Traditional engineering design employs the concept of a safety factor to prevent failure. In this article we are going to discuss, with a simple example, how probabilistic design can be used as an alternative to safety factors.
Background on Safety Factors
In engineering design, uncertainties in material strength, material dimensions, and loading is often accounted for with the use of a safety factor. This entails calculating applied stress assuming some appropriate worst case loading using nominal material strength and nominal dimensions. The factor of safety is then:
Factor of Safety = material strength/applied stress
Usually material strength is yield strength where material is not permanently deformed. The choice of a suitable safety factor involves experience, design codes, or company standards to name a few. The disadvantage of safety factors is that we don't get a complete picture of the risks involved, nor can we specify reliability to any degree.
Probabilistic design as you may have guessed involves using probabilities to determine the design. Probabilistic design allows for quantification of various sources of uncertainty, primarily loading, material strength, and material dimensions. There are other possibilities such as environmental conditions, but for the purposes of our discussion we will limit ourselves to loading, strength and dimensions. The design is specified so that a small, but acceptable level of failure is met. There are several variations on how we can model a probabilistic design:
- Stochastic loading, stochastic strength, stochastic dimensions
- Stochastic loading, stochastic strength, nominal dimensions
- Stochastic loading, nominal strength, nominal dimensions
- Assumed loading, stochastic strength, stochastic dimensions
- Assumed loading, nominal strength, stochastic dimensions
- Assumed loading, stochastic strength, nominal dimensions
A Simple Example
DISCLAIMER: The details of the example are intentionally vague and mostly made-up to illustrate probabilistic design. Do your own homework when assigning probability distributions.
Let's illustrate the concept with a simple beam loading example. We have a steel beam that is simply supported with a concentrated load at the center of the span.
The applied stress for this scenario is given by:
Applied stress = moment/section modulus
Section modulus is dependent on material shape and dimensions. Therefore, since the beam cross-sectional dimensions have variation, the section modulus will also have variation. Beam dimensions are assumed to be normally distributed with the nominal dimension as mean and standard deviation of 0.01. The material strength is assumed to be normally distributed with a mean of 43 ksi and standard deviation of 4.5. Loading is assumed to follow the extreme value maximum distribution with mode of 1000 and beta of 200.
The beam will have a rectangular cross-section and our goal is to design it such that we have a small percentage of failures due to overloading, and the resulting permanent deformation. To do this we make use of a spreadsheet to create the model and Monte Carlo simulation software, Simulation Master, to determine failure probability.
To simulate whether a loading scenario results in a safe condition or failure, we sample each random variable and subtract applied stress from yield strength. This is calculated in cell B7 and is the model output that is simulated. A positive condition means that yield strength is greater than applied stress. A negative condition means there is a failure.
We could make guesses for the nominal beam dimensions, run a simulation, and make adjustments to the dimensions until we get an acceptable failure probability. To avoid trial and error, we will make use of Simulation Master's optimizer which can be used to find the beam dimensions. To learn more about optimization with simulation, refer to this article.
To use the optimizer, we need to set up decision variables in cell E2 and E3 that will change the nominal dimensions (F2 and F3) during optimization.
To run the optimization we will perform 2500 simulation trials for each loop through the optimization routine. Simulation Master's can optimize for percentiles, but the smallest percentile is the first. We want a smaller failure probability than 1% so we set the objective such that the model output's (B7) 1st percentile value is 3 ksi. In other words, the objective is to set the beam dimensions so that 99% of the time yield strength - applied stress >= 3 ksi. This will result in a failure rate less than 1%. After running the optimization routine, we will run a final simulation for 100,000 trials for a more accurate estimate of failure probability.
After running the optimization, we get a beam height of 1.138 and a width of 1.491. Note that the model spreadsheet has these values already inserted.
The results of the 100,000 trial final simulation is shown below. Using the probability analysis tool, we see that the failure rate is .328%. Note that the 1st percentile of the final simulation is 4.39 ksi which is different than our original objective of 3 ksi. The difference is due to the fact that there were only 2500 simulation trials for each optimization loop (to save time) while the final simulation was 100,000 trials.
We can look at the sensitivity of failure to each random variable in the model by looking at the correlation between each variable and the model output. A tornado chart of the correlation coefficients for each variable to output is shown below.
The model output is most correlated with yield strength and the second most important factor is the negative correlation with load. Dimensional variation in height and width are relatively unimportant.