Analysis of developable shells with special reference to the finite element method and circular cylinders

The paper begins by noting that the practical and efficient numerical analysis of thin walled shells is far from a reality. Groundwork for the investigation starts with an examination of existing sufficiency conditions for convergence of the finite element method of analysis with refinement of mesh size; new and more practical conditions are then given specifically for shells. Working formulae of a suitable first approximation theory for the linear small deflexion behaviour are then given for arbitrary shells in lines of curvature and in geodesic coordinates. A variational principle is introduced which is more general than that for the well known assumed stress hybrid finite element model; its purpose is to provide a means to overcome the excessive rank deficiency which is sometimes encountered in the derive element stiffness matrix. , The formulae are next specialized to general developable shells for they are tne simplest to analyse and frequently occur in technology. Emphasis is given to the derivation of general formulae governing inextensional deformation, membrane action and rigid body movement because these constitute important factors in any adequate numerical analysis. . . , Specific application is made to circular cylindrical shells by first considering the interpolation of the kinematic continuity conditions along an arbitrary geodesic line. Details and numerical examples are provided for the first known fully compatible lines of curvature rectangular finite element which directly recovers arbitrary rigid body movements as well as inextensional deformations and membrane actions. The paper concludes with details and numerical examples of an arbitrarily shaped triangular finite element which employs the above mentioned variational principle m conjunction with linearly varying stress fields. All the rigid body movements are directly recovered as well as inextensional deformations and membrane actions. It is anticipated that this finite element and its derivatives will find widespread application.