Constrained optimization of thin-walled composite beams with coupling

A constrained optimization scheme is developed in order to analyze and design thin-walled, laminated composite, closed cross section beams with mechanical couplings. Both elliptical and rectangular cross section shapes are considered in order to isolate the influence of the geometry on the coupled modes. The design variables are expressed in terms of material properties, stacking sequence and number of layers. This work focuses on finding the optimal stacking sequence for a beam with extension-twist or bendingtwist coupling with no hygrothermal deformation, and with constraints on the natural frequencies. Introduction Elastically tailored thin-walled laminated composites provide flexibility to meet design requirements efficiently. Coupling between deformation modes such as extension-twist, bendingtwist, extension-flap bending, extension-sweep bending, and flap-sweep bending is created by an appropriate selection of fiber orientation, stacking sequence, materials, and geometric parameters. These mechanical couplings can be tailored to produce favorable static and dynamic response. For a given cross section, extension-twist coupling can be created by a layup which provides a Circumferencially Uniform Stiffness (CUS). Such a configuration is achieved by wrapping the composite layup using a winding technique. Bending-twist coupling is created by a Circumferencially Antisymmetric Stiffness (CAS) layup. A CAS beam has a (+9j) layup on one side of the cross section midplane, and (-dj on the other side. This approach provides a simple means to obtain the desired coupling, and the question remains as to the stacking sequences which maximize the extension-twist or bending-twist coupling. The existence of an optimum design may be ascertained by formulating the problem as a constrained optimization one. The objective of this work is to address the optimal design of a beam with maximized coupling and little or no hygrothermal warping while avoiding a specified natural frequency. The hygrothermal response of the beam is an important design factor since hygrothermal stresses could induce fracture and precipitate failure of a part during manufacturing. Moreover, a hygrothermally stable configuration is desired in service. There are many applications where it may be advantageous to avoid specific natural frequencies to prevent resonance from occurring. The robustness of the optimum design configurations are assessed by investigating their sensitivity to variations in material constants and manufacturing tolerances as expressed by variation in fiber angles. 1 Student Member, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA. 2 Associate Fellow, School of Aerospace Engineering, Georgia Institute of Technology, Atlanta, GA. 3 Member, Department of Engineering, Clark Atlanta University, Atlanta, GA. Copyright© 1996 by W. Karl Lentz. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. 2326 American Institute of Aeronautics and Astronautics The coupling response is predicted from the variational asymptotic anisotropic beam theory of Ref. 1. Closed form expressions for the displacement field are obtained and the influence of the material's anisotropy is explicitly identified. Closed form expressions for the compliance coefficients in terms of material and geometric parameters are provided through the constitutive relationships