Оптимальное Проектирование Статически Определимой Балки При Ограничении На Максимальный Прогиб

Abstract

Here is solved the optimization problem for the longitudinal depth distribution in the beam with a limitation on the maximum value of deflection. A review of the references is done, and it is shown that the known solutions are either erroneous, because they are based on false hypotheses, or have a narrow field of application, limited only to symmetrical constructions for which the point of the maximum deflection is known a priori. The paper considers a beam of the rectangular cross-section of constant width. The beam is assumed to be statically determinate, and the load is arbitrary and asymmetric and multidirectional as well. The points (or point) of the beam maximum deflections are unknown in advance and would be determined in the problem-solution procedure. A linear problem is considered. The optimization criterion is the mass of the beam. To find the deflections of the beam, i.e. to solve the differential equation of a variable cross-section beam bending the finite difference method is used. The design problem is reduced to the required beam depths obtaining in the system of nodal points. In this case, the desired solution must satisfy the restriction system for the nodal points shift and the sign of variables as well. Since the restrictions of the shift of each node are considered separately and independently, so the proposed method allows flexible control of the beam shift restrictions. Using the change of variables proposed in the paper, the problem to be solved is reduced to a nonlinear programming problem where the criterion function is separable and restrictions are linear functions. Using linearization, this problem can be reduced to the linear programming problem relatively to new variables. The model problem is solved, and it is shown that the proposed algorithm efficiently allows us to solve the problems of the beam optimal design with the restrictions of the maximally allowed deflection. The proposed approach can be spread for the strength limitations, for beams of variable width, I-beam cross-section, etc.