Show that the shearing stress developed at the neutral axis of a beam with circular cross section is τmax = (4/3)(F/πr2). Assume that the shearing stress is uniformly distributed across the neutral axis.
Solution :
Let us consider the circular section of a beam as displayed in following figure. We have assumed one layer EF at a distance y1 from the neutral axis of the circular section of the beam
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)
Q = 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)
For circular cross-section,
Moment of inertia, I = ПR4/4
b = Width of the given section where shear stress is to be determined.
Let us consider one strip of thickness dy and area dA at a distance y from the neutral axis of the section of the beam.
The width b will be dependent over the value of y
(b/2)2 = R2- y2
b = 2x SQRT(R2 -y2)
The area dA of the strip of thickness dy is
dA = b.dy = 2.SQRT(R2 -y2).dy
By substitute the values of (A. ȳ ) and width b. in the shear stress equation, we will have following expression for shear stress 
When y1 = R , the fiber is in outer surface,
the shear stress is τ = 0
If y1 = 0 ,the fiber is at neutral axis,and I= ПR4/4
The shear stress maximum τmax = 
Average shear stress, τav= Shear force/ Area
Average shear stress, τav= F/ПR2
The max. shear stress τmax = 4/3 τave
Therefore we can say that for a circular section, value of maximum shear stress will be equal to the 4/3 times of mean shear stress. The shear stress distribution in the circular section as shown in below
Thank you for the article :)
ReplyDeleteThank you for this insightful explanation of problems in shearing stresses—this topic often challenges students, and your step-by-step breakdown makes it much easier to grasp. Understanding the behavior of materials under various stress conditions is critical in structural and civil engineering.
ReplyDeleteThis knowledge plays a significant role in modern architectural solutions like Tensile Structure, where precise material behavior under tension and minimal shear is essential for both safety and aesthetics. These structures rely on the proper balance of forces to deliver strength with elegance.
Appreciate the clarity and depth of your content—it’s a valuable resource for learners and professionals alike!