Design for flexure and shear (rectangular & square section)

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 ??????
�=
??????�
??????
and shear formula is ??????
??????=
????????????
??????�

Where:
??????
� = flexure/bending stress
??????
?????? = 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
�ℎ
3
12
the bending stress becomes
??????
�=
??????�
??????

Putting these values in formula
= ??????
ℎ
2
x
12
�ℎ
3

=
6??????
�ℎ
2

So for rectangle bending stress becomes this, incase of
square as b=h so the denominator term will be ℎ
3


for shearing stress
??????
??????=
????????????
??????�
the value of ‘I’ remains same while Q is (
ℎ
2
×??????)(
ℎ
4
) =
�ℎ
2
8

Putting values in the formula
=
??????�ℎ
2
8
×
12
�
2
×1
ℎ
3

=
3??????
2�ℎ
for rectangular section the shearing stresses becomes this, for square it will be
3??????
2ℎ
2