Axisymmetric membrane elements

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

This page discusses:

ProductsAbaqus/Standard

Elements tested

MAX1

MAX2

MGAX1

MGAX2

Problem description



Model:

Thickness of membrane is 0.05.

Material:

For tests without orientation: linear elastic, Young's modulus = 105, Poisson's ratio = 0.3, thermal expansion coefficient = 10−7.

For tests using orientation: linear elastic, engineering constants with E1=​ 102, E2=​ 108, E3=​ 102, ν12=ν13=ν23=​ 0, and G12=G13=G23=​ 102.

Boundary conditions:

Degree of freedom 2 is fixed for the bottom node. In addition, degree of freedom 5 is fixed for the bottom node for elements supporting twist.

Initial conditions

For tests without orientation an initial stress field of σ11=​ 0.001 and σ22=​ 0.001 is applied to all elements. The temperature of all nodes is set to 0 initially.

History definition I (for all element types)

Step 1 (geometrically nonlinear):

Loading: All degrees of freedom at all nodes are constrained. This step is recommended to apply the initial stresses. In subsequent steps all the necessary boundary conditions are applied.

Step 2 (perturbation):

Loading: A concentrated force (in direction 2) of magnitude 314 is applied to the top node.

Analytical solution: u2 at top node = 0.04998.

Step 3 (perturbation):

Loading: Internal pressure of magnitude 500.

Analytical solution: Hoop stress = 10000.

Step 4 (perturbation):

Loading: The temperature at all nodes is increased to 5000.

Analytical solution: ϵ11=ϵ22=​ 0.0005.

History definition II (for element types MGAX1 and MGAX2)

Step 1 (geometrically nonlinear):

Loading: All degrees of freedom at all nodes are constrained. This step is recommended to apply the initial stresses. In subsequent steps all the necessary boundary conditions are applied.

Step 2 (perturbation):

Loading: A concentrated moment (in degree of freedom 5) of magnitude 200 is applied to the top node.

Analytical solution: Shear stress = 636.22.

History definition III (for element types MGAX1 and MGAX2 using a local coordinate system)

Step 1 (geometrically nonlinear):

Loading: A concentrated load of magnitude 2 is applied to the top node.

Step 2 (geometrically nonlinear):

Loading: A concentrated moment of magnitude 2 is applied to the top node.

Results and discussion

History definition I:

All elements yield exact solutions.

History definition II:

The deviation from the analytical solution is approximately 0.06%.

History definition III:

The results are compared with those from a similar well-refined model using CGAX4R (axisymmetric continuum elements that support twist) elements. Since the strain in the thickness direction is very small in the continuum model, the section Poisson's ratio is set to 0 for the membrane model. The results match very well.