In simplest terms, an item fails when a stress to which it is subjected exceeds the corresponding strength. In this sense, strength can be viewed as “resistance to failure.” Good design practice is such that the strength is always greater than the expected stress. The safety factor “η” can be defined in terms of strength and stress as follows:

**Figure 1: Stress and Strength Distribution Example**

If stress and strength are normally distributed random variables and are independent of each other, then the standard normal distribution and Z tables can be used to “quantitatively” determine the probability of failure. First, the Z-statistic is calculated as follows:

_{S}” is the mean of the strength distribution, “μ

_{sapp}” is the mean of the stress distribution, “σ

_{S}

^{2}” is the variance of the strength distribution and “σ

_{sapp}

^{2}” is the variance of the stress distribution. Using the Z value table for a standard normal distribution, the area above the calculated Z-statistic is the probability of failure.

If stress and strength are not normally distributed, other techniques (such as Monte Carlo simulation) may be used to determine the probability of failure.

Calculating the probability of failure using the distributions shown in Figure 1 and the related data we have:

The Interference Stress/Strength Analysis calculator in the Quanterion Automated Reliability Toolkit – Enhanced Reliability (QuART-ER) tool can be used to perform the calculations. The mean and standard deviation of the stress and strength distributions are entered in the appropriate fields and the resulting probability of failure and reliability are displayed as shown in Figure 2.

**Figure 2: QuART-ER Interference Stress/Strength Analysis Calculator**

For example, suppose it was possible through careful control of manufacturing processes to reduce the standard deviation of the component’s strength from 400 psi to 200 psi, with the other parameters remaining the same. The stress and strength curves would then be as shown in Figure 3.

**Figure 3: Stress and Strength Distribution Example – Reduced Strength Variability**

**Summary**

A brief overview of Interference Stress/Strength Analysis has been provided here, along with sample calculations using Quanterion’s QuART-ER Stress/Strength Calculator. This topic is also covered in the Quanterion-authored Reliability Information Analysis Center (RIAC) publication “System Reliability Toolkit” and is included in the “Reliability-101” training program.