Topic: Strength of Materials
A simply supported beam of length L is subjected to a varying distributed load $sin(\pi x/L)$ N/m, where distance x is measured from the left support. The magnitude of the vertical reaction in N at the left support is
- A. $3L/\pi$
- B. $2L/\pi$
- C. $L/2\pi$
- D. $L/\pi$
Answer: D. $L/\pi$
Explanation: Total load on the beam is $\int_{0}^{L}\sin\left(\frac{\pi x}{L}\right)dx = \left[-\frac{L}{\pi}\cos\left(\frac{\pi x}{L}\right)\right]_{0}^{L} = -\frac{L}{\pi}[-1 - 1] = \frac{2L}{\pi}$. Due to the symmetric nature of the sine wave load over the length $L$, the reactions at both supports will be equal. $R_{A} = R_{B} = \frac{1}{2} \times \text{Total Load} = \frac{1}{2} \times \frac{2L}{\pi} = \frac{L}{\pi}$.