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A cantilever beam of span l, carrying udl of w on entire span, slope at free end will be

$\frac{Wl^3}{3EI}$

$\frac{Wl^2}{2EI}$

$\frac{wl^3}{6EI}$

$\frac{wl^4}{8EI}$

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilStructural Mechanics-II

Option C

$\frac{wl^3}{6EI}$

Quick Summary:

For a cantilever beam of span $l$ subjected to a uniformly distributed load (UDL) of intensity $w$ per unit length, the slope at the free end is determined by integrating the bending moment equation $M_x = -\frac{w(l-x)^2}{2}$ . The resulting expression for the slope $\theta$ at the free end ( $x=0$ ) is $\theta = \frac{wl^3}{6EI}$ .

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💡 Explanation

🔢 Key Formulas

$\theta = \frac{wl^3}{6EI}$ — Slope at the free end of a cantilever with UDL

$\delta = \frac{wl^4}{8EI}$ — Deflection at the free end of a cantilever with UDL

⚙️ Working Principle

The slope is found by integrating the $\frac{M}{EI}$ diagram over the length of the beam. Since $M(x) = -\frac{w(l-x)^2}{2}$ , the rotation $\theta$ at the free end is $\int_{0}^{l} \frac{M}{EI} dx = \int_{0}^{l} \frac{w(l-x)^2}{2EI} dx$ . Performing this definite integral yields $\frac{w}{2EI} [\frac{(l-x)^3}{-3}]_0^l = \frac{w}{2EI} [0 - (-\frac{l^3}{3})] = \frac{wl^3}{6EI}$ .

📌 Key Points

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The slope at the fixed end of a cantilever beam is always zero.
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The maximum slope and maximum deflection for a cantilever with UDL occur at the free end.
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Flexural rigidity ( $EI$ ) represents the beam's resistance to bending; higher values result in lower slopes and deflections.

✅ Advantages

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Simple analytical calculation for standard loading cases
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Applicable to structural design of overhangs and cantilevers

❌ Disadvantages / Limitations

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Does not account for shear deformation (usually neglected in Euler-Bernoulli beam theory)
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Assumes linear elastic material behavior

🛠️ Applications / Uses

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Structural analysis of balcony projections
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Design of cantilever retaining walls and canopy structures

📄 Additional Information

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Option A ( $\frac{Wl^3}{3EI}$ ) is the deflection at the free end for a point load $W$ at the free end.
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Option B ( $\frac{Wl^2}{2EI}$ ) is the slope at the free end for a point load $W$ at the free end.
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Option D ( $\frac{wl^4}{8EI}$ ) is the deflection at the free end for a cantilever with UDL $w$ .

📊 Diagram / Illustration

✅

C is correct — The slope at the free end of a cantilever beam carrying a UDL of $w$ over its entire length is $\frac{wl^3}{6EI}$ .

Core Concepts Used

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Euler-Bernoulli Beam Theory Slope and Deflection Integration of Bending Moment Equation

💡 EXAM TIP

Remember the mnemonic 'Slope is 1 degree less than deflection' ( $\theta \propto l^3$ and $\delta \propto l^4$ ) for standard cantilever cases.

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