Non-linear analysis of two-layer beams with interlayer slip and uplift A. Kroflic ˇ, M. Saje, I. Planinc ⇑ University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, SI-1115 Ljubljana, Slovenia article info Article history: Received 17 February 2011 Accepted 23 June 2011 Available online 20 July 2011 Keywords: Non-linearity Reissner beam Slip Uplift Strain-based finite element abstract A new mathematical model for the non-linear analysis of two-layer planar beams considering flexible connections is introduced and an effective, strain-based finite element numerical solution method derived. The model and the solution method account for the exact geometrically non-linear behaviour in each separate layer. Material is assumed homogeneous but can be different in each layer. The shear strains are neglected. The laws of contact in both tangent and normal directions are taken non-linear. Numerical examples verify the proposed approach. The comparisons with numerical and experimental results from literature are made and the effects of uplift on ductility and stress distribution in beams are systematically explored. The theoretical model, combined with the present numerical formulation, has been found to result in realistic behaviour, while the numerical method proves to be accurate, reliable and computationally effective. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction New building technologies, materials and structural elements are invented on a daily basis in civil engineering. A great deal of these inventions emerge in the field of composite structures. Yet only a profound understanding of their behaviour may lead to an optimised combination of materials, geometry and building technology. The key in understanding the behaviour of composite structures is to perform extensive experimental and/or computational tests to assess effects of various parameters. The parameter of utmost importance is the stiffness of the contact between the layers, which may dramatically change the mechanical performance of a struc- ture, including its stiffness, ductility and load capacity. For that reason, much of the research on composites attempts to find out what is the effect of the contact properties on both global and local behaviour [1–8]. The majority of analyses have been performed by computer methods rather than experimentally. The early numerical modelings of multi-layer composite struc- tures date back to the middle of the previous century [9–12]. Researchers attempted to describe the partial interface connection with relatively simple mathematical models. With the increase of computer power, complex numerical models were developed for the analysis of composite beams [8,13–17]. These models ne- glected uplift at the contact and focused primarily on different non-linear layer material models and contact slip laws [18–26]. The models based on the geometrically non-linear beam theory have been very rarely discussed [1,21,27,28]. The particular exam- ples studying the effect of slip on the buckling capacity of two- layer composite beams are given in [6,29,30]. Adekola [13] was probably the first to discuss analytically the combined effect of both slip and uplift on the behaviour of two- layer composite beams. Robinson and Naraine [3] presented the solution in the form of explicit expressions of a somewhat modified Adekola’s system of differential equations. The above mentioned authors considered only a geometrically and materially linear mod- el. The models that account for a bilinear or fully non-linear contact model for the uplift have been given only recently, see [31–33]. When employing a finite element type of numerical solution, one has to select the optimal set of basic variables of the problem. There are several solutions available that consider displacement- based formulations [20,31,34]. Salari et al. [5] and Ayoub [2] considered a finite element formulation based on the force interpo- lation. Dall’Asta and Zona [35] and Ayoub and Filippou [36] employed mixed elements, where both the displacements and forces have been interpolated. Here a new finite element formulation for fully geometrically and materially non-linear analysis of two-layer beams is presented whose basic variables are strains. Hence, the only unknown func- tions of the formulation are strains. The Galerkin-type of the finite element formulation is employed as in Planinc et al. [37]. The mathematical model of the composite beam considers the follow- ing assumptions: the composite structure, an external loading and deformations are planar; the material of each layer is taken to be non-linear and homogeneous, yet it can differ from layer to layer; the geometrically and materially non-linear Reissner’s beam the- ory is assumed for each layer; shear strains are neglected. 0045-7949/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.compstruc.2011.06.007 ⇑ Corresponding author. E-mail address: igor.planinc@fgg.uni-lj.si (I. Planinc). Computers and Structures 89 (2011) 2414–2424 Contents lists available at ScienceDirect Computers and Structures journal homepage: www.elsevier.com/locate/compstruc