Development and applications of a quadratic isoparametric finite element for axisymmetric stress and deflection analysis

The theory and computer program for an axisymmetric finite element for static stress and deflection analysis is presented. The element is an eight noded isoparametric quadrilateral based on the displacement method which is capable of representing quadratic variation of element boundaries and displacements. Element stiffness properties are developed for linear elastic small displacement theory using homogeneous isotropic material. Test cases are compared with theoretical solutions from the theory of elasticity to identify program capabilities and limitations. Ability to analyse axisymmetric problems and to represent curved element boundaries has been demonstrated. Example problems including a cylindrical pressure vessel, a disk of uniform thickness subjected to centrifugal body force, and stress concentrations in a cylindrical rod due to a spherical inclusion are presented. In each of these cases program predicted deflection and stress values were within 2% of theoretical values. Limitations which have been identified include the prediction of discontinuous stresses at adjacent element boundaries, failure to match original element boundary stress conditions in substructure analyses, and the necessity of double precision calculations to correctly analyse problems whose theoretical solutions obey small displacement plate theory. Analysis of a spherical pressure vessel resulted in predicted displacements within 4% of theoretical values while stresses on element boundaries varied by 60% from theoretical values. Substructure analysis for the spherical inclusion problem resulted in prediction of boundary stresses which were incompatible with those originally obtained. Techniques to overcome this difficulty are proposed but are not tested. The inability to obtain reasonable results for flexural problems was found to be due to round off error in the single precision technique used for solving the structure equilibrium relations. Use of double precision calculations resulted in displacements and stresses within .25% and 4.% respectively of theory for the case of a clamped circular plate loaded by a uniform pressure normal to its surface