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CivilStructural Mechanics-II

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A cantilever beam of span l, carrying point load W at free end, slope at free end will be

$\frac{Wl³}{3EI}$

$\frac{Wl²}{2EI}$

$\frac{wl³}{6EI}$

$\frac{wl⁴}{8EI}$

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilStructural Mechanics-II

Option B

$\frac{Wl²}{2EI}$

Quick Summary:

For a cantilever beam of length $l$ carrying a concentrated point load $W$ at its free end, the slope $\theta$ at the free end is determined by integrating the bending moment equation or using the area-moment method. The slope is the result of the accumulation of curvature due to the bending moment induced by the load.

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$\theta = \frac{Wl²}{2EI}$ — Slope at the free end of a cantilever beam with point load

$\delta = \frac{Wl³}{3EI}$ — Deflection at the free end of a cantilever beam with point load

⚙️ Working Principle

The bending moment $M_x$ at a section $x$ from the free end is $W \cdot x$ . Using the governing differential equation $EI \frac{d^2y}{dx²} = M_x$ , the slope equation is derived by integrating once with respect to $x$ . Since the slope at the fixed end ( $x=l$ ) is zero, the integration constant is determined, yielding the maximum slope at $x=0$ as $\frac{Wl²}{2EI}$ .

📌 Key Points

▸
The slope is maximum at the free end and zero at the fixed support.
▸
The deflection is maximum at the free end.
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The result is derived using Macaulay's method or the Area-Moment Method.
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The units of slope are radians (dimensionless).

✅ Advantages

▸
Standardized formula for quick calculation
▸
Applicable for linear elastic analysis

❌ Disadvantages / Limitations

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Assumes material follows Hooke's Law
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Neglects shear deformation (Euler-Bernoulli beam theory assumption)

🛠️ Applications / Uses

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Structural analysis of cantilever overhangs
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Calculation of deflection in machine tool spindles
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Structural health monitoring of beams

📄 Additional Information

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Option A represents the deflection at the free end, not the slope.
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Option C is for a cantilever beam carrying a Uniformly Distributed Load (UDL) of intensity $w$ , which is $\frac{wl³}{6EI}$ for slope.
▸
Option D is for the deflection of a cantilever beam with UDL, which is $\frac{wl⁴}{8EI}$ .

📊 Diagram / Illustration

✅

B is correct — The slope at the free end of a cantilever beam carrying a point load $W$ at its tip is $\frac{Wl²}{2EI}$ .

Core Concepts Used

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Euler-Bernoulli Beam Theory Slope-Deflection Relationship Cantilever Beam Mechanics

💡 EXAM TIP

Always verify if the load is a point load or UDL and whether the question asks for 'Slope' (quadratic in $l$ ) or 'Deflection' (cubic/quartic in $l$ ).

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