against the surface of the backing material. The committee’s approach was to define a shape, which presumably means that any

non-planar armor models submitted for testing would have to be made with

curvatures that conformed to the pre-defined surrogate shapes, assuming the

manufacturer believes that close contact between armor and backing material

is important. Alternatively, the manufacturer may not hold such a view, and

instead opt to submit armor with non-planar shapes that do not conform to

the surrogate. Therein lies a key point: during compliance testing under the

envisioned new test protocol, there is no assurance that body armor will be

in firm contact with the backing material. Similarly, in field use, there is no

assurance that body armor will be in contact with the body. This observation

suggests that the development of a bust surrogate may not be as important

as understanding the impact of non-supported areas on armor performance.

In an effort to inform the committee about some of the questions raised,

a limited research effort began with an examination of available data to

understand common bust dimensions for females in the United States. The

next part was to identify the best sizes and shapes for the surrogates. The

final part was to specify practical methods of making these surrogates for

use in testing.

4
3 DETERMINATION OF COMMON BUST SIZES OF FEMALES IN

THE UNITED STATES

3	 Determination of common bust sizes of females in the United States

While many media articles frequently discuss topics such as the most common bust size in the United States, finding real data on this topic is difficult.

Several reports [7, 8, 9, 10], in addition to internet searches, were used to

gather information about this topic. The most helpful of these reports was

the United States Air Force Research Laboratory’s (AFRL) “CAESAR: Summary Statistics for the Adult Population (Ages 18-65) of the United States

of America” [7]. This report was a survey of the civilian populations of

three countries, the US, the Netherlands, and Italy. The data gathered was

weighted using data from the 1990 US Census. The data related to bust

size is reported as a bust/chest measurement and an under-band measurement, which is the torso circumference measured under the breasts. Using

common bra sizing charts [11], these measurements were converted into bra

sizes1. The under-band circumference is determined and is denoted here as

Cub, (see Figure 1)2. Traditionally, 5 inches (12.7 cm) are added to this

number, and the result in inches is simply rounded (e.g., 31.5 is rounded to

32). If the measurement is even, then it is used as the band size, Lb. If

the measurement results in an odd number, 1 inch (2.54 cm) is added to

the measurement to generate an even number for Lb. (US band sizes are

all even numbers.) Next, the circumference of the fullest part of the bust

area is measured (see Figure 2), in inches. The cup size is determined from

Equation 1, where Cb is the circumference of the fullest part of the bust, Lb

is the band size as determined previously, and Lc is the cup size [11].

Lc = Cb − Lb	 (1)

2 The quantity Lc can be converted to common “letter” cup sizes using

Table 1.

If Lc is 0 1 2 3 4 5

Then cup size is AA A B C D DD
 

Table 1: Cup size conversion chart.

1Certain commercial equipment, instruments, or materials are identified in this paper

in order to specify the experimental procedure adequately. Such identification is not

intended to imply recommendation or endorsement by the National Institute of Standards

and Technology, nor is it intended to imply that the materials or equipment identified are

necessarily the best available for this purpose.

2Units: The units in this document are primarily given in SI units. As the US industry

standard for body armor and other clothing measurements is inch-pound units, in many

cases the numbers reported here are conversions from inch-pound units to SI units. To

avoid confusion with the discussion of the determination of band sizes in this paragraph,

inch-pound and SI units are both reported.

5
3 DETERMINATION OF COMMON BUST SIZES OF FEMALES IN

THE UNITED STATES

Using Equation 1 and Table 1, the measurements reported in the AFRL

study were converted into cup size measurements. The results of this effort

are shown in Table 2 [7].

