Product: ABAQUS/Standard
, outer radius 
, and length 
is subjected to both internal and external pressure loads that are asymmetric. The pressure stresses take the following forms: 
at 
and 
at 
, where p is a pressure value and r and 
are the cylindrical coordinates. Assuming plane strain conditions and a linear elastic material with Young's modulus E and Poisson's ratio 
, the small-displacement solutions for stress and displacement are as follows: 
Only a slice of the cylinder is considered. Plane strain conditions are applied by setting 
0 everywhere. In the r-direction 10 elements are used in the second-order element models. In models using the first-order elements, 20 and 40 elements are used in the full- and reduced-integration models, respectively.
Linear elastic, Young's modulus = 30 × 106, Poisson's ratio = 0.3.
Boundary conditions:
0 everywhere; 
–9.9854 × 10–4 at and 
0°, as obtained from the equation for 
above. These constraints eliminate the rigid body motions in the global z- and x-directions, respectively.
Loading:
The asymmetric pressure loads are prescribed by applying the appropriate nonuniform distributed load types on the inside and outside surfaces of the cylinder with the *DLOAD option and defining the pressure stress equations for 
in user subroutine DLOAD. In the user subroutine, the value at each integration point, which is stored in COORDS(3), is expressed in degrees.
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 cylinder with these parameters: 
6, 
2, 
6, and 
10 × 103. The output locations are at points 
and 
on the 0° plane, where z can be any value along lines 
and 
in the figure shown on the previous page since the solution is independent of z, and at points E and G, which are at the corresponding locations on the 180° plane. The solutions predicted by ABAQUS agree well with the exact solution. Closer agreement is anticipated if a denser mesh is used.
| Variable | Exact | CAXA8n | CAXA8Rn | CAXA4n | CAXA4Rn |
|---|---|---|---|---|---|
| at A | –30000.0 | –29610.0 | –29760.0 | –28617.0 | –29132.0 |
 at A | –7890.4 | –7702.7 | –7849.6 | –7885.1 | –7722.9 |
 at A | 6089.6 | 6268.2 | 5973.4 | 6034.6 | 5729.2 |
 at A | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| at A | –9.9854 × 10–4 | –9.9854 × 10–4 | –9.9854 × 10–4 | –9.9854 × 10–4 | –9.9854 × 10–4 |
| at C | –10000.0 | –9988.9 | –9992.4 | –10101.0 | –10205.0 |
| at C | –3969.9 | –3964.4 | –3967.9 | –3952.2 | –3978.2 |
| at C | –2029.9 | –2024.3 | –2031.5 | –2013.4 | –1902.1 |
| at C | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| at C | –2.9222 × 10–3 | –2.9222 × 10–3 | –2.9222 × 10–3 | –2.9207 × 10–3 | –2.9221 × 10–3 |
| at E | 30000.0 | 29610.0 | 29760.0 | 28617.0 | 29132.0 |
| at E | 7890.4 | 7702.7 | 7849.6 | 7885.1 | 7722.9 |
| at E | –6089.6 | –6268.2 | –5973.4 | –6034.6 | –5729.2 |
| at E | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| at E | 9.9854 × 10–4 | 9.9854 × 10–4 | 9.9854 × 10–4 | 9.9854 × 10–4 | 9.9854 × 10–4 |
| at G | 10000.0 | 9988.9 | 9992.4 | 10101.0 | 10067.0 |
| at G | 3969.9 | 3964.4 | 3967.9 | 3952.2 | 3978.2 |
| at G | 2029.9 | 2024.3 | 2031.5 | 2013.4 | 1987.9 |
| at G | 0.0 | 0.0 | 0.0 | 0.0 | 0.0 |
| at G | 2.9222 × 10–3 | 2.9222 × 10–3 | 2.9222 × 10–3 | 2.9207 × 10–3 | 2.9221 × 10–3 |
Note:
The results are independent of n, the number of Fourier modes. The 
variable is not compared, since is treated as an internal variable in these elements and is not available for output. The accuracy of may be assumed to be comparable to the accuracy of .
Figure 1.3.35–1 through Figure 1.3.35–4 show plots of the undeformed mesh, the deformed mesh, the contours of , and the contours of , respectively, for the CAXA8R3 model.
CAXA41 elements.
User subroutine DLOAD used in ecnssfsm.inp.
CAXA42 elements.
User subroutine DLOAD used in ecntsfsm.inp.
CAXA43 elements.
User subroutine DLOAD used in ecnusfsm.inp.
CAXA44 elements.
User subroutine DLOAD used in ecnvsfsm.inp.
CAXA4R1 elements.
User subroutine DLOAD used in ecnssrsm.inp.
CAXA4R2 elements.
User subroutine DLOAD used in ecntsrsm.inp.
CAXA4R3 elements.
User subroutine DLOAD used in ecnusrsm.inp.
CAXA4R4 elements.
User subroutine DLOAD used in ecnvsrsm.inp.
CAXA81 elements.
User subroutine DLOAD used in ecnwsfsm.inp.
CAXA82 elements.
User subroutine DLOAD used in ecnxsfsm.inp.
CAXA83 elements.
User subroutine DLOAD used in ecnysfsm.inp.
CAXA84 elements.
User subroutine DLOAD used in ecnzsfsm.inp.
CAXA8R1 elements.
User subroutine DLOAD used in ecnwsrsm.inp.
CAXA8R2 elements.
User subroutine DLOAD used in ecnxsrsm.inp.
CAXA8R3 elements.
User subroutine DLOAD used in ecnysrsm.inp.
CAXA8R4 elements.
User subroutine DLOAD used in ecnzsrsm.inp.
