SUMMARYA 48 degrees-of-freedom (d.0.f.) quadrilateral thin elastic shell finite element using variable-order polynomial functions, B-spline functions and rational B-spline functions to model the shell surface is developed. This development may allow the stiffness formulation of the shell element to be linked to the geometry data bases created by computer aided design systems. The displacement functions are that of bicubic Hermitian polynomials. The displacement functions and d.0.f. are expressed and investigated in both the curvilinear and Cartesian forms. The curvilinear form is simpler and can provide the proper solution for a certain class of shell problems. For certain highly curved shells such as bellows, however, the curvilinear form fails to properly model some rigid body modes even with either the explicit inclusion of rigid body terms or the high order displacement functions. It is suggested in this study that such difficulty can be circumvented and the rigid body modes can be properly included if a Cartesian form is used for displacement functions. The straindisplacement equations are expressed in curvilinear co-ordinates. Thus, the Cartesian displacement functions require a transformation to curvilinear displacement at each numerical integration point. Examples include a pinched cylinder, a translational shell under central load, a uniformly loaded hypar shell, a pressurized ovel shell, a semi-toroidal bellows and a U-shaped bellows. For the first four examples, geometric modellings consist of polynomials of second-order (subparametric), third-order (isoparametric), and fourth and fifth-order (both superparametric) as well as B-spline functions of fourth-and fifth-order. The geometries of the pinched cylinder, the semi-toroidal bellows, and the U-shaped bellows were modelled exactly using rational B-spline functions. All the results obtained are in good agreement with alternative existing solutions.
