A triangular six-node shell element Do-Nyun Kim, Klaus-Jürgen Bathe * Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA article info Article history: Received 15 February 2009 Accepted 1 May 2009 Available online 3 June 2009 Keywords: Shells Finite element Triangular element Spatial isotropy MITC method Six-node element abstract We present a triangular six-node shell element that represents an important improvement over a recently published element [1]. The shell element is formulated, like the original element, using the MITC procedure. The element has the attributes to be spatially isotropic, to pass the membrane and bending patch tests, to contain no spurious zero energy mode, and is formulated without an artificial constant. In particular, the improved element does not show the instability sometimes observed with the earlier published element. We give the convergence behavior of the element in discriminating membrane- and bending-dominated benchmark problems. These tests show the effectiveness of the element. Ó 2009 Elsevier Ltd. All rights reserved. 1. Introduction A large amount of research has been expended over the past four decades on the development of shell finite elements, and yet more effective triangular shell elements are still much needed, see Refs. [1–3] and the references therein. In particular, the search for a general and uniformly effective six-node triangular shell ele- ment continues, and indeed the development of such an element represents one of the remaining key challenges in finite element analysis. While such an element is, in the first instance, sought for linear analysis, of course, the formulation should, as well, be di- rectly extendable to general nonlinear analysis. Numerous shell analyses are conducted routinely but very fine discretizations and quadrilateral elements are typically used [4]. An effective general curved six-node shell element would be very useful in that: (i) it can be employed to discretize virtually any shell geometry, (ii) it can be used to model shells overlaid on three-dimensional solids that are represented in free-form mesh- ing by 10 or 11-node tetrahedral solid elements, and (iii) it would give accurate solutions when using relatively coarse meshes. Originally, to a large extent, shell elements were developed by simply superimposing plate bending and in-plane membrane behavior, and flat facet-shell elements were proposed. As now well known, such elements are not truly representing shell behavior and indeed may not even converge depending on which shell prob- lem is solved [2]. The most promising formulation approach for a general shell element is based on the use of the ‘‘basic shell model” [2,5,6]. This mathematical model is obtained from the 3D contin- uum by introducing the Reissner–Mindlin kinematical hypothesis and the plane stress assumption for the mid-surface and the mate- rial layers parallel to that surface. Ideally, the shell element should then converge reliably and optimally to the exact solution of the mathematical model and for any well-posed shell problem. How- ever, the usual displacement interpolation leads to locking and a scheme needs to be used to alleviate this detrimental behavior. Successful quadrilateral general shell elements have been devel- oped using the mixed-interpolated-tensorial-component ap- proach, that is, the MITC procedure [7–11]. The advantage of this approach is that the elements are general, that is, they can be used for general shell geometries in linear and nonlinear analyses, and the elements have only the degrees of freedom of displacement- based elements with negligible additional computational cost. The MITC4 element is now widely used [4] and can also be em- ployed in a hierarchical manner to model additional 3D effects [12]. While tight mathematical convergence proofs of the MITC shell elements are not available, and indeed for general geometries may be out of reach, the elements have been thoroughly tested on appropriate ‘discriminating and revealing’ test problems [2,11–16]. However, these studies largely focused on the use of quadrilateral elements, equally successful general triangular shell elements are more difficult to develop. On the other hand, the family of MITC plate bending elements contains quadrilateral and triangular elements that are very effec- tive, and for plate bending solutions practically optimal [17–19]. Thorough mathematical convergence analyses and results of numerical studies have been published, see e.g. Refs. [20–23]. However, except for the MITC4 element, the elements contain internal nodes with rotational degrees of freedom only, which ren- ders them not effective for extension to shell analyses and general 0045-7949/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2009.05.002 * Corresponding author. Tel.: +1 617 253 6645. E-mail address: kjb@mit.edu (K.J. Bathe). Computers and Structures 87 (2009) 1451–1460 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc