Tuesday, 2 May 2017

SHEAR STRESS DISTRIBUTION IN I SECTION

SHEAR STRESS DISTRIBUTION IN I SECTION

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


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, 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 (mm4)
b= Width of the given section where shear stress is to be determined (m)

 
We must have to understand here that when we talk about I section of beam, we will have to draw the shear stress distribution in web and shear stress distribution in flange separately.

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.

 

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

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