Question

A simply supported beam of length L is loaded by two symmetrically applied point loads P at \( L/3 \) from each support. Both the loads are then shifted to new points which are at a distance \( L/4 \) from each support. The bending moments at the mid-section of the beam in both the cases are same. The magnitude of \( P_1 \) in terms of P is: \includegraphics[width=0.5\linewidth]{image7.png}

• A. \( \frac{P}{4} \)
• B. \( \frac{8P}{3} \)
• C. \( \frac{4P}{3} \)
• D. \( \frac{P}{3} \)

Hint:

When comparing bending moments in different load configurations, ensure the distances are consistent and solve for the unknown force using the moment equilibrium condition.

Answer 1

The Correct Option is C

For this problem, we are given a simply supported beam of length \( L \) with point loads \( P \) applied symmetrically at distances \( L/3 \) from each support. The bending moment at the mid-section of the beam is calculated for the two different load configurations (one at \( L/3 \) and one at \( L/4 \) from the supports). The bending moment at the mid-section of the beam is the same in both cases. This means that the load magnitudes must be adjusted accordingly to maintain the same moment. - The moment \( M \) at the mid-section due to a point load is given by: \[ M = P \cdot d, \] where \( d \) is the distance from the point of application to the mid-point of the beam. For the initial load position (at \( L/3 \)): \[ M_1 = P \cdot \frac{L}{3}. \] For the shifted load position (at \( L/4 \)): \[ M_2 = P_1 \cdot \frac{L}{4}. \] Since the bending moments are the same, we equate the two: \[ P \cdot \frac{L}{3} = P_1 \cdot \frac{L}{4}. \] Solving for \( P_1 \), we get: \[ P_1 = \frac{4P}{3}. \] Hence, the correct answer is \( (C) \frac{4P}{3} \). Final Answer: \( \frac{4P}{3} \)