Question

A point in a circular beam of diameter D is subjected to a shear force of F. The ratio of average shear stress to maximum shear stress is

• A. 0.75
• B. 1.33
• C. 1.50
• D. 0.67

Hint:

Be careful with the wording of the ratio.
- Max to Average for a circle is $\tau_{max} / \tau_{avg} = 4/3 \approx 1.33$.
- Average to Max for a circle is $\tau_{avg} / \tau_{max} = 3/4 = 0.75$.
Read the question carefully to provide the correct ratio.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
The question asks for the ratio of the average shear stress to the maximum shear stress ($\tau_{avg} / \tau_{max}$) for a circular cross-section.

Step 2: Key Formula or Approach:
For a beam with a circular cross-section, the shear stress distribution is parabolic, with the maximum value at the neutral axis (the center) and zero at the top and bottom.
The relationship between the maximum shear stress ($\tau_{max}$) and the average shear stress ($\tau_{avg}$) for a circular section is:
\[ \tau_{max} = \frac{4}{3} \times \tau_{avg} \]

Step 3: Detailed Explanation:
We are asked to find the ratio $\tau_{avg} / \tau_{max}$.
We can rearrange the formula to find this ratio:
\[ \frac{\tau_{avg}}{\tau_{max}} = \frac{1}{4/3} \] \[ \frac{\tau_{avg}}{\tau_{max}} = \frac{3}{4} \] \[ \frac{\tau_{avg}}{\tau_{max}} = 0.75 \]

Step 4: Final Answer:
The ratio of average shear stress to maximum shear stress for a circular beam is 0.75.