Deflection of Beams MCQ with answer and solution

Concept

Deflection of a cantilever beam under multiple loads using standard deflection formulas and superposition.

Formula

Deflection at free end due to point load W at distance x from fixed end = W_x²(3L - x)/(6EI); Deflection at free end due to load R at free end = RL³/(3EI)

Step-by-step solution

1. Identify the given beam: A cantilever of length L and flexural rigidity EI. 2. Calculate the downward deflection at the free end (x = L) due to the concentrated load P acting at the mid-span (x = L/2). Using the standard formula for a point load at distance a from the fixed support: δ_P = (P × (L/2)² × [3L - (L/2)]) / (6EI) = (P × L²/4 × [5L/2]) / (6EI) = 5PL³ / 48EI. 3. Calculate the upward deflection at the free end due to the upward load R acting at the free end: δ_R = R × L³ / (3EI). 4. Equate the two deflections for zero net deflection at the free end: 5PL³ / 48EI = RL³ / 3EI. 5. Solve for R: R = P × (5/48) × (3) = 5P / 16.

Explanation

To find the upward load R at the free end such that the net deflection at the free end is zero, we apply the principle of superposition. The downward load P is applied at the mid-span (distance L/2 from the fixed end), causing a downward deflection at the free end. The upward load R is applied at the free end (distance L), causing an upward deflection. Equating the two deflections at the free end allows us to solve for R.

Why the other options are wrong

A: Incorrect coefficient derived from wrong superposition. B: Incorrect coefficient, assuming simple support or midpoint symmetry. D: Incorrect coefficient from arithmetic error.

Shortcut

Remember the free-end deflection formula for a load at mid-span: 5PL³/(48EI), and equate it to RL³/(3EI) to quickly get R = 5P/16.

Common mistake

Using the formula for deflection at the load point instead of at the free end, or incorrectly calculating the cube of the distance for mid-span loading.