OS-V: 0085 Plane Strain: Analysis of Pressure Vessel

This problem examines the expansion of a pressure vessel due to an internal pressure.

The plane strain formulation assumes that the strains in the plane perpendicular to the axis (or simply out-of-plane) are zero or irrelevant.

Model Files

Before you begin, copy the file(s) used in this problem to your working directory.

Pressure_Vessel_LS_PPLANE.fem

When running a CAE analysis, using the plane strain assumption can lead to considerable savings in computational time and file storage. The simplification to a 2D mesh allows you to use a finer mesh resulting in a more accurate analysis than a coarsely 3D meshed full model. A typical example of plane strain application is for the analysis of pressure vessels.

The following example examines the expansion of a pressure vessel due to internal pressure. The OptiStruct results for principal stresses and the theoretical values for principal stresses are compared.

Benchmark Model

Plane Strain elements are used to model the quarter symmetric slice of the pressure vessel with a radius of 100 mm and thickness of 20 mm. The internal pressure of 10 MPa is applied on the nodes of the inner surface of the pressure vessel and a Linear Static analysis is performed.

The following is required for model setup:

The model must be in the XY or XZ plane, meaning for the XY plane all nodes must have Z coordinates equal to zero and for the XZ plane all Y coordinates are zero.
All element normals are pointed in the positive Z or Y direction.
Figure 3.
CTPSTN and CQPSTN are used as the element property.
PPLANE is used as the card image for the property.
The 10 MPa load is applied using PLOADE1 type and selecting the internal edge of the model.

For thick-walled cylinders, stresses can be expressed as:

Circumferential Direction - Hoop Stress

σ_{c} = \frac{p_{i} r_{i}^{2} - p_{o} r_{o}^{2}}{r_{o}^{2} - r_{i}^{2}} - \frac{r_{i}^{2} r_{o}^{2} (p_{o} - p_{i})}{r^{2} (r_{o}^{2} - r_{i}^{2})}

Radial Direction

σ_{r} = \frac{p_{i} r_{i}^{2} - p_{o} r_{o}^{2}}{r_{o}^{2} - r_{i}^{2}} + \frac{r_{i}^{2} r_{o}^{2} (p_{o} - p_{i})}{r^{2} (r_{o}^{2} - r_{i}^{2})}

Where,

$σ_{c}$: Stress in circumferential direction (MPa, psi).
$σ_{r}$: Stress in radial direction (MPa, psi).
$p_{i}$: Internal pressure in the tube or cylinder (MPa, psi).
$p_{o}$: External pressure in the tube or cylinder (MPa, psi).
$r_{i}$: Internal radius of tube or cylinder (mm, in).
$r_{o}$: External radius of tube or cylinder (mm, in).
$r$: Cylinder wall radius where stress is calculated (mm, in) ( $r_{i}$ < $r$ < $r_{o}$ ).

Linear Static Analysis Results


Model	Hoop Stress (Pa)	Radial Stress (Pa)
Theoretical	55455	-10000
OptiStruct	54710	-9205.6
Normalized	1.013	1.086

Figure 4. First Principle Stress (Hoop Stress)

Figure 5. Third Principle Stress (Radial Stress)

¹ MacDonald, Bryan J., "Practical Stress Analysis with Finite Elements" (2nd Ed), page 327-329