Question

A rectangular section beam subjected to a bending moment M varying along its length is required to develop the same maximum bending stress at any cross section. If the depth of the section is constant, then its width will vary as:

• A. \( M \)
• B. \( M^{1/2} \)
• C. \( M^2 \)
• D. \( \frac{1}{M} \)

Hint:

Considering the variation of section properties along the length of beams can lead to more efficient designs and material usage.

Answer 1

The Correct Option is B

To maintain constant bending stress (\(\sigma\)), the section modulus (\(S = \frac{I}{y}\)) must be proportional to the bending moment \(M\). If \(d\) is constant and \(I = \frac{1}{12} w d^3\), \(w\) must vary as \(M^{1/2}\) to keep \(\sigma\) constant: \[ \sigma = \frac{M}{S} = \frac{M}{\frac{wd^2}{6}} \quad \Rightarrow \quad w \propto M^{1/2} \]