MEGN 537 – Probabilistic Biomechanics Ch.6 – Randomness in Response Variables

1. MEGN 537 – Probabilistic BiomechanicsCh.6 – Randomness in Response Variables Anthony J Petrella, PhD

2. General Approach • Biomechanical system with many inputs • Functional relationship Y = f(Xi) may be unknown • Input distributions may be unknown • What is the impact of input uncertainty on output? • Two Approaches • Analytical (Ch.6) • Closed form solutions in some cases • Numerical (Ch.7-9) • Robust solutions to all problems

3. Goal of Prob Analysis • Understand impact of input uncertainty on output • Characterize CDF of output • Two kinds of output we will discuss: • Performance function, response functionY = Z = g(X1, X2,…, Xn) • Limit state functionZ = g(Xi) = 0, defines boundary between safe zone and failure: POF = P(g ≤ 0) • Note: text mixes these terms at times

4. Randomness in Response Variables • Note “randomness” is meant to convey uncertainty • The inputs or outputs are not truly random, but rather can be represented by distributions • The literature also refers to “non-deterministic”, which really means using a stochastic version of the deterministic model – can exhibit different outcomes on different runs

5. Considering Various Functional Relationships betweenOutput & InputsExact Solutions

6. Functional relationship: Can show that Y will have same distribution as X Mean: Standard deviation: Single Input, Known Function: Linear

7. Functional relationship: Mean & variance, computed from PDF: Single Input, Known Function: Non-Linear

8. Can show that Y will have same distribution as X Mean & variance, computed from PDF Can be done for normal and lognormal More difficult to integrate PDF for other distributions Single Input, Known Function: Non-Linear

9. Multiple Inputs, Known Function: General (may be Non-Linear) • Functional relationship: • Can be done with similar approach for single input but in general… • Functional relationship g( ) seldom known in practice • Joint PDF for inputs is needed but seldom known • Often difficult to find g-1( )

10. Multiple Inputs, Known Function: Sum of Normal Variables • Functional relationship: • Xi’s are statistically independent • Mean • Variance

11. Example • Consider a weight that is hung by a cable • The load carrying capacity or resistance of the cable (R) is a normal RV with mean = 120 ksi, SD = 18 ksi • The load (S) is also a normal RV with mean = 50 ksi,SD = 12 ksi • Assume that R and S are statistically independent • Define limit state function, g = R – S • Find POF

12. Solution

13. Multiple Inputs, Known Function: Product of Lognormal Variables • Functional relationship: • Xi’s are statistically independent • Lambda • Zeta

14. Example • Hoop stress in a thin walled pressure vessel is given by, shoop=pr/t • p = lognormal distribution, mean = 60 MPa,SD = 5 MPa • r = 0.5 m, t = 0.05 m • What is the probability that the hoop stress exceeds 700 MPa?

15. Solution

16. Central Limit Theorem • Sum of a large number of random variables tends to the normaldistribution • Product of a large number of random variables tends to the lognormal distribution → X will be normal for large n → X will be lognormal for large n

17. Considering Various Functional Relationships betweenOutput & InputsApproximate Solutions

18. Multiple Inputs, Known Function, Unknown Distributions • Functional relationship: • Distributions of Xi are unknown • Assume mean and variance of Xi are known • Assume Xi are statistically independent • Let us approximate g(Xi) with a 1st order Taylor series expansion about the mean values mXi,

19. Multiple Inputs, Known Function, Unknown Distributions • Approximate functional relationship: • Now we have,

20. Multiple Inputs, Known Function, Unknown Distributions • If the Xi are uncorrelated, then the variance simplifies to…

21. Multiple Inputs,Unknown Function & Distributions • In the most general case, even the form of the functional relationship is unknown • Function evaluations are done by experiment – either physical or computational • In this case, we can use finite difference equations to estimate the partial derivatives Forward difference Central difference

22. Multiple Inputs,Unknown Function & Distributions • The values of Yi+ and Yi- are found by simply perturbing each input one at a time by +1 or -1 SD… • Efficiency → fewest function evaluations possible • Note that Ym = E(Y), so you will already have it • Forward difference requires Yi+ • Central difference additionally requires Yi- (more function evals = more time = more cost)

23. Multiple Inputs,Unknown Function & Distributions • By Central Limit Theorem we will often assume the response is normal or lognormal • In the absence of sufficient information, we will assume it is normal • Once E(Y) and Var(Y) are estimated, we have an estimate for the entire CDF of the response function or limit state function • Advanced techniques (Ch.8) can then be used to improve the above estimate