Content uploaded by Elart von Collani
Author content
All content in this area was uploaded by Elart von Collani on Nov 11, 2014
Content may be subject to copyright.
c
Heldermann Verlag Economic Quality Control
ISSN 0940-5151 Vol 17 (2002), No. 1, 113 – 122
Modelling of Explosives Sensitivity
Part 1: The Bruceton Method
Roland Wild and Elart von Collani
Abstract: Considerable work has been done to investigate the sensitivity of explosive materi-
als. Various methods are developed for determining statistically the “explosion characteristics”.
Any statistical method is based on a stochastic model and, clearly, an inappropriate model
yields generally meaningless results, independently of the applied method. The most commonly
applied method in the field of explosive material is known as Bruceton method. In this paper
the Bruceton method and the underlying model are analyzed critically and the conclusion is
drawn that the Bruceton method cannot be recommended for determining the sensitivity of
explosive material. In a subsequent paper a different approach to solve the problem will be
suggested.
1 Introduction
Scientific investigation of real phenomena yielding quantified statements must necessarily
be based on a mathematical model of the aspects of interest. Mathematical modelling
consists of translating known information about the behavior of the phenomenon into
mathematical relations, e.g., in equations. In the following the proceeding is illustrated
by means of explosive sensitivity.
The first question which has to be addressed is how sensitivity of an explosive material
shall be described and the next how the actual sensitivity of a given material shall be
determined.
The sensitivity of an explosive material shall reflect its tendency to explode under stress.
Thus, for describing sensitivity one has to include a variable “stress” and a function
“tendency” defined on the stress. On the basis of stress and tendency a random experiment
and an estimating procedures must be developed for determining the unknown actual
values of the material in question.
The stress is given by energy and the amount of stress by the applied energy level. The
explosion tendency of a material is appropriately modelled by the probability to explode,
if it is exposed to a certain stress level. There is no tendency, if the stress falls below
a lower limit and there might be a maximum explosion probability if the stress level
exceeds an upper limit. Thus, the lower and upper limit and the run of the probability
curve between these two limits are of interest.
114 Roland Wild and Elart von Collani
2 Modelling
The stress level E may be measured in Joule (or any other unit) and its actual value is
denoted by e ∈ IR
+
0
. Each time the material is exposed to the stress level e it may explode
or not, i.e., has an outcome which occurs at random. For describing the future outcome
of the experiment a random variable is needed whose probability distribution depends on
e. Therefore, the random variable is denoted by X|e and defined by
X|e =
1 if an explosion occurs
0 if no explosion occurs
(1)
X|e is a simple Bernoulli variable with the following probability distribution:
P
X|e
{1}
= p(e)=1− P
X|e
{0}
(2)
The explosion probability p(e) as a function of the variable E with actual value e describes
the tendency, i.e., sensitivity of the material to explode.
Next, relevant features of p(e) have to be formulated:
• p(e)=p
min
for e<e
(3)
• p(e)=p
max
for e>e
u
(4)
• 0 ≤ p(e) ≤ p
max
for e
≤ e ≤ e
u
(5)
Very often, we have p
min
=0andp
max
= 1 and from experience there are rather good
information about the two limits e
and e
u
.
Moreover, the explosion tendency and, hence, the explosion probability is monotonously
increasing with the exposed stress:
• e
1
<e
2
⇒ p(e
1
) ≤ p(e
2
)(6)
For investigating the sensitivity of an explosive material at energy level e
0
, one would
define a sample of size n by
X
1
|e
0
,...,X
n
|e
0
(7)
and assume that (7) consists of independent and identically distributed Bernoulli vari-
ables. The underlying assumption is that the specimen of the explosive material exhibit a
more or less identical sensitivity. Consequently, the random process starts with exposing
the specimen to stress under identical conditions. The fact that sometimes the outcome
is a reaction and sometime no reaction is due to omnipresent randomness.
