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

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 ${\sigma }_{xx}={\sigma }_{yy}={\sigma }_{xy}=$ −1000 and, for plane strain elements, ${\sigma }_{zz}=$ −600.

Strains

Plane strain elements:

${\epsilon }_{xx}={\epsilon }_{yy}=$ −1.7333 × 10⁻⁵, ${\gamma }_{xy}=$ −8.6667 × 10⁻⁵.

Plane stress and shell elements:

${\epsilon }_{xx}={\epsilon }_{yy}=$ −2.3333 × 10⁻⁵, ${\gamma }_{xy}=$ −8.6667 × 10⁻⁵.

Displacements

$u_x=x{\epsilon }_{xx}+y{\gamma }_{xy}$, $u_y=y{\epsilon }_y$.

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

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

${\sigma }_{xx}=$ −1000(2 − y), ${\sigma }_{yy}={\sigma }_{xy}=$ −1000, and, for plane strain elements, ${\sigma }_{zz}=\nu \left({\sigma }_{xx}+{\sigma }_{yy}\right)$.

Strains

Plane strain elements:

${\epsilon }_{xx}=$ (−3.0333 (2 − y) + 1.3) × 10⁻⁵, ${\epsilon }_{yy}=$ (1.3(2 − y) − 3.03333) × 10⁻⁵, ${\gamma }_{xy}=$ −8.66667 × 10⁻⁵.

Plane stress and shell elements:

${\epsilon }_{xx}=$ (−3.333 (2 − y) + 1) × 10⁻⁵, ${\epsilon }_{yy}=$ ((2 − y) − 3.3333) × 10⁻⁵, ${\gamma }_{xy}=$ −8.6667 × 10⁻⁵.