Section Analysis

Plane Sections Remain Plane

Resultant forces on section with arbitrary stress profile

$M = \int_{z_{2}}^{z_{1}} t \sigma z \; dz\label{b}\tag{Eq.1} \\ F = \int_{z_{2}}^{z_{1}} t \sigma \; dz$

Linear stress profile on rectangular section

Let $\sigma = mz + c\label{a}\tag{Eq.2}$ and $t = const$ where $m = \frac{\sigma_{1} - \sigma_{2}}{z_{1} - z_{2}}$

$c = \sigma_{1} - z_{1}m$ Substitute the expression for $$\sigma$$ ($$\ref{a}$$) into the expression for moment for arbitrary stress profile ($$\ref{b}$$) and hold $$t$$ constant

\begin{align} M & = \int_{z_{2}}^{z_{1}} t (mz + c) z \; dz \\ M &= t \int_{z_{2}}^{z_{1}} mz^{2} + cz \; dz \\ M &= t \biggl[ \frac {mz^{3}}{3} + \frac{cz^{2}}{2} \biggr]_{z2}^{z1} \\ M &= t \biggl( \frac{m}{3}(z_{1}^{3} - z_{2}^{3})+\frac{c}{2}(z_{1}^{2}-z_{2}^{2})\biggr) \end{align}