HS-5020: Stochastic Study of a Cantilever I-beam

In this tutorial, you will complete a simple Stochastic study to investigate uncertain parameters of a cantilever I-beam model defined with four variables and four functions.

Before you begin, copy HS-5020.hstx from <hst.zip>/HS-5020/ to your working directory.
Figure 1.


Run a Stochastic Study

In this step, you will run a Stochastic study and review the evaluation scatter.

In this study, you will consider w_th, f_l and f_th as Random parameters (uncertain, not controllable by design) and h as Design with Random parameter (uncertain but controllable by design).

  1. Add a Stochastic.
    1. In the Explorer, right-click and select Add from the context menu.
      The Add dialog opens.
    2. For Definition from, select an approach.
    3. Select Stochastic, then Setup and click OK.
  2. Modify input variables.
    1. Go to the Stochastic > Definition > Define Input Variables step.
    2. Click the Distributions tab.
    3. In the Distribution Role column, select Design with Random for Total Height.
      Figure 2.


    4. In the Distribution column, verify Normal Variance is selected for all active variables.
      The Distribution column allows you to select the Distribution type. Normal Variance (shown in Figure 3) means parameters could take random values following normal distribution around their nominal values (µ).
      Figure 3.


  3. Go to the Stochastic > Specifications step.
  4. In the work area, set the Mode to Simple Random.
  5. In the Settings tab, change the Number of Runs to 500.


  6. Click Apply.
  7. Go to the Stochastic > Evaluate step.
  8. Click Evaluate Tasks.
    The evaluations are randomly sampled in the space and the designs are evaluated.
  9. Click the Evaluation Scatter tab.
  10. Review the scatter plots for Total Height and Web Thick.
    The Evaluation scatter shown in Figure 4 presents the sampling for Total Height and Web Thick. Due to the Design with Random distribution role the sampling for Total Height is truncated to only samples which fall between the upper and lower bounds.
    Figure 4.


Review Post-Processing Results

In this step, you will review the evaluation results within the Post-Processing step.

  1. Go to the Stochastic > Post-Processing step.
  2. Click the Distribution tab.
  3. From the Channel Selector, click Histogram.
  4. Review the histograms of the stochastic results.
    Figure 5 shows the histograms of Total Height and Web Thick values distributions. Each blue bin represents the frequency of runs yielding a sub-range of response values. Notice the Total Height histogram has no tails at the extremities because of the truncated sampling. The probability densities (red curves) indicate the relative likelihoods of the variables to take particular values. A higher value indicates that the values are more likely to occur. The cumulative distributions (green curves) indicate what percentage of the data falls below the value threshold.
    Figure 5.
  5. From the Channel Selector, click Box Plot.
  6. Identify the eventual outliers.
    Note: There are no outliers for Total Height because of the truncated sampling.
    Figure 6.


  7. Click the Reliability tab.
  8. Add a reliability.
    1. Click Add Reliability.
    2. Set Response to ly (r_1).
    3. Set Bound Type to >=.
    4. For Bound Value, enter 75.596800.
  9. Create another reliability by repeating step 8 with the following changes:
    • Set Response to lz (r_2).
    • Set Bound Type to >=.
    • For Bound Value, enter 14.404800.
  10. Create another reliability by repeating step 8 with the following changes:
    • Set Response to Disp (r_4).
    • Set Bound Type to <=.
    • For Bound Value, enter 4.41e-05.
    Considering the parameter uncertainties, there is 49% probability for the moments of inertia and the displacement to have values different than the nominal design.
    Figure 7.


  11. Click the Reliability Plot tab.
  12. Review the graphs.
    Figure 8 show the reliabilities for the values Web Thick and Total Height could take.
    Figure 8.