Buckling Length in Mixed-Integer Linear Frame Optimization

In structural optimization of trusses and frames, the member profiles have to be selected from material supplier's selection. This means that the optimization problem becomes discrete. The discrete frame optimization problem can be formulated as mixed-integer linear program (MILP) and thus solved for global optimality using well-known deterministic methods such as branch-and-cut. Within the formulation it is possible to include member buckling constraints. When using design standards as basis for member buckling resistance evaluation, the critical forces or buckling lengths of the members are required. Buckling length can be determined using many methods, both numerical and analytical. Regardless of the method, buckling length of a single member is dependent on surrounding members' stiffness which makes it practically impossible to include the correct buckling lengths in MILP formulation directly. In general, the question of buckling length in frame optimization has rarely been discussed in the structural optimization literature. Therefore, in this paper, an iterative approach to determine the correct buckling lengths is presented. In this approach, the MILP optimization is run several times. Linear stability analysis is performed between MILP runs to update buckling length data. The performance of the proposed method is illustrated in example calculations. The example structures are steel frames and Eurocode 3 is used as basis for member resistance constraints. In the examples, the method converges with a relatively low number of iterations.