Presentation on theme: "Approximate methods for calculating probability of failure"— Presentation transcript:
1Approximate methods for calculating probability of failure Working 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 Canfield’s Reliability Based Structural Design Chapter 4, Section 4.1.
2The normal distribution is attractive It 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.
3Approximation about mean Predecessor of FORM called first-order second-moment method (FOSM)
4Beam under central load example Probability of exceeding plastic moment capacityAll normalPLW (plastic section modulus)T (yield stress)mean10kN8m0.0001m^3600,000 kN/m^2Standard deviation2kN0.1mm^3100,000kN/m^2
5Reliability index for example Using the linear approximation getExample 4.2 of CGC shows that if we change to g=T-0.25PL/W we get 3.48 ( , exact is 2.46 or Pf=0.0069)
6Top Hat questionIn the beam example, the error in estimating the reliability index was due toLinear approximationg is not normalboth
7Most 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.
8Response minus capacity illustration r=randn(1000,1)* ; c=randn(1000,1)*1.5+13; x; fplot(f,[5,20]) hold on plot(r,c,'ro') xlabel('r') ylabel('c')
9Recipe for finding MPP with independent normal variables Transform 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.
10First order reliability method (FORM) Limit state g(X). Failure when g<0.Linear approximation of limit state together with assumption that random variables are normal.Approximate around most probable point.Then limit state is also normal variable.Reliability index is the distance of the mean of g from zero measured in standard deviations.
12Linear Example For linear example Then The failure boundary Distance from origin
13Check by MCS r=randn(1000,1)*1.25+10; c=randn(1000,1)*1.5+13; >> s=0.5*(sign(r-c)+1);pf=sum(s)/1000=0.0550Six repetitions gave: , , ,0.0450, ,With million samples gotSo even with a million samples, accurate only to two digits.
14General 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 of CGC (It is called the Rosenblatt transformation)Murray Rosenblatt, Remarks on a Multivariate Transformation, Ann. Math. Statist. Volume 23, Number 3 (1952),
15Approximate transformation If 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.