Percentile Cb Cub Lb Lc Cb Cub Lb Lc Cup size

(cm) (cm) (cm) (cm) (in) (in) (in) (in)

1 78.3 66.4 81.3 0.0 30.82 26.13 32 0 AA

2 79.3 67.5 81.3 0.0 31.23 26.58 32 0 AA

3 80.3 68.2 81.3 0.0 31.60 26.86 32 0 AA

5 81.4 69.0 81.3 0.0 32.05 27.17 32 0 AA

10 83.5 70.7 86.4 0.0 32.86 27.84 34 0 AA

20 86.4 72.8 86.4 0.0 34.02 28.66 34 0 AA

25 87.4 73.6 86.4 0.0 34.42 28.98 34 0 AA

50 93.2 77.7 91.4 2.5 36.68 30.61 36 1 A

75 102.2 84.0 96.5 5.1 40.25 33.09 38 2 B

80 105.8 86.1 101.6 5.1 41.64 33.91 40 2 B

90 114.1 92.9 106.7 7.6 44.91 36.56 42 3 C

95 120.8 100.2 111.8 10.2 47.54 39.43 44 4 D

97 127.5 104.6 116.8 10.2 50.21 41.19 46 4 D

98 131.5 109.6 121.9 10.2 51.78 43.13 48 4 D

99 137.4 114.2 127.0 10.2 54.11 44.95 50 4 D

Table 2: AFRL study bust size calculations [7].

In addition to the bust measurements presented in Table 2, the Appendix

to the AFRL study also included data on self-reported bra sizes. These are

summarized in Table 3 and simplified to include only cup sizes that were

represented in more than 1 % of the population surveyed.

The data from the AFRL study indicate the population’s mean bra size

is a 36B, and the self-reported sizes show 34B, 36B, and 36C as representing

the greatest percent of the population surveyed, however this study was performed over 10 years ago in three countries, and was weighted for the 1990

Census [7]. So it is possible that it is not representative of today’s female

population. The popular media has recently reported larger average bust

sizes for US women. For example, a 2012 USA Today article [12] reported

that ten years ago, the most common bra size in the US was a 36C, and today

it is a 36DD. Unfortunately, very little good data exist to objectively determine the current average bra size of US women. The bulk of the scientific

literature is focused on breast cancer research and not on defining average

bust size. Further complicating the matter is that the same bra size may

sometimes be numbered differently by different manufacturers [13]. Even

less is available for law enforcement and corrections officers, the target audience for this work. Law enforcement practitioners serving on the committee

provided some data, the most useful being size information from 103 female

6
3 DETERMINATION OF COMMON BUST SIZES OF FEMALES IN

THE UNITED STATES

Figure 1: Demonstration of measurement of under-band circumference [11].
 

Figure 2: Demonstration of measurement of circumference of fullest portion

of bust [11].

7
3 DETERMINATION OF COMMON BUST SIZES OF FEMALES IN

THE UNITED STATES

Size Percentage

(%)

32 B 2.20

34 A 7.39

34 B 15.70

34 C 7.39

34 D 2.94

36 A 4.62

36 B 12.99

36 C 12.07

36 D 2.29

36 DD 1.12

38 B 3.11

38 C 5.72

38 D 2.11

38 DD 1.35

40 B 1.40

40 C 1.40

40 D 1.30

42 B 1.10

SUM 88.13

Table 3: AFRL study self-reported bra sizes representing greater than 1 %

of the reporting population. Sizes representing greater than 10 % appear in

bold [7].

8
4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

correctional officers, including self-reported cup size [14]. These data indicated that B, C, and D were the most popular cup sizes for this population.

The data are summarized in Table 4.

Cup Size Percentage

A 6.80

B 31.07

C 27.18

D 23.30

DD or greater 11.65

Table 4: Self-reported cup size for 103 female correctional officers.

Another correctional agency provided data on vest sizes, but determining

the relationship between the body armor manufacturer’s size nomenclature

and bra or other clothing size is difficult. While the letter-scale bra cup size

convention is ubiquitious, it is not standardized. Individual bra cup sizes

and shapes vary among manufacturers and bra styles. Further complicating

the problem is the fact that cup size and band size are related, but not in

a well defined way. For example, an individual who wears a 36C bra may

also find other sizes comfortable to wear. There is a relationship between

between band size and cup size, for example, a slightly smaller band size

with a larger cup, 34D, or a slightly larger band size and a smaller cup, 38B,

may be preferred. The implication is that the same breast volume could

be considered a B, C, or D cup size, depending on body build and size, and

the design approach taken by the bra manufacturer. Therefore, based on the

available data, the Committee was comfortable with using two cup sizes, a B

and something larger, for testing purposes [15]. The suggestion of this work

for a larger size is to select a D cup because it represents a more extreme

case. The next section of this report will discuss possible sizes and shapes of

the surrogates to use in testing.

