Stress can refer to any effect impinging on the component or system that could cause it to fail, for example: pressure, temperature, applied voltage, torque.

Strength is the limit at which the component can withstand the applied stress. It has the same units as the stress variable, of course. The figure below shows how both of these can be random variables. The stress applied to a component or system can be a random variable dependent on weather and other operating conditions, the mode of use, etc. The strength of the component will vary somewhat from one component to another due to age, amount of use, manufacturing variability, etc. Thus, for any randomly selected component, its strength is also a random variable.

Here we pose the question: What is the probability that the applied stress is greater than the strength of the component? Scenarios of interest occur in the shaded overlap area in the figure above. In formal mathematics this requires doing an algebraic integration, which may not be possible depending on the distributions of stress and strain. However, with simulation we can determine this very easily. Model Stress and Strength shows an example.

The links to the Stress and Strength software specific models are provided here:

Crystal Ball

Stress_and_strength

In cells G7 and G8 we input distributions for stress and strength. These distributions may come from data or manufacturers statistics. Cell G9 is a flag that comes up with a 1 every time the simulation produces a stress higher than the component's strength, and a 0 otherwise. Finally cell G10 calculates the running mean of cell G9, which is equivalent to the proportion of times a generated stress value is greater than a generated strength value. Running a few thousand iterations stabilizes this value to the theoretical probability of failure of the component given the stress and strength distributions. You will need to run more iterations the less the distributions overlap – keeping Crystal Ball's Update display option on will let you see if the value is stabilizing. This model is one example of __numerical integration__.

@Risk

Stress_and_strength

In cells G7 and G8 we input distributions for stress and strength. These distributions may come from data or manufacturers statistics. Cell G9 is a flag that comes up with a 1 every time the simulation produces a stress higher than the component’s strength, and a 0 otherwise. Finally cell G10 calculates the running mean of cell G9, which is equivalent to the proportion of times a generated stress value is greater than a generated strength value. Running a few thousand iterations stabilizes this value to the theoretical probability of failure of the component given the stress and strength distributions. You will need to run more iterations the less the distributions overlap – keeping @RISK’s Update display option on will let you see if the value is stabilizing. This model is one example of __numerical integration__.