A continuum based three-dimensional shell element for laminated structures S. Klinkel*, F. Gruttmann, W. Wagner 1 Institut fu Èr Baustatik, Universita Èt Karlsruhe, Kaiserstr. 12, D-76131, Karlsruhe, Germany Received 30 April 1998; accepted 6 November 1998 Abstract In this paper a continuum based three-dimensional shell element for the nonlinear analysis of laminated shell structures is derived. The basis of the present ®nite element formulation is the standard eight-node brick element with tri-linear shape functions. Especially for thin structures under certain loading cases, the displacement based element is too sti and tends to lock. Therefore we use assumed natural strain and enhanced assumed strain methods to improve the relatively poor element behaviour. The anisotropic material behaviour of layered shells is modeled using a linear elastic orthotropic material law in each layer. Linear and nonlinear examples show the applicability and eectivity of the element formulation. # 1999 Elsevier Science Ltd. All rights reserved. Keywords: Nonlinear 3D shell formulation; Composite material; Enhanced assumed strains; Assumed natural strains 1. Introduction In structural mechanics the ecient computation of thin structures requires reliable and robust elements. In the past several shell elements have been developed, where the normal stresses in the thickness direction have been included in the underlying variational for- mulation. For these types of elements a discretization of the reference surface is necessary. The nodal par- ameters are the displacement vector and the extensible director vector of the reference surface. The direct in- terpolation of the extensible director vector is proposed in several papers, see e.g. BuÈchter et al. [1], Betsch and Stein [2], Eberlein and Wriggers [3], Bischo and Ramm [4]. The multiplicative decomposition of the director ®eld into a rotational part and a scalar stretching part has been investigated e.g. by Simo et al. [5], Betsch et al. [6] or Steinmann et al. [7]. The stresses are computed from a three-dimensional material law. This feature is especially useful for complicated non- linear constitutive equations. The condition of vanish- ing thickness stresses for thin structures is approximately ful®lled within the weak formulation. Locking eects which occur when using a three-dimen- sional material law along with constant normal thick- ness strains can be avoided by application of the enhanced assumed strain method to the thickness strains, see [1]. The associated variational formulation of this method has been developed by Simo and Rifai [8]. For linear membrane elements the formulation is identical to the method of incompatible modes introduced by Wilson et al. [9]. Further aspects for linear applications are discussed in Andel®nger and Ramm [10] and Korelc and Wriggers [11]. Geometrical nonlinearity has been included for two-dimensional problems by Simo and Armero [12] and for three-dimensional ®nite Computers and Structures 71 (1999) 43±62 0045-7949/99/$ - see front matter # 1999 Elsevier Science Ltd. All rights reserved. PII: S0045-7949(98)00222-3 * Corresponding author. E-mail address: baustatik@bau-verm.uni-karlsruhe.de (S. Klinkel) 1 Current address: Institut fuÈr Statik, TU Darmstadt, Alexander Str. 7, D-64283, Darmstadt, Germany.