Deflection of Beams MCQ with answer and solution

Concept

In Macaulay's method, the first integration of EI y'' = M gives the slope equation with constant C₁. Slope boundary conditions in this equation give C₁.

Formula

EI d²y/dx² = M_x ; EI dy/dx = ∫M_x dx + C₁ ; EI y = ∫∫M_x dx dx + C₁ x + C₂

Step-by-step solution

1. Write the moment equation: EI d²y/dx² = M_x. 2. Integrate once to get the slope equation: EI dy/dx = ∫M_x dx + C₁. 3. The first integral is the slope equation, so apply a slope boundary condition (e.g., dy/dx = 0 at a fixed end or at midspan of a symmetric beam). 4. This gives C₁. So the answer is A.

Explanation

The first integral of the elastic curve equation gives slope. So C₁ is found by using a known slope condition, such as zero slope at a fixed support. Factor of safety and load factor are design terms, not boundary conditions.

Why the other options are wrong

B: Partial factor of safety is a design (limit state) term. C: Load factor is a design term, not used in integration. D: Deflection conditions are applied to the second integral (deflection equation), mainly to find C₂.

Shortcut

1st integral = slope → slope condition; 2nd integral = deflection → deflection condition.

Common mistake

In simply supported beams, students forget that both C₁ and C₂ are often found together from two deflection conditions (y = 0 at both supports).