Shear Stress Ratio Explained

Shear stress is the stress component parallel to a material's cross-section. In structural members like beams, the shear stress distribution is not uniform across the cross-section. It varies depending on the shape of the section and the applied load. The question asks about the specific ratio between the maximum shear stress ($\tau_{max}$) and the average shear stress ($\tau_{avg}$) for a particular shape, which is given as a circular section.

Shear Stress Ratio Comparison

The distribution of shear stress and the resulting ratio of maximum to average shear stress differ for various cross-sectional shapes. Here’s a comparison for common shapes:

Circular Section Shear Stress Analysis

For a circular cross-section subjected to shear force (as in beam bending), the shear stress distribution is parabolic. The shear stress is zero at the extreme top and bottom points of the circle and reaches its maximum value at the neutral axis, which passes through the center of the circle.

The average shear stress ($\tau_{avg}$) is calculated by dividing the total shear force ($V$) by the total cross-sectional area ($A$):

$$ \tau_{avg} = \frac{V}{A} $$

For a circular section, the maximum shear stress ($\tau_{max}$) occurs at the neutral axis and is given by the formula:

$$ \tau_{max} = \frac{4V}{3A} $$

To find the ratio of the maximum shear stress to the average shear stress, we divide the expression for $\tau_{max}$ by the expression for $\tau_{avg}$:

$$ \frac{\tau_{max}}{\tau_{avg}} = \frac{\frac{4V}{3A}}{\frac{V}{A}} $$

Simplifying this expression, we get:

$$ \frac{\tau_{max}}{\tau_{avg}} = \frac{4V}{3A} \times \frac{A}{V} = \frac{4}{3} $$

Therefore, the ratio of maximum shear stress to average shear stress in a circular section is indeed $\frac{4}{3}$.

Conclusion

Based on the standard formulas for shear stress distribution in beams, the ratio of maximum shear stress to average shear stress for a circular section is $\frac{4}{3}$. This confirms the premise of the question.