Engineering Structures 32 (2010) 1617–1630 Contents lists available at ScienceDirect Engineering Structures journal homepage: www.elsevier.com/locate/engstruct Non-linear analysis of two-layer timber beams considering interlayer slip and uplift A. Kroflič a , I. Planinc a,∗ , M. Saje a , G. Turk a , B. Čas a,b a University of Ljubljana, Faculty of Civil and Geodetic Engineering, Jamova 2, SI-1115 Ljubljana, Slovenia b Building and Civil Engineering Institute ZRMK, Dimičeva 12, SI-1000 Ljubljana, Slovenia article info Article history: Received 18 September 2009 Received in revised form 10 December 2009 Accepted 1 February 2010 Available online 26 February 2010 Keywords: Non-linearity Composite beam Slip Uplift Timber Strain-based finite element abstract A new mathematical model and its finite element formulation for the non-linear analysis of mechanical behaviour of a two-layer timber planar beam is presented. A modified principle of virtual work is employed in formulating the finite element method. The basic unknowns are strains. The following assumptions are adopted in the mathematical model: materials are taken to be non-linear and can differ from layer to layer; interacting shear and normal contact tractions between layers are derived from the non-linear shear contact traction–slip and the non-linear normal contact traction–uplift characteristics of the connectors; the geometrically linear and materially non-linear Bernoulli’s beam theory is assumed for each layer. The formulation is found to be accurate, reliable and computationally effective. The suitability of the theory is validated by the comparison of the numerical solution and the experimental results of full-scale laboratory tests on a simply supported beam. An excellent agreement between measured and calculated results is observed for all load levels. The further objective of the paper is the analysis of the effect of different normal contact traction–uplift constitutive relationships on the kinematic and static quantities in a statically determined and undetermined structure. While the shear contact traction–slip constitutive relationship dictates the deformability of the composite beam and has a substantial influence on most of the static and kinematic quantities of the composite beam, a variable normal contact traction–uplift constitutive relationship is in most cases negligible. © 2010 Elsevier Ltd. All rights reserved. 1. Introduction Composite structures may be a highly efficient structural form. If properly composed, they exhibit better bearing capacity and are easier to build. A particularly strong increase in the research and application of composite structures has been observed in recent years in the rehabilitation of buildings and bridges. The earliest theories dealing with composite planar beams were introduced in the middle of the last century after a num- ber of experimental observations had confirmed the beneficial connected behaviour of layers. The first mathematical theories of beams, composed of flexibly connected layers, were developed in- dependently in Sweden [1], the Soviet Union [2], Switzerland [3] and the United States of America [4]. Most subsequent theories consider linear elastic behaviour and small displacements (Girham- mar and Gopu [5], Kroflič et al. [6], Ranzi et al. [7], see also Schn- abl et al. [8,9]). A number of theories also consider non-linearity, as, e. g. Ayoub [10], who considered material non-linearities, Betti and Gjelsvik [11], Čas et al. [12], Gattesco [13], Hirst and Yeo [14], Ranzi and Bradford [15], who took into account both material and ∗ Corresponding author. E-mail address: iplaninc@fgg.uni-lj.si (I. Planinc). geometric non-linearities, Rassam and Goodman [16], Salari et al. [17], Seracino et al. [18], Thompson et al. [19]. Čas et al. [20] and Hozjan [21] seem to be the first to introduce a fully consistent materially and geometrically non-linear model of composite engi- neering beams. The majority of the analysis procedures take into consideration a solely interlayer slip between the layers while neglecting uplift. Mathematical models which consider both slip and uplift at the contact have rarely been proposed, e. g. in Adekola [22], Robinson and Naraine [23], who consider geometrically and materially linear behaviour, whereas Gara et al. [24] consider a bilinear constitutive law of materials. The present paper proposes a finite element formulation for the materially non-linear analysis of two-layer timber beams. Our for- mulation employs a modified principle of virtual work where the basic unknown functions are strains, i. e. deformation quantities. The Galerkin-type finite element formulation is employed, as in Planinc et al. [25]. The present paper is focused on the effect of slip and uplift at the contact interface on the mechanical behaviour of two-layer timber beams. The model considers the following as- sumptions: a composite structure, applied loading and deforma- tions are planar; the material of each layer is taken to be non-linear and homogeneous and can differ from layer to layer; interact- ing shear and normal contact tractions between the layers follow 0141-0296/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.engstruct.2010.02.009