Mechanical Stress/Strength Interference Theory

Often when evaluating the reliability of a part subjected to mechanical loading, the probability of failure under operating conditions is of interest. A useful method for this purpose is Mechanical Stress/Strength Interference Analysis. The method uses probability distributions of strength and stress and graphs to identify the conditions under which a part will fail, as well as the probability of these conditions.

Most parts under mechanical loading in normal operating conditions experience naturally scattered stress. These stresses can be represented probabilistically in a distribution graph as shown in Figure 1. Logically, the more extreme minimum and maximum stress levels are not as probable as values closer to the mean. The stress distribution can be determined from actual stress measurement or simulated stress measurement using finite element analysis. An extensive record of actual measured stresses on the component is useful in determining the stress distribution. Most often, a normal distribution is assumed for the stress distribution for purposes of simplification. The accuracy of the assigned stress distribution greatly affects the precision of the stress/strength interference analysis.

Figure 1. The Natural Variation of Stress Under Normal Operating Conditions can be Represented in a Probability Distribution Graph (Click to Zoom)

The strength distribution is now determined. A part may also undergo variances in strength during its use showing some type of probability distribution as does stress. Variances in strength may be caused by variances in manufacturing processes, raw materials, etc. Certain data items are required to find the proper strength distribution. The properties and conditions listed in Table 1 should be determined before the analysis.

Table 1. Properties and Conditions Needed for Strength Distribution
 · Alloy · Design life · Loading type · Surface finish · Heat treatment · Operating temperature · Stress concentration factor(s)

Strength distributions can be derived from references such as RADC-TR-66-710 and RADC-TR-68-403, or by using test data to determine a distribution curve to match (Weibull, normal, etc.). Figure 2 is a typical probability graph of the strength distribution of a component under mechanical loading.

Figure 2. The Natural Variation of Strength can be Represented in a Probability Distribution Graph (Click to Zoom)

Separately the stress and strength graphs tell nothing of the reliability of the component in its operating environment. Only when superimposed upon each other do they give meaning to the analysis, as shown in Figure 3.

Figure 3. Superimposing the Stress Graph on the Strength Graph Shows the Stress/Strength Interference (Click to Zoom)

Stress and strength are treated as random variables in this analysis. The area where the two distributions intersect is the stress/ strength interference. It is in this interference area that a mechanical component will fail. Failure occurs whenever a randomly selected stress value from the stress distribution exceeds a randomly selected strength value from the strength distribution.

Mathematically, failure is defined as the condition when Stress > Strength. Evaluated properly, the area of interference represents the probability of failure or P (Stress > Strength). Accordingly, since interference represents probability of failure, then the reliability is expressed as R = 1 - interference.

Logically, the smaller the interference region is, the more reliable the component. To increase reliability, either the mean stress and strength values must have a greater difference (the two plots move away from each other horizontally) or the standard deviations must be decreased (the two plots cover less horizontal area). It is obvious from the graphs that under normal operating conditions the component would usually not fail because the mean strength is significantly greater than the mean stress. Failure will only occur when extreme values of stress and strength occur at the same time representing a point within the interference region.

The simplest example for calculating interference would be for a normal distribution of both stress and strength. For a calculation of this type the following formula is used

The area of interference is determined from tabulated values of `z' (available in probability texts). Mathematical formulas and parameters vary from case to case, depending on the type of strength distribution used. The strength distributions used vary greatly but, unless actual stress data is available, usually a normal distribution is assumed for mechanical stress.

For an unusual distribution of strength or stress, a Monte-Carlo Technique can be used to randomly select a sample from each distribution and compare them. After hundreds or thousands of samples have been simulated, the probability of failure can be estimated from the results.

Interference theory has other engineering applications. For example, it may be used to analyze two parts that must, when assembled, fit together tightly. Variances in the geometry of the parts, such as length, width, thickness, and other physical features, could cause interference fit, where the parts would not be compatible with each other. Distribution formulas can also be used to determine the probability that a part will be manufactured to a certain size.

Example. An F-14 jet missile hook is used to secure a 500-lb Sparrow missile to the aircraft until it is fired. Due to several mechanical failures of this part, an investigation was performed in 1989 to determine a solution for the problem. The stress and strength data produced from this investigation can be used to perform a calculation of interference and reliability. A tensile strength test was performed on the missile hook and several readings of strength were recorded in the report. The strength distribution chart is shown in Figure 4. Using these readings, a mean strength of 280.6 ksi and standard deviation of 8.98 ksi were computed assuming a normal distribution.

Figure 4. Strength Distribution for Example (Click to Zoom)

Next the stress distribution is determined. For this example, only average operational stress was available, so a standard deviation was assumed. To assess the effects of this assumption, three different deviations were used in the calculations. The formulas used are consistent with those found in Section 2.1 of RADC-TR66-710. Values of `z' were calculated using the mean stress and strength variances. Then the interference was interpolated using the calculated values of z and Table 2.1 of RADC-TR-66-710.

Table 2 summarizes the calculations.

Table 2. Summary of Calculations for Example
Stress Deviation (as % of mean) Stress Deviation z Interference (R)* Reliability (1-R)
25 40.1347 2.92 0.00175 0.99825
30 48.1616 2.45 0.00714 0.99286
35 56.1886 2.11 0.0179 0.9821