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

2. Basics of Reliability
Recall that we can find exact CDF of a system response variable if…
All inputs are statistically independent normal variables and the functional relationship g(X) is a linear function of Xi’s
All inputs are statistically independent lognormal variables and the functional relationship g(X) is a multiplicative function of the Xi’s
Risk-based reliability employs the above concepts
Option 1: limit state Z = R – S, then POF = P(Z ≤ 0)
Option 2: limit state Y = R/S (safety factor), then POF = P(Y ≤ 1)
These two options produce different results for POF (see Haldar, Example 7.3), which is undesirable
Also, in practice we often want to consider different known or unknown limit states, different input distributions, and correlations among inputs

3. Monte Carlo Simulation
Given: 1. input distributions 2. deterministic model (can be “black box” so g(X) is not known in closed form)
Find: POF
Method:
Randomly generate numbers (ui) 0 to 1
Generate input values from CDF’s, xi = F-1(ui)
Input xi into deterministic model
Evaluate limit state g(X)
POF = # of failures / # of trials
Repeat until POF converges

4. Monte Carlo Simulation
Accuracy is related to the number of trials
Example: six sigma requires at least 1 million 30 seconds/trial  347 days
Advantages
Guaranteed to work every time
Guaranteed to converge to the correct solution
Disadvantages
Inefficient
Requires a large number of trials
Results in large computation times

5. 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…

6. Example: ws8
Recall that if we wish to consider a complex limit state that is a function of many inputs, then we can simplify the problem by estimating the limit state with a first-order Taylor series expansion…
Furthermore, if we do not have a closed form expression for the limit state, then finite difference approximations can be used to estimate the partial derivatives
Then we can easily find…

7. Example: ws8
The limit state for axial stress in a spinal rod was given as,
which is a simple linear expression
It was assumed that all inputs were uncorrelated variables with unknown distributions
The form of the functional relationship was assumed unknown
The mean and standard deviation of each input was given

8. Example: ws8 Forward difference 1 2 3 5 4 6 7 𝛽= 20 10.99652 =1.818757
𝛽= =
POF =Φ −𝛽 =1−Φ 𝛽 =

9. Example: ws8 Conclusions: MC provides excellent results
MC requires many trials to provide accurate results
MC cannot capture CDF tails without trials
MV method more accurate than MC for few trials, much more efficient
MV based on a linearization of the limit state about the mean values of the inputs – only accurate for linear or approximately linear functions

10. Advanced Mean Value Method (AMV)
We would like to overcome linear limitation of MV method
Try second (or higher) order Taylor series expansion…(SORM, Ch.8)
There is a simpler way to do it…
We would also like to overcome lack of invariance exhibited by basic risk-based reliability methods…
Give different answers depending on form of limit state
Let us consider the AMV method, which includes improvements on the MV method to address both of the above limitations
To develop the AMV method, we must introduce the… Most Probable Point (MPP) of failure

11. Most Probable Point (MPP)
Inputs must first be transformed to standard or “reduced” variates
Computed in the same way as standard normal variates, but the variable need not be normal

12. Most Probable Point (MPP)
Failure
Safe

13. Most Probable Point (MPP)
Closest to the origin → highest frequency, most likely

14. Most Probable Point (MPP)
Consider the curve that defines a fixed value of the limit state, there are many combinations of inputs that represent points on that curve
MPP is the most probable combination of the input variables that satisfy the limit state equation – that is the most probable combination of input values leading to a specified value of the limit state (e.g., in reliability theory, g = 0 defines failure)
Values of the input variables can be put into a column vector, where the prime symbol indicates each variable has been transformed to a reduced or standard variate
The MPP can then be written, where the star superscript identifies the MPP in the reduced or standard variate coordinate system

15. AMV Method (NESSUS)
AFOSM Method uses MPP to compute the Hasofer-Lind reliability index, bHL, developed for a linear performance function (limit state)
Similar to the AMV method in NESSUS
Joint PDF

16. AMV Method (NESSUS)
The AFOSM / AMV Method can compute the reliability index exactly in the case of a linear limit state
AFOSM / AMV Method can also be used to compute reliability index for a non-linear limit state
When limit state is non-linear, finding the reliability index, bHL (minimum distance to the origin), becomes an optimization problem…
Minimize the distance,
subject to the constraint,
Note, const. is usually zero for finding POF, but one may also wish to find performance at certain P-levels, in which case const. ≠ 0

17. 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 unit vector passes through the origin
MPP → closest to the origin → highest likelihood on joint PDF in the reduced coord. space *
gradient direction
gradient → perpendicular to tangent direction

18. Geometry of MPP
If we can compute/estimate the gradient, then we can look along that direction (from the origin) for an intersection with the limit state to estimate the MPP
MPP → closest to the origin → highest likelihood on joint PDF in the reduced coord. space *
gradient direction
gradient → perpendicular to tangent direction
function gradient unit vector

19. AMV Method (NESSUS)
The optimization problem can be solved in closed form using the Lagrange multiplier method (see Haldar, p.201)
From this we find the reliability index, bHL, which is the magnitude of the minimum distance from the origin to the limit state in the reduced coordinate system
The MPP can be expressed as,
where
* alpha is a unit vector in the direction of the limit state gradient

20. 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…

21. 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( ) for each P-level using the relevant MPP’s computed in step 6

22. AMV Example Consider the recumbent cycling analysis
MATLAB: Fap = pedal_trial_nessus(rqk,rtx,rty,rhky,angle)

23. AMV Example A/P force at knee, Fap, is assumed deterministic
Estimate bending stress at the hip with simple expression
where,

24. AMV Example Let the limit state be defined as,
Find the 90% performance level for the limit state
Step 1: the limit state equation is defined above
Step 2: complete the MV method to make a first order estimate of g( ) at the 90% level

25. AMV Example
Step 2: MV method (continued)

26. AMV Example Step 3: assume the MPP is initially at,
Step 4: compute the partial derivatives at the MPP in the reduced coordinate system, then find alpha

27. AMV Example
Step 4: compute the partial derivatives at the MPP in the reduced coordinate system, then find alpha

28. AMV Example
Step 4: the geometry…showing Linear and MV results with alpha in the reduced coordinate system
Recall: generally, the true form of g( ) will not be known, so the solid curves in the plot would also not be known, but they are shown here for comparison

29. AMV Example
Step 5: estimate MPP at 90%: bHL = F-1(0.9) = 1.28

30. AMV Example
Step 6: convert MPP from reduced coordinates back to original coordinates
Step 7: Update estimate for g( )…

31. AMV Example: Final Results

32. AMV Example Conclusions
Improved accuracy compared to MV method for non-linear limit state functions
Based on MPP in reduced coordinate space, which eliminates invariance problem
Designed to work with uncorrelated normal variables
We will explore later how to expand to non-normal variables, advanced methods are also available to consider correlated non-normal inputs (NESSUS can do this)