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This study demonstrates, as an illustration of the influence of stress state susceptibility of material properties on the behavior of composite structures, the solution of problem of buckling and postbuckling analysis of stiffened thin-walled composite cylinder, where straightforward approach with linear elastic theory leads to inaccurate results even on initial stage of loading. During buckling process, a stress state in material point is able to change significantly and it should be taken into account, for example, for materials with different elastic moduli in tension and compression and for other stress state types, too. Nonlinear elastic model based on the use of triaxiality parameter is considered and implemented in FEA software to capture effects related with sensitivity of material elastic properties to the stress state.