We present a mixed-integer nonlinear programming (MINLP) formulation to achieve minimum-cost designs for reinforced concrete (RC) structures that satisfy building code requirements. The objective function includes material and labor costs for concrete, steel reinforcing bars, and formwork according to typical contractor methods. Restrictions enforce correct geometry of the cross-section dimensions for each element and relative sizes of cross-section dimensions of elements within the structure. Other restrictions define a stiffness and displacement correlation among all structural elements via finite element analysis. The design of minimum cost RC structures introduces a new class of optimization problems, namely, mixed-integer nonlinear programs with complementarity constraints. The complementarity constraints are used to model RC element strength and ACI code-required safety factors. We reformulate the complementarity constraints as nonlinear equations and show that the resulting ill-conditioned MINLPs can be solved by using an off-the-shelf MINLP solver. Our work provides discrete-valued design solutions for an explicit representation of a process most often performed implicitly with iterative calculations. We demonstrate the capabilities of a mixed-integer nonlinear algorithm, MINLPBB, to find optimal sizing and reinforcing for cast-in-place beam and column elements in multistory RC structures. Problem instances contain up to 678 variables, of which 214 are integer, and 844 constraints, of which 582 are nonlinear. We solve problems to local optimality within a reasonable amount of computational time, and we find an average cost savings over typical-practice design solutions of 13 percent.

## Citation

Mathematics and Computer Science Division Preprint ANL/MCS-Pnnnn-1109, Argonne National Laboratory, November 2009