Axial Loads and Normal Strain

When axial load is applied in the x direction: ${ϵ}_x=\frac{{\sigma }_x}{E}$
When there is stress in all directions:
${ϵ}_y={ϵ}_z=-v\frac{{\sigma }_x}{E}$

Poissons Ratio

This is a topic I, for some reason, struggle to apply. Basically, an axial force (stress) causes cdeformation in its perpendicular axes.

Sometimes there’s only one force in one direction, the rest of the stresses just become 0 in that case :)

Poisson's Ratio

v (typically ~0.3 for most elements)

${ϵ}_x=\frac{1}{E}\left[{\sigma }_x-\nu \left({\sigma }_y+{\sigma }_z\right)\right]$
${ϵ}_y=\frac{1}{E}\left[{\sigma }_y-\nu \left({\sigma }_x+{\sigma }_z\right)\right]$
${ϵ}_z=\frac{1}{E}\left[{\sigma }_z-\nu \left({\sigma }_y+{\sigma }_x\right)\right]$

Directly Relating Axial and Transverse Strain

$v={\frac{-{ϵ}_y}{{ϵ}_x}}_{}$

Direct Shear Stress

Applied force to cut through a member (like scissors) $\tau =\frac{F}{A_s}$

Pure Shear Stress + Deformation

The Shear strain y = Change in right angle
Hooks law for shear:
$\tau =G\gamma$
$E=2G\left(1+\nu \right)$

Deflection / Deformation Equation

$\delta =ϵL$