Thick composite cylinder subjected to internal pressure

This example provides verification of the composite solid (continuum) elements in Abaqus. The problem consists of an infinitely long composite cylinder, subjected to internal pressure, under plane strain conditions. The solution is compared with the analytical solution of Lekhnitskii (1968) and with a finite element model where each layer is discretized with one element through the thickness. A finite element analysis of this problem also appears in Karan and Sorem (1990).

This page discusses:

Problem description
Geometry and model
Results and discussion
Input files
References
Tables
Figures

ProductsAbaqus/Standard

Most composites are used as structural components. Shell elements are generally recommended to model such components. Illustrations of composite shell elements in bending can be found in Analysis of an anisotropic layered plate, Composite shells in cylindrical bending, and Axisymmetric analysis of bolted pipe flange connections. In some cases, however, the analyst cannot avoid the use of continuum elements to model structural components. In these problems careful selection of the element type is usually essential to obtain an accurate solution. The performance of continuum elements for the analysis of bending problems is discussed in Performance of continuum and shell elements for linear analysis of bending problems. The discussion considers only the behavior of structures composed of homogeneous materials, but the same considerations apply when modeling composite structures with continuum elements. In other cases the deformation through the thickness of the composite may be nonlinear—for example, when material nonlinearities are present—and several elements may be required through the thickness for an accurate analysis. Such a discretization can be accomplished only with continuum elements. Other problems where the use of continuum elements may be preferred include thick composites where transverse shear effects are predominant, composites where the normal strain cannot be ignored, and when accurate interlaminar stresses are required; i.e., near localized regions of complex loading or geometry. In these problems the solutions obtained by solid elements are generally more accurate than those obtained by shell elements. An exception is the distribution of transverse shear stress through the thickness. The transverse shear stresses in solid elements usually do not vanish at the free surfaces of the structure and are usually discontinuous at layer interfaces. A discussion of the transverse shear stress calculations for solid and shell elements can be found in Composite shells in cylindrical bending.

In this problem the normal strain cannot be ignored since the displacement field due to the internal pressure is nonlinear through the cylinder thickness. At least two quadratic elements through the thickness are required to obtain accurate results. The example, therefore, demonstrates the use of composite solid elements for a problem where a shell element analysis would be inadequate.

Problem description

The cylinder configuration and material details are shown in Figure 1. The inside radius, $d_{i}$ , is 60 mm, and the outside radius, $d_{o}$ , is 140 mm. The structure consists of eight orthotropic layers of equal thickness, arranged in a stacking sequence of [0°, 90°]₄. The laminae are stacked in the radial direction, with the material fibers oriented along the circumferential and axial directions. In other words, the fibers are rotated 0° or 90° about the radial direction, where a 0° rotation implies primary fibers oriented along the circumferential direction. For this purpose we define a local coordinate system where the 1, 2, and 3 directions refer to the radial, circumferential, and axial directions, respectively. The fiber composite with the primary fibers along the circumferential direction has the following orthotropic elastic properties in this coordinate system:


$E_{1} = $ 10.0 GPa,	$E_{2} = $ 250.0 GPa,	$E_{3} = $ 10.0 GPa,
$G_{12} = G_{23} = $ 5.0 GPa,		$G_{13} = $ 2.0 GPa,
$ν_{12} = $ 0.01,		$ν_{23} = ν_{13} = $ 0.25.

We also define the composite with the primary fibers along the axial direction of this local coordinate system. Recognizing that the Poisson's ratios, $ν_{i j}$ , must obey the relations $\frac{ν_{i j}}{E_{i}} = \frac{ν_{j i}}{E_{j}}$ for an orthotropic material with engineering constants, the rotated material properties are


$E_{1} = $ 10.0 GPa,	$E_{2} = $ 10.0 GPa,	$E_{3} = $ 250.0 GPa,
$G_{12} = $ 2.0 GPa,		$G_{13} = G_{23} = $ 5.0 GPa,
$ν_{12} = $ 0.25,		$ν_{23} = ν_{13} = $ 0.01.

Each of these sets of elastic material properties is specified by giving the engineering constants. The name of each material is referred to in the composite solid section definition. This material definition ensures that the output components in the different layers are provided in the same coordinate system.

There is another method in Abaqus that can be used to define the ply orientation of the composite material. In this method only one definition of the material properties is used, but a separate orientation definition is given for each layer. This layer orientation is specified, together with the material name, in the solid section definition. The orientation can be specified by referring to a local coordinate system or by specifying an angle relative to the section orientation definition. The section orientation is specified in the solid section definition. Since the material properties of each layer in this case are specified in a different local coordinate system, the output variables are provided in different coordinate systems. Input files illustrating both methods are provided.

In addition to the material description for each layer, the stacking direction, the thickness of each layer, and the number of section points through the layer thickness required for the numerical integration of the element matrices to complete the description of the composite arrangement are defined. Three section integration points are specified in each layer. Since the analysis is linear elastic, this is sufficient to describe the stress distributions through the section. The layers can be stacked in any of the three isoparametric element coordinate directions, which—in turn—are defined by the order in which the nodes are given on the element data line. In this example the element connectivity is specified so that the first isoparametric direction lies along the radial direction.

Geometry and model

Because of symmetry, only a segment of the body needs to be analyzed. For simplicity of boundary condition application a quarter segment is chosen and is discretized with four elements in the circumferential direction and one element in the axial direction. One, two, four, or eight elements are used in the radial direction. Figure 2 shows the finite element discretization for the case where two elements are used in the radial direction. A nonuniform mesh, with two material layers in the inside element and six layers in the outside element, is used to capture the variation of the radial displacement through the section.

