EGGN 598 – Probabilistic Biomechanics Ch.7 – First Order Reliability Methods

1. EGGN 598 – Probabilistic BiomechanicsCh.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, b • 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 Safe Safe Failure Failure

4. Geometry of MPP • 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) MPP → closest to the origin → highest likelihood on joint PDF in the reduced coord. space * gradient → perpendicularto tangent direction gradientdirection

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 circleof certain radius b • We start with the linearization ofg(X) and compute its gradient • We look outward along the gradientuntil we reach the desired prob level • This is the MPP because the linearg(X) is guaranteed to be tangent tothe prob contour at that point

7. Review: AMV Geometry • The red dot is (l_fem’*,h_hip’*), the tangency point for glinear_90% • When we recalculate g90% 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 updatedcurve may not be exactly tangentto the 90% prob circle, so theremay 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: • Define the limit state equation • 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 • Assume an initial value of the MPP, usually the means • Compute the partial derivatives and find alpha (unit vector in direction of the function gradient) • 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 bHL 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): • Convert the MPP from reduced coordinates back to original coordinates • Obtain an updated estimate of g(X) for each P-level using the relevant MPP’s computed in step 6 • 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. AMV+ Example • We will continue with the AMV example already started and extend it with the AMV+ method Mean Value Method AMV Method – Iteration 1 g(X) X = MPP-1

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

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

14. AMV+ Example AMV Method – Iteration 2 AMV Method – Iteration 2

15. AMV+ Example