Question:
What is the ratio of maximum shear stress to average shear stress for a circular section?
What is the ratio of maximum shear stress to average shear stress for a circular section?
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For a circular cross-section, the shear stress distribution is parabolic, with maximum shear stress occurring at the neutral axis and zero at the extreme fibers. The maximum shear stress is 4/3 times the average shear stress.
Updated On: May 27, 2025
- $\frac{4}{3}$
- $\frac{3}{4}$
- $\frac{3}{2}$
- $\frac{2}{3}$
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The Correct Option is A
Solution and Explanation
Step 1: Define average shear stress.
For any cross-section, the average shear stress ($\tau_{avg}$) is defined as the total shear force ($V$) divided by the cross-sectional area ($A$).
$\tau_{avg} = \frac{V}{A}$
Step 2: Define maximum shear stress for a circular section.
For a circular cross-section, the maximum shear stress ($\tau_{max}$) occurs at the neutral axis. Its value is given by:
$\tau_{max} = \frac{4}{3} \times \frac{V}{A}$
Step 3: Calculate the ratio of maximum shear stress to average shear stress.
Ratio = $\frac{\tau_{max}}{\tau_{avg}}$
Substitute the expressions from Step 1 and Step 2:
Ratio = $\frac{\frac{4}{3} \times \frac{V}{A}}{\frac{V}{A}}$ Ratio = $\frac{4}{3}$
Step 4: Evaluate the options.
The calculated ratio is $\frac{4}{3}$, which matches option 1. The final answer is $\boxed{\text{1}}$.
For any cross-section, the average shear stress ($\tau_{avg}$) is defined as the total shear force ($V$) divided by the cross-sectional area ($A$).
$\tau_{avg} = \frac{V}{A}$
Step 2: Define maximum shear stress for a circular section.
For a circular cross-section, the maximum shear stress ($\tau_{max}$) occurs at the neutral axis. Its value is given by:
$\tau_{max} = \frac{4}{3} \times \frac{V}{A}$
Step 3: Calculate the ratio of maximum shear stress to average shear stress.
Ratio = $\frac{\tau_{max}}{\tau_{avg}}$
Substitute the expressions from Step 1 and Step 2:
Ratio = $\frac{\frac{4}{3} \times \frac{V}{A}}{\frac{V}{A}}$ Ratio = $\frac{4}{3}$
Step 4: Evaluate the options.
The calculated ratio is $\frac{4}{3}$, which matches option 1. The final answer is $\boxed{\text{1}}$.
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