4 Options for bust surrogate sizes and shapes

4.1 Surrogate shape determination

The previous section concluded with a suggestion for two cup sizes to be

selected for standardized testing, assuming that the standard should be prescribed with regard to the bust surrogate, but the matter of determining a

suitable shape and dimension of bust surrogates for these two bra cup sizes

also had to be addressed. After reviewing available literature [16, 17] from

researchers focused on post-mastectomy reconstruction, several shapes were

logical choices for further consideration and consistent with the committee’s

objective for the shapes to be of relatively simple geometry. The profile

9
4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

shapes of the various options were an inverted ‘V’ (sloped lines), a circular

arc, a parabola, and a catenary. When these curves are described as a function of x, rotated around the y-axis, and then cut by a plane perpendicular to

the y-axis, they form a right circular cone, a spherical cap, a paraboloid, and

a solid of revolution of a catenary, respectively, all of which have a circular

base. The size of the solids generated are determined by the radius, r, of the

circular base, and the height, h, of the solid. These sizing parameters would,

ideally, correspond to typical values from anthropomorphic studies or common bra cup sizes and produce volumes that were consistent with anatomical

studies of breast volume.

4.1.1 Right circular cone

The right circular cone is one of the simplest idealized shapes produced by

rotating a profile described by Equation (2) around the y-axis. The slopes

of the line segments, ±h

r , establish the constraints to ensure that the height

and base radius conditions are met.

⎧

⎪⎪⎨

⎪⎪⎩
 

−h

x + h 0 ≤ x ≤ r

r f(x) = (2) h

x + h −r ≤ x ≤ 0

r

Expressions were derived for the volume and surface area of the right

circular cone.

π 2 V = r h (3) 3

√ S = πr r2 + h2

4.1.2 Spherical cap and hemisphere

The circular profile shape considered is described by Equation (4), where R

represents the radius of the circle, as well as the radius of the sphere produced

after rotation around the y-axis.

x2 + y2 = R2 (4)

Note the nomenclature: an uppercase R denotes the radius of the parent

sphere, and if a spherical cap of height, h, is taken from the sphere, the

circular base of the spherical cap has a radius denoted by a lowercase r. The

spherical geometry is constrained by Equation (5), which requires that for

any given specification of r and h, the parent sphere from which the spherical

cap is taken must have a radius R.

r2 + h2

R = (5) 2h

10
4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

Expressions were derived for the volume and surface area of the spherical

cap in terms of r and h.

πh 2 V = 3r + h2 (6) 6

 2 + h2 S = π r

When h=R, then r=R, and these expressions reduce to the common formulas for a hemisphere:

2πR3 V = (7) 3

S = 2πR2

For certain combinations of r and h, the required value of h may be

greater than R, which implies that the spherical cap is more than half a

sphere. Such a shape would have a smaller circular base than its largest

circular cross-section. Due to this constraint, the spherical cap is not a

preferred geometry.

4.1.3 Paraboloid

The parabola profile shape is described by Equation (8), where a is a constant

that adjusts the focal length, or flatness, of the parabola.

f(x) = ax2 (8)

To ensure that the paraboloid will satisfy requirements for height and

base radius, a is defined by Equation (9).

h

a = (9) 2 r

Expressions were derived for the volume and surface area of the paraboloid.

V = πr2

h (10) 2

 3 πr S = r2 + 4h2 2 − r3

6h2

The paraboloid geometry does not suffer from the same constraints as the

spherical cap.

