Optimisation - Using the lagrange method

[lak91]

Homework Statement

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).

Homework 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.

 

[Valkarie]

I think I need to set up the lagrange expression for this problem. The Attempt at a SolutionL = Length of beamF= Force at end of beamx = length of small shoulderD = diameter of large shoulderd = diameter of small shoulderObjective function: V = (pi*D^2/4)*(L-2*x) +2*(pi*d^2/4)*x Constraint: σA = (32*F*L)/(pi*D^3) =σ Lagrange Expression: L = F(V - (pi*D^2/4)*(L-2*x) +2*(pi*d^2/4)*x) + λ((32 * F * L) / (pi * D^3 ) - σ) Now I need to take the derivative with respect to D, d and x and set them equal to 0 dL/dD = -(pi*D^2/2)*(L-2*x) - (96 * F * L * λ)/ (pi * D^4) = 0ddL/dd = 2*(pi*d^2/4)*x - (96 * F * L * λ)/ (pi * D^4) = 0dL/dx = - (pi*D^2/2) + (pi*d^2/2) = 0 Solve for x = (pi*D^2/2)/(pi*d^2/2) Substitute back in the constraint equation and solve for dσA = (32 * F * L) / (pi * D^3) d = (32 * F * L) / ((4 * pi * σ)^1/3)