Minimum weight design of truss structures with geometric nonlinear behavior

The paper presents an optimization method based on optimality criterion for minimum weight design of structures with geometric nonlinear behavior. The nonlinear critical load is determined by finding the load level at which the Hessian of the potential energy ceases to be positive definite. A recurrence relation based on the criterion that at optimum the nonlinear strain energy density be equal in all the members is used to develop an algorithm. Sample problems are given to illustrate the application of the method to truss type structures with a large number of design variables. NUMBER of algorithms based on the optimality criterion approach have been developed to design a minimum weight structure with specified constraints on nodal displacements, element stresses, system stability, etc.1'2 The optimization algorithm consisted of analyzing the structure by the finite element method and using a recurrence relation derived from the appropriate optimality criterion to modify the design variables. The optimality criterion was derived by, differentiating the Lagrangian with respect to the design variables. The displacements were assumed to be small and, in the finite element analysis, linear equilibrium equations were solved to determine the response of the structure to the ap- plied loads. In the case of system stability the constraints were defined (see Refs. 3-6) by the associated linear eigenvalue problem. This definition of system stability for some structures may not be valid because of the nonlinear behavior of the structure which may be due to the geometry of the structure or the presence of geometric imperfections. The correct procedure for these structures would be to analyze the structure* by using nonlinear equilibrium equations. This will be particularly true for large space structures (LSS). In this paper an optimization method based on the optimality criterion approach is presented for structures with geometric nonlinear behavior. In the case of a structure optimized with constraints on linear stability the optimum structure can have more than one critical buckling mode. This design tends to become im- perfection sensitive and a small deviation in the geometry of the structure can reduce its load carrying capacity sub- stantially. The imperfection sensitivity of the optimized structure can be reduced by designing the structure so that buckling loads associated with the critical buckling modes are not equal (see Ref. 7). A real structure has geometric im- perfections and these structures tend to have nonlinear