Beam theory: Tension/compression and shear

Engineering Fundamentals

Tension/compression

Stress

$\sigma=\frac{F_z}{A}=\frac{F_z}{w\cdot h}$

Strain

$\epsilon=\frac{\Delta l}{l}$

Young’s modulus (modulus of elasticity)

The Young’s modulus is only dependent on the type of material and not on its dimensions.
$E=\frac{\sigma}{\epsilon}=\frac{F_z}{A\cdot\epsilon}$

Stiffness

In contrast with the Young’s modulus, stiffness is dependent on both the type of material and its dimensions.
$C_z=\frac{F_z}{\Delta l}=\frac{EA}{l}$

Poisson’s ratio

The Poisson’s ratio is a measure of the ratio between the transverse strain and axial strain during tension or compression.
$\nu=\frac{\frac{\Delta h}{h}}{\frac{\Delta l}{l}}=\frac{\Delta h\cdot l}{\Delta l\cdot h}\$
Note: for metals $\nu\approx0.3$

Shear

Average shear stress

$\tau=\frac{F_y}{A}=\frac{F_y}{w\cdot h}$

Shear strain

The shear strain is defined as the deformation per length. The deformation is parallel to the cross section of the beam.
$\gamma=\frac{\delta}{l}=\tan{\alpha}$

Shear modulus

$G=\frac{\tau}{\gamma\ }=\frac{F_y}{A\cdot\gamma}=\frac{E}{2\left(1+\nu\right)}$
Note: for metals $\nu\approx0.3$ thus $G\approx0.385E$

Shear stiffness

$C_y=\frac{F_y}{\delta}=\frac{G\cdot A}{l}$

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