An overhanging beam ABC is pinned at A and on a roller at B (span AB = 6 m, overhang BC = 3 m) and carries a 10 kN downward point load at C. Macaulay's method gives EIy = −(5/6)x³ + C₁x + (5/2)[x − 6]³, with x measured from A (kN, m). Find the slope at A if EI = 10 × 10¹³ N·mm².

Asked in SSC JE 07/06/2024 (9 am–11 am)

Detailed Explanation

Note on this question: the exam printed the deflection equation with "0·x" in place of the C₁x term, and the commission cancelled the question. Here the equation is shown with the constant C₁, which is what the method actually gives, and the slope is worked out from it.

Reactions first: taking moments about B, R_A × 6 = −10 × 3, so R_A = 5 kN acting downward, and R_B = 15 kN upward. Macaulay's method writes one bending-moment expression for the whole beam and integrates it twice.

\[ EI\frac{d^2y}{dx^2} = -5x + 15[x-6] \]

\[ EI\frac{dy}{dx} = -\frac{5x^2}{2} + C_1 + \frac{15}{2}[x-6]^2,\qquad EIy = -\frac{5x^3}{6} + C_1x + C_2 + \frac{5}{2}[x-6]^3 \]

At A (x = 0), y = 0 ⇒ C₂ = 0. At B (x = 6), y = 0 (bracket term is zero):

\[ -\frac{5(6)^3}{6} + 6C_1 = 0 \Rightarrow C_1 = 30\ \text{kN·m}^2 = 30\times10^{9}\ \text{N·mm}^2 \]

Slope at A (x = 0): \[ \theta_A = \frac{C_1}{EI} = \frac{30\times10^{9}}{10\times10^{13}} = 3\times10^{-4}\ \text{rad} \]

Why “0.0003 radian” Is Correct

At x = 0 every other term of the slope equation vanishes, so EIθ_A = C₁. The support condition at B fixes C₁ = 30 kN·m², which gives θ_A = 0.0003 radian. Because R_A acts downward, span AB hogs and the beam rises from A, so this slope is upward (anticlockwise).

Overhanging beam: R_A = 5 kN downward, R_B = 15 kN upward. The whole beam hogs (BMD −30 kN·m at B); span AB bows upward and the overhang tip deflects downward.

Why the Other Options Are Wrong

B. 0.00015 radian: 0.00015 rad is half the correct value (C₁ = 15 kN·m² would be needed).
C. 0.0015 radian: 0.0015 rad is ten times too large — a unit slip between kN·m² and N·mm².
D. 0.003 radian: 0.003 rad is a hundred times too large.

Key Point

In Macaulay's method, slope at the left support = C₁/EI, with C₁ found from y = 0 at the other support.

Key Concepts for Students

Macaulay brackets [x − a] are taken as zero when x < a.
Integrate brackets as a whole: ∫[x − a]ⁿ dx = [x − a]ⁿ⁺¹/(n + 1).
Convert units before dividing: 1 kN·m² = 10⁹ N·mm².

Practice more: this is question 1 in Slope and Deflection MCQs – Part 1. Try the whole set and check every answer instantly.