11
 

4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

4.1.4 Catenary rotated around the y-axis

The catenary profile shape is described by Equation (11), where a is a scaling

parameter that adjusts the flatness of the catenary. The catenary, which is

generally described as the shape that a chain or cable assumes under its

own weight when supported on the ends [18, 16, 17], has been the focus of

much research related to breast reconstruction because the family of catenary

curves provides a great deal of flexibility to describe spans and positive and

negative curvatures.

 a −x

f(x) = e

x

a + e a (11) 2

For any combination of r and h, the value for a must first be determined

from the implicit Equation (12).

 a −r

a + h = e

r

a + e a (12) 2

Expressions were derived for the volume and surface area of the catenary

rotated around the y-axis:

V = πr a a 2 (a + h) − πa2 (r − a) e

r

− (r + a) e

−r

+ 2a (13)

⎡ ⎛ ⎞ ⎤ √ ⎝

a + h a + h

 2

S = 2πa⎣ h2 + 2ah ln + − 1⎠ − h⎦ a a

4.1.5 Other options

All of the shapes described thus far assume that two discrete breast surrogates would be used for body armor testing. Another option for consideration

was also developed, and based on the idea that a single continuous ‘plank’

could be used as a surrogate for a bust that was already supported by a

bra. Conceptually, the plank would be described as a wedge or a frustrum of

wedge. The sloped sides of these shapes would loosely approximate the desired bust form and the length would span the width of the non-planar bust

region of the body armor. Similar to the approach that will be described for

the discrete breast surrogates, the sizing parameters for the wedge or frustrum of wedge would be guided by measured breast volumes and typical bra

cup dimensions. This approach would not, of itself, produce a region where

body armor was unsupported, such as that possibly produced by the intermammary cleft (cleavage). Variations on this approach could be considered

to address this matter by either notching the wedge or frustrum of wedge or

using two shorter units separated by some distance; however the committee

rejected this concept in favor of the discrete rounded shapes.

Elliptic versions of the cone and paraboloid, which are formed by a cutting

plane that is not perpendicular to the y-axis, were also considered but not

put forth to the committee because of their added complexity.

12
4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

4.2 Determination of surrogate dimensions

Specifications for underwires, the reinforcing wires often used under a bra cup

to provide additional support and shape provide an indication of the average

diameter of bra cups for various combinations of cup and band size [19].

In order to estimate reasonable sizing parameters for the B cup and D cup

surrogates, two approaches were taken to investigate the loose relationship

between bra size and bra cup dimensions. One relied on measurements of

actual bras, while the other considered breast volume measurements found

in scientific literature.

Average cup diameter, measured across the underwire, and depth measurements for a selection of different bra sizes are summarized in Table 5.

Reported values are the mean of no less than 7 measurements of different

bras, with size recorded as reported by the manufacturer. The diameter was

measured across the widest point of the cup for width and the deepest point

of the cup for depth.

Mean Diameter Mean Depth

Bra Cup Standard Cup Standard

Size Diameter Deviation Depth Deviation

(cm) (cm) (cm) (cm)

36B 13.34 0.37 4.81 0.96
 

36D 15.32 0.22 9.45 0.22
 

38D 15.97 0.44 9.43 0.57
 

Table 5: Critical dimensions measured from actual bras.

Inspection of Table 5 justifies the rationale for avoiding a hemispherical

surrogate shape. Such a shape has a cup depth equal to the cup radius (one

half the value of the cup diameter) regardless of the bra size. Inspection of

actual bra cup dimensions reported in Table 5 reveals that the ratio of cup

depth (the height of the cup) to cup radius ranges from 0.72 to 1.23, whereas

evidence of a hemispherical geometry would yield a ratio of 1.0. The wide

range reinforces the need for a surrogate shape whose sizing parameters can

be adjusted independently for any given cup volume.

The mean bra cup dimensions reported in Table 5 were used to determine

the volumes of various geometrical shapes possessing those dimensions, where

the mean cup diameter was taken as 2r and the mean cup depth was taken as

h in the volume equations presented earlier. Table 6 and Table 7 present the

results for the hemispherical and non-hemispherical geometries, respectively.

Other parameters derived from sizing constraints mentioned earlier for the

non-hemispherical shapes are also shown in Table 7.

Note that two sets of values are shown in Table 6 for the hemispherical

geometry, depending on whether the radius of the hemisphere, R, is taken as

the mean bra cup depth, h, or the radius of the bra cup base circle (half of the

13
4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

mean cup diameter), r. Due to the constraint imposed by the hemispherical

geometry, h = r, the height and diameter dimensions in Table 5 cannot be

satisfied simultaneously.

