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.