On Carter’s statistical design method for mechanical components
Hussam Ahmed and P.J. Clark
Heriot Watt University, School of the Built Environment, Edinburgh, UK.
Brief abstract
Carter’s statistical design method for mechanical components [1,2] is based on
meeting a specified hazard function threshold value due to repeated loads in a
structural reliability model. This paper calculates several design curves for use in
Carter’s design method. In particular design curves are presented for normally
distributed strength and load, and for combinations of Weibull probability
distributions. The design curves depend on hazard function values of 10-9, 10-8, and
10-7. The design method is illustrated by application to some simple mechanical
design problems.
Extended abstract
The use of structural reliability theory to design simple mechanical components such
of FOSM methods, based on the concept of a safety index, β, is well developed in the
reliability literature [1]. By simple mechanical components we mean bolts, connecting
rods, drive shafts, etc. This model ensures that the probability of failure of the
component due to the application of a single load lies below a tolerable threshold.
However, if quality control measures ensure that the manufactured components are of
adequate strength, and assuming that the components retain their strength during their
service life, then component failure is due to stress rupture caused by an excessive
external load. In this case the reliability of the component is characterized by a
constant failure rate or hazard function per load, h(n), defined by Carter [1,2] as:
n
s

R(n)   S ( s )   L( ) d  ds
0
0

R(n)  R(n  1)
h( n) 
R ( n)

where R(n) is the reliability after n loads, S(s) is the probability density function of the
strength of the component with respect to the random variable s, a measure of the
strength, and L(ξ) is the probability density function of the external load with respect
to the random variable ξ, the magnitude of load.
The hazard function acquires a constant value after about 20-100 load cycles, and so
we are considering components that should withstand many more load cycles than this
during their useful life. A component is described as “intrinsically reliable” if the
hazard function has a threshold value of 10-9. In Carter’s “statistical design method”
the component is designed to meet this hazard function value rather than a specified
reliability. Such components are suitable for high reliability applications in, for
example, aviation, defence, medicine, and safety systems.
The statistical parameters defining normally distributed strength and load are the
mean and standard deviation of strength ( S , σS) and load ( L , σL) respectively,
though other probability distributions may be used in some applications. In the design
problem the load statistics are known, but the strength statistics are not. Instead the
designer specifies the coefficient of variance of the strength γS=σS/ S (similarly γL).
He is able to determine the resisting strength of the component from the safety margin
SM and the mean load. Carter establishes a relationship between the safety margin
and a parameter called load roughness LR:
LR 
L
 
2
L
2
S
SM 
S L
 
2
S
2
L

1  LR 2
S

LR
L
Carter’s statistical design curve is obtained by finding the values of LR and SM
satisfying the equation for the specified hazard function. The resulting curves for the
normal distribution for selected values of hazard function are illustrated in the figure:
Similar curves have been derived for the Weibull distribution with different shape
parameters. These curves are new. Some examples of application of this method to
design of simple components are treated in the paper. Although Carter’s method is
described here in relation to simple components, more elaborate structures with welldefined failure modes and repetitive loading mechanisms (e.g. stiffened panels or
shells) can be designed using the same ideas.
References
[1] A.D.S.Carter, Mechanical Reliability and Design, Macmillan, 1997.
[2] A.D.S.Carter, Reliability and Design, Macmillan, 1986.