The aim of this work is to present a formulation and corresponding computational implementation for the sizing optimization of structures. To achieve this goal, the formulation considers the structural geometric nonlinear behavior and include a constraint related to the collapse load. Plane frame finite elements and Updated Lagrangian approach are used for the geometric nonlinear analysis. The standard Newton-Raphson method, in connection with different load increment strategies and iteration, such as use the arch length method and strategies based on the control of generalized displacements, which allow the algorithm to transpose the critical points that happen to appear along the equilibrium path. The mathematical programming algorithms applied in this work make use of the gradients of the objective function and of the constraints, which depend on the gradients of the structural response. Starting from general equilibrium equations for the Update Lagrangian approach, valid for any finite element, approximate analytical expressions for the sensitivity analysis whit respect of design variables were developed taking advantage of the structural characteristics.