Product: ABAQUS/Standard
, outer radius 
, and length 
is subjected to a bending moment, M, applied to its end planes. For a linear elastic material with Young's modulus E and Poisson's ratio 
, the solutions for stress and displacement are as follows:
is the moment of inertia of the cylinder and r, 
, and z are the cylindrical coordinates.Only one-half of the structure is considered, with a symmetry plane at 
0. The form of the displacement solution, which is a quadratic function in both r and z, suggests that a single second-order element should model the structure accurately. The full- and reduced-integration second-order elements do use a single element mesh, but an 8 × 12 mesh is used for the fully integrated first-order elements and a 16 × 24 mesh is used for the reduced-integration first-order elements.
Linear elastic, Young's modulus = 30 × 106, Poisson's ratio = 0.33.
Boundary conditions:
0 on the 0 plane; at 
on the 0 plane, 
at 
0° is set equal to at 180° with the *EQUATION option to remove the rigid body motion in the global x-direction.
Loading:
The bending load is simulated by applying a surface traction of the form 
on the 
plane of the cylinder. This is done by applying the appropriate nonuniform pressure load with the *DLOAD option and defining the variation of the pressure in both the r- and -directions with 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 structure with 
1 and dimensions 
6, 
2, and 
6. The output locations are at points 
, 
, 
, and 
on the 0° plane, as shown in the figure on the previous page, and at points 
, and H, which are at the corresponding locations on the 180° plane. The CAXA8n elements match the exact solution precisely.
| Variable | Exact | CAXA8n | CAXA8Rn | CAXA4n | CAXA4Rn |
|---|---|---|---|---|---|
 at A | 2 | 2 | 2.040 | 2.102 | 2.124 |
| at A | 0 | 0 | 0 | 0 | 0 |
 at A | 0 | 0 | 0 | 0 | 0 |
| at B | 2 | 2 | 2 | 2.098 | 2.091 |
| at B | –6 × 10–7 | –6 × 10–7 | –5.927 × 10–7 | –6.000 × 10–7 | –6.015 × 10–7 |
| at B | 4 × 10–7 | 4 × 10–7 | 4.164 × 10–7 | 3.996 × 10–7 | 3.984 × 10–7 |
| at C | 6 | 6 | 5.979 | 5.895 | 5.877 |
| at C | –1.76 × 10–7 | –1.76 × 10–7 | –1.881 × 10–7 | –1.757 × 10–7 | –1.762 × 10–7 |
| at C | 0 | 0 | 0 | 0 | 0 |
| at D | 6 | 6 | 6 | 5.898 | 5.908 |
| at D | –7.76 × 10–7 | –7.76 × 10–7 | –7.954 × 10–7 | –7.757 × 10–7 | –7.779 × 10–7 |
| at D | 1.2 × 10–6 | 1.2 × 10–6 | 1.211 × 10–6 | 1.200 × 10–6 | 1.203 × 10–6 |
| at E | –2 | –2 | –2.040 | –2.102 | –2.124 |
| at E | 0 | 0 | 0 | 0 | 0 |
| at E | 0 | 0 | 0 | 0 | 0 |
| at F | –2 | –2 | –2 | –2.098 | –2.091 |
| at F | 6 × 10–7 | 6 × 10–7 | 5.927 × 10–7 | 6.000 × 10–7 | 6.015 × 10–7 |
| at F | –4 × 10–7 | –4 × 10–7 | –4.164 × 10–7 | –3.996 × 10–7 | –3.984 × 10–7 |
| at G | –6 | –6 | –5.979 | –5.895 | –5.877 |
| at G | 1.76 × 10–7 | 1.76 × 10–7 | 1.881 × 10–7 | 1.757 × 10–7 | 1.762 × 10–7 |
| at G | 0 | 0 | 0 | 0 | 0 |
| at H | –6 | –6 | –6 | –5.898 | –5.908 |
| at H | 7.76 × 10–7 | 7.76 × 10–7 | 7.954 × 10–7 | 7.757 × 10–7 | 7.779 × 10–7 |
| at H | –1.2 × 10–6 | –1.2 × 10–6 | –1.211 × 10–6 | –1.200 × 10–6 | –1.203 × 10–6 |
Note: The results are independent of n, the number of Fourier modes.
Figure 1.3.33–1 through Figure 1.3.33–4 show plots of the undeformed mesh, the deformed mesh, the contours of , and the contours of , respectively, for the CAXA4R4 model.
CAXA41 elements.
User subroutine DLOAD used in ecnssfsk.inp.
CAXA42 elements.
User subroutine DLOAD used in ecntsfsk.inp.
CAXA43 elements.
User subroutine DLOAD used in ecnusfsk.inp.
CAXA44 elements.
User subroutine DLOAD used in ecnvsfsk.inp.
CAXA4R1 elements.
User subroutine DLOAD used in ecnssrsk.inp.
CAXA4R2 elements.
User subroutine DLOAD used in ecntsrsk.inp.
CAXA4R3 elements.
User subroutine DLOAD used in ecnusrsk.inp.
CAXA4R4 elements.
User subroutine DLOAD used in ecnvsrsk.inp.
CAXA81 elements.
User subroutine DLOAD used in ecnwsfsk.inp.
CAXA82 elements.
User subroutine DLOAD used in ecnxsfsk.inp.
CAXA83 elements.
User subroutine DLOAD used in ecnysfsk.inp.
CAXA84 elements.
User subroutine DLOAD used in ecnzsfsk.inp.
CAXA8R1 elements.
User subroutine DLOAD used in ecnwsrsk.inp.
CAXA8R2 elements.
User subroutine DLOAD used in ecnxsrsk.inp.
CAXA8R3 elements.
User subroutine DLOAD used in ecnysrsk.inp.
CAXA8R4 elements.
User subroutine DLOAD used in ecnzsrsk.inp.
