Acta Mech 225, 523–541 (2014)

DOI 10.1007/s00707-013-0972-5

Leo Škec · Gordan Jelenic´

Analysis of a geometrically exact multi-layer beam with a rigid interlayer connection

Received: 24 January 2013 / Revised: 5 July 2013 / Published online: 17 September 2013 © Springer-Verlag Wien 2013

Abstract A finite-element formulation for geometrically exact multi-layer beams is proposed in the present work. The interlayer slip and uplift are not considered. The number of layers is arbitrary, and the basic unknown functions are the horizontal and vertical displacements of the composite beam’s reference axis and the cross-sectional rotation of each layer. Due to the geometrically exact definition of the problem, the governing equations are nonlinear in terms of basic unknown functions and the solution is obtained numerically. In general, each layer can have different geometrical and material properties, but since the layers are rigidly connected, the main application of this model is on homogeneous layered beams. Numerical examples compare the results of the present model with the existing geometrically nonlinear sandwich beam models and also with the 2D plane-stress elements and, where applicable, with the results from the theory of elasticity. The comparison with 2D plane-stress elements shows that the multi-layer beam model is very efficient for modelling thick beams where warping of the cross-section has to be considered. 1 Introduction

Research and application of layered composite structures using beam elements in many areas of engineering have increased considerably over the past couple of decades and continue to be a topic of undiminished interest in the computational mechanics community. Layered composite beam models can be divided into three basic categories: two-layer beams, sandwich beams and multi-layer beams. The analysis can be linear, providing an analytical solution, or geometrically or/and materially nonlinear, which leads to numerical solutions. Girhammar and Pan [3] proposed an analytical solution for a geometrically and materially linear two-layer composite beam with interlayer slip using the Bernoulli beam theory, while Schnabl et al. [12] used the Timoshenko beam theory. In addition to interlayer slip, Kroflicˇ et al. [7] introduced interlayer uplift to a geometrically and materially linear and nonlinear two-layer beam model [8]. For the sandwich beam model with partial interaction, Schnabl et al. [11] proposed an analytical solution and Vu-Quoc and Deng [20] proposed a geometrically exact formulation for sandwich beams with a rigid interlayer connection. Sousa and da Silva [17] proposed an analytical solution for geometrically and materially linear multi-layer beams allowing for interlayer slip, while the model proposed by Škec et al. [18] introduced both the slip and the interlayer uplift. Vu-Quoc et al. [21]

L. Škec · G. Jelenic´ (B)

University of Rijeka, Faculty of Civil Engineering, Radmile Matejcˇic´ 3, 51000 Rijeka, Croatia

E-mail: gordan.jelenic@gradri.hr

Tel.: +385-51-265955

Fax: +385-51-265998

L. Škec

E-mail: leo.skec@gradri.hr

Tel.: +385-51-265956

Fax: +385-51-265998 524 L. Škec, G. Jelenic´ and Vu-Quoc and Ebciog˘lu [22] proposed a geometrically exact formulation for multi-layer beams with a rigid interconnection, which is also the topic of the present work. Vu-Quoc et al. [21] and Vu-Quoc and Ebciog˘lu [22] used the Galerkin projection (see [15] and [20] for details) to obtain the computational formulation of the resulting nonlinear equations of equilibrium in the static case (the formulation of the equations of motion in the general dynamic case was proposed, too), while in the present work, the equilibrium equations are derived from the principle of virtual work. While the resulting numerical procedure is of necessity equal, here, we focus on the actual transformation of the displacement vector for each layer to the displacement vector of the beam reference line and show that it may be written in a remarkably elegant form allowing for simple numerical implementation. Furthermore, we specifically analyze problems with large number of layers and thick beam problems with pronounced cross-sectional warping, and we compare the performance of the elements derived to the analytical results and to the finite-element results obtained using 2D plane-stress elements. In the proposed finite-element formulation, the number of layers is arbitrary and they are assembled in a composite beam with the interlayer connection allowing only for the occurrence of independent rotations of each layer. 2 Problem description 2.1 Position of the composite beam in the material coordinate system

Consider an initially straight composite beam of length L and a cross-section composed of n parts with heights hi and areas Ai , where i ∈ [1, n] is an arbitrary layer (see Fig. 1).

The layers are made of linear elastic material with Ei and Gi acting as Young’s and shear moduli of each layer’s material. Each layer has its own material coordinate system defined by an orthonormal triad of vectors

E1,i , E2,i and E3,i , with axes X1,i , X2,i and X3,i . The axes X1,i coincide with the reference axes of each layer, which are chosen arbitrarily (they can pass through the corresponding layer, but also fall outside of it) and are mutually parallel. Thus, we can introduce a base vector E1 = E1,i and a coordinate X1 = X1,i . The cross-sections of the layers are symmetric with respect to a common vertical principal axis X2 defined by a base vector E2 = E2,i (a condition for a plane problem). However, for any chosen point on the beam, the coordinate Xi,2 changes for each layer i . The axes X3,i are mutually parallel but do not necessarily correspond with the horizontal principal axes of the layers’ cross-sections, and from Fig. 1, it follows that X3 = X3,i and

E3 = E3,i . The distance from the bottom of a layer to the layer’s reference axis is denoted by ai . The first and the second moment of area of the cross-section Ai with respect to axis X3,i are defined as