Question

A cantilever beam of length \(L\), and flexural rigidity \(EI\), is subjected to an end moment \(M\), as shown in the figure. The deflection of the beam at \(x = \frac{L}{2}\) is

• A. \(\frac{ML^2}{2EI}\)
• B. \(\frac{ML^2}{4EI}\)
• C. \(\frac{ML^2}{8EI}\)
• D. \(\frac{ML^2}{16EI}\)

Hint:

When calculating deflection for a cantilever beam subjected to an end moment, remember to use the appropriate formula for deflection and adjust for the position of the applied load.

Answer 1

The Correct Option is C

For a cantilever beam with a moment \(M\) applied at the free end, the deflection at a point located at a distance \(x = \frac{L}{2}\) from the fixed support can be calculated using the beam deflection formula. For a cantilever beam subjected to a moment \(M\) at its free end, the deflection at any point \(x\) is given by the following equation: \[ \delta(x) = \frac{M x^2}{2EI} \left( \frac{3L - x}{L} \right) \] For the deflection at the midpoint \(x = \frac{L}{2}\), substituting \(x = \frac{L}{2}\) into the equation: \[ \delta\left(\frac{L}{2}\right) = \frac{M \left( \frac{L}{2} \right)^2}{2EI} \left( \frac{3L - \frac{L}{2}}{L} \right) \] Simplifying the expression: \[ \delta\left(\frac{L}{2}\right) = \frac{M L^2}{8EI} \] Thus, the correct answer is (C).