The model is bounded in the axial direction to impose plane strain conditions.

The load is a constant internal pressure of $P = $ 50 MPa applied in a linear perturbation step.

Results and discussion

All displacements and stresses reported here are normalized with respect to pressure, using

\bar{u} = \frac{u}{p} \times 10^{3}; \bar{σ} = \frac{σ}{p} .

The predicted displacements and stresses at the inside and outside surfaces of the cylinder are compared with the analytical results in Table 1 and Table 2. Results are shown for different element types and for different mesh densities. The tables show that a model discretized with one solid element (linear or quadratic) in the radial direction is inadequate to model the nonlinear variation of the displacement field. A substantial improvement is obtained with two elements through the thickness. The tables further show that the convergence of the finite element results onto the analytical solution is slow with mesh refinement. A mesh with two nonuniform quadratic elements through the thickness predicts remarkably accurate results, with the exception of the circumferential stress at the outside surface of the cylinder. The outside stress is, however, more than two orders of magnitude smaller than the inside stress and is, therefore, not a good measure of the accuracy of the solution.

The displacement and stress fields through the thickness are shown in Figure 3 through Figure 5. The figures compare the normalized radial displacement, the circumferential stress, and the radial stress with the analytical solution for the case where the cylinder is discretized with two C3D20R elements (of different sizes) in the radial direction. The figures show that the radial displacement and circumferential stress are in good agreement with the analytical solution. The radial stress, especially near the inside of the cylinder, is not quite as accurate. For example, the analytical solution at the inside surface is ${\bar{σ}}_{r} = $ −1.0 ( $σ_{r} = - p$ ). The finite element result for this mesh is ${\bar{σ}}_{r} = $ −0.741 (25.9% error). This result must be seen in light of mesh refinement; no improvement in the radial stress at the inside surface is obtained with four elements through the thickness, and it only improves to ${\bar{σ}}_{r} = $ −0.926 (7.4% error) when eight elements are used through the thickness (the results for the four-element and eight-element meshes are not shown in the figures). It is clear from these figures why quadratic elements and a refined mesh are required for an accurate analysis.

Input files

thickcompcyl_2el_nonuniform.inp: Model discretized with two nonuniform elements in the radial direction.
thickcompcyl_1el_sectorient.inp: Model in which the ply orientation is specified with a rotation relative to the section orientation. This model is discretized with one element in the radial direction.
thickcompcyl_4el_orient.inp: Model in which the ply orientation is specified with an orientation reference. This model is discretized with four elements in the radial direction.
thickcompcyl_8el.inp: Model in which each layer is discretized with one homogeneous element through the thickness.

References

Karan, S. S., and R. M. Sorem, “Curved Shell Elements Based on Hierarchical p-Approximation in the Thickness Direction for Linear Static Analysis of Laminated Composites,” International Journal for Numerical Methods in Engineering, vol. 29, pp. 1391–1420, 1990.
Lekhnitskii, S. G., Anisotropic Plates, translated from second Russian edition by S. W. Tsai and T. Cheron, Gordon and Breach, New York, 1968.

Tables

Table 1. Normalized radial displacement at inside and outside of cylinder. Analytical solution: ${\bar{u}}_{r}^{(i)} = $ 1.4410; ${\bar{u}}_{r}^{(o)} = $ 0.1476.
Element type	Elements in radial direction	Inside		Outside
Element type	Elements in radial direction	$u_{r}$	% error	$u_{r}$	% error
C3D8	1	1.1825	17.9	−0.2407	263.0
C3DI	1	1.2227	15.2	0.1004	32.0
C3DI⁽¹⁾	2	1.4231	12.4	0.1876	27.1
C3DI⁽²⁾	2	1.5526	7.74	0.1828	23.8
C3D20R	1	1.2581	12.7	0.1646	11.5
C3D20R⁽¹⁾	2	1.3609	5.56	0.1448	1.90
C3D20R⁽²⁾	2	1.3869	3.75	0.1481	0.34
C3D20R	4	1.3922	3.39	0.1447	1.95
C3D20R	8	1.4161	1.73	0.1496	1.35
1 - Uniform mesh
2 - Nonuniform mesh

Table 2. Normalized circumferential stress at inside and outside of cylinder. Analytical solution: ${\bar{σ}}_{θ}^{(i)} = $ 5.7060; ${\bar{σ}}_{θ}^{(o)} = $ 0.0103.
Element type	Elements in radial direction	Inside		Outside
Element type	Elements in radial direction	$σ_{θ}$	% error	$σ_{θ}$	% error
C3D8	1	3.608	36.8	−0.0307	397.0
C3DI	1	3.912	31.4	0.0362	251.1
C3DI⁽¹⁾	2	4.686	17.9	0.004	60.8
C3DI⁽²⁾	2	4.838	15.2	−0.0081	179.1
C3D20R	1	5.132	10.1	0.0414	300.0
C3D20R⁽¹⁾	2	5.496	3.68	0.0134	30.0
C3D20R⁽²⁾	2	5.548	2.77	0.0192	85.6
C3D20R	4	5.574	2.31	0.0119	15.1
C3D20R	8	5.606	1.75	0.0107	3.90
1 - Uniform mesh
2 - Nonuniform mesh

Figures