3 The Bruceton Method
3.1 The Model
The model used as basis of the Bruceton method differs greatly from the approach outlined
in the previous section. The approach described e.g. in [5] uses the Bernoulli experiments
Modelling of Explosives Sensitivity 115
only implicitly. Instead of regarding the Bernoulli variables X|e, a quantity called critical
energy level is introduced and a random process is regarded assigning to each fictitious
specimen of explosive material one specific critical energy level.
Let Ω denote the set of all fictitious specimen and e
ω
be the critical energy level of
specimen ω ∈ Ω. The assigning process is defined by a random variable, say ξ with
ξ :Ω→ IR
+
(8)
For a given specimen ω with critical energy level e
ω
the Bernoulli variable (2) degenerates:
P
X(e)
({1})=
1ife ≥ e
ω
0ife<e
ω
(9)
Next, it is assumed that ξ or a known function of ξ has a normal distribution N (µ, σ
2
).
Consequently,
F
ξ
(e)=
1
√
2πσ
2
e
−∞
e
−
(e−µ)
2
2σ
2
dx (10)
gives the probability that the critical energy level assigned to the specimen in question is
less than or equal to e.Ofcourse,µ is the expectation of ξ and σ its variance.
Next consider the two-stage experiment of firstly assigning a critical energy level to spec-
imen ω and then exposing this specimen to an energy level e. The event of explosion is
equivalent to the event of assigning a critical energy level to ω which is less or equal to e.
Hence, we obtain:
P
X|e
({1})=p(e)=F
ξ
(e)=
1
√
2πσ
2
e
−∞
e
−
(e−µ)
2
2σ
2
dx (11)
The approach used as a basis of the Bruceton method differs fundamentally from the one
described in the previous section. The explosion experiments are thought to be purely
deterministic. Randomness comes in when assigning the critical energy levels to each
specimen. In contrast to the previous model which assumes more or less homogenous
specimen, the actual model assumes comparatively large inhomogeneity among the spec-
imens.
However, one should note that either approach leads to a function p(e) which can be in-
terpreted as explosion probability. We conclude that the Bruceton up-and-down method,
as described in [5], is based on an explosion probability p(e) which is given by the normal
distribution integral. The parameter µ can be identified as the 50%-point of p(e), i.e.:
p(e
0.5
)=0.5=p(µ) (12)
and the parameter σ
2
determines how fast the explosion probability increases. A small σ
2
means a steep rise of p(e) and a large value of σ
2
a slow rise. The latter effect is illustrated
by Figure 1 (small σ
2
) and Figure 2 (large σ
2
).
As to the features listed in the previous section, the model (10) assumed for the Bruceton
method meets the requirements (5) and (6), but violates (3) and (4).
116 Roland Wild and Elart von Collani
The run of the explosion probability p(e) as function of the energy level e is illustrated in
Figure 1 and 2.
5
10 15 20 25 30
e
0.2
0.4
0.6
0.8
1
explosion probability
Figure 1: The explosion probability (Bruceton method) for µ =10andσ
2
=1.
5
10 15 20 25 30
e
0.2
0.4
0.6
0.8
1
explosion probability
Figure 1: The explosion probability (Bruceton method) for µ =10andσ
2
=6.
Evidently, model (10) should only be applied,
• if it is known that the explosion probability has a symmetric S-shaped course with
symmetry-center at (µ, 0.5) and
• if the analysis does not refer to small or large values of p(e).
Thus, if the behavior of the explosive material is of importance for small values of p(e)
(which is always true in the case of explosives), model (10) is worthless, because require-
ment (3) is not met and because the assumed model is not capable of reflecting sufficiently
well a possible steep (concave) rise of the explosion probability in an neighborhood of e
.
Modelling of Explosives Sensitivity 117
Moreover, if it is to be expected that the explosion probability is not symmetric with
respect to (µ, 0.5), then again it should not be applied, as the conclusions drawn could be
wrong.
