Machine Design

FE Update - Probability analysis goes beyond linear static

FEA seems something of a crap shoot when you consider that published material properties should be presented in bell-shaped curves to show averages and distributions.

Probability analysis goes beyond linear static

A close-up of the tank frame shows that skin thickness Ts, base B, and height H could incrementally vary to test different designs in a least-weight study.


A section from an aircraft wing tank shows several design features that could be turned into variables.


One output from Veros for the tank problem is a graph of system reliability versus weight reduction. It shows that only a little more weight reduction is possible before reliability drops.

And loads assigned to FEA simulations should be in a range to reflect their uncertainty. Most users know this and run dozens of analyses, each with slightly different tweaks to the model and loads. This makes simulation a what-if game with lots of manual intervention.

A more-efficient way to handle variations in dimensions, loads, and material properties is to use software that accepts inputs as variables with distributions. VeroSolve, a statistical-analysis program from Veros Software, Irvine, Calif. (, provides this probabilistic capability. The statistical program's output, more than just maximum stress values, includes several plots for probability of success and a sensitivity chart that pinpoints characteristics or parameters most influential in the design.

The developer can provide links from their software to many different solvers. For mechanical engineers, the solver is likely to be FEA and CFD programs.

Linking is necessary because after each analysis, the statistical program selects another set of parameters based on one of several user-selected functions. "One way to select variables would be a mean-value-based method," says Eric Fox, vice president of technology with the developer. "Selecting variables by this method assumes everything behaves linearly. Sometimes that's not realistic, but it's faster than others. Another way to select variables is with high-order approximations which are more accurate but take longer. There is the random or Monte Carlo method. It's most accurate but slowest. The point is they allow trade-offs to balance the degree of complexity needed for the problem," says Fox.

"The real analysis challenge comes from the large number of variables, which might include a few different material properties, several loads, many dimensions, and a range of temperatures," says Fox. "In cases with many variables, a design of experiments or response surface is a better way to create a grid of possible variable values. The software juggles possible combinations to get the proper sampling for the grid."

Sensitivity analysis, one of the more important outputs, highlights the item with the most influence on the system. "That item is not always obvious," he adds. "The interesting and surprising ones are in vibration problems because these are influenced by unsuspected part geometry."

Users can approach analyses several ways. For example, the statistical program can tell users what information it needs, and they respond with variables to plug into the analysis software. Or, users can write stress equations into the software as subroutines. But most engineers working with complex problems run FEA programs. Fox says most users link FEA software to the statistical program so it can get information after each run and instruct the FEA program to make appropriate changes.

When VeroSolve is linked to an FEA program, users would first set up FEA models they are confident will provide good predictions. The model could be single parts or assemblies. "For example, when working on a turbine blade, the variables include dimensions in areas where high-stress concentrations are expected, and things that influence the stress, such as blade weight, material properties, temperature, and notch geometry."

Hasty users faced with many analyses, but without access to the statistical software, might assume a worst-case value for each characteristic. "They might assign the lowest-ultimate-tensile strength, highest temperature, worst-case coefficient of expansion, and so on," says Fox. "If the design comes in with a max-stress lower than the spec'd allowable, the analyst might think the task is done. But that person wouldn't know whether the failure probability is 1 in 100 or 1 in a million. Moreover, that analyst has no clue as to what variable is most sensitive or driving the design. Now assume this worst-case design fails. What can the analyst adjust?" asks Fox.

With the statistical software, analysts can use real-world distributions for properties. "Instead of setting the temperature at 700°F, it can be a random temperature between 600 and 700°F," suggests Fox. "And instead of worst-case ultimate-tensile strength, it could be a distribution with the worst case at the extreme of the distribution. And so on with other variables. Running this model gives a distribution of stress and the probability of their occurring. For the turbine blade, stress will span some range, and the probability of being greater than the allowable specification might be one in a billion."

"Expect more time to complete analyses using this method," says Fox. "The time consuming part is the stress analyses. Completing most models take as few as a dozen runs while larger models may need over 300, so it's usually an overnight job."

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