Page 43 - Shigley's Mechanical Engineering Design

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18 Mechanical Engineering Design
The stress and strength terms in Eq. (1–3) must be of the same type and units. Also, the
stress and strength must apply to the same critical location in the part.

EXAMPLE 1–2 A rod with a cross-sectional area of A and loaded in tension with an axial force of P
2000 lbf undergoes a stress of σ = P/A. Using a material strength of 24 kpsi and a
design factor of 3.0, determine the minimum diameter of a solid circular rod. Using
Table A–17, select a preferred fractional diameter and determine the rod’s factor of safety.

Solution Since A = πd /4,σ = P/A, and from Eq. (1–3), σ = S/n d , then
2
P P S
σ = = =
2
A πd /4 n d
Solving for d yields
1/2 1/2
4(2000)3
Answer d = 4Pn d = = 0.564 in
πS π(24 000)
5
From Table A–17, the next higher preferred size is in 0.625 in. Thus, when n d is
8
replaced with n in the equation developed above, the factor of safety n is
πSd 2 π(24 000)0.625 2
Answer n = = = 3.68
4P 4(2000)
Thus rounding the diameter has increased the actual design factor.

1–12 Reliability
In these days of greatly increasing numbers of liability lawsuits and the need to conform to
regulations issued by governmental agencies such as EPA and OSHA, it is very important
for the designer and the manufacturer to know the reliability of their product. The reliabil-
ity method of design is one in which we obtain the distribution of stresses and the distribu-
tion of strengths and then relate these two in order to achieve an acceptable success rate.
The statistical measure of the probability that a mechanical element will not fail in
use is called the reliability of that element. The reliability R can be expressed by
(1–4)
R = 1 p f
where p f is the probability of failure, given by the number of instances of failures per
total number of possible instances. The value of R falls in the range 0 R 1. A reli-
ability of R = 0.90 means that there is a 90 percent chance that the part will perform
its proper function without failure. The failure of 6 parts out of every 1000 manufactured
might be considered an acceptable failure rate for a certain class of products. This rep-
resents a reliability of
6
R = 1 − = 0.994
1000
or 99.4 percent.
In the reliability method of design, the designer’s task is to make a judicious selec-
tion of materials, processes, and geometry (size) so as to achieve a specific reliability