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

2. 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

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 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

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 between Output & Inputs Exact Solutions

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

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

8. Single Input, Known Function: Non-Linear
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

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
Natural log of Y is normal and ln(g) becomes a sum of normal variables, then…
Lambda
Zeta

14. 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?

15. Solution

16. 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

17. Considering Various Functional Relationships between Output & Inputs Approximate 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