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Patterned Load Analysis

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

fA=AIwdxdyf_{A} = \iint_{A}^{}{Iwdxdy}

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

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

fA=AI[pwmax+(1p)wmin]dxdyf_{A}=\iint_{A} I\left[p \cdot w_{\max }+(1-p) \cdot w_{\min }\right] dxdy

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

p={1I>00I0p = \left\{ \begin{matrix} 1 & I > 0 \\ 0 & I \leq 0 \\ \end{matrix} \right.

And thus the equation can further be rewritten as

fA=AI(wmax+wmin2)dxdy+AI(wmaxwmin2)dxdyf_{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 AA can always be separated into a series of smaller and non-overlapping area aia_{i}, which exclusively covers the entire area. Assume the sign of II in each individually separated area aia_{i} does not change, i.e. II is always positive or negative across the xx-yy plane within an area aia_{i}, then the equation can be expanded as

fA=AI(wmax+wmin2)dxdy+iaiI(wmaxwmin2)dxdyf_{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

fAmax =fmean +iΔfif_{\text {Amax }}=f_{\text {mean }}+\sum_{i}\left|\Delta f_{i}\right|

where by definition

fmean =12AIwmaxdxdy+12AIwmindxdyΔfi=12aiIwmaxdxdy12aiIwmindxdy\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 fA,minf_{A,min} can be derived as

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

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

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

the equations can be simplified as

fmean =smax+smin2AIwdxdyΔfi=smaxsmin2AIwdxdy\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 fmeanf_{mean} can be evaluated directly from the analysis with all area fully loaded, and Δfi\Delta f_{i} can be evaluated directly from the analysis with load being only applied to the area aia_{i}, which means the equations can be further simplified as

fmean =smax+smin2fAΔfi=smaxsmin2fA\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