# Risk Analysis and Probabilistic Design with Monte Carlo Simulation

Monte Carlo simulation can be used to perform simple relationship-based simulations. This type of simulation has many applications in probabilistic design, risk analysis, quality control, etc. The Monte Carlo utility includes a *User Defined* distribution feature that allows you to specify an equation relating different random variables. The following example uses the Life Comparison tool to compare the *pdf* of two user-defined distributions. A variation of the example that demonstrates how to obtain the joint *pdf* of random variables is available in the Weibull++/ALTA Help file. A demonstration on how to perform the example using ReliaSoft's advanced stochastic event simulation software, **RENO**, is also available (view it in html).

**Monte Carlo Simulation: A Hinge Length Example**

A hinge is made up of four components A, B, C, D, as shown next. Seven units of each component were taken from the assembly line and measurements (in cm) were recorded.

The following table shows the measurements. Determine the probability that D will fall out of specifications.

[math]\begin{matrix} \text{Dimensions for A} & \text{Dimensions for B} & \text{Dimensions for C} & \text{Dimensions for D} \\ \text{2}\text{.0187} & \text{1}\text{.9795} & \text{30}\text{.4216} & \text{33}\text{.6573} \\ \text{1}\text{.9996} & \text{2}\text{.0288} & \text{29}\text{.9818} & \text{34}\text{.5432} \\ \text{2}\text{.0167} & \text{1}\text{.9883} & \text{29}\text{.9724} & \text{34}\text{.6218} \\ \text{2}\text{.0329} & \text{2}\text{.0327} & \text{30}\text{.192} & \text{34}\text{.7538} \\ \text{2}\text{.0233} & \text{2}\text{.0119} & \text{29}\text{.9421} & \text{35}\text{.1508} \\ \text{2}\text{.0273} & \text{2}\text{.0354} & \text{30}\text{.1343} & \text{35}\text{.2666} \\ \text{1}\text{.984} & \text{1}\text{.9908} & \text{30}\text{.0423} & \text{35}\text{.7111} \\ \end{matrix}\,\![/math]

**Solution**

In a Weibull++ standard folio, enter the parts dimensions measurements of each component into separate data sheets. Analyze each data sheet using the normal distribution and the RRX analysis method. The parameters are:

[math]\begin{matrix} \text{A} & \text{B} & \text{C} & \text{D} \\ \hat{\mu }=2.0146 & \hat{\mu }=2.0096 & \hat{\mu }=30.0981 & \hat{\mu }=34.8149 \\ \hat{\sigma }=0.0181 & \hat{\sigma }=0.0249 & \hat{\sigma }=0.1762 & \hat{\sigma }=0.7121 \\ \end{matrix}\,\![/math]

Next, perform a Monte Carlo simulation to estimate the probability that (A+B+C) will be greater than D. To do this, choose the **User Defined** distribution and enter its equation as follows. (Click the **Insert Data Source** button to insert the data sheets that contain the measurements for the components.)

On the Settings tab, set the number of data points to **100**, as shown next.

Click **Generate** to create a data sheet that contains the generated data points. Rename the new data sheet to "Simulated A+B+C."

Follow the same procedure to generate 100 data points to represent the D measurements. Rename the new data sheet to "Simulated D."

Analyze the two data sets, "Simulated A+B+C" and "Simulated D," using the normal distribution and the RRX analysis method.

Next, open the Life Comparison tool and choose to compare the two data sheets. The following picture shows the *pdf* curves of the two data sets.

The following report shows that the probability that "Simulated A+B+C" will be greater than "Simulated D" is 16.033%. (Note that the results may vary because of the randomness in the simulation.)