1.3.1 Membrane loading of plane stress, plane strain, membrane, and shell elements

Product: ABAQUS/Standard

Elements tested

CPS3 CPS4 CPS4I CPS4R CPS4RT CPS6 CPS6M CPS6MT CPS8 CPS8R

CPE3 CPE3H CPE4 CPE4H CPE4I CPE4IH CPE4R CPE4RH CPE4RHT CPE4RT CPE6 CPE6H CPE6M CPE6MH CPE6MHT CPE6MT CPE8 CPE8H CPE8R CPE8RH

CPEG3 CPEG3H CPEG4 CPEG4H CPEG4I CPEG4IH CPEG4R CPEG4RH CPEG6 CPEG6H CPEG6M CPEG6MH CPEG8 CPEG8H CPEG8R CPEG8RH

M3D3 M3D4 M3D4R M3D6 M3D8 M3D8R M3D9 M3D9R

S4 S4R S4R5 S8R S8R5 S9R5 STRI3 STRI65 SC8R

Problem description

Material:

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

For the coupled temperature-displacement elements dummy thermal properties are prescribed to complete the material definitions.

Boundary conditions:

and, for shell elements, at all nodes.

Step 1

A distributed pressure of 1000/length is applied on each edge (for shell elements, equivalent concentrated loads). Equivalent concentrated shear forces corresponding to distributed shear loading of 1000/length are applied on each edge in the directions shown.

Response:

Stresses

At every integration point –1000 and, for plane strain elements, –600.

Strains

Plane strain elements:

–1.7333 × 10^–5, –8.6667 × 10^–5.

Plane stress and shell elements:

–2.3333 × 10^–5, –8.6667 × 10^–5.

Displacements

, .

For lower-order elements the test description is complete. For higher-order elements another step definition is included.

Step 2

Hydrostatic pressure loading along the two vertical faces, varying from 0 at the top to 1000/length at the bottom, is added to the loads already applied in Step 1.

Response:

Stresses

–1000(2 – y), –1000, and, for plane strain elements, .

Strains

Plane strain elements:

(–3.0333 (2 – y) + 1.3) × 10^–5, (1.3(2 – y) – 3.03333) × 10^–5, –8.66667 × 10^–5.

Plane stress and shell elements:

(–3.333 (2 – y) + 1) × 10^–5, ((2 – y) – 3.3333) × 10^–5, –8.6667 × 10^–5.

Results and discussion

The results for generalized plane strain elements depend on the boundary constraints applied to the generalized plane strain reference node. In these tests the reference nodes in the lower-order generalized plane strain elements are constrained such that the results are the same as their plane strain counterparts. For the higher-order generalized plane strain elements these nodes are unconstrained, so the results are the same as their plane stress counterparts.

Elements using reduced integration may have additional boundary conditions to those specified above. All elements yield exact solutions.