this document includes the derivation of short formula for bending and shear stresses which are observed in beams of different sections.

1. University of Engineering and technology Peshawar, Pakistan Prepared by: Muhammad Bilal
Design for Flexure and Shear
For any cross-sectional shape, flexure and shear are given in the following formulas:
Flexure formula 𝑓𝑏=
𝑀𝑐
𝐼
and shear formula is 𝑓𝑣=
𝑉𝑄
𝐼𝑏
Where:
𝑓𝑏 = flexure/bending stress
𝑓𝑣 = shearing stress
M = maximum moment applied to the beam
V = maximum shear applied to the beam
I = moment of inertia about the neutral axis
Q = moment of area above or below neutral axis
b = breadth
h= height/depth
c= outermost fiber distance from neutral axis
Now consider a rectangular section
in such case we know the value of ‘c’ i.e. h/2 outermost distance from neutral axis and also we know
the value of ‘I’ moment of inertia of rectangle is
𝑏ℎ3
12
the bending stress becomes
𝑓𝑏=
𝑀𝑐
𝐼
Putting these values in formula
= 𝑀
ℎ
2
x
12
𝑏ℎ3
=
6𝑀
𝑏ℎ2
So for rectangle bending stress becomes this, incase of
square as b=h so the denominator term will be ℎ3
for shearing stress
𝑓𝑣=
𝑉𝑄
𝐼𝑏
the value of ‘I’ remains same while Q is (
ℎ
2
× 𝑏)(
ℎ
4
) =
𝑏ℎ2
8
Putting values in the formula
=
𝑉𝑏ℎ2
8
×
12
𝑏2
×1
ℎ3
=
3𝑉
2𝑏ℎ
for rectangular section the shearing stresses becomes this, for square it will be
3𝑉
2ℎ2