1. EGGN 598 – Probabilistic Biomechanics Ch
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, 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
Failure
Safe

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 → perpendicular to tangent direction
gradient 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 b
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 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 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:
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