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Continuing the development of the equations in Transfer of Inherent Torsion, lets consider the affect of accidental eccentricity ae, in which the center of loading is arbitrarily shifted by that amount in each direction.
Constant Accidental Eccentricity
Assuming the shift is in the same direction on both levels, the torsion on the lower level now becomes
TL = FU ( CLU + ae – CRL) + FL ( CLL + ae - CRL )
= FU CLU + FL CLL + (FU + FL ) (ae - CRL)
= (FU + FL ) [ (FU CLU + FL CLL ) / (FU + FL) + ae - CRL ]
= F ( CLF + ae - CRL )
where F = FU + FL and CLF is the centroid of the forces FU and FL. In other words, we can treat the upper level force FU as a load on the level below positioned at the unshifted center of loading CLU, on the level above, determine the center of loading of the combined upper force and lower loads, and apply the accidental eccentricity to the total force on the level below. This is shown in the diagram below
Variable Accidental Eccentricity
Accidental eccentricity can from one level to another due to variable building width D, .
The above equation is modified for differing accidental eccentricities on adjacent levels as follows
TL = FU ( CLU + aeU – CRL) + FL ( CLL + aeL - CRL )
so the resulting torsion is
TL= F ( CLF + aeF - CRL )
where aeF is the effective accidental eccentricity given by
aeF = ( aeU FU + aeL FL ) / ( FU + FL )
To demonstrate the effect of using aeF for typical cases, for a penthouse 40% as wide as the level below with the same mass per unit width, aeF is roughly 60% of aeL. For a 6-storey tower where 5 upper storeys are 1/3 the width of the a podium base, aeF is roughly half (14/27) of aeL. Except for the unusual case of a structure shaped like an inverted pyramid, accidental eccentricities are reduced on lower levels when applied to the combined upper/lower level force.
When different from the design-code mandated aeL, a line showing the effective accidental eccentricity aeF appears in the Torsional Analysis Details output.
Design Accidental Eccentricity
As Shearwalls designs worst case of accidental eccentricity in each direction, this method conservatively assumes that the accidental eccentricity occurs in the same direction and at the same intensity on each level of the structure. It is equivalent to examining 2n design cases representing all combinations of positive and negative accidental eccentricity on each of n levels in the structure, and designing for the most severe loading of all these cases.
Although this may seem unlikely, the ASCE 7 Commentary C12.8.4.2 says that it is "typically conservative to assume that the center of mass offsets of all floors and roof occur simultaneously and in the same direction." That is, as we cannot know for certain which levels and directions the accidental eccentricities occur in, it is best practice to assume they occur on all levels.