The calculation of reliability consists of the comparison between stress and strength. If we consider stress and strength as deterministic quantities, failure occurs when the stress exceeds strength. In reality, stress and strength are stochastic variables (Figure 1a). If the two curves overlap (Figure 1b), failure may occur. Let us suppose that s and S are continuous random variables with probability density functions f(s) and f(S), respectively.

In what follows, the stress-strength interference model is explained according to work done by An-Min Lee (1). Letâ€™s have a closer look at the interference region (Figure 2).

The probability that a stress value lies in an interval of width ds is:

$$ \displaystyle \begin{equation} P[s_{0}-\frac{ds}{2}\le s\le s_{0}+\frac{ds}{2}]=f_{S}(s_{0})ds \end{equation} $$The probability that the strength is greater than a certain stress s_{0} is given by:

The probability that the strength is greater than a certain stress s_{0} is given by:

The reliability is the probability that the strength S is greater than the stress s for all possible values of the stress s.

$$ \displaystyle \begin{equation} R=\int_{0}^{\infty}fs\left(s\right)\left(\int_{s}^{\infty}\left(S\right)dS\right)ds \end{equation} $$The failure probability is calculated as:

$$ \displaystyle \begin{equation} F=1-R \end{equation} $$Alternatively, reliability is the probability that the stress s is lower than the strength S for all possible values of the strength S.

$$ \displaystyle \begin{equation} R=\int_{0}^{\infty}fs\left(S\right)\left(\int_{0}^{S}\left(s\right)ds\right)dS \end{equation} $$Letâ€™s consider an example where stress and strength are normal distributed. **Î¼** and **Ïƒ** refer to the mean and the standard deviation respectively.

- Î¼
_{stress}= 50 MPa - Ïƒ
_{stress}= 15 MPa - Î¼
_{strength}= 75 MPa - Ïƒ
_{strength}= 20 Mpa

According to eq. 4, the reliability **R** is calculated as the integral of the function $- fs\left(s\right)\left(\int_{s}^{\infty}\left(S\right)dS\right)ds -$. Since failure probability **F** is calculated as **F = 1- R, **F is calculated as the integral of $- fs\left(s\right)\left(\int_{0}^{S}fs\left(S\right)dS\right) -$. Interestingly, f_{unreliability} and f_{reliability} are not probability density functions because the area under these curves is smaller than 1.

The application of eq.(4) and eq.(5) deliver the following results:

- Failure probability: F = 0.159
- Reliability: R = 0.841

It is worthwhile noting that the area under the failure probability curve in Figure 2a does not correspond to the interference area. I just want to remark this point since you may find a lot illustrations in the web showing the failure probability as the interference area between stress and strength curves. Figure 4 shows that the value of interference area may significantly deviate from the correct one.

## References

- Lee, An-Min. “Stress-strength interference models in reliability.” (1989). (link)
- Haibach E (2006) Betriebsfestigkeit â€“ Verfahren und Daten zur Bauteilberechnung. Springer-Verlag, Berlin/Heidelberg. https://doi.org/10.1007/3-540-29364-7
- Reliawiki â€“ Reliasoft. Stress-Strength Analysis â€“ ReliaWiki

Bhavesh says

Understanding the delicate balance between stress and strength in materials is akin to choosing the proper phone case â€“ it’s not just about appearances but the intricate interplay between form and function. Just as stress and strength aren’t fixed values, a perfect phone case isn’t just about style; it’s about the reliability it offers amidst daily wear and tear, much like the intricate calculations described here. Remember, the right fit ensures your phone, like a well-calculated reliability model, stays secure in every situation.

https://acfootballcases.com/product/jordan-henderson-14-black-panzer/

Mitch F. says

Enrico, thanks for this reminder that even though the distributions overlap one needs to select a ‘poor’ strength ball and a ‘high’ stress ball to get a failure. One of the bigger takeaways of this for me is that even with modest unreliability, you may have considerably more strength/stress distribution overlap than one may have thought.

Enrico Belmonte says

Dear Mitch,

Thank you for your comment. Indeed it would be nice to see the relationship between interference area and reliability. I’m going to investigate it in a following post.

Kevin Walker says

Thank you, Enrico. I was one of the people with the misconception that the unreliability was the interference area. This clarified it for me.

Enrico Belmonte says

Great! I appreciate your feedback.

Steve Wachs says

I have found it useful to use the concept of the differences distribution (stress – strength) which is also normal if both the stress and strength distributions are normal. Then it’s very straight forward to simply calculate the area under this curve where the the differences distribution is negative. The math is very straightforward. Also, I like to show how simulation can be easily used to estimate the failure probability (especially if the underlying distributions are not both well described by a normal distribution.