Hemisphere (R = h) Hemisphere (R = r)

Bra r h V V

Size (cm) (cm) (cm3) (cm3)

36B 6.67 4.80 232 621
 

36D 7.66 9.45 1767 941
 

38D 7.99 9.42 1753 1068
 

Table 6: Volumes of hemispherical shapes based on mean dimensions in

Table 5.

Spherical Cap Catenary Rotated Paraboloid Cone

Bra r h R V a V V V

Size (cm) (cm) (cm) (cm3) (cm) (cm3) (cm3) (cm3)

36B 6.67 4.80 7.03 393 5.2786 349 335 223
 

36D 7.66 9.45 7.83 1312 4.1109 946 870 580
 

38D 7.99 9.42 8.10 1383 4.4160 1022 945 630
 

Table 7: Volumes of non-hemispherical shapes based on mean dimensions in

Table 5.

Published breast volume measurements [20, 21] for various bra cup sizes

were relied upon to assess the idealized surrogate geometries. The median

values, as well as the minimum and maximum values of measured breast volumes are denoted by subscripts med, min, and max, respectively, and shown

in Table 8. The cited studies reveal considerable variability in measured

breast volumes for each of the reported bra cup sizes.

Cup

Size

Vmed

(cm3)

Vmin

(cm3)

Vmax

(cm3)

A 225 100 500

B 425 150 1100

C 700 350 1800

D 950 250 2000

Table 8: Median, minimum, and maximum measured breast volumes reported by Ringberg [20].

Comparison of actual breast volumes (Table 8) with volumes based on

idealized geometrical shapes whose dimensions were based on bra cup measurements (Table 6 and Table 7) reveals that the hemispherical geometry is

14
4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

unreliable, either producing too little volume or too much volume depending

on whether the desired bra cup height or bra cup base radius is given preference. The cone geometry appears to produce too little volume consistently.

The spherical cap geometry produces too much volume for the larger cup

size, while the rotated catenary and paraboloid geometries both yield comparable results and provide reasonably good matches to the median values

of the measured breast volumes.

A second analysis was performed by assuming that the breast surrogate

volumes should be consistent with the median values for measured breast

volumes shown in Table 8. With the exception of the hemispherical geometry, this analysis also relies on an assumed height, h, of the breast surrogate

shapes. To establish the surrogate heights for the non-hemispherical geometries, Table 1 and Table 5 were consulted. Table 5 suggests that the depth

of the deepest point of the bra cup is consistent with the Lc values shown

in Table 1; however, a better approximation of actual bra cup depths, which

would correspond to the height of the idealized geometrical shape surrogate,

can be obtained by multiplying the Lc values in Table 1 by a factor of 0.93.

For the non-hemispherical geometries, the subsequent analyses used both

sets of values, Lc and 0.93Lc, for the heights of the surrogate shapes. By

using the volumes in Table 8 and assuming various geometrical shapes, the

base diameter of the surrogates, d, and where appropriate, any other sizing

constraints, were determined for each of the idealized geometrical shapes.

The geometrical constraints mentioned previously for the hemisphere determine the radius (height) for a given volume, so it is not possible to assign

a height of Lc and 0.93Lc while still satisfying the volume goals. Instead,

for the hemispherical geometry, the base diameter, d, (which is equal to 2h)

that provides the desired volume was determined. As before, subscripts on

the size parameters denote whether the parameters are associated with median, minimum, or maximum volumes. Results are shown in Table 9 for the

hemisphere, Table 10 for the paraboloid, Table 11 for the cone, Table 12 for

the spherical cap, and Table 13 for the rotated catenary.

The conclusion reached earlier about the unsuitability of the hemispherical geometry is further supported by the size parameters in Table 9, where

hemisphere heights are too large for the smaller cup sizes and too small for the

larger cup sizes when compared with bra cup dimensions in Table 5. Table 11

reveals that the right circular cone size parameters lead to base diameters

that are too large relative to the bra cup dimensions in Table 5. Table 12

indicates that the spherical cap geometry suffers from a shape constraint

issue mentioned earlier: for combinations of r and h, typically those associated with the larger cup sizes, h tends to be greater than R, which means

that the spherical cap is more than half a sphere. Under this condition, the

surrogate shape woud have a smaller circular base than its largest circular

cross-section. Of the geometries considered in depth, only the paraboloid

and the rotated catenary offer the flexibility to adjust height and circular

base radius independently. Importantly, when either of these two shapes are

15
4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

sized to realistic heights and volumes, the resulting circular base diameters

are similar to bra cup diameters. Of these two geometries, preference is given

to the paraboloid because of its computational simplicity.

