Simple and accurate two-noded beam element for composite laminated beams using a refined zigzag theory E. Oñate a,b,⇑ , A. Eijo a , S. Oller a,b a International Center for Numerical Methods in Engineering (CIMNE), Campus Norte UPC, 08034 Barcelona, Spain b Universitat Politècnica de Catalunya (UPC), Campus Norte UPC, 08034 Barcelona, Spain article info Article history: Received 7 October 2010 Received in revised form 2 November 2011 Accepted 24 November 2011 Available online 3 December 2011 Keywords: Two-noded beam element Zigzag kinematics Timoshenko theory Composite Sandwich beams abstract In this work we present a new simple linear two-noded beam element adequate for the analysis of com- posite laminated and sandwich beams based on the combination of classical Timoshenko beam theory and the refined zigzag kinematics proposed by Tessler et al. [22]. The element has just four kinematic variables per node. Shear locking is eliminated by reduced integration. The accuracy of the new beam ele- ment is tested in a number of applications to the analysis of composite laminated beams with simple sup- ported and clamped ends under point loads and uniformly distributed loads. An example showing the capability of the new element for accurately reproducing delamination effects is also presented. Ó 2011 Elsevier B.V. All rights reserved. 1. Introduction It is well known that both the classical Euler–Bernouilli beam theory [1] and the more advanced Timoshenko theory [2] produce inadequate predictions when applied to relatively thick composite laminated beams with material layers that have highly dissimilar stiffness characteristics. Even with a judiciously chosen shear cor- rection factor, Timoshenko theory tends to underestimate the axial stress at the top and bottom outer fibers of a beam. Also, along the layer interfaces of a laminated beam the transverse shear stresses predicted exhibit erroneous discontinuities. These difficulties are due to the higher complexity of the ‘‘true’’ variation of the axial displacement field across a highly heterogeneous beam cross- section. Indeed to achieve accurate computational results, 3D finite element analyses are often preferred over beam models. For com- posite laminates with hundred of layers, however, 3D modelling becomes prohibitively expensive, specially for non linear and progressive failure analyses. Improvements to the classical beam theories have been ob- tained by the so called equivalent single layer (ESL) theories that assume a priori the behavior of the displacement and/or the stress through the laminate thickness [3,4]. Despite being computation- ally efficient, ESL theories often produce inaccurate distributions for the stresses and strains (in particular the transverse shear stress) across the thickness. The need for composite laminated beam theories with better predictive capabilities has led to the development of the so-called higher order theories. In these theories higher-order kinematic terms with respect to the beam depth are added to the expression for the axial displacement and, in some cases, to the expressions for the deflection. A review of these theories can be found in [3,4]. Accurate predictions of the correct shear and axial stresses for thick and highly heterogenous composite laminated and sandwich beams can be obtained by using layer-wise theory. In this theory the thickness coordinate is split into a number of analysis layers that may or not coincide with the number of laminate plies. The kinematics are independently described within each layer and cer- tain physical continuity requirements are enforced [3,4]. A drawback of layer-wise theory is that the number of kine- matic variables depends on the number of analysis layers. The layer displacements can be condensed at each section in terms of the axial displacement for the top layer during the equation solu- tion process [5,6]. The displacement condensation processes can be however expensive for problems involving many analysis layers. 0045-7825/$ - see front matter Ó 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.cma.2011.11.023 ⇑ Corresponding author at: Universitat Politècnica de Catalunya (UPC), Campus Norte UPC, 08034 Barcelona, Spain. E-mail address: onate@cimne.upc.edu (E. Oñate). URL: http://www.cimne.com/eo (E. Oñate). Comput. Methods Appl. Mech. Engrg. 213–216 (2012) 362–382 Contents lists available at SciVerse ScienceDirect Comput. Methods Appl. Mech. Engrg. journal homepage: www.elsevier.com/locate/cma