A functional for shells of arbitrary geometry and a mixed finite element method for parabolic and circular cylindrical shells

Abstract

In this study a higher-order shell theory is proposed for arbitrary shell geometries which allows the cross-section to rotate with respect to the middle surface and to warp into a non-planar surface. This new kinematic assumption satis"es the shear-free surface boundary condition (BC) automatically. A new internal force expression is obtained based on this kinematic assumption. A new functional for arbitrary shell geometries is obtained employing Ga( teaux di!erential method. During this variational process the BC is constructed and introduced to the functional in a systematic way. Two di!erent mixed elements PRSH52 and CRSH52 are derived for parabolic and circular cylindrical shells, respectively, using the new functional. The element does not su!er from shear locking. The excellent performance of the new elements is veri"ed by applying the method to some test problems.(i) The theory of shells used in the formulation of thin or thick shells theories (Kirchho!, Reissner and higher-order shell theories) (ii) The chosen element, independent parameters, element interpolation function and special techniques such as reduced/selective integration, discrete Kirchho! theory, etc. (iii) The method used to derive FEM such as potential energy theorem, Hu}washizu and Hellinger}Reissner theory, variational formulation, weak formulation, Ga( teaux derivatives.Kirchho! hypothesis states that the normal remains straight and normal to the middle surface. The conventional treatment of shell structures based on Kirchho! hypothesis de"nes fully the displacement pattern by the middle surface. A great di$culty arises in satisfying the necessary continuity of slopes at interface. Also, Kirchho! hypothesis cannot take into account the transverse shear [6].Despite these di$culties, an extensive literature exists on applications of the FEM to Kirchho! shells. Detailed references to these approaches can be found in Zienkiewicz and Cheung [10], Przemieniecki et al. [11], De Veubeke [12], Zienkiewicz and Holister [13], Argyris [14], Clough and Tocher [15], and Jones and Strome [16]. In Reissner plate theory only C shape function continuity is required; hence, an interpolation "eld is more easily constructed [17,18].Most of the C shell elements that have appeared in the literature, have been derived by means of the potential energy theorem assuming the displacement and two rotations as independent parameters [19]. These elements, often exhibit the well-known shear-locking phenomenon [5,6]. Shear-locking mechanism has been studied and explained by numerous authors [19,20].To alleviate shear locking, Zienkiewicz et al. [20] have introduced the reduced integration, followed by Pawsey and Clough who have introduced the selective integration [22}26]. In dynamic analysis the last method leads to zero energy modes to avoid the di$culty. Belytschko [27] has developed the stabilization matrix.Also discrete Kirchho!}Mindlin element has been developed by numerous authors [27}29]. Discrete Kirchho! elements impose the Kirchho! conditions are at a discret...

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