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Math Help - Bending moment for a Beam

  1. #1
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    Bending moment for a Beam

    a simply supported beam of a span 8.0m carriers a design load of 30kN/m.

    a)write an expression for the bending moment, M, at a section which is at a distance of x from a support

    b)Express the engineers bending eqaution as a differential equation relating the vertical deflection, y, of the beam to the bending moment at x.

    c)obtain the particular soultion of this equation.

    i could do with some help especially part b and c

    for the expression i have:

    Mx = 120-10x/2

    Mx= 120x-5x sqaured this may not be correct.

    any help be great thank you
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  2. #2
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    Quote Originally Posted by size View Post
    a simply supported beam of a span 8.0m carriers a design load of 30kN/m.

    a)write an expression for the bending moment, M, at a section which is at a distance of x from a support

    b)Express the engineers bending eqaution as a differential equation relating the vertical deflection, y, of the beam to the bending moment at x.

    c)obtain the particular soultion of this equation.

    i could do with some help especially part b and c

    for the expression i have:

    Mx = 120-10x/2

    Mx= 120x-5x sqaured this may not be correct.

    any help be great thank you
    Dear size,

    If we denote the shearing force by S and the bending moment by M, using the differential equations for a beam we have,

    \frac{dS}{dx}=w where w is the weight per unit length.

    \frac{dS}{dx}=30\Rightarrow s=30x+C A is an arbitrary constant.

    Also, \frac{dM}{dx}=-S\Rightarrow \frac{dM}{dx}=-30x-C\Rightarrow M=\frac{-30x^2}{2}-Cx+D

    Using the boundary conditions x=0;~M=0\Rightarrow D=0

    x=8;~M=0\Rightarrow C=-120

    Therefore, M=-15x^2+120

    The bending moment and the vertical deflection is related by the Euler-Bernoulli equation,

    M=(EI)\frac{d^{2}y}{dx^2}

    Substituting the value obtained for M and using the boundary conditions you can obtain,

    y=\frac{1}{(EI)}\left(-\frac{5x^4}{4}+20x^3-640x\right)

    Hope this will help you.
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