Strain on bending

\[\varepsilon_x - -\frac{y}{\rho}\]

Where

• \(\varepsilon_x\) is normal strain
• \(y\) is distance of longitudinal line from neutral surface (where the beam length does not change under bending)
• \(\rho\) is radius of curvature (defining the arc of the bent beam)

Reminder, for an arc (part of a circle):

\[L = \theta r\]

Where

• \(L\) is length of arc
• \(\theta\) is angle
• \(r\) is radius

Sometimes a quantity called curvature is defined as \(\kappa = \frac{1}{\rho}\), and the equations on this page are expressed in terms of \(\kappa\) instead of \(\rho\).

Flexure formula

\[\sigma_x = -\frac{M y}{I}\]

Where

• \(\sigma_x\) is normal stress
• \(M\) is bending moment
• \(y\) is distance of plane from neutral surface
• \(I\) is inertial moment of the cross section (with respect to the neutral axis)

Axial loads

If axial loads are present, we can add such force to the flexure formula to get the normal stress:

\[\sigma_x = \frac{N}{A} -\frac{M y}{I}\]

Where

• \(N\) is axial force
• \(A\) is cross-section area

This assumes the axial load acts in the centroid of the cross area. If this is not the case, we can decompose it as a central force and a moment caused by the offset:

\[\sigma_x = \frac{N}{A} - \frac{Ney}{y} -\frac{M y}{I}\]

Where

• \(e\) is distance from centroid to where N is applied

In this last equation, \(M\) is any other bending moments from external forces.

Maximum stress in cross section

From the previous equations, we can see that \(\sigma_x\) is maximum when \(y\) is maximum (aka. the furthest point from the center of a beam).

Let \(c = y_{\text{max}}\) be the furthest point from neutral axis in a beam. Assuming a beam is symmetric about the neutral axis, \(c\) is the same from up and down the beam. Then we can rewrite flexure formula as:

\[\sigma_{\text{max}} = \frac{M c}{I}\]

Let \(S = \frac{I}{c}\) be the section moduli. Then the maximum bending stress becomes:

\[\sigma_{\text{max}} = \frac{M}{S}\]

Where the moment can be calculated as:

\[M = \frac{EI}{\rho}\]

Where

• \(E\) is Young’s Modulus
• \(I\) is inertial moment of the cross section (with respect to the neutral axis)
• \(\rho\) is radius of curvature

Shear stress in cross section

\[\tau = \frac{VQ}{Ib}\]

Where

• \(\tau\) is shear stress in a cross section for a bent beam
• \(V\) is shear force
• \(Q = \int y dA\) is first moment of area above the level of \(\tau\) section
• \(I\) is inertial moment of the cross section
• \(b\) is width of beam in the cross section

Q is annoying to do by hand, for which it is common to have some tables with values of Q for different shapes (much like for moment of inertia).