Minimum cost design of reinforced concrete beams using continuum-type optimality criteria

This paper outlines a general procedure for obtaining, on the basis of continuum-type optimality criteria (COC), economic designs for reinforced concrete beams under various design constraints. The costs to be minimized include those of concrete, reinforcing steel and formwork. The constraints consist of limits on the maximum deflection, and on the bending and shear strengths. However, the formulation can easily cater for other types of constraints such as those on axial strength. Conditions of cost minimality are derived using calculus of variation on an augmented Lagrangian. An iterative procedure based on optimality criteria is applied to a test example involving a reinforced concrete propped cantilever beam whose cross-section varies continuously. Numerical examples are presented in which the design variables are both the width and the depth or the depth alone, and the optimal costs are compared. The solution of the test example with depth alone as the design variable is confirmed by an alternative approach using discretized continuum-type optimality criteria (DCOC).