Stiffness and deformation

\[F = k\delta\]

Where

• \(F\) is force
• \(k\) is stiffness
• \(\delta\) is deformation

\[\delta = \frac{FL}{EA}\]

• \(\delta\) is deformation
• \(F\) is force
• \(L\) is length
• \(E\) is Young’s Modulus
• \(A\) is area

If non-uniform loading:

\[\delta_i = \frac{N_i L_i}{EA}\]

Where

• \(N_i\) is an internal force
• \(L_i\) is length from reference where the internal force is measured

Note: you need to consider ALL forces in the beam that act in the same axis of deformation.

In differential form:

\[d\delta = \frac{N(x) dx}{E A(x)}\]

Stresses on inclined sections

\[\sigma_{\theta} = \frac{F}{A} \cos^2{\theta}\]

Where

• \(\sigma_{\theta}\) is normal stress on angle \(\theta\)
• \(F\) is force
• \(A\) is area
• \(\theta\) is angle of inclination

\[\tau_{\theta} = -\frac{F}{A} \sin{\theta} \cos{\theta}\]

Where \(\tau_{\theta}\) is shear stress on angle \(\theta\)