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A new 48 d.o.f. quadrilateral shell element with variable‐order polynomial and rational B‐spline geometries with rigid body modes
Author(s) -
Moore Carleton J.,
Yang T. Y.,
Anderson David C.
Publication year - 1984
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.1620201114
Subject(s) - curvilinear coordinates , quadrilateral , mathematics , geometry , cartesian coordinate system , bellows , shell (structure) , tetrahedron , rigid body , finite element method , displacement (psychology) , mathematical analysis , physics , classical mechanics , structural engineering , engineering , mechanical engineering , psychology , civil engineering , psychotherapist
Abstract A 48 degrees‐of‐freedom (d.o.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.o.f. are expressed and investigated in both the curvilinear and Cartesian forms. The cuivilinear 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 strain–displacement 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.