Optimality Conditions for Multiple Loaded Structures — Integrating Control and Finite Element Method

The minimum cost design of structures (beam, truss or frame) is discussed. The structure has unknowns: area of cross section and location of some points (for example: supports). For the same structure multiple load is considered: vibration, buckling and static load — with prescribed frequency, buckling force and compliance, respectively.Optimality conditions for the unknowns are derived by variational approach. The problem is formulated in the optimal control form, so obtained results are valid for an arbitrary structure, load and boundary conditions. Optimality equations for specific structure can be solved numerically: by combination of the finite element method and the iterative algorithm. An illustrative example is presented and efficiency of the optimal design is judged by comparing it with a uniform cross-sectional area design.