Vmed Vmin Vmax

Cup hmed dmed hmin dmin hmax dmax

Size (cm) (cm) (cm) (cm) (cm) (cm)

A 4.75 9.51 3.63 7.26 6.20 12.41

B 5.88 11.75 4.15 8.31 8.07 16.14

C 6.94 13.88 5.51 11.02 9.51 19.02

D 7.68 15.37 4.92 9.85 9.85 19.69

Table 9: Hemisphere parameters for Vmed, Vmin, and Vmax in Table 8.

h = Lc h = 0.93Lc

Cup

Size

h

(cm)

dmed

(cm)

dmin

(cm)

dmax

(cm)

h

(cm)

dmed

(cm)

dmin

(cm)

dmax

(cm)

A

B

C

D

2.54

5.08

7.62

10.16

15.02

14.60

15.29

15.43

10.01

8.67

10.82

7.92

22.39

23.48

24.53

22.39

2.36

4.72

7.09

9.45

15.57

15.14

15.86

16.00

10.38

8.99

11.21

8.21

23.22

24.35

25.43

23.22

Table 10: Paraboloid parameters for Vmed, Vmin, and Vmax in Table 8.

h = Lc h = 0.93Lc

Cup

Size

h

(cm)

dmed

(cm)

dmin

(cm)

dmax

(cm)

h

(cm)

dmed

(cm)

dmin

(cm)

dmax

(cm)

A

B

C

D

2.54

5.08

7.62

10.16

18.39

17.88

18.73

18.90

12.26

10.62

13.25

9.69

27.42

28.76

30.04

27.42

2.36

4.72

7.09

9.45

19.07

18.54

19.42

19.60

12.72

11.01

13.74

10.05

28.43

29.82

31.15

28.43

Table 11: Right circular cone parameters for Vmed, Vmin, and Vmax in Table 8.

Based on these studies, the recommended geometry for discrete

surrogates is a paraboloid whose dimensions are similar to those

shown in Table 10 for h = 0.93Lc. A reasonable choice for the B

cup surrogate has a base diameter of 15.14 cm (or 15 cm) and a

height of 4.72 cm (or 4.7 cm), and the D cup surrogate has a base

diameter of 16 cm and a height of 9.45 cm (or 9.5 cm).

16
h

=

Lc h

=

0.93

Lc

Vmed

Vmin

Vmax

Vmed

Vmin

Vmax

Cup

h

Rmed

dmed

Rmin

dmin

Rmax

dmax

h

Rmed

dmed

Rmin

dmin

Rmax

dmax

Size (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm)

A 2.54 11.95 14.73 5.78 9.57 25.52 22.20 2.36 13.62 15.33 6.49 10.02 29.31 23.06

B 5.08 6.94 13.37 3.54 6.39 15.26 22.74 4.72 7.64 14.12 3.71 7.15 17.26 23.73

C 7.62 6.38 12.51 4.46 6.29 12.41 22.89 7.09 6.80 13.59 4.58 7.67 13.77 24.08

D 10.16 6.32 10.02 null null 9.55 19.07 9.45 6.54 11.70 null null 10.28 20.49

Table 12: Spherical cap parameters for

Vmed, Vmin, and

Vmax in Table 8.

h

=

Lc h

=

0.93

Lc

Vmed

Vmin

Vmax

Vmed

Vmin

Vmax

Cup

h

amed

dmed

amin

dmin

amax

dmax

h

amed

dmed

amin

dmin

amax

dmax

Size (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm) (cm)

A 2.54 11.3714 14.93 5.1915 9.89 24.9457 22.33 2.36 13.0885 15.50 5.9480 10.28 28.7807 23.16

B 5.08 5.7246 14.30 2.2499 8.32 14.0952 23.27 4.72 6.5216 14.88 2.5294 8.67 16.1838 24.17

C 7.62 4.4808 14.79 2.4643 10.29 10.6113 24.11 7.09 5.0603 15.41 2.7558 10.73 12.1161 25.07

D 10.16 3.6838 14.73 1.2569 7.28 7.0562 21.73 9.45 4.1277 15.35 1.3826 7.60 7.9884 22.63

Table 13: Rotated catenary parameters for

Vmed, Vmin, and

Vmax in Table 8.

