We were discussing shear stress distribution in rectangular section and shear stress distribution in circular section in our last sessions. We have also
discussed bending stress of composite beam in pure bending during our
previous session.

Now we are going ahead to start new topic i.e. Shear
stress distribution in I- 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 I section of a beam.

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 shear stress distribution in rectangular
section, circular section, I section and T section.

In this post, we will see shear stress distribution in
I section of a beam. We will understand here the variation of shear stress in I
section of a beam through the height of I section.

Let us consider I section 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 = Overall width of I section of the beam

D = Overall depth of I section of the beam

b= Thickness of web of I section of the beam

d= Depth of web of I section of the beam

F = Shear force acting on the I section of the beam

N.A: Neutral axis of the beam section

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

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

ȳ = Distance of C.G of the area, where
shear stress is to be determined, 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)

τ = Shear stress (N/mm

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

^{2})
ȳ = Distance of C.G of the area, where
shear stress is to be determined, from neutral
axis of the beam section (m)

A. ȳ = Moment of the whole shaded
area about the neutral axis

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 first deal with shear stress
distribution for flange of I section of the beam. We will see shear stress distribution for web of I section just after completing this post.

###
**Shear stress distribution in the flange of I section beam**

Let us determine first the value of shaded area of
flange i.e. A and distance of C.G of the shaded area from the neutral axis i.e.
ȳ from above figure and after that we will use
the expression for shear stress at section.

Now we will put the above value in the equation of
shear stress and we will have

As we can see from above equation
that shear stress will vary according to parabolic equation for I section beam and
shear stress will decrease if y will be increased.

####
**Shear stress at upper edge of flange **

Let us consider the upper edge of
the flange of I section beam, for upper edge we will have value of y = D/2 and
after putting the value of y = D/2 in equation of shear stress for I section,
we can easily determined that shear stress at upper edge of the flange and it
will be zero.

####
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Shear stress at lower edge of flange

Let us consider the lower edge of
the flange of I section beam, for lower edge we will have value of y = d/2 and
after putting the value of y = d/2 in equation of shear stress for I section,
we can easily determined that shear stress at lower edge of the flange and it
will be as mentioned here

####
**Shear stress distribution in I section**

###
**Reference:**

Strength
of material, By R. K. Bansal

Image
Courtesy: Google