"In theory there is no
difference between Theory and Practice.
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
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 Pfail=10%. This has the effect of moving the joint
probability density to a new mean =
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 withinb 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.