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Homework Help: Optimisation - Using the lagrange method

  1. Sep 8, 2013 #1
    1. The problem statement, all variables and given/known data
    The problem asks to design a cantilever beam of a minimum weight consisting of 2 steps.

    Given: total length (L), Force (F) at the end of the beam and allowable stress (σ)
    Need to find the diameters D and d, the length of the smaller shoulder of the beam (x).

    2. Relevant equations
    MB = F*x
    MA= F*L

    σB = MB/ZB = (32 * F* x)/ (pi * d^3)
    σA = MA/ZA = (32 * F* L)/ (pi * d^3)

    Where zA = (pi*D^3)/32 zB = (pi*D^3)/32 section modulus

    So this is what I have done so far :

    Solve for D = ((32*F*L)/(4*pi*(σ))^1/3
    Solve for V = (pi*D^2/4)*(L-2*x) +2*(pi*d^2/4)*x = pi*D^2*x/2+ pi*d^2*x/2

    Im not sure how to implement this into the lagrange expression.

    I now that the volume represents the objective function and the the condition of strength at point A represents a constraint.
  2. jcsd
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