1. MEGN 537 – Probabilistic Biomechanics Ch.7 – First Order Reliability Methods Anthony J Petrella, PhD

2. A Summary of AMV We desire to find various probability levels of a certain outcome metric for a biomechanical system We have a computational model for the system, but cannot write a closed-form expression for the response function (limit state) In other words, we do not know g(X) for the response of interest Begin with the MV method and sample the limit state, g(X), to build a linear model of the system using a first-order Taylor series expansion about the means, call this g linear (X) The linear model gives us some idea of g(X) and allows us to estimate the first and second moments of g(X)…

3. A Summary of AMV If we assume g linear (X) is normally distributed, then the first and second moments allow us to find values of the function at various probabilities Values of g linear (X) are then taken as first order estimates for values of the actual limit state, g(X), at various probabilities… If g(X) is linear, then g linear (X) is an exact representation and the probability levels will be accurate If g(X) is non-linear, then g linear (X) will only be accurate near the expansion point (all inputs set to mean values) and probability levels other than 0.5 (mean for normal variable) will exhibit error

4. A Summary of AMV For example, consider the non-linear limit state, where,

5. AMV Geometry Starting with the MV method we can perturb/sample g(X) to develop the linear approximation, g linear (X) We can then estimate the mean & standard deviation, and we can compute the value of g linear (X) at various probability levels (see table) Then we can plot g linear (X) in the reduced variate space (see plot)

6. AMV Geometry In the reduced variate space probability levels are circles – think of the plot below as a top view of the joint PDF of all inputs To find the value of g(X) corresponding to a specific probability level, one must find the g(X) curve tangent to the desired probability circle If g(X) is non-linear, then for a given fixed value of the response g(X) may be a very different curve than g linear (X) (see plot) Recall the origin in reduced variate space corresponds to 50% probability = means of the inputs, which were normal Notice g linear (X) and g(X) agree perfectly at the origin because the Taylor series expansion was centered there Realize g(X) is normally not known!

7. AMV Geometry Note that g linear_90% = g 90% That is, the value of the response is identical, but the values of l_fem’ and h_hip’ are very different for each curve The point where g linear_90% is tangent to the 90% prob circle is a good guess for the values of l_fem’ and h_hip’ that will produce an accurate estimate of g 90% AMV involves finding that tangent point (l_fem’*,h_hip’*) and then recalculating g(X)

8. AMV Geometry The red dot is (l_fem’*,h_hip’*), the tangency point for g linear_90% When we recalculate g 90% at (l_fem’*,h_hip’*) we obtain an updated value of g(X) and the curve naturally passes through (l_fem’*,h_hip’*) Note however that the updated curve may not be exactly tangent to the 90% prob circle, so there may still be a small bit of error (see figure below)

9. Homework 6 - AMV Your execution commands should look like this… copy..\..\pedal_data.mat copy..\..\pedal_trial_nessus.m copy..\..\unit.m matlab -wait -nodesktop -nojvm -nosplash -minimize -r pedal_prob_nessus