AIAA JOURNAL Vol. 34, No. 10, October 1996 Governing Equations of a Stiffened Laminated Inhomogeneous Conical Shell Zahit Mecitoglu* Istanbul Technical University, Maslak, Istanbul 80626, Turkey This study presents the dynamic equations of a stiffened composite laminated conical thin shell under the influence of initial stresses. The governing equations of a truncated conical shell are based on the Donnell-Mushtari theory of thin shells including the transverse shear deformation and rotary inertia. The extension-bending coupling is considered in the derivation. The composite laminated conical shell is also reinforced at uniform intervals by elastic rings and/or stringers. The stiffening elements are relatively closely spaced, and therefore the stiffeners are smeared out along the conical shell. The inhomogeneity of material properties because of temperature, moisture, or manufacturing processes is taken into account in the constitutive equations. A generalized variational theorem is derived so as to describe the complete set of the fundamental equations of the conical shell. Next, the uniqueness is examined in solutions of the dynamic equations of the conical shell, and the boundary and initial conditions are shown to be sufficient for the uniqueness in solutions. The equations of the laminated composite conical shell are solved by the use of the finite difference method as an illustrative example. The accuracy of results is tested by certain earlier results, and a good agreement is found. Introduction A STIFFENED laminated conical shell is one of the common structural elements used in modern airplane, missile, booster, and other space vehicles. The dynamic behavior of stiffened lam- inated conical shells under the dynamic loads and initial stresses is of considerable engineering importance for determination of the failure and fatigue life of the shell. The inhomogeneity as a result of environmental effects such as temperature and moisture may de- grade the mechanical properties of anisotropic materials and affect the dynamic behavior of the shell and should be taken into account in the constitutive equations of the conical shell. Studies of the vibrations of cylindrical and truncated conical shells made of isotropic and anisotropic materials are referenced in Ref. 1. Although much literature exists on the free vibration of isotropic conical shells, e.g., Refs. 2-5, fewer studies were found about stiffened, orthotropic, or anisotropic conical shells. Singer 6 and Weingarten 7 derived the equations of motion of a Donnell- Mushtari-type orthotropic shell theory. Bacon and Bert 8 showed the effect of changing the ratio of orthotropic constants upon the fun- damental frequencies of shells of revolution. Cohen 9 investigated the asymmetrical free vibrations of ring-stiffened orthotropic shells of revolution. Mecitoglu and Dokmeci 10 developed a shell finite el- ement that includes smeared stringers and rings and examined the uniqueness on the solutions. Weingarten, 1 ' Goldberg et al., 12 - l3 and Schneider et al. 14 studied the vibration of the conical shells sub- jected to initial stresses. Newton 15 obtained results for the conical shells having orthotropic material properties that vary in the merid- ional direction. Martin 16 studied the free vibrations of anisotropic conical shells. Heyliger and Jilani 17 considered the inhomogene- ity in the free vibrations of elastic layered cylinders and spheres. Inhomogeneity of the material properties as a result of tempera- ture was studied by Mecitoglu 18 for a truncated isotropic conical shell. Sivadas and Ganesan, 19 and Sankaranarayanan et al. 20 stud- ied the laminated conical shells with variable thickness. Kayran and Vinson 21 examined the effect of transverse shear and rotary inertia on the free vibrations of truncated conical shells. Tong stud- ied the free vibration of orthotropic 22 and laminated 23 ' 24 conical shells using a simple and exact solution technique. Langley 25 ap- plied a dynamic stiffness technique to the vibration of the stiffened shell structures. Ley et al. 26 examined the buckling of the stiffened Received Nov. 30, 1994; revision received Jan. 16, 1996; accepted for publication May 11, 1996. Copyright © 1996 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. * Associate Professor, Aeronautics and Astronautics Faculty. anisotropic conical shells. Although the uniqueness in solutions of the elastodynamic problems is very important, only a few works examining this subject are found in the literature. 27 The purpose of this paper is to derive all of the dynamic equations of a composite laminated conical thin shell reinforced by stringers and rings and to examine the uniqueness in solutions of the govern- ing shell equations. The dynamic equations of a stiffened composite laminated conical shell are derived within the frame of the Donnell- Mushtari theory of elastic thin shells. A generalized variational the- orem is given so as to describe the complete set of the fundamental equations of the conical shell. The geometric nonlinearities and ini- tial stresses are taken into account in the derivation of governing equations. The rings and/or stringers are smeared out along the con- ical shell. The inhomogeneity of material properties as a result of temperature, moisture, or manufacturing processes is taken into ac- count in the constitutive equations. The uniqueness is examined in solutions of the dynamic governing equations of stiffened shells, and a theorem of uniqueness is given that enumerates the initial and boundary conditions sufficient for the uniqueness. The dynamic equations of the stiffened laminated conical shell are solved by the finite difference method to obtain the vibration characteristics, and a good agreement is obtained with certain earlier results. Governing Equations Consider a conical shell as shown in Fig. 1, where R indicates the radius of the cone at the large end, a denotes the semivertex angle of the cone, and L is the cone length along its generator. The terms h s and h R are the height of the stringers and rings, respectively; b s and b R are the width of the stringers and rings, respectively; 0 is the angular spacing of the stringers; and S is the spacing of the rings. We use the s-0 coordinate system; s is measured along the cone's generator starting at the cone vertex, and 9 is the circumferential coordinate. Displacement Field The Weierstrass theorem states that any function that is continu- ous in an interval may be approximated uniformly by polynomials in this interval. Thus, the displacement field in the shell can be represented by the following relationships: U(s, 0, z) = u(s, 0) + zp x (s, 0) + rxvCs, 0) + • • • V(s, 0, z) = v(s, 0) + z&Cs, 9) + z 2 y f) (s, 0) + • • • (1) W(s, 6, z) = w(s, 0) + zp z (s, 0) + z 2 y,(s, 0) + • • • 2118