Because of the limited usefulness of (10), it is proposed in [5] to use transformations
h(e) instead of e aiming in coming closer to a symmetric S-shaped explosion probability
p(h(e)). Particularly, the use of the transformation y =log(e) is recommended and
investigated.
However, a reasonable selection of the transformation must be based on some knowledge
about the actual course of the explosion characteristic p(e). But, if there are information
available then one could use them directly for developing a more realistic model p(e)than
to try to find a model by rather obscure means. The transformation itself represents
another potential source of error without guaranteeing any essential advantages.
3.2 The Method
Following [5] we assume that the original energy level e is transformed into y =loge and
the explosion probability refers to p(y).
The Bruceton method aims at determining the explosion probability p(y) by estimating
the actual values of the two parameters µ and σ
2
of model (10) based on a sample which
is defined by a sampling plan, i.e., an instruction how many experiments have to be made
on which energy levels.
The Bruceton method refers to a so-called adaptive sampling plan, where each subsequent
experiment is determined by the outcome of the actual experiment. First, a starting
energy level, say y
0
=loge
0
, and the total sample size N are chosen.
Next, a sample must be defined. To this end consider the random variable
Y
i
=
1 if the experiment on trensformed energy level y
i
yields a reaction
0 if the experiment on transformed energy level y
i
yields no reaction
(13)
with p
i
=Φ
y
i
−µ
σ
being the probability of a reaction and 1 − p
i
the probability of no
reaction.
Assume that the following energy levels are fixed:
...,y
−k
,y
−k+1
,...,y
−1
,y
0
,y
1
,...,y
k
,y
k+1
,... (14)
with
y
−k
= y
0
− kd and y
k
= y
0
+ kd (15)
and d suitably chosen.
Then a Bruceton sample of size N is defined in the following way:
Y =
Y
i
0
,Y
i
1
,...,Y
i
N−1
(16)
118 Roland Wild and Elart von Collani
where
i
0
=0 and i
j
= i
j−1
+(−1)
Y
i
j−1
for j =1,...,N − 1 (17)
Note that each sample element depends on the outcome of the previous sample element.
The sampling plan given by
Y (16) has the following meaning:
• Start with an experiment on level y
0
.
• If a reaction is observed (Y
0
= 1), then perform the second experiment on level
y
0+(−1)
1
= y
−1
. If no reaction is observed (Y
0
= 0), then perform the second experi-
ment on level y
0+(−1)
0
= y
1
.
• Let the experiment number j − 1, j =1,...,N − 2, be performed on level y
i
j−1
.
If a reaction is observed (Y
i
j−1
= 1), then perform experiment number j on level
y
i
j−1
+(−1)
1
= y
i
j−1
−1
. If no reaction is observed (Y
i
j−1
= 0), then perform experiment
number j on level y
i
j−1
+(−1)
0
= y
i
j−1
+1
.
In other words, in case of a reaction the energy level for the next experiment is decreased
by d, and in case of no reaction the level is increased by d.
Define
Z
i
(Y
i
j
)=i
j
(18)
then the sequence of random variables Z
i
, i =0, 1,...,N − 1 defines a finite Markov
chain. For k
1
,k
2
=0, ±1, ±2,..., the transition probabilities are given by
P (Z
i+1
= k
2
|Z
i
= k
1
)=
1 − p
k
1
for k
2
= k
1
+1
p
k
1
for k
2
= k
1
− 1
0otherwise
(19)
and the initial state probabilities by
P (Z
i
= k)=
1 for k =0
0 for k = ±1, ±2,...
