FORM : First Order Reliability Analysis

In the most general case, the state of a structure is described by a structural model characterized by many stochastic and/or deterministic variables X (the stresses in the masonry panel sy, txy, the material properties, ...), that can vary in function of some independent parameters t (time or place). Moreover, design parameters W (the choice of the used material, dimensions, ...), chosen by the designer, can have an influence on the probability distribution and/or deterministic value of X. Probabilistic design means that a structure is designed so that the probability of failure pf does not exceed a certain threshold pfs during a prescribed period tL (life cycle): pf < pfs.
In determining the safety of a structure mostly more than one failure mode has to be reviewed. failure mode i can be described mathematically in a limit state function: gi(X,W,t). failure is defined as exceeding a certain limit state function, resulting in the following three situations (figure 1):
  1. gi(X,W,t) < 0 failure in failure mode i,
  2. gi(X,W,t) = 0 the structure is in a critical situation,
  3. gi(X,W,t) > 0 a certain safety margin does exist due to failure mode i.
Most generally, variables such as loads and stresses are time-dependent. Commonly these stochastic values are transformed into time-independent variables, set up for a certain reference period (tL). The problem is than reduced to : gi(X,W,t) --> gi(X,p) for a certain time tL.
For the limit state function under consideration and a preset reference period, the probability of failure is calculated as: pf,i = P[gi(XWp) <0]. To perform the calculation a FORM-algorithm is used. the algorithm, as implemented in MATLAB, calculates the probability of failure pf or the reliability index ( b = - F-1(pf), in which F is the standardized cumulative normal distribution ) as a direct measure for the reliability of the structure.



b represents a measure for the shortest distance from the mean values of the parameter x (x1, x2) and the failure point x*, in units of standard deviations. the method accounts for the probability distribution of the different variables but linearises the limit state function at the failure point x*, which is why the method is called a "first order" reliability method.

The probability of failure pf,i, calculated for the limit state function gi(x,p), needs to be combined with the other possible failure criteria of the structural element.
The theory of system reliability provides a method to obtain a sharp lower and upper bound for the reliability or probability of failure pf in terms of mode failure probabilities, as: P(F) = P(F1) P(F2 S1) P(F3 S1 S2) . . . , where Fi denotes the event "failure of the structure due to failure in the i-the mode, under all loading" and Si denotes the complementary event "survival in the i-the mode under all loading."


Copyright 1996, Katholieke Universiteit Leuven
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URL: http://www.bwk.kuleuven.ac.be/bwk/materials/ls02.htm