For a circular cross-section, the relationship between the maximum shear stress (qmax) and average shear stress (qav) is gives as
Circular Cross-Section Shear Stress Relationship
In structural mechanics, understanding how shear stress distributes across different cross-sectional shapes is fundamental. For a beam with a circular cross-section, the shear stress is not uniform across its area.
Defining Average Shear Stress ($q_{av}$)
The average shear stress ($q_{av}$) is simply the total shear force ($F$) acting on the cross-section divided by the total area ($A$) of that cross-section.
The formula is:
\(\rm q_{av} = \frac{F}{A}\)
For a circle of radius $r$, the area is \(A = \pi r^2\). Therefore,
\(\rm q_{av} = \frac{F}{\pi r^2}\)
Shear Stress Distribution in a Circle
When a circular cross-section is subjected to shear forces, the shear stress distribution is parabolic. The stress is zero at the top and bottom surfaces of the circle and reaches its maximum value at the neutral axis, which is the horizontal line passing through the center of the circle.
Calculating Maximum Shear Stress ($q_{max}$)
The shear stress at any point within the cross-section can be calculated using the shear formula, which involves the shear force, the first moment of area, the moment of inertia, and the width of the cross-section at that point.
The general formula for shear stress $q(y)$ at a distance $y$ from the neutral axis is:
\(\rm q(y) = \frac{V Q}{I b}\)
Where:
- \(V\) represents the shear force.
- \(Q\) is the first moment of area about the neutral axis of the portion of the cross-section lying above or below the point where stress is calculated.
- \(I\) is the moment of inertia of the entire circular cross-section about the neutral axis. For a circle, \(I = \frac{\pi r^4}{4}\).
- \(b\) is the width of the cross-section at the distance $y$. For a circle, \(b(y) = 2\sqrt{r^2 - y^2}\).
Through detailed integration of this formula over the circular area, the maximum shear stress ($q_{max}$) at the neutral axis ($y=0$) can be determined.
The Relationship Between $q_{max}$ and $q_{av}$
Analysis of the shear stress distribution in a circular cross-section reveals a specific relationship between the maximum shear stress and the average shear stress. The maximum shear stress is consistently higher than the average shear stress.
The derived relationship for a circular cross-section is:
\(\rm q_{max} = \frac{4}{3} q_{av}\)
This indicates that for a circular cross-section, the maximum shear stress is 4/3 times (or approximately 1.333 times) the average shear stress.
Important Questions from Shear Stress and Bending Stress
- For a beam to be classified as a beam of uniform strength, which of the following conditions must be met?
A block is of dimensions of the upper surface 100 mm x 100 mm. The height of the block is 10 mm. A tangential force of 10 kN is applied at the centre of the upper surface. The block is displaced by 1 mm with respect to the lower face. Direct shear stress in the element is:
The ratio of moment carrying capacity of a square cross-section beam of dimension D to the moment carrying capacity of a circular cross-section of diameter D is:
The maximum shear stress in a rectangular cross section _________ per cent greater than the average shear stress on the cross section.
Maximum flexural stress for a cast iron pipe having maximum bending moment 125000 N-mm and section modulus 17017 mm3 will be
