The optimum shape of a bending beam (Q2761375)

The authors consider the problem of minimizing the static bending of an elastic beam with variable cross-section: \(Au(x) = (EI(x)u''(x))'' = g(x),\quad x\in (-l,l),\quad l>0.\) Here \(u(x)\) is bending, \(I(x)\) is inertia moment, \(E\) is Young modulus, and \(g(x)\) is the distribution of transversal force applied to the beam. Two cases of ends fixing are discussed, namely freely supported and rigidly clamped ends. For the case of beam with clamped ends, the authors construct analytical and numerical solutions.