Modelling of multi-layered band plates with trapezoidal corrugated cores: stability analysis

A sandwich structure is the answer for the need of a light and rigid construction. Depending on the application sandwich structures may take many different forms. A classical solution is the composition of two thin and rigid faces with a light, relatively thick, core closed between them. The core can be made of the foam, polyurethane or metal one, wood or corrugated sheet. Since the total thickness of a sandwich beam or plate is high and the properties of particular layers are distinctly different, a proper theoretical description of the behaviour of such structures is needed.

The theoretical bases for sandwich beams are described in the literature [1, 2]. Computational models for sandwich plates and shells, predictor-corrector procedures, and the sensitivity of the response of a sandwich structure to variations in geometric and material parameters have been studied by Noor et al. [3]. Carlsson et al. [4] derived tension, shear, bending and twisting rigidities for sandwich structures with a corrugated core. Analytical description of the shear stiffness of a corrugated core can be found in [5‐7]. Carrera [9] formulated the zigzag hypotheses for multilayered plates. The paper by Buannic et al. [10] is devoted to the computation of the effective properties of sandwich panels with a corrugated core. Porous plates with mechanical properties which vary through the thickness are analysed by Magnucki et al. [11]. Zenkour  [8] applied the simple and mixed first-order shear deformable plate theories to develop analytical solutions for deflections and stresses of rectangular plates. Talbi et al. [12] presented an analytical homogenization model for corrugated cardboard and its numerical implementation in the FE method with the use of shell elements. In the work by Cheng et al. [13] authors employed the finite element method (FEM) to derive equivalent stiffness properties of sandwich structures with various types of cores. Similar results can be found in [14, 15]. Magnucki et al. [16] presented analytical and numerical (FEM) calculations as well as experimental verification of the obtained results devoted to a sandwich beam with a crosswise or lengthwise corrugated core. The considered beam was made of an aluminium alloy. The plane faces (outer layers) and the corrugated core were glued together. Global buckling of sandwich beam-rectangular plates and columns with metal foam core has been analysed by Jasion et al. [17] and Jasion and Magnucki [18]. Local phenomena in beams and columns have been described by Hadi [19] and Jasion and Magnucki [20], to mention a few.

The current work was inspired by the results obtained in the following papers [21‐26]. The subject of the study is an orthotropic sandwich band plate. The plate is composed of seven layers as shown in Fig. 1—the main core and two three-layer faces. The main core is a crosswise corrugated sheet with the corrugation in the form of trapezoid. The core of the faces, closed between two flat sheets—internal and external—is also a corrugated sheet with the same type of corrugation, yet the direction of corrugation is perpendicular to the direction of the corrugation of the main core. The material of all seven layers is an isotropic one.

The goal of the present paper is to formulate the mathematical and physical model of the band plate. The field of displacements has been defined in the way to take into account the shear effect which appear in the core of the faces. This hypothesis makes a basis for formulation of displacements, strains, stresses and equilibrium equations. Moreover the finite element model (FE model) of the band plate has been prepared, and the results from the FE analyses were compared to the results of the analytical solution. Two types of analyses have been performed. First, the buckling analysis is to determine the buckling modes and buckling load for a family of band plates. Second, the modal analysis is to investigate the influence of the length of the band plate on the natural frequencies as well as on the modes of vibrations.

The field of displacements of the cross section of the band plate has been assumed as it is presented in Fig. 2. The following symbols have been introduced: the thickness of the main core \(t_{\mathrm{c}1}\), the thickness of the core of the faces \(t_{\mathrm{c}2}\), and the thickness of the flat sheets of the faces \(t_\mathrm{s}\).

In this section the following field of displacements is introduced:where \(x_1=t_\mathrm{s}/t_\mathrm{c1}\)—dimensionless parameter, \(\phi (y)=v_1(y)/t_{\mathrm{c2}}\)—dimensionless function determines the field of displacement. Strains in all layers of the band plate are as followsThe physical relations, according to Hooke’s law, arewhereand E—the Young’s modulus of metal sheets, \(\widetilde{G}_{yz}^{(\mathrm{c2})}\)—the shear modulus introduced in [7], denotations—indexes \(\mathrm c1\), 01—main core, \(\mathrm{c2}\), 02—inner corrugated layer of facings, s—sheets.

On the basis of Hamilton’s principlewherethe equations of motion are analytically derivedwhere \(\rho _s\)—the mass density of the sheet, b—the width of the band plate.

The system of Eq. (13) will serve as the basis in the following analytical investigation of buckling and vibrations of the band plate.

Numerical studies have been conducted on a family of band plates of the width \(b=230\) mm and the length L which takes the following values: 1000, 1200, 1400, 1600, 1800, 2000, 2200, 2400 mm. The thickness of the plate \(t=67\) mm. All layers are made of a sheet of the thickness \(t_\mathrm{s}=0.8\) mm. Other dimensions are given in Fig. 4. In the following subsections the results obtained from FE analyses are compared with these given by analytical formulae.

The results of the analyses presented in the previous section concern only one family of band plates. Nevertheless they show clearly the behaviour of this kind of structures. First of all the deformation of the cross section assumed in the analytical solution seams to be correct since it follows the deformation observed in the FE analyses as shown in Fig. 9a. The main core does not deform, whereas a considerable deformation of the core of the faces can be seen.

Second feature that characterize the analysed structures is the susceptibility to the local phenomena. The local deformation, as shown in Fig. 9b, appears on the narrow thin stripes of sheets which are not connected to other elements of the band plate. The local deformation is observed during the loss of stability of the band plate (Fig. 7b).

Analysing the selected results of the analytical and numerical investigation presented in Tables 1 and 2 a good agreement can be observed between the obtained values. The relative errors describe the discrepancy between the analytical and both numerical solution. As for the buckling load the relative error is in the range between 2.0 and 4.5 % for the ANSYS and between 1.6 and 5.0 % for the ABAQUS, taking into account only global modes of buckling. Even better consistency is observed for the modal analysis: the relative error for the ANSYS is up to 3.0 % and for the ABAQUS up to 3.5 %.

It should be noted that for plates of the length smaller than \(L=1600\) mm the buckling load obtained analytically—global mode, is higher than this obtained from ANSYS and ABAQUS—local mode. The difference is higher for shorter beams. It comes from the fact that for a global mode of buckling the band plate behaves as an Euler column—the buckling load increases with the decrease of the length; the whole structure undergoes bending. The local mode, however, has similar shape for all shorter beams; only some of narrow stripes of sheet between trapezoids buckles. This suggest that the local phenomena may decrease the buckling resistance of this kind of band plate considerably. The numerical analysis of local phenomena will probably need a finer mesh.

The proposed seven-layer band plate is a light and very stiff structure. The stiffness can be easily modified by changing the hight of the corrugation of the main core and the core of the faces. The stiffness can be increased by using thicker external sheets in which, according to the foregoing results, deformations are the highest and which are subjected to the highest stresses. The band plate can be made of easily available materials like steel and assembled with the use of laser welding or an adhesive. The band plates can be used to design of many different plated structures like containers, floor in railway carriages. The high stiffness and the susceptibility to local buckling make the global loss of stability unlikely to appear.

The presented analyses were limited to linear ones, and a perfect geometry of the plate has been assumed. However, due to a complex shape of the plate there is a high possibility to introduce many geometrical imperfections during manufacturing process. These may significantly influence the buckling behaviour of the plate. Then the nonlinear study should be carried out to validate the results. The experiments are also planned as a further step of the project.