Flexure Formula

Stresses caused by the bending moment are known as flexural or bending stresses. Consider a beam to be loaded as shown.

Consider a fiber at a distance [math]y[/math] from the neutral axis, because of the beam’s curvature, as the effect of bending moment, the fiber is stretched by an amount of [math]cd[/math]. Since the curvature of the beam is very small, [math]bcd[/math] and [math]Oba[/math] are considered as similar triangles. The strain on this fiber is

By Hooke’s law, [math]\varepsilon = \sigma / E[/math], then

which means that the stress is proportional to the distance [math]y[/math] from the neutral axis.

Considering a differential area [math]dA[/math] at a distance [math]y[/math] from N.A., the force acting over the area is

The resultant of all the elemental moment about N.A. must be equal to the bending moment on the section.

[math]\displaystyle M = \int dM = \int y\, dF = \int y \, \left( \frac{E}{\rho}y \, dA \right)[/math]

[math]\displaystyle M = \frac{E}{\rho} \int y^2 \, dA[/math]

but [math]\int y^2 \, dA = I \,\,[/math], then

[math]M = \dfrac{EI}{\rho} \,\, \text{or} \,\, \rho = \dfrac{EI}{M}[/math]

substituting [math]\rho = Ey / f_b[/math]

[math]\dfrac{Ey}{f_b} = \dfrac{EI}{M}[/math]

then

and

The bending stress due to beams curvature is

[math]f_b = \dfrac{Mc}{I} = \dfrac{\dfrac{EI}{\rho}c}{I}[/math]

The beam curvature is:

where [math]\rho[/math] is the radius of curvature of the beam in mm (in), [math]M[/math] is the bending moment in N·mm (lb·in), [math]f_b[/math] is the flexural stress in MPa (psi), [math]I[/math] is the centroidal moment of inertia in mm⁴ (in⁴), and [math]c[/math] is the distance from the neutral axis to the outermost fiber in mm (in).