Monday, 1 May 2017

SHEAR STRESS DISTRIBUTION IN RECTANGULAR SECTION

SHEAR STRESS DISTRIBUTION IN RECTANGULAR SECTION

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
ȳ = 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

Where,
F = Shear force (N)
τ = Shear stress (N/mm2)
A = Area of section, where shear stress is to be determined (mm2)
ȳ = 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 (mm4)
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, ȳ could be written as mentioned here
ȳ = y + (d/2-y)/2
ȳ = (d/2+ y)/2

I = bd3/12

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, τav= 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

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