On optimal geometrically non-linear trusses

The paper is devoted to the investigation of regularities inherent to optimal geometrically non-linear trusses. The single static loading case is considered, a single structural material is used (except specially indicated cases) and buckling effects are neglected. The so-called small strains and large rotations case is investigated. Some regularities inherent to the kinematic and static variational principles for geometrically non-linear trusses are considered. Then the strain compatibility conditions resulting from the static variational principle are obtained and explored. It is shown that 1) the conditions are linear with respect to subcomponents of rod Green strains such as rotations and geometrically linear strains, 2) strains (in particular, rotations and geometrically linear strains) within rods which are not members of the so-called basic structure are fully determined by geometrically linear strains in rods of the basic structure. Extensions of some theorems (Maxwell's theorem, Michell's theorem, theorems on the stiffness properties of equally-stressed structures, etc.) known for geometrically linear structures are proved. Conditions assuring better or worse quality of equally-stressed geometrically non-linear truss as compared to geometrically linear ones are obtained. It is shown that in numerical optimization of geometrically non-linear trusses in the case of negligible rotations of compressed rods some updated analytical optimization algorithms (derived earlier for geometrically linear case) are monotonic. A simple numerical example confirming the features is presented.