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

$f_{A} = \iint_{A}^{}{Iwdxdy}$

where, $A$ is the floor area domain across the $x$-$y$ plane, $I$ is the influence surface function across the $x$-$y$ plane, and $w$ is an un-factored distributed load function varying across the $x$-$y$ plane.

For the maximum internal force in a section $f_{Amax}$ resulted under a range of distributed load $w_{\max}$ and $w_{\min}$ can be calculated as

$f_{A}=\iint_{A} I\left[p \cdot w_{\max }+(1-p) \cdot w_{\min }\right] dxdy$

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

$p = \left\{ \begin{matrix} 1 & I > 0 \\ 0 & I \leq 0 \\ \end{matrix} \right.$

And thus the equation can further be rewritten as

$f_{A}=\iint_{A} I\left(\frac{w_{\max }+w_{\min }}{2}\right) d x d y+\iint_{A}\left|I\left(\frac{w_{\max }-w_{\min }}{2}\right)\right| dxdy$

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

$f_{A}=\iint_{A} I\left(\frac{w_{\max }+w_{\min }}{2}\right) d x d y+\sum_{i} \iint_{a_{i}}\left|I\left(\frac{w_{\max }-w_{\min }}{2}\right) d x d y\right|$

which can be further simplified as an absolute sum function

$f_{\text {Amax }}=f_{\text {mean }}+\sum_{i}\left|\Delta f_{i}\right|$

where by definition

\begin{aligned} f_{\text {mean }}&=\frac{1}{2} \iint_{A} I w_{\max } d x d y+\frac{1}{2} \iint_{A} I w_{\min } d x d y \\ \Delta f_{i}&=\frac{1}{2} \iint_{a_{i}} I w_{\max } d x d y-\frac{1}{2} \iint_{a_{i}} I w_{\min } d x d y \end{aligned}

And similarly, the minimum internal force in a section $f_{A,min}$ can be derived as

$f_{A \min }=f_{\text {mean }}-\sum_{i}\left|\Delta f_{i}\right|$

In most situations, $w_{\max}$ and $w_{\min}$ differ only by a scalar factor, which is related to the load factor of safety in ultimate limit state design. Putting

\begin{aligned} &w_{\min }=s_{\min } w \\ &w_{\max }=s_{\max } w \end{aligned}

the equations can be simplified as

\begin{aligned} &f_{\text {mean }}=\frac{s_{\max }+s_{\min }}{2} \iint_{A} I w d x d y \\ &\Delta f_{i}=\frac{s_{\max }-s_{\min }}{2} \iint_{A} I w d x d y \end{aligned}

By comparing these equations to first equation, it can be seen that $f_{mean}$ can be evaluated directly from the analysis with all area fully loaded, and $\Delta f_{i}$ can be evaluated directly from the analysis with load being only applied to the area $a_{i}$, which means the equations can be further simplified as

\begin{aligned} &f_{\text {mean }}=\frac{s_{\max }+s_{\min }}{2} f_{A} \\ &\Delta f_{i}=\frac{s_{\max }-s_{\min }}{2} f_{A} \end{aligned}

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