Template:Stress-strength analysis

Stress-Strength Analysis

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 probability of failure:

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

The probability of success or Reliability is calculated as:

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

The above calculation assumes both stress and strenght have positive domain.

For general cases, the reliability can be calcualted using the following equation:

[math]\displaystyle{ 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] and:

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

Both the stress and strenght distributions can be estimated from actual data or specified by engineers based on engineering knowledge or existing references. Based on the sources of the distribution, there are two types of variations associated with the calculated probability.

Variation in Model Parameters

Variation in Probability Values

Weibull++'s Stress-Strength Wizard allows you to perform such calculations.

The Stress-Strength Calculator is accessible from the Tools folder of the Project Explorer. The tool first asks for the locations of the stress and strength data from among the folios available, as shown below.

File:Firstchoice.jpg

Once the tool knows where to find data for stress and strength, it automatically calculates the reliability and graphs the distributions of the stress and strength data, as shown below.

Once the reliability has been calculated, the user has two different options for performing sensitivity analysis (seeing how the system performance changes with changes in parameter values). The user can manually alter the distribution parameters used to calculate the reliability by clicking the “Alter Parameters” link for the appropriate distribution, or the user can use the Parameter Estimator to find what parameters result in a specified target reliability. The “Alter Parameters” link allows the user to manually set what she would like the distribution parameter(s) to be. Alternatively, the Parameter Estimator utility allows the user to select a target reliability to reach and a distribution parameter to adjust. The utility then adjusts the parameter value until the specified target is reached and outputs the parameter value that does so, as shown below.