Background· Option Il: (i) The limit state function isevaluated at a number of points. (i) Asmooth analytical function is.fitted to theseReliabilityanalysisPasemter(m) Based on th eomanth rogrdiorfunctioncQmbinatiapiilyehalysis is performed.userFinite ElementEvaluation of limit statefunction for givencomputerparameter combinationprogram
Background • Option II: (i) The limit state function is evaluated at a number of points. (ii) A smooth analytical function is fitted to these points. (iii) Based on the smooth function, the reliability analysis is performed. Reliability analysis computer program Evaluation of limit state function for given parameter combination Finite Element computer program Parameter combinations given by user
Background A third option is also possible, where thepoints which define the response surface aredefined iteratively during the reliabilityReliabilityanalys'sPaas. This is referredomst eptivecombipatispsaiyanbymethod.TuserFinite ElementEvaluation of limit statefunction for givencomputerparameter combinationprogram
Background • A third option is also possible, where the points which define the response surface are defined iteratively during the reliability analysis. This is referred to as an iterative response surface method. Reliability analysis computer program Evaluation of limit state function for given parameter combination Finite Element computer program Parameter combinations given by user
Background· The failure function, G(x), is evaluated for a set of points inthe space of basic variables (i.e. X-space)We then seek a function G,(x) which best fits the discrete setof values of G(x). Typically, is taken to be an nth orderpolynomial.The difference between G(x) and Gz(x) can e.g.be quantifiedby squaring the difference at a number of control points andsummingthesesquares
Background • The failure function, G(x), is evaluated for a set of points in the space of basic variables (i.e. X-space) • We then seek a function G2(x) which best fits the discrete set of values of G(x). Typically, is taken to be an nth order polynomial. • The difference between G(x) and G2(x) can e.g. be quantified by squaring the difference at a number of control points and summing these squares
Second orderpolynomial responsesurfaces. A second order polynomial is very much applied fortheresponse surface (see e.g.Faravelli(1989),Bucher andBourgund (1990), Rajashekhar and Ellingwood (1993)·This hasthefollowing formG2(x) = A + XTB + XTCXwhere A is a constant, BT = [By, B2,...B,] is a vector ofconstants, and Cis an (n x n) matrix containing thecoefficientsforthe secondorderterms:11nC=SVn
Second order polynomial response surfaces • A second order polynomial is very much applied for the response surface (see e.g. Faravelli (1989), Bucher and Bourgund (1990), Rajashekhar and Ellingwood (1993)) • This has the following form: G2(x) = A + X TB + X TCX where A is a constant, B T = [B1 , B2 ,.Bn ] is a vector of constants, and C is an (n x n) matrix containing the coefficients for the second order terms: n n 1 1 1n sym.C . . C .C C
Selection of points where failure functionisevaluated: Coefficients of second order popynomial areobtained by conducting a series of “numericalexperiments" One simple approach is to select poins aroundthe mean point, parallell to the axis of eachvariablé, see circles in figture belowMeanvalue Additid nar of-aiagonal points can also beadded,|see stars in figure below+
Selection of points where failure function is evaluated • Coefficients of second order popynomial are obtained by conducting a series of “numerical experiments” • One simple approach is to select poins around the mean point, parallell to the axis of each variable, see circles in figure below • Additional off-diagonal points can also be added, see stars in figure below Mean value point