Cylinder subjected to an asymmetric pore pressure field: CAXA elements

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

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ProductsAbaqus/Standard

Elements tested

CAXA8Pn

CAXA8RPn

(n = 1, 2, 3, 4)

Problem description



A hollow cylindrical soil column of circular cross-section, inner radius Ri, outer radius Ro, and length 2⁢L is subjected to an asymmetric pore pressure distribution of the form

U=Uo⁢r⁢cos⁡θRo,

where Uo is the constant pore pressure at the outside surface of the cylinder at θ=​ 0° and r and θ are the cylindrical coordinates. The presence of pore pressure gradients in the radial and circumferential directions causes the pore fluid in the soil to flow in these directions, and bending of the cylinder results. This is a coupled problem in which the stress equilibrium and fluid continuity equations must be solved simultaneously with the pore pressure CAXA elements. For illustration purposes we consider only the steady-state coupled problem, and we assume that the material is linear with constant permeability and is made up of incompressible grains and fluid. The results predicted by the pore pressure CAXA models will be compared with those predicted by the corresponding three-dimensional model.

Only one-half of the structure is considered, with a symmetry plane at z=​ 0. A mesh convergence study indicates that a single second-order CAXA element can model the structure adequately and yield accurate results. However, two elements are used in the radial direction so that direct comparison of results obtained with the three-dimensional model can be made. In the three-dimensional model the C3D20P element is used in a finite element mesh with 2 elements in the radial direction, 1 in the axial direction, and 12 in the circumferential direction. To facilitate comparison of results with the CAXA models, all nodes in the three-dimensional model are transformed to a local cylindrical system, and a cylindrical orientation is applied to the material so that displacement, stress, and strain components are output in the same cylindrical system.

Material:

Linear elastic, Young's modulus = 1 × 108, Poisson's ratio = 0.3, permeability = 1 × 10−5, initial void ratio = 1.0 everywhere.

Boundary conditions:

uz=​ 0 on the z=​ 0 plane; ur=​ 0 is applied at r=Ri and z=​ 0 to eliminate the rigid body motion in the global x-direction.

Loading:

A pore pressure field of the form U=Uo⁢r⁢cos⁡θ/Ro is applied. The pore pressure at each corner node on the inside and outside walls of the cylinder is calculated, and the pore pressure values are prescribed as boundary conditions via degree of freedom 8.

Results and discussion

The results obtained with the CAXA8Pn and CAXA8RPn (n = 1, 2, 3 or 4) elements and those obtained with the C3D20P elements are tabulated below for a structure with these parameters: L=​ 6, Ri=​ 2, Ro=​ 6, and Uo=​ 3 × 106. The output locations are at points A=(Ri,0,0), B=(Ri,L,0), C=(4,0,0), D=(4,L,0), E=(Ro,0,0), and F=(Ro,L,0) on the θ=​ 0° plane, as shown in the figure on the previous page. Results that are exactly equal and opposite to those shown below are obtained at the same locations on the θ=​ 180° plane. It is apparent that the results of the CAXA models match closely with the results of the three-dimensional model. The stress solution, which is shown in the table below, reveals that the effective stress components are identical to the pore pressure everywhere so that the total stress is zero everywhere in the cylinder. The results obtained from the CAXA models are independent of n, the number of Fourier modes, and appear to be more accurate than the three-dimensional model because the applied asymmetric pore pressure field can be prescribed precisely in the CAXA models. In the three-dimensional model more elements are needed in the θ-direction to get results with higher accuracy. Note the similarity between the solution to this problem and the asymmetric temperature analysis described in Cylinder subjected to an asymmetric temperature field: CAXA elements.

VariableC3D20P CAXA8Pn CAXA8RPn
ur at A 0  0  0 
uz at A 0  0  0 
U at A 1 × 106 1 × 106 1 × 106
σr⁢r at A 9.9066 × 105 1 × 106 1 × 106
σz⁢z at A 9.9397 × 105 1 × 106 1 × 106
σθ⁢θ at A 9.9751 × 105 1 × 106 1 × 106
ur at B −3.5791 × 10−2 −3.6 × 10−2 −3.6 × 10−2
uz at B 2.3853 × 10−2 2.4 × 10−2 2.4 × 10−2
U at B 1 × 106 1 × 106 1 × 106
σr⁢r at B 9.9066 × 105 1 × 106 1 × 106
σz⁢z at B 9.9397 ×105  1 × 106 1 × 106
σθ⁢θ at B 9.9751 × 105 1 × 106 1 × 106
ur at C 1.1926 × 10−2 1.2 × 10−2 1.2 × 10−2
uz at C 0  0  0 
U at C 1.9987 × 106 2 × 106 2 × 106
σr⁢r at C 1.9944 × 106 2 × 106 2 × 106
σz⁢z at C 1.9947 × 106 2 × 106 2 × 106
σθ⁢θ at C 2.0038 × 106 2 × 106 2 × 106
ur at D −2.3864 × 10−2 −2.4 × 10−2 −2.4 × 10−2
uz at D 4.7711 × 10−2 4.8 × 10−2 4.8 × 10−2
U at D 1.9987 × 106 2 × 106 2 × 106
σr⁢r at D 1.9943 × 106 2 × 106 2 × 106
σz⁢z at D 1.9945 × 106 2 × 106 2 × 106
σθ⁢θ at D 2.0036 × 106 2 × 106 2 × 106
ur at E 3.1819 × 10−2 3.2 × 10−2 3.2 × 10−2
uz at E 0  0  0 
U at E 3 × 106 3 × 106 3 × 106
σr⁢r at E 3.004 × 106 3 × 106 3 × 106
σz⁢z at E 2.9961 × 106 3 × 106 3 × 106
σθ⁢θ at E 3.0105 × 106 3 × 106 3 × 106
ur at F −3.9718 × 10−2 −0.4 × 10−2 −0.4 × 10−2
uz at F 7.1581 × 10−2 7.2 × 10−2 7.2 × 10−2
U at F 3 × 106 3 × 106 3 × 106
σr⁢r at F 3.004 × 106 3 × 106 3 × 106
σz⁢z at F 2.9965 × 106 3 × 106 3 × 106
σθ⁢θ at F 3.0105 × 106 3 × 106 3 × 106

Figure 1 through Figure 4 show plots of the undeformed and deformed meshes, the applied asymmetric pore pressure field, the contours of ur, and the contours of uz, respectively, in the CAXA8P4 model.

Figures

Figure 1. Deformed mesh.

Figure 2. Contours of pore pressure.

Figure 3. Contours of r-displacement.

Figure 4. Contours of z-displacement.