The results of semianalytical calculation of the critical buckling loads and load-bearing capacity of laminated composite panels having different geometric shapes in plan in the course of loading with compressive and tangential forces in the plane of the layers are presented. The stability problems of orthotropic layered plates with variable geometric parameters have been solved by the Rayleigh–Ritz method with approximation of the deflection function by orthogonal Krylov polynomials. The effect of shape, the method of fastening the panels, and the arrangement pattern of reinforcing fibers in the layers of the composite has been studied. Rational reinforcement patterns that correspond to the highest critical buckling load for nonrectangular and shallow curved panels have been revealed. The load-bearing capacity under supercritical strain has been assessed based on the Tsai–Wu strength criterion.