7.3.2 Derating of Mechanical and
For mechanical and structural components, such failure rate
versus stress data may be obtainable from the manufacturer or users, but time
rate data may not be available. In using a manufacturer's rating and single
design stress values, the design engineer must keep in mind that they are
really distributions, not single values. Either the worst case “tolerances”
for both stress and strength or a plot of
the distributions must be utilized. When there is time dependency for the
distributions (e.g., degradation, wear out), the stress and strength
distributions must be related in the appropriate manner to the cyclic or time
operation in the intended environment.
The classical approach to
mechanical and structural design is to give every part enough strength to
handle the worst stress it will encounter. Several references, such as
MIL-HDBK-5 are available, providing data on the strength of materials. Some of
these provide limited data on strength degradation with time, resulting from
fatigue. Effective design procedures should provide for evaluating alternative
configurations with respect to reliability. Since failure is not always
related to time, the designer needs techniques for comparing stress vs.
strength, and determining the quantitative reliability measure of the design.
The traditional use of safety factors and safety margins is inadequate for
providing a reliability measure of design integrity. The concept of stress
strength in design recognizes the reality that loads or stresses and strengths
of particular items subjected to these stresses cannot be identified as a
specific value but have ranges of values with a probability of occurrence
associated with each value in the range. The ranges of values (variables) may
be described with appropriate statistical distributions for the item.
Stress/strength design requires knowledge of these distributions. After the
strength and stress distributions are determined, a probabilistic approach can
be used to calculate the quantitative reliability measure of the design,
including confidence limits.
To illustrate the concept of stress
and strength distributions related to reliable design, assume that a large
number of tests of the strength of a given manufactured item have been run,
with each test being run to failure. A relationship (frequency distribution)
between the number failing at any particular value of strength (or band of
values) and the value can be determined. Figure 7.3-1(a) shows a generalized
frequency distribution of the results. If the exact relationship were known,
the probability of a randomly selected specimen failing at a particular value
of stress F’ could be predicted. It would be that fraction of the population,
whose strength was equal to or less than a stress F’. Similarly if a large
number of experiments were conducted, and the stress was recorded on each
experiment, a relationship between the relative frequency of stresses and the
stress can be established. This relationship is shown in Figure 7.3-1(b). If
the exact relationship were known, the probability that on any randomly
selected trial the stress would exceed a strength S’ could be predicted. This
would be the fraction of the population (of possible trials) in which the
stress exceeded the strength S’. With both of these distributions defined,
unreliability is determined as the probability that the stress is greater than
the strength. Unreliability can be determined analytically, graphically, by
numerical integration or by probabilistic techniques such as "Monte Carlo"
provided the form or shape of the two probability distribution functions are
known. The curves from Figure 7.3-1(a) and 7.3-1(b) are combined in Figure
7.3-1(c) to illustrate the region of the unreliability given by the shaded
area where stress exceeds strength. Figure 7.3-2 illustrates normal (gaussian)
stress and strength distributions, where the stress and strength variables are
identified as Kips (a thousand pounds).
Looking at Figure 7.3-2, two things
may happen with time and repeated stress. The variance of the strength
distribution may change; for example the curve may extend from 13 to 23 Kips
rather than the original 16 to 20 Kips. This would result in an increased
unreliability since the shaded area would now extend from 13 to 20 Kips. This
is shown in Figure 7.3-3(a). The other factor that could change with time and
stress is that the mean of the strength distribution might be lowered, to say
15 Kips. This, in turn, would result in a decreased reliability as shown by
the shaded area of Figure 7.3-3(b).
The purpose of stress strength
analysis is to improve the reliability of the design. That is, to find the
optimum comparison of stress and strength that will have an acceptable
probability of success and compete favorably with other constraints such as
weight, cost, and availability of material.
There are four basic procedures the
designer may use to increase reliability.
||Increase Average Strength: This
approach is tolerable if size, weight, and cost increases can be
accepted or if a stronger material is available.|
||Decrease Average Stress:
Occasionally the average allowable stress on a component can be
reduced without greatly affecting its performance.|
||Decrease Stress Variation: The
variation in stress is usually hard to control. However, the stress
distribution can be effectively truncated by putting limitations on
||Decrease Strength Variation: The
inherent part-to-part variation in strength can be reduced by
improving the basic process, holding tighter control over the process,
or by utilizing tests to eliminate the less desirable
References , 
and  provide more details on this procedure and its application to
mechanical and structural components.