# The factor of safety: stress-rupture modes

• A. D. S. Carter
Chapter

## Abstract

Before evaluating the factor of safety statistically, the reader is reminded of the obscurantism surrounding the factor discussed in Chapter 2. Assumptions will have to be made to remove ill-defined parameters before any quantification can be undertaken. Thus it will be assumed that a single role factor is to be evaluated. In doing so we are not restricting the approach in any way. Secondary functions of the factor can be introduced into the load and strength distributions as described in section 3.2 and treated simultaneously with the primary function, but these secondary functions would have to be defined. Second, it is necessary to define ‘nominal’. This is done by identifying any nominal value as lying knom standard deviations from the mean, where knom has to be chosen to agree with whatever definition of nominal is being used for the particular application in hand. It is then possible to write
$${L_{nom}} = \bar L + {k_{nomL}}{\sigma _L}$$
(5.1)
and
$${S_{nom}} = \bar S - {k_{nomS}}{\sigma _S}$$
(5.2)
from which it follows that
$${\phi _L} = \frac{{\bar L + {k_L}{\sigma _L}}}{{\bar L + {k_{nomL}}{\sigma _L}}} = \frac{{1 + {k_L}{\gamma _L}}}{{1 + {k_{nomL}}{\gamma _L}}}$$
(5.3)
$${\phi _S} = \frac{{\bar S - {k_s}{\sigma _s}}}{{\bar S - {k_{nomS}}{\sigma _L}}} = \frac{{1 - {k_S}{\gamma _S}}}{{1 - {k_{nomS}}{\gamma _S}}}$$
(5.4)
$$\phi = \frac{{{\phi _L}}}{{{\phi _S}}}$$
(5.5)
where γ L and γ S are the coefficients of variation of the load and strength respectively. It is thus very simple to express the factor of safety in terms of statistically defined load and strength distributions and use it in a worst-case design, provided of course that the parameters that define the design are completely known.

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### References

1. American Society of Metals (1960) Metals Handbook, Vol. 1, Properties and Selection, 8th edn.Google Scholar
2. Gordon, J.E. (1988) The New Science of Strong Materials: Or why you don’t fall through the floor, 2nd edn, Penguin Books, Harmondsworth.Google Scholar