Wednesday, 3 May 2017

SHEAR STRESS DISTRIBUTION IN THE WEB OF I-SECTION

SHEAR STRESS DISTRIBUTION IN THE WEB OF I-SECTION

We have seen shear stress distribution in rectangular section and shear stress distribution in circular section in our previous sessions. We were discussing shear stress distribution in a beam of I section in our last post, where we have seen that when we talk about shear stress distribution for I section of beam, we will have to draw the shear stress distribution in web and shear stress distribution in flange separately.
Before going ahead, I will request you to please find the post “ Shear stress distribution in flange of I section” for better understanding of current post which is based on the shear stress distribution in web of I section.

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 web of I section of a beam.

 
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)
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)

A. ȳ = Moment of the whole shaded area about the neutral axis
A1 ȳ1 = Moment of shaded area of the flange about the neutral axis
A2 ȳ2 = Moment of shaded area of the web about the neutral axis

We must have to note it here that when we need to draw the shear stress distribution for web of I section, we will have to secure the formula for the moment of the whole shaded area about the neutral axis i.e. A. ȳ and it will be equal to the summation of moment of shaded area of the flange about the neutral axis and moment of shaded area of the web about the neutral axis of the beam section.

 
Moment of the whole shaded area about the neutral axis = (Moment of shaded area of the flange about the neutral axis) + (moment of shaded area of the web about the neutral axis of the beam section)
A. ȳ = A1 ȳ1 + A2 ȳ2
Now we will first figure out the value for A1 ȳ1 i.e. Moment of shaded area of the flange about the neutral axis.
Similarly, let us find out the value for A2 ȳ2 i.e. Moment of shaded area of the web about the neutral axis.
Let us find the moment of the whole shaded area about the neutral axis i.e. A. ȳ
 
Let us use the value of moment of the whole shaded area about the neutral axis i.e. A. ȳ in the equation of shear stress for a section and we will have following expression as displayed here

As we can see here from above result that shear stress will vary with the value of y by following parabolic law and we can also conclude here that value of shear stress will be reduced with increase in the value of y.

Hence we can draw the shear stress distribution diagram for I section beam. Before drawing the shear stress distribution diagram for I section, we must have to secure the value of shear stress at y = d/2 and at y = 0.

Shear stress at neutral axis, y = 0

Shear stress at neutral axis, y = d/2
Shear stress distribution in I-section 

We will discuss bending stress of composite beam in our next post in the category of strength of material. Please comment your feedback and suggestions in comment box provided at the end of this post.

Reference:

Strength of material, By R. K. Bansal
Image Courtesy: Google

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