Fibre-reinforced laminated composite tubes with free ends under uniform internal pressure

Abstract

The stress and deformation fields in a fibre-reinforced composite tube under uniform internal pressure are discussed in some detail. In the interior region far from the ends classical laminate theory delivers rather poor results and has to be adjusted to include effects due to lateral contraction and to curvature. In the region near the ends boundary conditions (here stress-free ends will be assumed) require more elaborate methods of calculation. The use of finite element methods may prove to be problematic because in some parts of the boundary region very large gradients are expected. The problem is particularly acute in lay-ups with angle-plies where 3D-elements would be needed. In the following study analytical solutions based on asymptotic approximations of the three-dimensional equations of linear elasticity for homogeneous orthotropic materials will be presented. One “small” parameter ε R characterising thin shell geometry and another “small” parameter ε G following from homogenised material properties of the shell structure and whose order of magnitude is comparable with ε R are used to derive asymptotically consistent approximate solutions according to the following pattern: The adjusted laminate theory leads to stress distributions in each ply of the laminated tube which do not satisfy zero stress boundary conditions at the stress-free ends. In terms of asymptotic theory this is a typical problem of singular perturbations and can be solved by considering boundary layers near the free ends where stress and deformation fields satisfy the boundary conditions and match conveniently with stress and deformation distributions calculated with the adjusted laminate theory in the interior zone. To derive boundary layer equations which are easy to handle analytically and still obtain fairly accurate results, we replace the laminated structure by its homogeneous orthotropic equivalent. The boundary layer solutions are then obtained following the main ideas developed by Sayir [(1985) Local Effects in the Analysis of Structures, Elsevier Science Publishers; (1986) Bull. Tech. Univ. Istanbul 39, 515–528]. We first study as a leading example a hypothetical cross-ply structure and concentrate on transverse shear and axial normal stress distributions across the wall thickness and along the axial direction in the boundary layer region. We show that all boundary conditions near the tube ends can be fulfilled by introducing first a “normal-stress” boundary layer starting at the tube end and extending in the axial direction, then a shorter “shear-stress” boundary layer contained in the previous one and finally two lateral boundary layers starting at the inner and outer surface of the tube wall inside the shear stress boundary layer. We compare analytical results obtained with the help of the asymptotic method described above with those obtained from three-dimensional finite-element calculations for each ply. Then we discuss two cases of symmetric angle-ply lay-ups corresponding to tubes actually tested in the laboratory by the second author and enhance similarities and differences with respect to the cross-ply lay-up. We also show how the interlaminar shear stresses which may cause delaminations and failure depend strongly on the lay-up.