4 OPTIONS FOR BUST SURROGATE SIZES AND SHAPES
 

17
5 OPTIONS FOR CREATING BUST SURROGATES FOR TESTING
 

The methods described can be used to develop sizing recommendations

for other cup sizes, some of which are shown in Table 10. Depending on

surrogate materials and manufacturing methods, some adjustments to these

dimensions may be necessary. For example, manufacturing a surrogate from

multiple layers of materials having a uniform thickness may benefit from

adjusting the total height of the surrogate to some multiple of the singlelayer thickness. A surrogate whose size is adjusted may be reassessed using

the equations and tables presented to confirm that the surrogate dimensions

are a reasonable representation of actual bra cup sizes and breast volumes.

For comparison purposes, profiles of the various shapes are superimposed

on each other in Figure 3 and Figure 4. The parameters selected for the

profiles are as discussed above for the B and D cup. The catenary and the

parabola appear nearly identical in this comparison, as would be expected

given the similarities noted earlier.

5	 Options for creating bust surrogates for testing

Many different ideas were discussed and debated for making the foam surrogates. This section will discuss several of these efforts.

5.1 Molding the foam

Efforts to mold the foam using heat were unsuccessful. An aluminum cylindrical mold (negative of cylindrical shape) was heated in an oven to 120 ◦C.

After reaching temperature, the mold was manually pressed into the polyethylene foam to create a foam cylinder. While the mold was able to shape the

foam into a cylindrical shape, the process created a significant skin layer and

it was difficult to carefully control the final shape of the sample. The presence

of the skin layer indicates that the foam structure has been destroyed and

there is a solid layer of polymer on the outside of the foam. There is concern

that the inconsistent thickness of this layer and its presence on such a small

foam part would unduly influence test results. In addition, the inability to

create a consistent geometry would provide difficulties in mating armor to

the foam for testing.

5.2 Cutting the foam

A hot wire foam cutting apparatus (Figure 5) was located and purchased

for the purpose of cutting the foam into the desired shape. The blade could

be curved to the appropriate shape and the wedge of foam could be rotated

beneath the knife to cut the foam into the desired parabolic shape. However,

once the device was obtained it was determined that the foam could not be

18
5 OPTIONS FOR CREATING BUST SURROGATES FOR TESTING
 

B cup

Base (cm)

-10 -5 0 5 10

Height (cm)

0

2

4

6

8

Catenary 

Parabola

Figure 3: Comparison of possible geometric shapes for B cup surrogates. In

this figure, the base was set to 15.1 cm and the height was set to 4.7 cm.

19
5 OPTIONS FOR CREATING BUST SURROGATES FOR TESTING
 

D cup

Base (cm)

-10 -5 0 5 10

Height (cm)

0

2

4

6

8

10

12

Catenary 

Parabola

Figure 4: Comparison of possible geometric shapes for D cup surrogates. In

this figure, the base was set to 16 cm and the height was set to 9.5 cm.

20
5 OPTIONS FOR CREATING BUST SURROGATES FOR TESTING
 

Figure 5: Hot wire foam cutter used to cut foam.
 

21
5 OPTIONS FOR CREATING BUST SURROGATES FOR TESTING
 

cut repeatably in this manner. The hot knife blade melted and caught in the

foam, producing jagged, melted edges and a nonuniform shape.

Figure 6: Prototype foam stacks cut with scissors. Final cut stacks would

use cutting dies for uniform shape and size.

5.3 Other efforts

Other ideas that were considered were using mold-able floral foam or craft

foam to support the non-planar areas of the armor and create a flat surface for

mounting to a typical foam pack. Another option was to fill the back of the

armor in a similar manner with spray foam such as that used for insulation.

