Stability of anisotropic shells of revolution of positive or negative Gaussian curvature

Abstract

A mixed variational principle is derived by Hamilton's method from the principle of minimum potential energy for thin anisotropic shells of revolution and is then used to derive a normal system of equations with complex coefficients. Discrete orthogonalization is used to solve this homogeneous system and the nonlinear system of equations that describes the precritical state of shells. A shell generated by revolving a circular arc around the axis parallel to its chord is analyzed for stability. The solution is compared with the approximate solution obtained assuming that the precritical state is membrane. It is established that the approximate problem formulation gives incorrect results for shells of negative Gaussian curvature Keywords: anisotropic shell of revolution, stability, mixed variational principle, Hamilton's method, nonlinear system of equations, discrete orthogonalization, Gaussian curvatureIntroduction. Cylindrical and conical composite shells are analyzed for stability in the publications [12, 13, 17, 19, 21, etc.], which suggest that the anisotropy due to the misalignment between the reinforcement directions and the coordinate axes of the shell has a significant effect on the critical axial load, external pressure, and torque. It was established [2] that the critical loads obtained by modeling a composite by an orthotropic body can only be approached from below by decreasing the thickness and increasing the number of elementary layers. Anisotropic shells generated by revolving a shallow circular arc around the axis parallel to its chord was analyzed for stability in [18] using the theory of shallow anisotropic shells and assuming membrane precritical state. This approach was earlier applied to isotropic shells of similar geometry [7,15,20].The present paper continues the publications [14,18,19]. We will outline a procedure for stability analysis of anisotropic shells of revolution with an arbitrary meridian and a nonlinear precritical state. It is based on the nonlinear system of equations derived from the stationarity condition for the functional obtained from the potential energy functional by Hamilton's method [9]. We will also use the numerical discrete-orthogonalization method [3] to solve inhomogeneous and homogeneous boundary-value problems. This method [5,6] was widely used in calculations [4,8].We will use the procedure for stability analysis of the same shells as in [18]. This will allow us to evaluate the applicability of the approximate method, which was also used in [7,15,20].