Cylinder subjected to asymmetric pressure loads: CAXA elements

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

The following topics are discussed:

ProductsAbaqus/Standard

Elements tested

CAXA4n

CAXA4Rn

CAXA8n

CAXA8Rn

(n = 1, 2, 3, 4)

Problem description



A hollow cylinder of circular cross-section, inner radius Ri, outer radius Ro, and length 2L is subjected to both internal and external pressure loads that are asymmetric. The pressure stresses take the following forms: σrr=-pcosθ at r=Ro and σrr=-RoRipcosθ at r=Ri, where p is a pressure value and r and θ are the cylindrical coordinates. Assuming plane strain conditions and a linear elastic material with Young's modulus E and Poisson's ratio ν, the small-displacement solutions for stress and displacement are as follows:

σrr=A[r+Ri2Ro2r3+(3-2ν)(1-2ν)(Ro2+Ri2)r]cosθσθθ=A[3r-Ri2Ro2r3-(Ro2+Ri2)r]cosθσzz=ν(σrr+σθθ)σrθ=A[r+Ri2Ro2r3-(Ro2+Ri2)r]sinθur=1+ν2EA[(1-4ν)r2-Ri2Ro2r2+2(3-4ν)(1-2ν)(Ro2+Ri2)lnr]cosθuθ=1+ν2EA[(5-4ν)r2-Ri2Ro2r2-2(3-4ν)(1-2ν)(Ro2+Ri2)lnr-2(Ro2+Ri2)(1-2ν)]sinθ,

where

A=-(1-2ν)Rop4(1-ν)(Ro2+Ri2).

Only a slice of the cylinder is considered. Plane strain conditions are applied by setting uz= 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.

Material:

Linear elastic, Young's modulus = 30 × 106, Poisson's ratio = 0.3.

Boundary conditions:

uz= 0 everywhere; ur= −9.9854 × 10−4 at r=Ri and θ= 0°, as obtained from the equation for ur 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 and defining the pressure stress equations for σrr in user subroutine DLOAD. In the user subroutine, the θ value at each integration point, which is stored in COORDS(3), is expressed in degrees.

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 cylinder with these parameters: L= 6, Ri= 2, Ro= 6, and P= 10 × 103. The output locations are at points A=(Ri,z,0) and C=(Ro,z,0) on the θ= 0° plane, where z can be any value along lines AA and CC 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 θ= 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
σrr at A −30000.0 −29610.0 −29760.0 −28617.0 −29132.0
σzz at A −7890.4 −7702.7 −7849.6 −7885.1 −7722.9
σθθ at A 6089.6 6268.2 5973.4 6034.6 5729.2
σrθ at A 0.0 0.0 0.0 0.0 0.0
ur at A −9.9854 × 10−4 −9.9854 × 10−4 −9.9854 × 10−4 −9.9854 × 10−4 −9.9854 × 10−4
σrr at C −10000.0 −9988.9 −9992.4 −10101.0 −10205.0
σzz at C −3969.9 −3964.4 −3967.9 −3952.2 −3978.2
σθθ at C −2029.9 −2024.3 −2031.5 −2013.4 −1902.1
σrθ at C 0.0 0.0 0.0 0.0 0.0
ur at C −2.9222 × 10−3 −2.9222 × 10−3 −2.9222 × 10−3 −2.9207 × 10−3 −2.9221 × 10−3
σrr at E 30000.0 29610.0 29760.0 28617.0 29132.0
σzz at E 7890.4 7702.7 7849.6 7885.1 7722.9
σθθ at E −6089.6 −6268.2 −5973.4 −6034.6 −5729.2
σrθ at E 0.0 0.0 0.0 0.0 0.0
ur at E 9.9854 × 10−4 9.9854 × 10−4 9.9854 × 10−4 9.9854 × 10−4 9.9854 × 10−4
σrr at G 10000.0 9988.9 9992.4 10101.0 10067.0
σzz at G 3969.9 3964.4 3967.9 3952.2 3978.2
σθθ at G 2029.9 2024.3 2031.5 2013.4 1987.9
σrθ at G 0.0 0.0 0.0 0.0 0.0
ur 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 uθ variable is not compared, since uθ is treated as an internal variable in these elements and is not available for output. The accuracy of uθ may be assumed to be comparable to the accuracy of ur.

Figure 1 through Figure 4 show plots of the undeformed mesh, the deformed mesh, the contours of σrr, and the contours of ur, respectively, for the CAXA8R3 model.

Input files

ecnssfsm.inp

CAXA41 elements.

ecnssfsm.f

User subroutine DLOAD used in ecnssfsm.inp.

ecntsfsm.inp

CAXA42 elements.

ecntsfsm.f

User subroutine DLOAD used in ecntsfsm.inp.

ecnusfsm.inp

CAXA43 elements.

ecnusfsm.f

User subroutine DLOAD used in ecnusfsm.inp.

ecnvsfsm.inp

CAXA44 elements.

ecnvsfsm.f

User subroutine DLOAD used in ecnvsfsm.inp.

ecnssrsm.inp

CAXA4R1 elements.

ecnssrsm.f

User subroutine DLOAD used in ecnssrsm.inp.

ecntsrsm.inp

CAXA4R2 elements.

ecntsrsm.f

User subroutine DLOAD used in ecntsrsm.inp.

ecnusrsm.inp

CAXA4R3 elements.

ecnusrsm.f

User subroutine DLOAD used in ecnusrsm.inp.

ecnvsrsm.inp

CAXA4R4 elements.

ecnvsrsm.f

User subroutine DLOAD used in ecnvsrsm.inp.

ecnwsfsm.inp

CAXA81 elements.

ecnwsfsm.f

User subroutine DLOAD used in ecnwsfsm.inp.

ecnxsfsm.inp

CAXA82 elements.

ecnxsfsm.f

User subroutine DLOAD used in ecnxsfsm.inp.

ecnysfsm.inp

CAXA83 elements.

ecnysfsm.f

User subroutine DLOAD used in ecnysfsm.inp.

ecnzsfsm.inp

CAXA84 elements.

ecnzsfsm.f

User subroutine DLOAD used in ecnzsfsm.inp.

ecnwsrsm.inp

CAXA8R1 elements.

ecnwsrsm.f

User subroutine DLOAD used in ecnwsrsm.inp.

ecnxsrsm.inp

CAXA8R2 elements.

ecnxsrsm.f

User subroutine DLOAD used in ecnxsrsm.inp.

ecnysrsm.inp

CAXA8R3 elements.

ecnysrsm.f

User subroutine DLOAD used in ecnysrsm.inp.

ecnzsrsm.inp

CAXA8R4 elements.

ecnzsrsm.f

User subroutine DLOAD used in ecnzsrsm.inp.

Figures

Figure 1. Undeformed mesh.

Figure 2. Deformed mesh.

Figure 3. Contours of radial stress.

Figure 4. Contours of r-displacement.