Calculation of Reliability Using the Stress Strength Interference Model

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). Let’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 s₀ is given by:

$$ \displaystyle \begin{equation} P\left(S>s_{o}\right)=\int_{s_{o}}^{\infty}f_{s}\left(S\right)dS \end{equation} $$

The probability that the strength is greater than a certain stress s₀ is given by:

$$ \displaystyle \begin{equation} fs\left(s_{o}\right)ds\ast\int_{s_{o}}^{\infty}fs\left(S\right)dS \end{equation} $$

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. 

$$ \displaystyle \begin{equation} R=\int_{0}^{\infty}fs\left(S\right)\left(\int_{0}^{S}\left(s\right)ds\right)dS \end{equation} $$

Let’s consider an example where stress and strength are normal distributed. μ and σ refer to the mean and the standard deviation respectively.

• μ_stress = 50 MPa
• σ_stress = 15 MPa
• μ_strength = 75 MPa
• σ_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

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