We were discussing section modulus of the beam and derivation for beam bending equation in our
previous session. We have also discussed assumptions made in the theory of simple bending and expression for bending stress in pure bending during our last
session.

Now we are going ahead to start new topic i.e. Shear
stress distribution in rectangular section in the strength of material with the
help of this post.

Let us go ahead step by step for easy understanding,
however if there is any issue we can discuss it in comment box which is
provided below this post. So let us come to the main subject i.e. Shear stress
distribution in rectangular section.

In our previous session, we were discussing the
bending stress produced in a beam which is subjected to a pure bending. We have
assumed there that beam will be subjected with a pure bending moment and shear
force will be zero and hence shear stress will also be zero.

In actual practice, beam will be subjected with
shear force also and therefore shear stress too. Shear force and hence shear
stress will vary section to section.

We will see here the shear stress distribution
across the various sections such as rectangular section, circular section, I
section and T section. In this post, we will see shear stress distribution in
rectangular section.

Let us consider the rectangular section ABCD of a
beam as displayed in following figure. We have assumed one layer EF at a
distance y from the neutral axis of the beam section.

We have following information from above figure.

b= Width of the rectangular section

d= Depth of the rectangular section

N.A: Neutral axis of the beam section

EF: Layer of
the beam at a distance y from the neutral axis of the beam section

A= Area of section CDEF, where shear stress is to be
determined

È³ = Distance of C.G of the area
CDEF from neutral axis of the beam section

####
*Shear stress at a section will be given by following formula
as mentioned here*

*Shear stress at a section will be given by following formula as mentioned here*

Where,

F = Shear force (N)

Ï„ = Shear stress (N/mm

^{2})
A = Area of section, where shear stress is to be determined
(mm

^{2})
È³ = Distance of C.G of the area
CDEF from neutral axis of the beam section (m)

I = Moment of inertia of the given section about the
neutral axis (mm

^{4})
b= Width of the given section where shear stress is
to be determined (m)

Let us secure the value of the area of section,
where shear stress is to be determined and we can write it as mentioned here

A= b x (d/2-y)

Distance of C.G of the area CDEF
from neutral axis of the beam section, È³ could be written as mentioned here

È³ = y + (d/2-y)/2

È³ = (d/2+ y)/2

I = bd

^{3}/12
Let us use the value of above parameters in equation
of shear stress and we will have

We can easily say from above equation that maximum
shear stress will occur at y = 0 or maximum shear stress will occur at neutral
axis and value of shear stress will be zero for the area at the extreme ends.

We will also find the value of maximum shear stress and
it could be easily calculated by using the value of y = 0 and therefore we will
have following formula for maximum shear stress as displayed here in following
figure.

As we know that average shear stress or mean shear
stress will be simply calculated by dividing shear force with area and
therefore we can say that

Average shear stress, Ï„

_{av}= Shear force/ Area
Therefore we can say that for a rectangular section,
value of maximum shear stress will be equal to the 1.5 times of mean shear
stress.

We can say, from equation of shear stress for a
rectangular section, that shear stress distribution diagram will follow
parabolic curve and we have drawn the shear stress distribution diagram for a
rectangular section as displayed in following figure.

We will
discuss another topic i.e. Shear stress distribution in circular section in
the category of strength of material in our next post.

###
**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google