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CivilStructural Mechanics-II

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A cantilever beam of span l, carrying point load W at free end, deflection 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 A

$\frac{Wl³}{3EI}$

Quick Summary:

For a cantilever beam of length $l$ subjected to a point load $W$ at its free end, the maximum deflection occurs at the free end. This deflection is derived using the double integration method or the moment-area method as $\delta_{max} = \frac{Wl³}{3EI}$ .

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

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

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

⚙️ Working Principle

The deflection is caused by the bending moment induced throughout the beam length. Since the bending moment $M_x$ at a distance $x$ from the free end is $Wx$ , integrating the differential equation of the elastic curve $EI \frac{d^2y}{dx²} = M_x$ with boundary conditions of zero slope and zero deflection at the fixed support yields the cubic variation of deflection.

📌 Key Points

▸
Maximum deflection always occurs at the free end of a cantilever beam.
▸
The deflection is directly proportional to the load $W$ and the cube of the length $l$ .
▸
The product $EI$ is known as the flexural rigidity of the beam.
▸
The slope at the free end is $\theta = \frac{Wl²}{2EI}$ .

✅ Advantages

▸
Simple linear relationship with load
▸
Easy to compute for standard load cases

❌ Disadvantages / Limitations

▸
Limited to small deflection theory
▸
Assumes linear elastic material behavior

🛠️ Applications / Uses

▸
Design of cantilever balconies
▸
Structural analysis of aircraft wings
▸
Mechanical spring and cantilever sensor design

📄 Additional Information

▸
Option B represents the slope at the free end, not the deflection.
▸
Option C ( $wl³/6EI$ ) is incorrect as it does not match standard cantilever deflection formulas.
▸
Option D ( $wl⁴/8EI$ ) represents the maximum deflection of a cantilever under a Uniformly Distributed Load (UDL).

📊 Diagram / Illustration

✅

A is correct — The deflection at the free end of a cantilever beam with a point load $W$ is $\frac{Wl³}{3EI}$ .

Core Concepts Used

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Flexural Rigidity ($EI$) Elastic Curve Equation Cantilever Boundary Conditions

💡 EXAM TIP

Always verify if the question asks for deflection (unit of length) or slope (dimensionless/radians) to avoid choosing the wrong formula.

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