Question

The maximum shear stress \((q_{\max})\) in a rectangular beam is ____

• A. \(1.25\) times the average
• B. \(1.50\) times the average
• C. \(1.75\) times the average
• D. \(2.0\) times the average

Hint:

For common sections: \[ \boxed{ \begin{aligned} \text{Rectangular section: }& \tau_{\max}=1.5\,\tau_{\text{avg}}\\ \text{Circular section: }& \tau_{\max}=\frac{4}{3}\tau_{\text{avg}} \end{aligned} } \]

Answer 1

The Correct Option is B

Step 1: Picture the shear stress diagram.
For a rectangular cross section, shear stress varies parabolically over the depth, being zero at the top and bottom fibres and reaching a peak right at the neutral axis.
Step 2: Use a property of the parabola.
For a parabolic shape, the average height works out to two thirds of the peak height, so \[ q_{avg} = \frac{2}{3}q_{max} \]
Step 3: Rearrange for the maximum. \[ q_{max} = \frac{3}{2}q_{avg} = 1.5\,q_{avg} \]
so the peak shear stress at the neutral axis is exactly one and a half times the average shear stress over the section.
\[ \boxed{1.50\ \text{times the average}} \]