MEGN 537 – Probabilistic Biomechanics Ch MEGN 537 – Probabilistic Biomechanics Ch.6 – Randomness in Response Variables Anthony J Petrella, PhD

General Approach Biomechanical system with many inputs Two Approaches 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

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 function Y = Z = g(X1, X2,…, Xn) Limit state function Z = g(Xi) = 0, defines boundary between safe zone and failure: POF = P(g ≤ 0) Note: text mixes these terms at times

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

Considering Various Functional Relationships between Output & Inputs Exact Solutions

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

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

Single Input, Known Function: Non-Linear Y may not have same distribution as X due to nonlinearity Mean & variance, computed from PDF Can be done for normal and lognormal More difficult to integrate PDF for other distributions

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( )

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

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

Solution

Multiple Inputs, Known Function: Product of Lognormal Variables Functional relationship: Xi’s are statistically independent Natural log of Y is normal and ln(g) becomes a sum/difference of normal variables, then… Lambda Zeta

Example Hoop stress in a thin walled pressure vessel is given by, shoop = p·r / 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?

Solution

Central Limit Theorem Sum of a large number of random variables tends to the normal distribution 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

Considering Various Functional Relationships between Output & Inputs Approximate Solutions

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,

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

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

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

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), if forward diff, you will already have it Forward difference requires Yi+ Central difference additionally requires Yi- (more function evals = more time = more cost)

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 (which has been approximated as linear) Advanced techniques (Ch.8) can then be used to improve the above estimate