Reliability from stress-strength interference

(a) Stress-strength interference

In probabilistic design, stress $s$ and strength $S$ are random variables with distributions. Failure occurs when stress exceeds strength. The region where the two distribution curves overlap is the interference; its area is related to the probability of failure.

 f
 |   stress     strength
 |    /\          /\
 |   /  \        /  \
 |  /    \__/\__/    \
 +-------------------------> MPa
       overlap = interference

Reliability $R = P(S > s)$. A larger gap between the means or smaller scatter gives higher reliability. The factor of safety based on means alone does not show this.

(b) Numerical

For normal variables the margin $M = S - s$ is also normal:

$$\mu_M = 350 - 250 = 100\ \text{MPa}$$ $$\sigma_M = \sqrt{30^2 + 25^2} = \sqrt{1525} = 39.05\ \text{MPa}$$

Standard normal variate (safety index):

$$z = \frac{\mu_M}{\sigma_M} = \frac{100}{39.05} = 2.56$$

From the normal table $\Phi(2.56) = 0.9948$, so

$$R = 0.9948,\qquad P_f = 1 - R = 0.0052\ (\text{about } 1 \text{ in } 190)$$

Mean strength for $R = 0.999$: $z = 3.09$.

$$\mu_S - 250 = 3.09 \times 39.05 = 120.7 \;\Rightarrow\; \mu_S = 370.7\ \text{MPa}$$

Answer: $R = 0.995$, $P_f = 0.0052$; required mean strength about 371 MPa.