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Chapter 1 of 12 • Page 1 of 248🔒 Protected PDF • Watermarked
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CivilStructural Mechanics-II
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At the free end of cantilever beam, deflection is ___________ and slope is ___________.

A

Maximum, Maximum

B

Zero, Maximum

C

Maximum, Zero

D

Zero, Zero

Correct Answer

⚙️ TE • Technical Concept & PrincipleCivilStructural Mechanics-II
Option A

Maximum, Maximum

Quick Summary:

In a cantilever beam fixed at one end and loaded at the other, both the deflection and the slope reach their maximum values at the free end. This is because the cumulative bending moment and shear force effects are greatest at the point furthest from the rigid support.

⚙️TETechnical SolutionConcept & Principle
💡 Explanation

In a cantilever beam fixed at one end and loaded at the other, both the deflection and the slope reach their maximum values at the free end. This is because the cumulative bending moment and shear force effects are greatest at the point furthest from the rigid support.

🔢 Key Formulas

δmax=PL33EI\delta_{max} = \frac{PL^3}{3EI}δmax​=3EIPL3​ — Maximum deflection at free end for a point load

θmax=PL22EI\theta_{max} = \frac{PL^2}{2EI}θmax​=2EIPL2​ — Maximum slope at free end for a point load

⚙️ Working Principle

The cantilever beam functions by transferring loads to the fixed support. As the load is applied at the free end, the internal bending moment is Mx=P(L−x)M_x = P(L-x)Mx​=P(L−x). Integration of the curvature d2ydx2=MEI\frac{d^2y}{dx^2} = \frac{M}{EI}dx2d2y​=EIM​ shows that the displacement yyy and slope θ\thetaθ increase monotonically along the length of the beam, reaching their peak at x=Lx=Lx=L.

📌 Key Points
  • ▸

    At the fixed support (x=0), both slope (θ\thetaθ) and deflection (δ\deltaδ) are zero.

  • ▸

    At the free end (x=L), the beam experiences its maximum curvature and displacement.

  • ▸

    The deflection profile of a cantilever under point load is a cubic parabola.

✅ Advantages
  • ▸

    Simple structural analysis model

  • ▸

    Easily determines the worst-case design load for beam members

❌ Disadvantages / Limitations
  • ▸

    High moment concentration at the fixed end

  • ▸

    Limited load-carrying capacity compared to simply supported beams

🛠️ Applications / Uses
  • ▸

    Balconies

  • ▸

    Bridge wings

  • ▸

    Aircraft wings

📄 Additional Information
  • ▸

    In structural mechanics, the slope is the first derivative of the deflection function with respect to the neutral axis.

  • ▸

    Option B, C, and D are incorrect because they violate the fundamental boundary conditions of a cantilever beam where displacement and rotation accumulate toward the free end.

📊 Diagram / Illustration
Cantilever Beam DeflectionFree EndFixed End
δmax\delta_{max}δmax​
✅

A is correct — The free end of a cantilever beam experiences the maximum cumulative rotation and displacement due to the applied external load.

Core Concepts Used
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Boundary Conditions Euler-Bernoulli Beam Theory Slope-Deflection Relationship
💡 EXAM TIP

Always verify boundary conditions first: fixed supports restrict both motion and rotation, while free ends allow both to reach their theoretical maximums.

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