Journal of Mechanical Science and Technology 25 (5) (2011) 1105~1117 www.springerlink.com/content/1738-494x DOI 10.1007/s12206-011-0305-3 Limit-point buckling analyses using solid, shell and solid–shell elements † Marc Killpack 1,2 and Farid Abed-Meraim 1,* 1 LEM3, UMR CNRS 7239, Arts et Métiers ParisTech, 4 rue A. Fresnel, 57078 Metz Cedex 03, France 2 Georgia Tech Lorraine, Georgia Institute of Technology, 2-3 rue Marconi, 57070 Metz, France (Manuscript Received June 23, 2010; Revised January 15, 2011; Accepted January 27, 2011) ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Abstract In this paper, the recently-developed solid-shell element SHB8PS is used for the analysis of a representative set of popular limit-point buckling benchmark problems. For this purpose, the element has been implemented in Abaqus/Standard finite element software and the modified Riks method was employed as an efficient path-following strategy. For the benchmark problems tested, the new element shows better performance compared to solid elements and often performs as well as state-of-the-art shell elements. In contrast to shell elements, it allows for the accurate prescription of boundary conditions as applied to the actual edges of the structure. Keywords: Assumed-strain; Benchmark problems; Boundary conditions; Limit-point buckling; Locking; Solid–shell ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- 1. Introduction Traditionally, shell finite elements are best suited for the numerical simulation of thin structure applications, while solid elements are regularly used for bulk structures. However, engineering structures often combine thin components with thick/bulk geometries in the same assembly. Thus, the finite element modeling of such applications would be considerably simplified if the same type of finite element could be success- fully used in both zones. Alternatively, solid and shell ele- ments can be mixed in the same model, but special effort is required for matching the translations in solid elements and rotations in shell elements. This laborious task generally in- volves defining algebraic constraints at the interface or intro- ducing solid-to-shell transition elements which often result in excessively stiff behavior. In addition, most shell formulations rely on specific kinematic descriptions along with particular constitutive assumptions, which require the additional effort of implementation of plane-stress behavior models. In order to overcome the aforementioned limitations, solid- shell elements have recently emerged as an interesting alterna- tive. Indeed, these elements combine a shell-like response with three-dimensional element geometry, thus naturally matching solid elements in the same mesh. Numerous devel- opments have been made in this direction during the last dec- ade (e.g. Domissy [1], Cho et al. [2], Hauptmann and Schweizerhof [3], Lemosse [4], Sze and Yao [5], Hauptmann et al. [6]). The SHB8PS element is one such recently devel- oped element that is based on a purely three-dimensional ap- proach (Abed-Meraim and Combescure [7, 8], Legay and Combescure [9]). Note that most of the methods developed earlier were based on the enhanced assumed strain method proposed by Simo and co-workers (Simo and Rifai [10], Simo and Armero [11], Simo et al. [12]), and consisted of either the use of a conventional integration scheme with appropriate control of all locking phenomena or the application of a re- duced integration technique with associated hourglass control. Both approaches have been extensively investigated and evaluated in various structural applications, as reported in recent contributions by Vu-Quoc and Tan [13], Chen and Wu [14], Kim et al. [15], Alves de Sousa et al. [16, 17], Reese [18], and Abed-Meraim and Combescure [19]. In spite of its three-dimensional geometry (eight-node hexahedron), the SHB8PS element has received specific treatments and enhancements so that it exhibits the desirable shell features in structural applications. The standard three- dimensional constitutive law is modified such that plane-stress assumptions are approached and the integration points are aligned along a preferential direction designated as the "thick- ness". Reduced integration is employed in order to improve the element's computational efficiency and to alleviate mem- brane and shear locking. The hourglass modes that are thus inherently induced are physically stabilized following the efficient approach of Belytschko and Bindeman [20]. In order to eliminate the various locking phenomena (transverse shear, membrane, thickness), the discrete gradient operator is pro- † This paper was recommended for publication in revised form by Editor Maeng- hyo Cho * Corresponding author. Tel.: +33 3 87 37 54 79, Fax.: +33 3 87 37 54 70 E-mail address: farid.abed-meraim@ensam.eu © KSME & Springer 2011