Simplified example, buckling of Euler beam, cont.:Approximate limit state function based on second order polynomial:(Approximationpointsarethemeanvalueandplus/minustwostandarddeviations)Second order approximatelimitstate function10-8-6-Limit state function4-2Normalized variable
Simplified example, buckling of Euler beam, cont.: • Approximate limit state function based on second order polynomial: (Approximation points are the mean value and plus/minus two standard deviations)
Simplified example, buckling of Euler beam, cont.:Approximate limitstatefunctionbasedonstraight line:(Approximation points are the mean value and the plus two standard deviations point.Theexact failure probability is obtained)Approximate lincar limit state function2.5-2-1.5Limitstatefunctian0.5-2Normalizedvariable-0.5-1-
Simplified example, buckling of Euler beam, cont.: • Approximate limit state function based on straight line: (Approximation points are the mean value and the plus two standard deviations point. The exact failure probability is obtained)
Simplified example, buckling of Euler beam, cont.:Approximatelimitstatefunctionbasedoncubicpolynomial:(Approximation points are the mean value, the plus/minus two standard deviations, and the plus one standard deviation. The exact failure probability is obtained)Approximatecubiclimitstatefunction10-8-6-Limit state function4-2Normalizedvariable
Simplified example, buckling of Euler beam, cont.: • Approximate limit state function based on cubic polynomial: (Approximation points are the mean value, the plus/minus two standard deviations, and the plus one standard deviation. The exact failure probability is obtained)
Simplified example, buckling of Euler beam, cont.:Wenext considerthe same case,but withthebucklinglengthbeingLognormal distributed instead of GaussianThe"truereliabilityindex"based onthe"true limit statefunction'now becomes 1.80 (i.e. which corresponds to a failure probabilityof 0.036)Thereliabilityindexbased onthesecond order approximationbecomes 1.44 (i.e. a failure probability equal to 0.075)Forthe linear approximation the exact result is again obtained (seefigure below)For the cubic approximation the exact result is also obtained (seefigure below)19
Simplified example, buckling of Euler beam, cont.: • We next consider the same case, but with the buckling length being Lognormal distributed instead of Gaussian • The ”true reliability index” based on the ”true limit state function” now becomes 1.80 (i.e. which corresponds to a failure probability of 0.036) • The reliability index based on the second order approximation becomes 1.44 (i.e. a failure probability equal to 0.075) • For the linear approximation the exact result is again obtained (see figure below) • For the cubic approximation the exact result is also obtained (see figure below) 19
Simplified example, buckling of Euler beam, cont.:"Exact"limitstatefunctionaftertransformationfromlognormaltothestandardnormalvariableu:(The reliability index corresponds tothepoint g(u)=0)Exactlimitstatefunction86LimitstatefunctionNormalizedvariable
Simplified example, buckling of Euler beam, cont.: • ”Exact” limit state function after transformation from lognormal to the standard normal variable, u: (The reliability index corresponds to the point g(u)=0)