Cylinder subjected to an asymmetric temperature field: CAXA elements

This problem contains basic test cases for one or more Abaqus elements and features.

This page discusses:

Elements tested
Problem description
Results and discussion
Input files
Figures

ProductsAbaqus/Standard

Elements tested

CAXA4n

CAXA4Rn

CAXA8n

CAXA8Rn

(n = 1, 2, 3, 4)

Problem description

A hollow cylinder of circular cross-section, inner radius $R_{i}$ , outer radius $R_{o}$ , and length $2 L$ , is subjected to an asymmetric temperature distribution that is a linear function of the spatial coordinates:

\begin{matrix} T = \frac{T_{o} r \cos θ}{R_{o}}, \end{matrix}

where $T_{o}$ is the constant temperature at the outside surface of the cylinder at $θ = $ 0° and r, $θ$ , and z (see displacement solution, below) are the cylindrical coordinates. For a linear elastic material of Young's modulus E, Poisson's ratio $ν$ , and thermal expansion coefficient $α$ , the solution for a structure subjected to such a temperature distribution is stress-free, with displacements as follows:

\begin{matrix} u_{r} & ​ = \frac{α T_{o} \cos θ}{2 R_{o}} (r^{2} - z^{2} + 2 R_{i} R_{o} - R_{i}^{2}) \\ u_{θ} & ​ = \frac{α T_{o} \sin θ}{2 R_{o}} (r^{2} + z^{2} - 2 R_{i} R_{o} - R_{i}^{2}) \\ u_{z} & ​ = \frac{α T_{o} \cos θ}{2 R_{o}} r z . \end{matrix}

Only one-half of the structure is considered, with a symmetry plane at $z = $ 0. The form of the displacement solution, which is a quadratic function in both r and z, indicates that a single second-order element can model the structure adequately and yield accurate results. This problem is also solved with an 8 × 12 mesh of fully integrated first-order elements and a 16 × 24 mesh of reduced integration first-order elements.

Material:

Linear elastic, Young's modulus = 30 × 10⁶, Poisson's ratio = 0.33, coefficient of thermal expansion = 1 × 10⁻⁴.

Boundary conditions:

$u_{z} = $ 0 on the $z = $ 0 plane; $u_{r} = $ 0.06 is applied at $r = R_{i}$ and $z = $ 0 to eliminate the rigid body motion in the global x-direction. This value of $u_{r}$ is obtained from the equation for $u_{r}$ above.

Loading:

A temperature field of the form $T = T_{o} r \cos θ / R_{o}$ is applied by calculating the temperature at each node and defining the temperature value.

Results and discussion

The analytical solution and the Abaqus results for the CAXA8n, CAXA8Rn, CAXA4n, and CAXA4Rn (n = 1, 2, 3 or 4) elements are tabulated below for a structure with these parameters: $L = $ 6, $R_{i} = $ 2, $R_{o} = $ 6, and $T_{o} = $ 300. The output locations are at points $A = (R_{i}, 0, 0)$ , $B = (R_{i}, L, 0)$ , $C = (R_{o}, 0, 0)$ , and $D = (R_{o}, L, 0)$ on the $θ = $ 0° plane, as shown in the figure on the previous page, and at points $E, F, G$ , and H, which are at the corresponding locations on the $θ = $ 180° plane. While both the CAXA8n and CAXA8Rn elements match the exact solution precisely with a zero state of stress, the models using the CAXA4n and CAXA4Rn elements fail to predict a stress-free state, even though the displacement solutions predicted are quite reasonable. However, the CAXA4Rn models give much more accurate results than the CAXA4n models. This example demonstrates that the fully integrated first-order elements do not handle bending problems very well.


