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"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
 

g-function as the line where capacity=demand exactly, an admittedly poor design having a 50% failure rate.

The density is centered at (0, 0). The bivariate density is bisected again by the b g-function. A standard normal density is plotted along that line, and the probability of the "most probable point" is indicated by the shaded region in the lower right.  Anything below the y=x line (i.e. Capacity < Demand) will fail.

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 oneOvrSqrt2pi.gif.   The density of the two-dimensional standard normal distribution at (x=0, y=0) is oneOvr2pi.gif, 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 oneOvrKroot2pi.gif.  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 Pfail=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."

 

Figure 2 plots the idealized joint bivariate normal probability density of demand and capacity, showing the g-function as the line where capacity=demand+.  The density is re-centered at (0, 0).  The bivariate density is bisected by the b vector through the origin to the closest point on the  g-function.  A standard normal density is plotted along that line, and the probability of the "most probable point" is indicated by the shaded region in the upper left.

 

 


The figure shows the  90%, 90%, 95% and 99% concentric circles of the bivariate normal density.  The red circle is at a radius of 1.282 standard deviations, and indicates that 10% of the observations would be in the cross-hatched failure region in the upper left formed by the tangent (which is the g-function) to the MPP (red dot). 

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.

 

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