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The layup optimization for maximizing buckling loads of long laminated composite cylindrical shells subjected to combinations of axial compression, external lateral pressure, and torsion is carded out on the basis of Flugge's theory. The layer angles, that is, fiber orientation angles, of the laminated composite cylindrical shells are restricted to the values of 0, 45, -45, and 90 deg, and then nine lamination parameters are used as design variables for the layup optimization. Explicit expressions relating the nine lamination parameters are derived and used to describe the feasible region in the design space of lamination parameters. Thus, the layup optimization problem becomes a constrained nonlinear optimization problem, and the optimum lamination parameters are determined by a mathematical programming method. The laminate configurations aiming to realize the optimum lamination parameters are obtained by analytic formulas or by an unconstrained optimization procedure. It is observed that, for the laminated composite cylindrical shells with the layer angles restricted to the values of 0, 45, -45, and 90 deg, the optimum buckling loads are less than 10% lower than the optimum buckling loads obtained for unrestricted laminate configurations.

In the classical lamination theory and the first-order shear deformation theory, the stiffnesses of laminated composites can be expressed as linear functions of 12 lamination parameters. A method is presented for determining the feasible region in general design space of 12 lamination parameters. In some optimization problems, the local optimum can be avoided by using lamination parameters instead of layer angles and thicknesses. Thus, the lamination parameters are useful design variables in the layup optimization for mechanical properties of laminated composites. To perform layup optimization, the feasible region of the lamination parameters needs to be known. The lamination parameters are functionals of the distribution function of fiber orientation angles through the thickness. In the determination of the feasible region, the laminate configurations are not restricted. In the method, a variational approach is applied to find the boundary of the feasible region in the general design space of 12 lamination parameters. The feasible region for any set of lamination parameters can be also obtained. With use of the method, the feasible regions in four different design spaces are examined as an example. The reliability and the validity of the method are confirmed.

Astudy was conducted to present a stiffness design method of symmetric laminates using lamination parameters. Results show that the stiffness components are a linear function with respect to the two lamination parameters, which are governed by four in-plane and four out-of-plane lamination parameters, respectively. In the stiffness optimization, the use of lamination parameters as design variables leads to an efficient optimization approach because of the existing relation between the lamination parameters.