At the beginning of the present paper, the post -buckled composite plates are examined. The plates have a symmetric lay -up. The layer orientation angles vary in a point-wise way or in a plate-wise way. The von Karman theory describes a plate behavior. For a plate stiffness maximization problem the first -order optimality conditions are derived. In the conditions, t he essential roles play the mid -plane principal strains and the principal curvatures. The optimality conditions may lead to a co -axiality of some structural tensors. Analytical examples illustrate the solutions of such types. Further, several nu merical examples are presented. In the examples, t he plates are loaded by shear and have the high buckling -resisting ability also. A significa nt conclusion of the paper is that in the properly designed lay-up the outermost layers and the innermost layers play different roles. The former ones withstand mainly buckling, and the latter ones withstand mainly post -buckling. The innermost layers have the orientation angle rather close to 0°. The weight reduction opportunities are demonstrated. Keywords: composite plate s; post-buckling ; lay -up. © Sergey Selyugin, 2019 1. INTRO DUCTION Post-buckling of thin composite plates attracts the attention of numerous researchers , working in various modern application areas. Among the se applications are the aerospace, the automotive, the marine , and the civil ones. According to design practice in composite structure s, the account of the load-carrying capability “ above buckling” gives an opportunity of extra weight saving , as compar ed to existing traditional design. Composite plates are part of the considerable number of structures in the above applications. The loading applied to the plates is a combination of compression and shear. The compression dominat es for the structures similar to the upper panels of an airplane wing structure. As the examples of known structure s designed for pure shear , one may indicate airplane wing ribs and spars . In some cases , the shear-loaded plates should withstand the same level of shear loading applied in the direct and opposite loading direction. Such c ases are inherent in the aerospace structures like the ribs and the spars of the vertical tail plane, the fuselage skin panels, etc . 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 o ptimiz ing the composite plate , an engineer , in fact , does his best to save the structural weight , the material cost , and the production t ime. Due to that , the optimization problems for the composite plates attract the considerable attention at the design phase. The lay-up 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 p apers dealt with the lay-up optimization of composite plates . The non -exhaustive list [1] -[22] contains the important publications devoted to the post -buckling analysis and optimization . In the list, there are also severa l latest papers. The foundations of the post -buckling theory are described in [23]. Other interesting publications may be found in the