Optimum design of structures subjected to follower forces

In this paper, shape optimization is used to optimize the critical load of an Euler–Bernoulli cantilever beam with constant volume subjected to a tangential compressive tip load and/or a tangential compressive load arbitrarily distributed along the beam. This is achieved by varying appropriately the beam cross-section, thus its stiffness and mass properties, along its length, so that the critical load reaches its maximum or a prescribed value. The problem is reduced to a nonlinear optimization problem under equality and inequality constraints as well as specified lower and upper bounds, which, together with large slenderness ratios, ensure the validity of the Euler–Bernoulli theory and the serviceability of the beam. The evaluation of the objective function requires the solution of the dynamic stability problem of a cantilever beam with variable cross-section. This problem is solved using the analog equation method (AEM) of Katsikadelis for the fourth-order hyperbolic differential equation with variable coefficients, together with a simple and direct iterative method for the evaluation of the critical load based on the eigenvalue sensitivity. Besides its accuracy, this method overcomes the shortcoming of a possible FEM solution, which would require resizing of the elements and re-computation of their stiffness properties during the optimization process. Example problems of various types of follower forces are presented, which illustrate the method and demonstrate its applicability and efficiency.