Question

Calculate the actual bending stress, fb using the flexure formula and the beam loading condition

Answer 1

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

 

ε=cdab=yρ

 

By Hooke's law, ε=σ/E, then

 

σE=yρ;σ=yρE

 

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

 

For this section, the notation fb will be used instead of σ.

 

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=fbdA=yρEdA=EρydA

 

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

 

M=∫dM=∫ydF=∫y(EρydA)

 

M=Eρ∫y2dA

 

but ∫y2dA=I, then

 

M=EIρorρ=EIM

 

substituting ρ=Ey/fb

 

Eyfb=EIM

 

then

fb=MyI

 

and

(fb)max=McI

 

The bending stress due to beams curvature is

fb=McI=EIρcI

fb=Ecρ

 

The beam curvature is:

k=1ρ

where ρ is the radius of curvature of the beam in mm (in), M is the bending moment in N·mm (lb·in), fb 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).