(20)
For illustration, consider a Bruceton sample (16) of size N = 5 and the following event
with respect to the 5-dimensional random vector
Y :
E
Y
= {(1, 1, 0, 1, 0)} (21)
The event E
Y
with respect to
Y corresponds to the following event with respect to
Z
E
Z
= {(0, −1, −2, −1, −2, −1)} (22)
The probability for E
Y
or likewise for E
Z
is given by:
P
Y
(E
Y
)=p
0
p
−1
(1 − p
−2
)p
−1
(1 − p
−2
)
=Φ
y
0
− µ
σ
Φ
y
−1
− µ
σ
2
1 − Φ
y
−2
− µ
σ
2
(23)
Modelling of Explosives Sensitivity 119
Consider a Bruceton sample
Y of size N with N
i
being the number of reactions and M
i
the
number of no reactions at level y
i
implying that N =
i
(N
i
+ M
i
), where the summation
goes over all possible levels y
i
. Note that the number of possible energy levels is bounded
by the sample size.
The log-likelihood function for µ and σ is given by:
log L(
Y )=
i
Φ
y
i
− µ
σ
N
i
+
i
1 − Φ
y
i
− µ
σ
M
i
(24)
The log-likelihood equations for µ and σ are obtained by equating the derivatives of
log L(
Y ) with respect to µ and σ with zero.
i
N
i
φ
y
i
−µ
σ
Φ
y
i
−µ
σ
− M
i
φ
y
i
−µ
σ
1 − Φ
y
i
−µ
σ
= 0 (25)
i
N
i
(y
i
− µ)
φ
y
i
−µ
σ
Φ
y
i
−µ
σ
− M
i
(y
i
− µ)
φ
y
i
−µ
σ
1 − Φ
y
i
−µ
σ
= 0 (26)
where
...,(N
−k
,M
−k
), (N
−k+1
,M
−k+1
),...,(N
o
,M
o
),...,(N
k
,M
k
), (N
k+1
,M
−k+1
),...
are the random numbers of the occurrences of reactions and no reactions at the trans-
formed energy levels y
i
, i =0, ±1, ±2,....
For determining a maximum-likelihood estimator for (µ, σ
2
) the two equations (25) and
(26) must be solved simultaneously for µ and σ
2
. Note that this is a purely numerical
problem. If not the normal integral but a different explosion probability would be selected,
the numerical problem would change, but the stochastic problem would be in principle
the same, namely to determine the actual values of the distribution parameters of the
assumed model by means of a sample.
The maximum-likelihood estimators of µ and σ
2
are functions of the random numbers N
i
and M
i
. The joint probability distribution of the N
i
and M
i
has been used for specifying
the log-likelihood function (24). Note that N
i
and M
i
are conditional random variables
under the condition of y
i
and that the normal integral is used only for parameterizing the
probability of the corresponding Bernoulli experiments.
Because the random variables (N
i
and M
i
) linked to the estimators for µ and σ are not
normally distributed, it is not at all justified to perform statistical analysis using methods
based on the normal distribution. Unfortunately this is done in data analysis referring to
the Bruceton method.
120 Roland Wild and Elart von Collani
4 Conclusion
In sensitivity analysis of explosive materials, specimen of the same explosive mixture
are exposed to energy and observed whether or not a reaction takes place. Evidently
each of these experiments is appropriately modelled by a Bernoulli variable, adopting
either the value 1 with a probability depending on the applied energy or the value 0 with
complementary probability.
Consequently, any series of such experiments is appropriately modelled by a sequence of
Bernoulli variables and these are the only random variables which can be analyzed and
exploited.
The aim of performing such experiments is generally to determine the unknown probability
of reaction or explosion p as a function of the energy level e. It makes sense to use a
parametric representation of p(e) and to determine the actual values by means of the
outcomes of the Bernoulli experiments. In order to obtain an appropriate estimate of the
explosion probability function p(e) it is of utmost importance to observe p(e) for energy
levels e covering the whole interval [e
,e
u
].
The approach used by the Bruceton method is based on the normal integral:
p(e)=
1
√
2πσ
2
e
−∞
e
−
(e−µ)
2
2σ
2
dx (27)
which violates two of the four necessary conditions. More over, the normal integral is
rather inflexible. To overcome this inflexibility additional transformations have to be
used leading to an increased complexity of the model and the subsequent analysis.