Variable	Exact	CAXA8n	CAXA8Rn	CAXA4n	CAXA4Rn
$σ_{r z}$ at A	0	0	0	−14071	0.0168
$u_{r}$ at A	6 × 10⁻²	6 × 10⁻²	6 × 10⁻²	6 × 10⁻²	6 × 10⁻²
$u_{z}$ at A	0	0	0	0	0
$σ_{r z}$ at B	0	0	0	11664	−3.2186
$u_{r}$ at B	−3 × 10⁻²	−3 × 10⁻²	−3 × 10⁻²	−2.9644 × 10⁻²	−2.9999 × 10⁻²
$u_{z}$ at B	6 × 10⁻²	6 × 10⁻²	6 × 10⁻²	6.0312 × 10⁻²	6.0001 × 10⁻²
$σ_{r z}$ at C	0	0	0	−14076	0.0162
$u_{r}$ at C	1.4 × 10⁻²	1.4 × 10⁻²	1.4 × 10⁻²	1.3993 × 10⁻²	1.4 × 10⁻²
$u_{z}$ at C	0	0	0	0	0
$σ_{r z}$ at D	0	0	0	11108	−3.5190
$u_{r}$ at D	5 × 10⁻²	5 × 10⁻²	5 × 10⁻²	5.0306 × 10⁻²	5.0001 × 10⁻²
$u_{z}$ at D	18 × 10⁻²	18 × 10⁻²	18 × 10⁻²	17.95 × 10⁻²	18 × 10⁻²
$σ_{r z}$ at E	0	0	0	−14071	−0.0168
$u_{r}$ at E	−6 × 10⁻²	−6 × 10⁻²	−6 × 10⁻²	−6 × 10⁻²	−6 × 10⁻²
$u_{z}$ at E	0	0	0	0	0
$σ_{r z}$ at F	0	0	0	−11664	3.2186
$u_{r}$ at F	3 × 10⁻²	3 × 10⁻²	3 × 10⁻²	2.9644 × 10⁻²	2.9999 × 10⁻²
$u_{z}$ at F	−6 × 10⁻²	−6 × 10⁻²	−6 × 10⁻²	−6.0312 × 10⁻²	−6.0001 × 10⁻²
$σ_{r z}$ at G	0	0	0	14076	3.5100
$u_{r}$ at G	−1.4 × 10⁻²	−1.4 × 10⁻²	−1.4 × 10⁻²	−1.3993 × 10⁻²	−1.4 × 10⁻²
$u_{z}$ at G	0	0	0	0	0
$σ_{r z}$ at H	0	0	0	11108	3.5100
$u_{r}$ at H	−5 × 10⁻²	−5 × 10⁻²	−5 × 10⁻²	−5.0306 × 10⁻²	−5.0001 × 10⁻²
$u_{z}$ at H	−18 × 10⁻²	−18 × 10⁻²	−18 × 10⁻²	−17.95 × 10⁻²	−18 × 10⁻²

Note:

The results are independent of n, the number of Fourier modes.

Figure 1 through Figure 4 show plots of the undeformed and deformed meshes, the applied asymmetric temperature field, the contours of $u_{r}$ , and the contours of $u_{z}$ , respectively, for the CAXA84 model.

Input files

ecnssfsl.inp: CAXA41 elements.
ecntsfsl.inp: CAXA42 elements.
ecnusfsl.inp: CAXA43 elements.
ecnvsfsl.inp: CAXA44 elements.
ecnssrsl.inp: CAXA4R1 elements.
ecntsrsl.inp: CAXA4R2 elements.
ecnusrsl.inp: CAXA4R3 elements.
ecnvsrsl.inp: CAXA4R4 elements.
ecnwsfsl.inp: CAXA81 elements.
ecnxsfsl.inp: CAXA82 elements.
ecnysfsl.inp: CAXA83 elements.
ecnzsfsl.inp: CAXA84 elements.
ecnwsrsl.inp: CAXA8R1 elements.
ecnxsrsl.inp: CAXA8R2 elements.
ecnysrsl.inp: CAXA8R3 elements.
ecnzsrsl.inp: CAXA8R4 elements.

Figures