Comparison of Numerical Results for Nonlinear Finite Element Analysis of Beams and Shells Based on 2-D Elasticity Theory and on Novel Finite Rotation Theories for Thin Structures

In recent years a novel approach to the derivation of geometrically nonlinear Kirchhoff-Love type theories for thin elastic shells was initiated by PIETRASZKIEWICZ [10–13]. Based on the polar decomposition theorem he developed an exact theory of finite rotations in thin-walled structures. In [10–13] the rotational part of the shell deformation was described by a finite rotation vector, which in turn was expressed in terms of displacements of the shell middle surface and their gradients. This provides a sound basis for the derivation of constrained kinematic relations for thin elastic shells undergoing small strains ( of 0(η), η«l) accompanied by small (0(η)), moderate (0(η1/2)), large (0(η1/4)) or unrestricted rotations. Based on this approach a complete family of geometrically nonlinear Kirchhoff-Love type shell theories has been systematically derived in the works of PIETRASZKIEWICZ [11,13–18] and SCHMIDT [22–29]. In [22–29] special interest was focussed on the derivation of fully variationally consistent theories and associated energy principles. The aforementioned hierarchical set of shell equations consists of theories for unrestricted rotations, large rotations (with variants for large rotations of the normal only accompanied by moderate or small in-surface rotations) or moderate rotations. A shell theory for unrestricted rotations has been also developed by IURA and HIRASHIMA [4], while a variant for large rotations of the normal accompanied by small in-surface rotations has been also given by NOLTE and STUMPF [9]. STEIN [30]derived geometrically nonlinear theories for beams undergoing moderate or large rotations, respectively, while several other ones obtained by specializing the aforenamed shell theories for the one-dimensional case have been given recently by NOLTE [8].