Dynamic Equations foe Spherical and Cylindrical Shells Using Power Series Method

Licentiatavhandling, 2013

Shells are commonly used in many branches of engineering and have therefore been investigated for a number of different types of shells. A shell can be considered as a curved plate having small thickness compared to the other geometrical dimensions as well as to the wavelengths of importance. The most important superiority of shells in comparison to plates is that shell structures can provide high strength and low weight because of their membrane stiffness. Spherical and cylindrical shells appear in many applications and some dynamic shell theories have thus been developed for these cases. All these theories seem to depend on more or less ad hoc kinematical assumptions and/or other approximations. In this thesis, dynamic equations for an isotropic spherical and an anisotropic cylindrical shell are derived by using a method developed during the last decade for bars, plates, and beams. The main advantage with the method is that it is very systematic and can be developed to any order. It also seems that the resulting structural equations are asymptotically correct to any order. First, dynamic equations are derived for a spherical shell made of a homogeneous, isotropic material. The starting point is a power series expansion of the displacement components in the thickness coordinate relative to the mid-surface of the shell. By using these expansions, the three-dimensional elastodynamic equations yield a set of recursion relations among the expansion functions. Applying the boundary conditions on the surfaces of the spherical shell and eliminating all but the six lowest order expansion functions give the shell equations as a power series in the shell thickness. In principle, the equations can be truncated to any order in the shell thickness, leading to very complicated expressions. Surface differential operators are introduced to decrease the length of the shell equations and tackle this complexity. the displacement field is split into a scalar (radial) part and a vector (tangential) part, and the shell equations are given in terms of the surface operators. After some manipulations, the final four shell equations are obtained in a more compact form which can be presented explicitly. Dynamic equations for an anisotropic cylindrical shell are also derived using the same technique. As a special case a 2D circular ring is considered and the eigenfrequencies are computed and are compared with the exact solution which is obtained by expressing the displacements in terms of Bessel and Neumann functions. Graphite epoxy is considered as an anisotropic material. For all cases, results are compared to exact three-dimensional theory. The computations for eigenfrequencies from the power series approximation are in good correspondence with results from the exact solution.