Keywords: steel space frames, optimum structural design, load and resistance factor design, combinatorial optimization, metaheuristic search techniques, ant colony optimizer.

Steel buildings are preferred in residential as well as commercial buildings due to their high strength and ductility particularly in regions which are prone to earthquakes. Some of these buildings have irregular shapes due to architectural considerations. Such buildings undergo twisting as a result of their unsymmetrical topology when they are subjected to lateral loads caused by wind or earthquakes. This occurs due to the fact that the resultant of lateral loads acting on the building does not pass through the centre of gravity of the structure. As a result the structure is subject to a torsional moment which must be resisted by the three dimensional frame system. The members of the frame are generally made out of steel I sections which are thin walled open sections. The effect of warping becomes important in the stress analysis of these sections when they are subjected to torsional moments. It is shown that normal stresses develop in addition to shear stresses when a thin walled open section is subjected to a torsional moment. These normal stresses can be computed from a simple formula which is similar to flexural stress formula derived by Vlasov. This expression requires the computation of the sectorial coordinate and warping moment of inertia of the thin walled open section. The consideration of the effect of warping in the analysis of three dimensional steel frames necessitates the consideration of an additional degree of freedom in a joint. This brings the total number of degrees of freedom to seven at a joint of the space frame. These are the usual three translations along global x, y, and z axes, three rotations about the global axes and additional warping deformation. Consequently, the member stiffness matrix in a local coordinate system has fourteen rows and fourteen columns. The global stiffness matrix of the structure is generated by accumulating the global stiffness matrices of each member. At the end of the analysis of a three dimensional structure under the lateral as well as vertical loads, member end forces are obtained. The member end forces vector contains the forces along the local x, y, and z axes as well as the moments about these axes. Furthermore, the bi-moments at both ends are also included in the member end force vector. The bi-moments are used to compute the normal stresses due to constrained warping of the member. In this study, an algorithm is developed for the optimum design of the three dimensional irregular steel frames which takes into account warping deformations of the thin walled section. The optimum design problem is formulated according to LRFD-AISC (Load and Resistance Factor Design, American Institute of Steel Construction). Design constraints include the displacement limitations, inter-storey drift restrictions of multi-storey frames, strength requirements for beams and beam-columns. Furthermore, additional constraints are considered to satisfy practical requirements. These include three types of inequalities. The first type ensures that the flange width of the beam section at each beam-column connection of each storey is less than or equal to the flange width of column section. The second and third type of constraints make sure that the depth and the mass per metre of the column section at each storey at each beam-column connection are less than or equal to width and mass of the column section at the lower storey. The mathematical model of the design problem turns out to be a discrete nonlinear programming problem. The ant colony optimization technique is employed to determine its optimum solution. The design algorithm developed selects optimum W sections for beams and columns of three dimensional steel frames such that the above constraints described are satisfied and the frame has the minimum weight. It is noticed that the ant colony optimization technique is an efficient algorithm in finding the optimum design of large scale three dimensional steel frames where the effect of warping is taken into account.

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