Template:Stress-Strength Expected Probability

The Expected Probability

Stress-strength analysis has been used in mechanical component design. The probability of failure is based on the probability of stress exceeding strength. The following equation is used to calculate the expected probability of failure:

[math]\displaystyle{ F=P[Stress\ge Strength]=\int_{0}^{\infty }{{{f}_{Strength}}(x)\cdot {{R}_{Stress}}(x)}dx }[/math]

The expected probability of success or the expected reliability, R , is calculated as:

[math]\displaystyle{ R=P[Stress\le Strength]=\int_{0}^{\infty }{{{f}_{Stress}}(x)\cdot {{R}_{Strength}}(x)}dx }[/math]

The above calculation assumes both stress and strength are in the positive domain. For general cases, the expected reliability can be calculated using the following equation:

[math]\displaystyle{ R=P[{{X}_{1}}\le {{X}_{2}}]=\frac{1}{{{F}_{1}}(U)-{{F}_{1}}(L)}\int_{L}^{U}{{{f}_{1}}(x)\cdot {{R}_{2}}(x)}dx }[/math]

where:

[math]\displaystyle{ L\le {{X}_{1}}\le U }[/math]

[math]\displaystyle{ \begin{align} & {{X}_{1}}:\text{ Stress } \\ & {{X}_{2}}:\text{ Strength } \\ \end{align} }[/math]

When [math]\displaystyle{ U = infinite }[/math] and [math]\displaystyle{ L = 0 }[/math], the above two equations are the same.