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Abstract or Summary 
 The Weibull theory, an attempt to account for the variability of
measured strength in brittle materials, is described in some detail.
An important and useful modification is derived, the normalized survival
probability distribution, given by the equation
[equation]
where S is the probability that the specimen will survive the fraction
[beta] of the mean strength, and m is ideally a material constant called
the Weibull modulus. The symbol [gamma] denotes the gamma function.
Some of the theoretical implications were investigated in a series
of strength tests made on tubular BeO specimens by pressurizing them
to failure. The results of these experiments were in good agreement
with the theory. The test apparatus developed for this study appears
to offer a useful addition to laboratory methods of measuring the tensile
strength of brittle materials.
Equation (1) is a simple analytic expression and can be used to
deduce many aspects of the strength behavior of brittle materials. For
example, the behavior of least values (that is, the strength of the weakest
specimen in a sample of size n) may be estimated by the formula
for the most probable strength of the weakest specimen. This least
strength [beta asterisk] is given by
[equation].
Since the behavior of the least values constitutes the engineering
limitation to the application of brittle materials, design must be based
on an understanding of least values. The possibilities of influencing
the least values by proof testing to eliminate the weak elements, or by
prestressing (bias), are examined. It appears that the benefits of
proof testing may be limited because mechanical damage may be induced
even at low stresses.
It is suggested that the safety margin be given as the "extreme"
safety factor, defined as the ratio of the most probable least strength
to the operational stress, which in terms of the requisite failure probability
F is given by
[equation].
Many of the useful formulas of the Weibull theory are approximated
by simple forms which may serve as practical rules of thumb.
Some general aspects of design philosophy are considered and a
number of design rules are proposed.
For purpose of general information, the variance of the Weibull
distribution and its least values are computed and plotted, demonstrating
the superior reliability of the extreme values and the interesting
doublevalued nature of standard deviation for certain combinations
of sample size n and Weibull modulus m. A table of stress
distributions and the corresponding "risks of rupture" is presented and
may be used in circumstances where a complex stress distribution is
to be approximated by more tractable forms.

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