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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

тЪЩя╕П TE тАв 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.

тЪЩя╕ПTETechnical SolutionConcept & Principle
ЁЯТб 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.

ЁЯФв Key Formulas

╬┤max=PL33EI\delta_{max} = \frac{PL┬│}{3EI}╬┤maxтАЛ=3EIPL3тАЛ тАФ Maximum deflection at free end for a point load

╬╕max=PL22EI\theta_{max} = \frac{PL┬▓}{2EI}╬╕maxтАЛ=2EIPL2тАЛ тАФ Maximum slope at free end for a point load

тЪЩя╕П 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(LтИТx)M_x = P(L-x)MxтАЛ=P(LтИТx). Integration of the curvature d2ydx2=MEI\frac{d^2y}{dx┬▓} = \frac{M}{EI}dx2d2yтАЛ=EIMтАЛ shows that the displacement yyy and slope ╬╕\theta╬╕ increase monotonically along the length of the beam, reaching their peak at x=Lx=Lx=L.

ЁЯУМ Key Points
  • тЦ╕

    At the fixed support (x=0), both slope (╬╕\theta╬╕) and deflection (╬┤\delta╬┤) are zero.

  • тЦ╕

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

  • тЦ╕

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

тЬЕ Advantages
  • тЦ╕

    Simple structural analysis model

  • тЦ╕

    Easily determines the worst-case design load for beam members

тЭМ Disadvantages / Limitations
  • тЦ╕

    High moment concentration at the fixed end

  • тЦ╕

    Limited load-carrying capacity compared to simply supported beams

ЁЯЫая╕П Applications / Uses
  • тЦ╕

    Balconies

  • тЦ╕

    Bridge wings

  • тЦ╕

    Aircraft wings

ЁЯУД Additional Information
  • тЦ╕

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

  • тЦ╕

    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.

ЁЯУК Diagram / Illustration
Cantilever Beam DeflectionFree EndFixed End
╬┤max\delta_{max}╬┤maxтАЛ
тЬЕ

A is correct тАФ 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
ЁЯТб 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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