"In theory there is no difference between Theory and Practice.
In practice, there is."

How FORM/SORM is Supposed to Work

part 2 of 3

FORM (First Order Reliability Method) has been used extensively by engineers for nearly two decades. What has been studiously ignored, however, is how well FORM/SORM assumptions hold up in describing real data.

It is standard practice to illustrate the idea with only a one-dimensional demand and a one-dimensional capacity so that the concept can be plotted in a n=2 dimensional plot, like Figure 1
 

The figure shows the  90%, 90%, 95% and 99% concentric circles of the bivariate normal density.  The MPP appears as a red dot.  While it may appear that the bivariate normal density is formed by rotating a univariate normal density, that is not the case.  The density of the one-dimensional standard normal distribution at x=0 is .   The density of the two-dimensional standard normal distribution at (x=0, y=0) is , as it must be so that its double integral equals one.  For an n-dimensional standard multivariate normal density the maximum ordinate at the origin is .  In summary: the MVN is not simply a rotation of the standard Normal.  The foundation of NESSUS is built on sand.

Of course a design having capacity exactly equal to its demands would have an unacceptably high failure rate of 50%.  To remedy this we must either increase the capacity or decrease the demand, in either case thus redefining the joint probability density such that the new g-function is now a line where demand=capacity-.  can be selected to provide the desired failure rate.  Since we are comparing the first 7 of 68 real laboratory specimen failures, we choose to make P_fail=10%.  This has the effect of moving the joint probability density to a new mean =
(0, ). It is convenient (but confusing) to re-define the origin to be
(0, 0) as before, which means that the g-function is no longer shown as a line partitioning failures from non-failures and where demand equals capacity, going through (0, 0), but rather going through (0, ) and having the desired failure rate (10% here).  In other words we have redefined failure to be "having as margin less than " rather than "fails."

Notice too that this red circle would contain fewer than half (0.473) of all observations, not 90% as might be guessed.  (And this is for a bivariate joint density.  As the number of dimensions increases, the hypervolume within b standard deviations of the centroid diminishes exponentially.  In 6 dimensions, less than 3% of the total hypervolume would be within 1.282 standard deviations from the centroid, while 3.09 standard deviations would envelop only 20% of the hypervolume, not 99.9%) 

Now, our situation is in three dimensions, the Paris Law parameters, C, n, that combine to define capacity, and the applied stress, .  The g-function is given by this equation. 

But these 68 tests were run with = constant, so we can still plot the FORM/SORM results in two dimensions, C, n.  Stress is normal to C, n but is constant.  The resulting plot is Figure 3, next page.