1 / 32

MEGN 537 – Probabilistic Biomechanics Ch.7 – First Order Reliability Methods

MEGN 537 – Probabilistic Biomechanics Ch.7 – First Order Reliability Methods. Anthony J Petrella, PhD. Basics of Reliability. Recall that we can find exact CDF of a system response variable if…

yamka
Download Presentation

MEGN 537 – Probabilistic Biomechanics Ch.7 – First Order Reliability Methods

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. MEGN 537 – Probabilistic BiomechanicsCh.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 distributions2. 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 trials,@ 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), or in terms of b …

  6. Example: TIC 7 • 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: TIC 7 • 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: TIC 7 1 2 Forward difference 3 5 4 6

  9. Example: TIC 7 Conclusions: • MC provides excellent results • MC requires many trials to provide accurate results • MC cannot capture CDF tails without 103-104 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 ValueMethod (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 • Give different answers depending on input distributions • 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) Safe Safe Failure Failure

  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 passes through the origin MPP → closest to the origin → highest likelihood on joint PDF in the reduced coord. space * gradientdirection gradient → perpendicularto 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 * gradientdirection gradient → perpendicularto tangent direction

  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( ) willnot be known, so the solid curves in the plotwould also not be known, but they are shownhere 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)

More Related