Improved procedure for determining the ductility of buildings under seismic loads

Displacement ductility is a parameter that characterizes the seismic response of structures. Moreover, displacement ductility can be used in order to determine whether a structural design, performed according to a specific seismic code or not, may achieve the main goal of the seismic design: to develop energy dissipation in a stable manner. Determination of displacement ductility is not an easy task, because the structural response usually does not show a clear location of the points that define yield and ultimate displacements. In this paper, some of the main procedures for ductility displacement are revised and compared, and then improvements are performed to such procedures in order to compute the displacement ductility. A new procedure is then introduced, leading to determine the ultimate displacement using the seismic collapse threshold and the yield displacement, achieving the balance of dissipated energy. The procedure has been used to calculate displacement ductility of reinforced concrete framed buildings.Peer Reviewe

Mixed-integer convex representability

Motivated by recent advances in solution methods for mixed-integer convex optimization (MICP), we study the fundamental and open question of which sets can be represented exactly as feasible regions of MICP problems. We establish several results in this direction, including the first complete characterization for the mixed-binary case and a simple necessary condition for the general case. We use the latter to derive the first non-representability results for various non-convex sets such as the set of rank-1 matrices and the set of prime numbers. Finally, in correspondence with the seminal work on mixed-integer linear representability by Jeroslow and Lowe, we study the representability question under rationality assumptions. Under these rationality assumptions, we establish that representable sets obey strong regularity properties such as periodicity, and we provide a complete characterization of representable subsets of the natural numbers and of representable compact sets. Interestingly, in the case of subsets of natural numbers, our results provide a clear separation between the mathematical modeling power of mixed-integer linear and mixed-integer convex optimization. In the case of compact sets, our results imply that using unbounded integer variables is necessary only for modeling unbounded sets