Optimization and optimality of frames with equally stable parts (links)

The problem of optimal design of transport overpass supports has the form of the problem of optimization of multi-storey frame system from the condition of stability. Analysis of the work of individual elements of frame systems in existing design solutions shows that not all parts of the frame are effectively involved in ensuring the overall stability of the structure. In this regard, it is important to solve the problem of optimization of frame elements with equal stability of individual parts. It is important to know the ratio of optimal frames and optimal frames with equally stable links. In an apparatus of forming a mathematical model of the critical state is adopted the method of displacement with the trigonometric functions, the effects of longitudinal forces on the reaction. The condition of the critical state of multi-storey frames, the stiffness matrix of which is the Jacobian matrix, is written by the determinant equation. The General solution of the problem of optimization of multi-storey frames in an analytical, closed form was obtained by the author earlier. For a complete analytical solution of optimization problems for equally stable parts, it is necessary to find solutions to two optimization problems for free frame links and two problems for non-free frame links. By a decision established that, forming the frame of the optimal revostock parts, we obtain a frame equivalent to the optimal non-segmented frame. And, therefore, the position stated by A. F. Smirnov that the most rational of the stability condition of the system will be such, all links (parts) of which have equal stability is true. The frame of the minimum mass with equal parts is optimal from the stability condition.