The p resent 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 plate - wise ways . The v on Karman theory is employed . The b oundary conditions are the simple support ones or the clamped one s. 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 the m 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 illustrati on of the optimality conditions is presented. Keywords: composite plate, postbuckling, lay -up, optimality, co -axiality. ________________________________________________________________________ © Sergey S elyugin, 2019 - 2 - 1. INTRODUCTION Postbuckling 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 s tructure s 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 lay - up optimizatio n 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 impo rtant publications devoted to the postbuckling analysis and optimization. The foundations of the postbuckling theory are described in [23]. - 3 - There are many reviews and textbooks, describing various aspects of the subject (see Turvey and Marshall [2], Kassa poglou [24], Weaver [25], Gürdal et al. [26], Bruyneel [27], Ghiasi et al. [28, 29]). Among journal papers, it s eems that one of the latest is the paper of Raju et al. [ 30]. Other interesting publications may be found in the