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CivilStructural Mechanics-II

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A cantilever beam of span l, carrying udl of w on entire span, 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 D

$\frac{wl⁴}{8EI}$

Quick Summary:

For a cantilever beam of span $l$ subjected to a Uniformly Distributed Load (UDL) of intensity $w$ over its entire length, the maximum deflection occurs at the free end. This deflection is determined by integrating the bending moment equation or applying the Moment Area Method, resulting in the standard formula $y_{max} = \frac{wl⁴}{8EI}$ .

⚙️TETechnical SolutionConcept & Principle

💡 Explanation

🔢 Key Formulas

$\delta_{max} = \frac{wl⁴}{8EI}$ — Max deflection for UDL

$\theta_{max} = \frac{wl³}{6EI}$ — Max slope for UDL

⚙️ Working Principle

The deflection arises due to the internal bending moment $M(x)$ induced by the UDL. At a distance $x$ from the free end, $M(x) = -\frac{w x²}{2}$ . By the double integration method, where $EI \frac{d^2y}{dx²} = M(x)$ , integrating twice with boundary conditions (slope and deflection are zero at the fixed support) yields the expression at the free end ( $x=l$ ).

📌 Key Points

▸
Maximum deflection occurs at the free end of the cantilever.
▸
The deflection is directly proportional to the load intensity $w$ and the fourth power of the span $l$ .
▸
The parameter $EI$ represents the flexural rigidity of the beam section.

✅ Advantages

▸
Standardized calculation for structural analysis.
▸
Essential for checking serviceability limit states in steel and concrete design.

🛠️ Applications / Uses

▸
Design of cantilever balconies.
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Analysis of retaining walls and overhang components in structural frames.

📄 Additional Information

▸
For a cantilever with point load $W$ at the free end, the deflection is $\frac{Wl³}{3EI}$ .
▸
Option A is the deflection for a point load at the free end.
▸
Option C is the slope at the free end for a UDL.

📊 Diagram / Illustration

✅

D is correct — the deflection at the free end of a cantilever beam carrying a UDL $w$ is given by $\frac{wl⁴}{8EI}$ .

Core Concepts Used

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Bending Moment Flexural Rigidity Cantilever Beam Mechanics

💡 EXAM TIP

Remember: Deflection for UDL (total load $wL$ is $W$ ) is $\frac{Wl³}{8EI}$ , which confirms $\frac{wl⁴}{8EI}$ as the correct form.

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