1. EGGN 598 – Probabilistic Biomechanics Ch.7 – First Order Reliability Methods Anthony J Petrella, PhD

2. Review: Reliability Index To address limitations of risk-based reliability with greater efficiency than MC, we introduce the safety index or reliability index,  Consider the familiar limit state, Z = R – S, where R and S are independent normal variables Then we can write, and POF = P(Z ≤ 0), which can be found as follows…

3. Review: MPP Failure Safe

4. Recognize that the point on any curve or n-dimensional surface that is closest to the origin is the point at which the function gradient passes through the origin Distance from the origin is the radius of a circle tangent to the curve/surface at that point (tangent and gradient are perpendicular) Geometry of MPP MPP→closest to the origin →highest likelihood on joint PDF in the reduced coord. space * gradient direction gradient → perpendicular to tangent direction

5. Review: AMV Example For example, consider the non-linear limit state, where,

6. Review: AMV Geometry Recall the plot depicts (1) joint PDF of l_fem and h_hip, and (2) limit state curves in the reduced variate space (l_fem’,h_hip’), To find g(X) at a certain prob level, we wish to find the g(X) curve that is tangent to a certain prob contour of the joint PDF – in other words, the curve that is tangent to a circle of certain radius  We start with the linearization of g(X) and compute its gradient We look outward along the gradient until we reach the desired prob level This is the MPP because the linear g(X) is guaranteed to be tangent to the prob contour at that point

7. Review: 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)

8. AMV+ Method The purpose of AMV+ is to reduce the error exhibited by AMV AMV+ simply translates to…“AMV plus iterations” Recall Step 3 of AMV: assume an initial value for the MPP, usually at the means of the inputs Recall Step 5 of AMV: compute the new value of MPP AMV+ simply involves reapplying the AMV method again at the new MPP from Step 5 AMV+ iterations may be continued until the change in g(X) falls below some convergence threshold

9. AMV+ Method (NESSUS) Assuming one is seeking values of the performance function (limit state) at various P-levels, the steps in the AMV+ method are: 1.Define the limit state equation 2.Complete the MV method to estimate g(X) at each P-level of interest, if the limit state is non-linear these estimates will be poor 3.Assume an initial value of the MPP, usually the means 4.Compute the partial derivatives and find alpha (unit vector in direction of the function gradient) 5.Now, if you are seeking to find the performance (value of limit state) at various P-levels, then there will be a different value of the reliability index  HL at each P-level. It will be some known value and you can estimate the MPP for each P-level as…

10. AMV+ Method (NESSUS) The steps in the AMV+ method (continued): 6.Convert the MPP from reduced coordinates back to original coordinates 7.Obtain an updated estimate of g(X) for each P-level using the relevant MPP’s computed in step 6 8.Check for convergence by comparing g(X) from Step 7 to g(X) from Step 2. If difference is greater than convergence criterion, return to Step 3 and use the new MPP found in Step 5.

11. We will continue with the AMV example already started and extend it with the AMV+ method AMV+ Example Mean Value MethodAMV Method – Iteration 1 X = MPP-1 g(X)

12. We will continue with the AMV example already started and extend it with the AMV+ method AMV+ Example AMV Method – Iteration 2AMV Method – Iteration 1 X = MPP-2 g(X)

13. AMV+ Example AMV Method – Iteration 2AMV Method – Iteration 1

14. AMV+ Example AMV Method – Iteration 2

15. AMV+ Example