Normal (axial) stress: $\sigma = \frac{F}{A}$

Direct (average) shear stress: $\tau_{ave} = \frac{V}{A}$

Normal (axial) strain: $\epsilon = \frac{\delta}{L}$ (also denoted as $\epsilon_{axial}$ )

Lateral strain: $\epsilon_{lateral} = -\nu \,\epsilon_{axial}$

Shear strain: $\gamma\approx \tan \gamma = \frac{\delta}{h}$ (note this is defined by a change in angle!)

Force-elongation-temperature relation: $\delta = \frac{F\,L}{E\,A} + \alpha\,L\,\Delta T \qquad k = \frac{E A}{L} = \frac{1}{f}$

Constitutive relations: $\sigma = E\,\epsilon \qquad \tau = G\, \gamma$

Isotropic materials: $G = \frac{E}{2(1+\nu)}$

Normal stresses on inclined plane: $\sigma_n = {\bf n}\cdot\ {\bf t}^{n}={\bf n}\cdot\ {\bf T}\,{\bf n} = \sigma_{x} \cos ^2(\theta) + 2 \,\tau_{xy} \sin(\theta) \cos(\theta)+\sigma_{y} \sin ^2(\theta)$

Shear stresses on inclined plane: $\tau_{ns} = {\bf s}\cdot\ {\bf t}^{n}={\bf s}\cdot\ {\bf T}\,{\bf n} = (\sigma_y - \sigma_{x}) \sin(\theta) \cos(\theta) + \tau_{xy} ( \cos^2(\theta) - \sin^2(\theta) )$

Torsion: $\tau = \frac{T\,r}{J} \qquad \tau = G \gamma \qquad \gamma = \frac{r \phi}{L} \qquad \phi = \frac{T\,L}{J\,G}$

For circular cross section: $J = \frac{\pi d^4}{32}$ and $I_{c'} = \frac{\pi r^4}{4} = \frac{\pi d^4}{64}$

Relations among distributed load, shear and bending moment: $\frac{dV}{dx} = -w$ and $\frac{dM}{dx} = V$

Parallel axis-theorem: $I_c = I_{c'} + A\, d_{cc'}^2$

Centroid of the semi-circle: $\bar{y} = \frac{4R}{3 \pi}$

Normal stress: $\sigma_x = \frac{F}{A} + \frac{M_y\,z}{I_y} - \frac{M_z\,y}{I_z}$

Shear stress (due to transverse shear force): $\tau = \frac{V\,Q}{I\,t}$

First moment: $Q = A \, \bar{y}$

Generalized Hooke's law:

$\epsilon_x = +\frac{1}{E} \sigma_x - \frac{\nu}{E} \sigma_y -\frac{\nu}{E} \sigma_z$

$\epsilon_y = -\frac{\nu}{E} \sigma_x + \frac{1}{E} \sigma_y -\frac{\nu}{E} \sigma_z$

$\epsilon_z = -\frac{\nu}{E} \sigma_x - \frac{\nu}{E} \sigma_y + \frac{1}{E} \sigma_z$

$\gamma_{xy} = \frac{\tau_{xy}}{G}$

$\gamma_{xz} = \frac{\tau_{xz}}{G}$

$\gamma_{yz} =\frac{\tau_{yz}}{G}$

Transformation of Plane-Stress:

$\sigma_{x'} = \frac{ \sigma_x + \sigma_y}{2} + \frac{ \sigma_x - \sigma_y}{2}\cos(2\theta) + \tau_{xy} \sin(2\theta)$

$\sigma_{y'} = \frac{ \sigma_x + \sigma_y}{2} - \frac{ \sigma_x - \sigma_y}{2}\cos(2\theta) - \tau_{xy} \sin(2\theta)$

$\tau_{x'y'} = - \frac{ \sigma_x - \sigma_y}{2}\sin(2\theta) + \tau_{xy} \cos(2\theta)$

Mohr's circle center: $\sigma_{ave} = \frac{ \sigma_x + \sigma_y}{2}$

Mohr's circle radius: $R = \sqrt{\left(\frac{ \sigma_x - \sigma_y}{2}\right)^2+ \left(\tau_{xy}\right)^2}$

Principal stresses: $\sigma_1 = \sigma_{ave} + R$ and $\sigma_2 = \sigma_{ave} - R$

Orientation of principal plane: $\tan(2 \theta_{p}) = \frac{\tau_{xy}}{(\sigma_x-\sigma_y)/2}$

Cylindrical pressure vessels: $\sigma_h = \frac{pr}{t}$ and $\sigma_a = \frac{pr}{2t}$

Tresca criterion:

$|\sigma_1| = \sigma_Y, \, |\sigma_2| = \sigma_Y \qquad \textrm{when} \,\sigma_1,\,\sigma_2 \, \textrm{have the same sign}$

$|\sigma_1 - \sigma_2| = \sigma_Y \qquad \qquad \,\textrm{when} \,\sigma_1,\,\sigma_2 \, \textrm{have opposite sign}$

Von-Mises criterion:

$\sigma_1^2 -\sigma_1 \,\sigma_2 + \sigma_2^2 = \sigma_Y^2$

Deflection: $y'' = M/(EI)$

Buckling:

$P_{cr} = \frac{\pi^2 \,EI}{(Le)^2}$

pinned-pinned: $Le = L$

pinned-fixed: $Le = 0.7\,L$

fixed-fixed: $Le = 0.5\,L$

fixed-free: $Le = 2\,L$