Keywords: thick laminated composites, three-point bending, beam theory, classical laminate theory, first order, refined higher order, finite element analysis.

This paper investigates the thickness effect in composite laminates. In plate structures that are loaded in the plane, the effect of the thickness is hard to evaluate. To clearly present this effect, the deflection of a simply supported laminated beam with different thicknesses is studied. Therefore, the results of a finite element analysis (FEA) and three other analytical theories are compared together. The first method is a beam theory based on the classical laminate theory (CLT) [1,2]; the second is based on a first order shear theory [1], with two different shear correction factor calculations [1,3]; and the third uses a refined higher order method [4].

The simply supported composite laminated beam is loaded in the middle with a linear force . The beam has the following dimensions: the length : 100 mm, the width : 20 mm, and the thickness varies from 0.5 to 20 mm. Two types of lay-up are tested: a cross-ply laminate ( ) and a quasi-isotropic laminate ( ). The thicknesses are changed by repeating each lay-up times. To keep the deflection in the same order of magnitude and to remain within the small deformation domain, the transverse load value is adjusted for each case. To allow a better comparison, dimensionless values are often used. For three-point bending problem, the ASTM proposes a span-to-depth ratio ( ) [5].

Several analytical theories have been developed for laminated beams [1,2], and most of them reduce 3D problems to 2D ones. The first method considered is based on the CLT [1,2], and this theory uses the Kirchhoff assumption, which results in neglecting the transverse shear and the out-of-plane normal effects. The displacement in the z direction is given by the following equation [1]:

The second method is the Timoshenko first-order beam theory (TFBT) for a simply supported beam. It extends the previous theory by including a transverse shear deformation term. The shear correction factor K is often difficult to calculate for composite laminated beams. Since a quasi-isotropic symmetric lay-up is used, is assumed to be 5/6, the value for isotropic materials. Another method for estimating the factor, developed by Madabhusi-Raman and Davalos [3], is used in this study. The deflection of the beam is defined by [1]:

Through the years, several higher order theories have been developed in order to avoid the calculation of the shear correction . For this research, a refined higher-order beam theory (RHBT), developed by Zenkour [4], is used as the third method for the comparison.

The FEA model was built in ANSYSR [6] using a SOLID46 element type. The element type is defined by eight nodes, each having three degrees of freedom (, , translations). Although up to 250 layers can be stacked by element, beam models are built with 4 to 16 elements through the thickness in order to achieve accurate results.

By comparing the results, a clear thickness effect can be outlined. The CLT is not available for thick laminated beams ( ). From a comparison of the other analytical methods, the results are even closer and then an additional effort in calculation could be avoided using the TFBT with a constant factor. Furthermore, there is a difference over 5%, between the theories and ANSYSRresults for thicknesses above 13 mm ().

The results show that for a low thickness ratio, there is an obvious thickness effect, and there remains a need for extensive experiments of thick laminates in order to be able to better handle this effect.

1

J.N. Reddy, "Mechanics of laminated composite plates theory and analysis", CRC Press, p. 782, Boca Raton, FL, USA, 1997.

2

I.M. Daniel, O. Ishai, "Engineering mechanics of composite materials", Oxford University Press, p. 395, New York, 1994.

3

P. Madabhusi-Raman, J.F. Davalos, "Static shear correction factor for laminated rectangular beams", Composites Part B: Engineering (Proceedings of the 2nd International Conference for Composites Engineering, Aug 21-24 1995, 27(3-4), p. 285-293, 1996. doi:10.1016/1359-8368(95)00014-3

4

A.M. Zenkour, "Transverse shear and normal deformation theory for bending analysis of laminated and sandwich elastic beams", Mechanics of Composite Materials and Structures, 6(3), p. 267-283, 1999. doi:10.1080/107594199305566

5

ASTM D790-03, Standard Test Methods for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials. 2003, ASTM International.

6

AnsysR, "Ansys Release 9.0 Documentation".

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