Flexure Formula

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

Flexure of a beam

 

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

$ \varepsilon = \dfrac{cd}{ab} = \dfrac{y}{\rho} $

 

By Hooke's law, $ \varepsilon = \sigma / E $, then
 

$ \dfrac{\sigma}{E} = \dfrac{y}{\rho}; \,\, \sigma = \dfrac{y}{\rho}E $

 

which means that the stress is proportional to the distance $ y $ from the neutral axis.
 

For this section, the notation $ f_b $ will be used instead of $ \sigma $.

 

flexure-analysis-in-a-section-of-beam.gif

 

Considering a differential area $ dA $ at a distance $ y $ from N.A., the force acting over the area is
 

$ dF = f_b \, dA = \dfrac{y}{\rho}E \, dA = \dfrac{E}{\rho}y \, dA $

 

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

$ \displaystyle M = \int dM = \int y\, dF = \int y \, \left( \frac{E}{\rho}y \, dA \right) $

$ \displaystyle M = \frac{E}{\rho} \int y^2 \, dA $
 

but $ \int y^2 \, dA = I \,\, $, then
 

$ M = \dfrac{EI}{\rho} \,\, \text{or} \,\, \rho = \dfrac{EI}{M} $
 

substituting $ \rho = Ey / f_b $
 

$ \dfrac{Ey}{f_b} = \dfrac{EI}{M} $
 

then

$ f_b = \dfrac{My}{I} $

 

and

$ (f_b)_{max} = \dfrac{Mc}{I} $

 

The bending stress due to beams curvature is

$ f_b = \dfrac{Mc}{I} = \dfrac{\dfrac{EI}{\rho}c}{I} $

$ f_b = \dfrac{Ec}{\rho} $

 

The beam curvature is:

$ k = \dfrac{1}{\rho} $

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

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