However, when this option was attempted, large gaps and air pockets were

created in the spray foam, meaning that the support structure was nonuniform and unsuitable for testing. Various methods of cutting stacks of stab

foam into the desired shapes were also discussed [15]. The consensus was that

using new materials added new and undesirable variability to the test, and

would require a significant validation effort. Therefore, the STC determined

that the best course of action was to prepare the foam surrogates from the

existing foam backing materials used in the test method. Working within

this constraint, NIST recommends using circles of foam cut and stacked to

approximate the desired size and shape (Prototype of this concept shown in

Figure 6). However, after this meeting additional work was done outside of

NIST to look at materials other than the foams specified during the STC.

Another option for more realistic and uniform bust surrogates would be to

22
have such surrogates machined from foam. Examining the feasibilty and cost

of this approach is an item for future work.

6 References

[1] NIJ. Stab Resistance of Personal Body Armor, NIJ Standard–0115.00,

September 2000.

[2]	 Stab Resistant Armor Compliant Product List.

http://www.justnet.org/, retrieved 13 May 2013.

[3] Bulletproof Vest Parternship Grant Act Applications.	 retrieved from:

http://www.ojp.usdoj.gov/bvpbasi/, June 2013.

[4] K. A. Cheeseman.	 Women Working in Corrections: Where We Have

Been and Where We Are Going. Corrections Today, December/January,

2012.

[5] J. B. Morton.	 ACA and Women Working in Corrections. Corrections

Today, 1 October 2005.

[6] F. DiMarino. Women as corrections professionals.	 Corrections.com, 06

July 2009.

[7] C. Harrison and K Robinette.	 CAESAR: Summary Statistics for the

Adult Population (Ages 18-65) of the United States of America. United

States Air Force Research Laboratory, AFRL-HE-WP-TR-2002-0170,

2002.

[8]	 Basic human body measurements for technological design. Part 1. Body

measurement definitions and landmarks., 2002. ISO 7250-1.

[9] ISO 7250-2.	 Basic human body measurements for technological design.

Part 2. Statistical summaries of body measurements from individual ISO

populations.

[10] J Hoke, editor.	 Architectural Graphic Standards. John Wiley and Sons,

Inc., 2000.

[11] R Shelasky and J Wilson.	 How to Measure Your Bra Size. Real

Simple, http://www.realsimple.com/beauty-fashion/clothing/shoppingguide/how-to-measure-bra-size-00100000077331/index.html, Retrieved

2012.

[12] Hadley Malcom.	 HerRoom’s universal sizing chart demystifies larger

bras. USA Today, 9 April 2012.

23
[13] Learn about sister sizing. Retrieved from: http://www.brasize.com/brasister-size.html, July 2013.

[14] Stab Reistant Armor STC. Small sampling of bust sizes for 103 female

correctional officers. Unpublished communication.

[15] Stab Reistant Armor STC. Meeting Minutes. Unpublished.

[16] G Catanuto, A Spano, Pennati A, Riggio E, Farinella G, Impoco G,

Spoto S, Gallo G, and Nava M. Experimental methodology for digital

breast shape analysis and objective surgical outcome evaluation. Journal

of Plastic, Reconstructive and Aesthetic Surgery, 61:314–318, 2008.

[17] G Farinella, G Impoco, G Gallo, S Spoto, and G Catanuto.	 Unambiguous Analysis of Woman Breast Shape for Plastic Surgery Outcome

Evaluation. In Eurographics Italian Chapter Conference, 2006.

[18]	 Merriam Webster Dictionary, online edition.

[19] Don McCunn. Underwire Charts. Design Enterprises of San Francisco.

[20] A Ringberg, Bageman E, Rose C, Ingvar C, and Jernstrom H. Of cup and

bra size: Reply to a prospective study of breast size and premenopausal

breast cancer incidence. Int. J. Cancer, 119:2242–2243, 2006.

[21] VL Katch, B Campaigne, P Freedson, S Sady, Katch F, and Behnke A.

Contribution of breast volume and weight to body fat distribution in

females. Am J Phys Anthropol, 53:93–100, 1980.

24