Slide 1

Approximate methods for calculating probability of failureWorking with normal distributions is appealingFirst-order second-moment method (FOSM)Most probable pointFirst order reliability method (FORM)

The material for this lectures relies on Choi, Grandhi and Canfields Reliability Based Structural Design Chapter 4, Section 4.1.1The normal distribution is attractiveIt has the nice property that linear functions of normal variables are normally distributed.Also, the sum of many random variables tends to be normally distributed.Probability of failure varies over many orders of magnitude.Reliability index, which is the number of standard deviations away from the mean solves this problem.

2Approximation about meanPredecessor of FORM called first-order second-moment method (FOSM)

3Beam under central load exampleProbability of exceeding plastic moment capacity

All normal

PLW (plastic section modulus)T (yield stress)mean10kN8m0.0001m^3600,000 kN/m^2Standard deviation2kN0.1m0.00002m^3100,000kN/m^2

4Reliability index for exampleUsing the linear approximation get

Example 4.2 of CGC shows that if we change to g=T-0.25PL/W we get 3.48 (0.00025, exact is 2.46 or Pf=0.0069)

5Top Hat questionIn the beam example, the error in estimating the reliability index was due toLinear approximationg is not normalboth6Most probable point (MPP)The error due to the linear approximation is exacerbated due to the fact that the expansion may be about a point that is far from the failure region (due to the safety margin).Hasofer and Lind suggested remedying this problem by finding the most probable point and linearizing about it.The joint distribution of all the random variables assigns a probability density to every point in the random space. The point with the highest density on the line g=0 is the MPP.

7Response minus capacity illustrationr=randn(1000,1)*1.25+10;c=randn(1000,1)*1.5+13;f=@(x) x;fplot(f,[5,20])hold onplot(r,c,'ro') xlabel('r')ylabel('c')

8Recipe for finding MPP with independent normal variablesTransform into standard normal variables (zero mean and unity standard deviation)

Find the point on g=0 of minimum distance to origin. The point will be the MPP and the distance to the origin will be the reliability index based on linear approximation there.

9First order reliability method (FORM)Limit state g(X). Failure when g> s=0.5*(sign(r-c)+1);pf=sum(s)/1000=0.0550Six repetitions gave: 0.0660, 0.0640, 0.0670,0.0450, 0.0550, 0.0640With million samples got 0.06258So even with a million samples, accurate only to two digits.

General caseIf random variables are normal but correlated, a linear transformation will transform them to independent variables.If random variables are not normal, can be transformed to normal with similar probability of failure. See Section 4.1.5 of CGC (It is called the Rosenblatt transformation)Murray Rosenblatt, Remarks on a Multivariate Transformation, Ann. Math. Statist. Volume 23, Number 3 (1952), 470-472.

14Approximate transformationIf we want the transformed variable u to have the same CDF as the original x at a certain value of x we would requireUnfortunately we cannot enforce that everywhere. If we focus on MPP we get the following transformation (Eqs. 4.38, 4.39), which must be used iteratively, starting from some guess.

X* replaced by xMPP in equations.