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⚙️ TE • Technical Concept & PrincipleTechnical AbilityCivil Engineering

Option A

Option 1

Quick Summary:

The maximum deflection in a cantilever beam occurs at the free end (at $x = L$ ). Given the slope equation $EI \frac{dy}{dx} = -P(Lx - \frac{x^2}{2})$ , we determine the deflection equation $y$ by integrating with respect to $x$ and applying boundary conditions.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$EI \frac{dy}{dx} = -P(Lx - \frac{x^2}{2})$ — Slope equation for cantilever beam with point load

$y_{max} = -\frac{PL^3}{3EI}$ — Maximum deflection at the free end

⚙️ Working Principle

The double integration method relates the bending moment to the deflection through the equation $EI \frac{d^2y}{dx^2} = M(x)$ . Given the slope $EI \frac{dy}{dx} = \int M(x) dx = -P(Lx - \frac{x^2}{2})$ , we integrate once more: $EI y = \int -P(Lx - \frac{x^2}{2}) dx = -P(\frac{Lx^2}{2} - \frac{x^3}{6}) + C_1$ . At the fixed end ( $x=0$ ), the deflection $y=0$ , implying $C_1=0$ . Thus, $y = -\frac{P}{EI}(\frac{Lx^2}{2} - \frac{x^3}{6})$ . Evaluating at $x=L$ gives $y_{max} = -\frac{P}{EI}(\frac{L^3}{2} - \frac{L^3}{6}) = -\frac{PL^3}{3EI}$ .

📌 Key Points

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Maximum deflection occurs at the free end of a cantilever beam.
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The constant of integration $C_1$ is zero due to the fixed support boundary condition $y(0) = 0$ .
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The negative sign indicates the deflection is downwards (opposite to the positive y-axis direction).

✅ Advantages

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Analytical solution provides precise results for simple loading.
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Double integration method is fundamental for understanding beam behavior.

❌ Disadvantages / Limitations

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Becomes complex for multiple point loads or discontinuous loading without singularity functions.

🛠️ Applications / Uses

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Structural design of cantilevers.
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Verification of deflection limits in steel and concrete members.

📄 Additional Information

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The result $-\frac{PL^3}{3EI}$ corresponds to Option A.
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Option B ( $-\frac{PL^3}{6EI}$ ) corresponds to a triangular load case (moment).
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Options C and D are dimensionally inconsistent for deflection as they represent slope or incorrect power laws.

📊 Diagram / Illustration

✅

A is correct — The integration of the slope equation at the boundary condition $x=L$ yields the maximum deflection of $-\frac{PL^3}{3EI}$ .

Core Concepts Used

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Cantilever beam deflection Double integration method Boundary conditions in structural analysis

💡 EXAM TIP

Always remember the standard deflection formula for a point load at the free end of a cantilever, $\delta = \frac{PL^3}{3EI}$ , to save time in competitive exams.

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