By the principle of superposition for linear elastic structural systems,
the internal force in a section can be calculated as

where, is the floor area domain across the - plane, is the
influence surface function across the - plane, and is an un-factored distributed load function
varying across the - plane.

For the maximum internal force in a section resulted under a
range of distributed load and can be calculated as

where, is a binary function related to the influence surface as

And thus the equation can further be rewritten as

The floor area domain can always be separated into a series of
smaller and non-overlapping area , which exclusively covers the
entire area. Assume the sign of in each individually separated area
does not change, i.e. is always positive or negative across
the - plane within an area , then the equation can be expanded
as

which can be further simplified as an absolute sum function

where by definition

And similarly, the minimum internal force in a section can be
derived as

In most situations, and differ only by a scalar
factor, which is related to the load factor of safety in ultimate limit
state design. Putting

the equations can be simplified as

By comparing these equations to first equation, it can be seen that
can be evaluated directly from the analysis with all area
fully loaded, and can be evaluated directly from the
analysis with load being only applied to the area , which means
the equations can be further simplified as

This item was written by Ir. Dr. Don Y.B. Ho of Ove Arup & Partners,
Hong Kong Ltd and is reproduced here with permission