The paper investigates the problems and the development of optimization algorithms for beams and frames made of work-hardening elasto-plastic materials. Minimization problems for functionals of stresses or strains with prescribed volume of material are discussed. It is proved that for a wide variety of functionals, which are proposed to be treated as a "compliance" of the structure, the optimal structure will be fully stressed. On the basis of the theoretical results obtained, optimization algorithms for statically indeterminate beams and frames are developed. A test model example is given. Rather rapid convergence rate of the proposed algorithms is demonstrated. It is noted that during the optimization process a statically indeterminate structure degenerates to a statically determinate one.

A composite von Karman plate in postbuckling is considered. Using the first Piola stress tensor and the displacement gradient tensor, a complementary energy variational theorem is proven. The proof is given in the case of symmetric lay-up. According to the theorem, at the actual stress state of the plate the complementary energy (as a function of the internal forces and of the moments) reaches its stationary value. The stationary feature of the actual state is valid as compared to other states satisfying the static equilibrium and the static boundary conditions. It is shown how the theorem may be generalized to the case of a non-symmetric lay-up.

The problems of the optimization of bar structures made of work-hardening elasto-plastic materials are investigated on the basis of the deformation theory of plasticity. Two optimization criteria are discussed -the minimum of the total complementary energy and the minimum of the total elasto-plastic strain potential energy. It is proved that these criteria are equivalent and lead to equally-stressed structures, if the structures are statically determinate. Furthermore, a problem of the minimization of the integral over the structural volume of an arbitraxy monotonously increasing strictly convex smooth function of the absolute value of a strain or a stress with prescribed volume of material for statically determinate bar structures also proves to lead to equally-stressed structures. On the basis of the results obtained, two algorithms for the optimization of statically indeterminate trusses are proposed and illustrated. It is noted that the example structure degenerates to a statically determinate one as a result of the optimization process.

The present paper deals with the optimization of post-buckled composite plates. The plates have a symmetric lay-up. The layer orientation angles vary in the point-wise or in the platewise ways. The von Karman theory is employed. The boundary conditions are the simple support ones or the clamped ones. The structural potential energy is treated as a measure of structural stiffness. For the plate stiffness maximization problem, the first-order necessary conditions of the local optimality are derived. The mathematical treatment of the conditions is performed. The conditions contain two terms. One of them corresponds to the mid-plane strains; another one corresponds to the plate curvatures. The optimality conditions may lead to a co-axiality of some structural tensors. An illustration of the optimality conditions is presented.-2 - INTRODUCTIONPostbuckling of thin composite plates attracts the attention of numerous researchers, working in various modern application areas. Among these applications, there are the aerospace, the automotive, the marine, and the civil ones. According to design practice in composite structures, the account of the load-carrying capability "above buckling" gives an opportunity of extra weight saving, as compared to existing traditional design.Composite plates are a part of a considerable number of structures in the above applications. The loading applied to the plates is a combination of compression and shear. The compression dominates for the structures similar to the upper panels of an airplane wing structure. As the examples of known structures designed for pure shear, one may indicate airplane wing ribs and spars, as well as some fuselage panels. As a rule, the shear-loaded plates have the so-called angle-ply symmetric lay-up. For example, buckling is not allowed in airplane structures in the case of loading up to the so-called limit loads (LL) and may be allowed for the loading level between the limit loads and the so-called ultimate loads (UL).The latter loads are 1.5 times higher than the limit loads.When optimizing the composite plate, an engineer, in fact, does his best to save the structural weight, the material cost, and the production time. Due to that, the optimization problems for the composite plates attract considerable attention at the design phase. The layup optimization problem is one of them requiring a solution at the beginning of the design process. Since the last decade of the previous century, numerous papers dealt with the lay-up optimization of composite plates. The non-exhaustive list [1]-[22] contains the important publications devoted to the postbuckling analysis and optimization. The foundations of the postbuckling theory are described in [23].

Thin elastic plates (metallic or composite) experiencing large deflections are considered. The plate deflections are much larger than the plate thickness. The geometrically nonlinear elasticity theory and the Kirchhoff assumptions are employed. The elongations, the shears and the in-plane rotations are assumed to be small. A kinematic variational principle leading to a boundary value problem for the plate is derived. It is shown that the principle gives proper equilibrium equations and boundary conditions. For moderate plate deflections the principle is transformed to the case of the von Karman plate.