Probabilistic design

Sources of variability

Though the laws of physics dictate the relationships between variables and measurable quantities such as force, stress, strain, and deflection, there are still three primary sources of variability when considering these relationships.

The first source of variability is statistical, due to the limitations of having a finite sample size to estimate parameters such as yield stress, Young's modulus, and true strain. Measurement uncertainty is the most easily minimized out of these three sources, as variance is proportional to the inverse of the sample size.

We can represent variance due to measurement uncertainties as a corrective factor ${\displaystyle B}$, which is multiplied by the true mean ${\displaystyle X}$ to yield the measured mean of ${\displaystyle \overline{X}}$. Equivalently, ${\displaystyle \overline{X}=\overline{B}X}$.

This yields the result ${\displaystyle \overline{B}=\frac{\overline{X}}{X}}$, and the variance of the corrective factor ${\displaystyle B}$ is given as:

${\displaystyle Var[B]=\frac{Var[\overline{X}]}{X}=\frac{Var[X]}{nX}}$

where ${\displaystyle B}$ is the correction factor, ${\displaystyle X}$ is the true mean, ${\displaystyle \overline{X}}$ is the measured mean, and ${\displaystyle n}$ is the number of measurements made.

The second source of variability stems from the inaccuracies and uncertainties of the model used to calculate such parameters. These include the physical models we use to understand loading and their associated effects in materials. The uncertainty from the model of a physical measurable can be determined if both theoretical values according to the model and experimental results are available.

The measured value ${\displaystyle \hat{H}(\omega )}$ is equivalent to the theoretical model prediction ${\displaystyle H(\omega )}$ multiplied by a model error of ${\displaystyle \varphi (\omega )}$, plus the experimental error ${\displaystyle \epsilon (\omega )}$. Equivalently,

${\displaystyle \hat{H}(\omega )=H(\omega )\varphi (\omega )+\epsilon (\omega )}$

and the model error takes the general form:

${\displaystyle \varphi (\omega )=\sum _{i=0}^n{a}_i{\omega }^n}$

where ${\displaystyle a_i}$ are coefficients of regression determined from experimental data.

Finally, the last variability source comes from the intrinsic variability of any physical measurable. There is a fundamental random uncertainty associated with all physical phenomena, and it is comparatively the most difficult to minimize this variability. Thus, each physical variable and measurable quantity can be represented as a random variable with a mean and a variability.

Comparison to classical design principles

Consider the classical approach to performing tensile testing in materials. The stress experienced by a material is given as a singular value (i.e., force applied divided by the cross-sectional area perpendicular to the loading axis). The yield stress, which is the maximum stress a material can support before plastic deformation, is also given as a singular value. Under this approach, there is a 0% chance of material failure below the yield stress, and a 100% chance of failure above it. However, these assumptions break down in the real world.

The yield stress of a material is often only known to a certain precision, meaning that there is an uncertainty and therefore a probability distribution associated with the known value. Let the probability distribution function of the yield strength be given as ${\displaystyle f(R)}$.

Similarly, the applied load or predicted load can also only be known to a certain precision, and the range of stress which the material will undergo is unknown as well. Let this probability distribution be given as ${\displaystyle f(S)}$.

The probability of failure is equivalent to the area between these two distribution functions, mathematically:

${\displaystyle P_f=P(R<S)=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(R)f(S)dSdR}$

or equivalently, if we let the difference between yield stress and applied load equal a third function ${\displaystyle R-S=Q}$, then:

${\displaystyle P_f=\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}\underset{-\mathrm{\infty }}{\overset{\mathrm{\infty }}{\int }}f(R)f(S)dSdR=\underset{-\mathrm{\infty }}{\overset{0}{\int }}f(Q)dQ}$

where the variance of the mean difference ${\displaystyle Q}$ is given by ${\displaystyle {\sigma }_Q^2=\sqrt{{\sigma }_R^2+{\sigma }_S^2}}$ .

The probabilistic design principles allow for precise determination of failure probability, whereas the classical model assumes absolutely no failure before yield strength. It is clear that the classical applied load vs. yield stress model has limitations, so modeling these variables with a probability distribution to calculate failure probability is a more precise approach. The probabilistic design approach allows for the determination of material failure under all loading conditions, associating quantitative probabilities to failure chance in place of a definitive yes or no.

Methods used to determine variability

In essence, probabilistic design focuses upon the prediction of the effects of variability. In order to be able to predict and calculate variability associated with model uncertainty, many methods have been devised and utilized across different disciplines to determine theoretical values for parameters such as stress and strain. Examples of theoretical models used alongside probabilistic design include:

• Finite element analysis
• Stochastic finite element method
• Boundary element method
• Meshfree methods
• Analytical methods (refer to classical design principles)

Additionally, there are many statistical methods used to quantify and predict the random variability in the desired measurable. Some methods that are used to predict the random variability of an output include:

• the Monte Carlo method (including Latin hypercubes);
• propagation of error;
• design of experiments (DOE)
• the method of moments
• Statistical interference
• quality function deployment
• Failure mode and effects analysis