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Second Order Differential Equations - Beam Deflections

  1. Jul 27, 2017 #1
    1. The problem statement, all variables and given/known data
    A cantilever of length ##L## is rigidly fixed at one end and is horizontal in the unstrainted position. If a load is added at the free end of the beam, the downward deflection, ##y##, at a distance, ##x##, along the beam satisfies the differential equation: [tex]\frac{d^2y}{dx^2}=k\left(L-x\right) \ for\ 0\le x\le L[/tex]
    Where ##k## is a constant. Find the deflection, ##y##, in terms of ##x## and hence find the maximum deflection of the beam.


    2. Relevant equations
    Basic knowledge of integration

    3. The attempt at a solution
    After double integrating both sides I'm left with ##y=\frac{k}{2}Lx^2-\frac{k}{6}x^3+cx+d## As the question tells us the beam is horizontal when their is no weight on it, we know that ##y=0## when ##x=0##.
    ##0=\frac{k}{2}L\left(0\right)^2-\frac{k}{6}\left(0\right)^3+c\left(0\right)+d##
    ##d=0##
    ##y=\frac{k}{2}Lx^2-\frac{k}{6}x^3+cx##
    This is where I get stuck - I can't find any part of the question which helps me calculate the value of the constant ##c##. The answer says that ##c## should be zero, but I don't understand how they know this.
     
  2. jcsd
  3. Jul 28, 2017 #2

    ehild

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    The beam is horizontal when unloaded. What do you think the direction of the tangent of the loaded beam is at the fixed end?
    images?q=tbn:ANd9GcQJcrbfWK8K62FnNax_ghIZRI-NfC0BBBU308CzuXHtzCglpddkxA.png
     
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