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F„( x )= F,,,( x„lx,,x2,…x,„, )• F„.,( x„,Ix,,x2,…x,„2 )…F„( x, ) (11.36) the transformation from the X-space to the U-space may be performed using the so-called Rosenblatt transformation:
4,(u, )= F„, (x, ) (1)( :42 ). F„,( x2lx, )
u. )= F„.( x„Ix,,x2,…x, ) where n is the number of random variables, F„,(x,lx,,x2 ….. x,,) is the conditional probability distribution function for the i’th random variable given the realisations of and 4t(.) is the standard normal probability distribution function. From the transformation given by Equation (11.37) the basic random variables X may be expressed in terms of standardised normal distributed random variables U by
x,= F;:(4)(u, )) x2 = F;12( (1)( u21x1 x„ = 4( u , x2,… x,„, )
In some cases the Rosenblatt transformation cannot be applied because the required conditional probability distribution functions cannot be provided. In such cases other transformations may be useful such as e.g. the Nataf transformation se e.g. Madsen et al. (1986). Standard commercial software for FORM analysis usually include a selection of possibilities for the representation of dependent non-normally distributed random variables.
11.7 Software for Reliability Analysis Several software packages are available for FORM analysis following the principles out lined in the forgoing sections. Most of the programs are more or less self-explanatory provided that the basic principles of FORM analysis are known. The reader is referred to software packages such as STRUREL and VaP for which more information is available on the web.
7 Assessment of Partial Safety Factors by FORM Analysis In code based design formats such as the Eurocodes and the Swisscodes, design equations are prescribed for the verification of the capacity of different types of structural components in regard to different modes of failure. The typical format for the verification of a structural component is given as design equations such as:
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