Besides these more strategic weaknesses, the indirect modelling approach used for ana-
lyzing Bruceton samples leads to some additional technical shortcomings.
• The Bruceton method concentrates on energy levels with explosion probability p(e)
close to 0.5 and does not yield any observations for small or large energy levels.
Thus, any implication referring to energy levels wich are not in a neighborhood of
e
0.5
with p(e
0.5
)=0.5 can be looked upon as a matter of faith without almost any
rational basement.
• It is numerically difficult to obtain the solutions of the log-likelihood equations
(25) and (26). The recommended approximate solutions are based on a number of
simplifications making a quality assessment of the resulting estimators more or less
impossible.
• The derivations of confidence intervals (see e.g. [5]) are based on the wrong assump-
tion that the applied (observed) energy levels visited during the up-and-down exper-
iments are the realizations of normally distributed random variables and, therefore,
the corresponding statements are doubtful.
Modelling of Explosives Sensitivity 121
In all it occurs that the up-and-down method may have merits for determining the ac-
tual value e
0.5
of explosive material, but cannot be used for making any statement on
the explosion probability p(e) as function of the energy level e. More over, the analysis
of the estimator for e
0.5
as described in literature is based on too many simplifications
and, therefore, doubtful. If the Bruceton method is good for determining e
0.5
, but not
appropriate for determining the function p(e), then the normal model for p(e) is unneces-
sary and can be omitted. Abandoning the normal integral would, of course, result in an
entirely different estimator for e
0.5
.
In summary we conclude that from a scientific and even more from a practical point
of view the Bruceton method cannot be recommended for studying the sensitivity of
explosive material.
References
[1] E. v. Collani, K. Dr¨ager (2001): Binomial Distribution Handbook for Applied Sci-
entists and Engineers.Birkh¨auser, Boston.
[2] v. Collani, E., Dumitrescu, M. & Lepenis, R. (2001): Neyman Measurement and
Prediction Procedures. Economic Quality Control 16, 109-132.
[3] v. Collani, E., M. Dumitrescu, (1999): Complete Neyman Measurement Procedure.
Research Report No. 85, W¨urzburg Research Group on Quality Control.
[4] v. Collani, E., R. Wild, (1999): Statistische Untersuchungen bei Go/NoGo-
Experimenten. Bericht Nr. 98/Y0038/50003, Wehrwissenschaftliches Institut f¨ur
Werk-, Explosiv- und Betriebsstoffe.
[5] W. J. Dixon & A. M. Mood (1948): A Method for Obtaining and Analyzing Sensi-
tivity Data. JASA 43, 109-126.
[6] H. J. Langlie (1962): A Reliability Test Method For “One-Shot” Items. Publication
No. U-1792, AERONUTRONIC, Newport Beach, California.
[7] C.-O. Leiber (1993): Zur Zuverl¨assigkeit von Testaussagen - Anz¨undsverzugszeit
und Fallhammerwerte. Sprengstoffe Pyrotechnik 30, 16 - 34.
[8] E. Strømsøe (1992): Prediction of Extremely Low and High Explosion Frequencies
from Sensitiveness Tests. Propellants, Explosives, Pyrotechnics 17, 295 - 297.
[9] Bundesinstitut f¨ur chemisch-technische Untersuchungen (BICT) (1980): AC 225,
Panel IV, Subpanel 2. Final Report: NATO Round-Robin Test on High Explosives.
Az.: 2.2-9/5154/80.
122 Roland Wild and Elart von Collani
Roland Wild
Wehrwissenschaftliches Institut f¨ur Werk-, Explosiv- und Betriebsstoffe (WIWEB)
Großes Cent
D-53912 Swisttal
RolandWild@bundeswehr.org
Elart von Collani
University of W¨urzburg
Sanderring 2
D-97070 W¨urzburg
collani@mathematik.uni-wuerzburg.de
