Optimal design of rectangular RC sections for ultimate bending strength

A minimum cost problem for ultimate strength in bending of rectangular reinforced concrete sections is investigated. The design variables are section depth and steel reinforcement areas. State equations are those of equilibrium with compression depth as state variable. The Kuhn-Tucker optimality conditions are solved analytically and formulas for nondimensional design and state variables are obtained in four cases: Two singly-reinforced solutions with either maximum allowable depth or smaller; Two doubly-reinforced with maximum allowable depth and either maximum compression depth or smaller. Each of the solutions is optimal in a region of the plane ‘nondimensional bending moment’–‘cost-effectiveness ratio of concrete to steel’. The formulas are for an arbitrary concrete constitutive law with tension cut-off and are specialized for the parabola-rectangle law of Eurocode 2.