Question

A simply supported beam is subjected to a concentrated moment \( M \) at the mid-span as shown in the figure. The magnitude of the bending moment at a distance of \( L/4 \) from the left support \( A \) is equal to: 

• A. \( M \)
• B. \( \frac{ML}{4} \)
• C. \( \frac{M}{4} \)
• D. \( \frac{M}{2} \)

Hint:

1. For concentrated moments, the bending moment is linearly distributed along the span of the beam.
2. Use equilibrium conditions to calculate reactions and moments for beams under various loading conditions.
3. Visualize the bending moment diagram to understand moment distribution along the beam.

Answer 1

The Correct Option is C

Step 1: Determine the reactions at the supports. For a simply supported beam with a concentrated moment \( M \) at the mid-span, the reactions at supports \( A \) and \( B \) are equal and opposite to maintain equilibrium. \[ R_A = \frac{M}{L}, \quad R_B = -\frac{M}{L} \] Step 2: Analyze the bending moment at a distance \( L/4 \) from \( A \). The bending moment \( M_x \) at any section \( x \) from the left support \( A \) is given by: \[ M_x = R_A \cdot x \] Substitute \( R_A = \frac{M}{L} \) and \( x = \frac{L}{4} \): \[ M_x = \frac{M}{L} \cdot \frac{L}{4} = \frac{M}{4} \] Conclusion: The bending moment at a distance \( L/4 \) from the left support \( A \) is \( \frac{M}{4} \). The correct option is \( \mathbf{(C)} \).