Flexure Formula


A D V E R T I S E M E N T


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.jpg

 

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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Re: Flexure Formula

Hi many thanks for this formula.

However, there is a mistake on one of the equations, M=|MdF, should read M|ydF.

Re: Flexure Formula

Thank you catching it, I made a correction to it and for easy reference, I copied the erroneous equation below:

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

And this is the correction being made:

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

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