A simply supported beam of span l, carrying point load W at centre of span, slope at both end will be

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Option C: $\frac{Wl^2}{16EI}$

Explanation & Concepts:
For a simply supported beam of span $l$ subjected to a central point load $W$, the slope $	heta$ at both supports is equal to $\frac{Wl^2}{16EI}$. This is derived from the standard elastic curve equation for structural members under static loading.

Working Principle: When a point load acts at the center of a simply supported beam, the beam undergoes bending, resulting in a maximum deflection at the center and maximum slope at the supports. Using the double integration method or Macaulay's method, the slope equation is $	heta = \frac{M \cdot L}{k cdot EI}$, where the specific distribution of the bending moment for a central load results in the factor of 16 at the supports.

Key Formulas: $	heta = \frac{Wl^2}{16EI}$ — Slope at supports; $\delta_{max} = \frac{Wl^3}{48EI}$ — Maximum deflection at center

Key Points: Slope is maximum at the ends (supports) and zero at the center.; Deflection is maximum at the center and zero at the supports.; The formula assumes the beam behaves elastically within the proportionality limit.; $E$ is the Young's Modulus and $I$ is the Area Moment of Inertia.

Advantages: Predictable structural behavior; Easy calculation for standard load cases

Disadvantages: Assumes material homogeneity; Applicable only for prismatic beams (constant EI)

Applications: Structural design of building floor beams; Bridge girders under specific point loads

Additional Info: The expression $\frac{Wl^3}{48EI}$ (Option B) represents the maximum deflection at the center, not the slope.. The expression $\frac{5wl^4}{384EI}$ (Option D) is the formula for maximum deflection in a simply supported beam with a Uniformly Distributed Load (UDL).

Answer: C is correct — The slope at the ends of a simply supported beam under a central point load is $\frac{Wl^2}{16EI}$.

Exam Tip: Always distinguish between slope (units of radians) and deflection (units of length); slope formulas will always have $l^2$ in the numerator, while deflection formulas have $l^3$ or $l^4$.

Examoogle · Answer

Option A: $\frac{Wl^3}{24EI}$

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Option B: $\frac{Wl^3}{48EI}$

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Option D: $\frac{5wl^4}{384EI}$

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