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> > Validation Case: Cylindrical Pressure Vessel

# 1 Problem Statement

In this case, A 1/8-symmetry model is built with three orthogonal symmetry planes, a uniform radial pressure is applied to the inner surface of the cylindrical vessel. This example demonstrates the calculation of membrane stresses in a simple thin-walled cylindrical pressure vessel.

Results are verified against hand calculations per Roark’s Formulas for Stress and Strain [1]. A derivation of these equations can be found in Ibrahim et al [2].

# 2 Geometry

Geometry Dimensions

• Inner radius of vessel R = 190~\text{mm}
• Height h = 20~\text{mm}
• Thickness t = 10~\text{mm}
• Internal pressure p = 2~\text{MPa}

# 3 Material:

Assign the material Structural Steel to the cylinder section. Material properties presented here are taken directly from the OnScale library of materials.

PropertySymbolValue
Young’s ModulusE200~\text{GPa}
Density\rho7850~\text{kg}\cdot\text{m}^{\text{-3}}
Poisson’s Ratio\nu0.30

Table 1. Material Properties for Structural Steel.

Note: All other material properties can be left at their default values.

# 4 Boundary Conditions and Loads:

Symmetry boundary conditions are applied to faces of the cylindrical section coplanar to the cardinal planes creating a 1/8 symmetry simulation.

• Symmetry 1
• Part 1 - Face 0
• Symmetry 2
• Part 1 - Face 1
• Symmetry 3
• Part 1 - Face 5
• Pressure Load 1 : 2~\text{MPa}
• Part 1 : Face 2

# 5 Meshing:

OnScale Solve automatically generates a 3D second-order tetrahedral mesh. Five simulations were conducted on meshes with densities ranging from very coarse to very fine.

Mesh DensityNumber of ElementsNumber of Degrees of Freedom
Very Coarse382912
Coarse7812325
Medium20277438
Fine405315610
Very Fine2041588501

Table 2. Mesh characterization.

# 6 Analytical Solution:

If an assumption is made that the wall of the vessel is sufficiently thin, the circumferential stresses \rho_{\text{hoop}} acting on the wall of the vessel can be calculated according to thin-wall theory [2], whereby:

\sigma_{\text{hoop}} = \frac{p\cdot R}{t}

where p is the internal pressure, R is the inner radius of the vessel and t is the thickness of the vessel wall. Because the top surface of the cylindrical vesel is unrestrained:

\sigma_{\text{axial}} = 0

The radial displacement \Delta R can be calculated as:

\Delta R = \frac{\sigma_{\text{hoop}}(R + \frac{t}{2})}{E}

with the longitudinal displacement \Delta y calculated as: \Delta y = -\frac{\sigma_{\text{hoop}} \cdot \nu \cdot h}{E}

# 7 Results Comparison:

ResultsAnalytical MethodOnScale Solve
\sigma_{\text{hoop}}~(\text{MPa})38.039.1
\sigma_{\text{axial}}~(\text{Pa})0.0061.9
\Delta R~(\mu\text{m})37.137.7
\Delta y~(\mu \text{m})-1.14-1.11

Table 3: Comparison of simulation and analytical results.

The analytical method solves a thin-walled problem where the assumptions is that the thickness of the wall is negligible with respect to the radius and shear stresses are assumed insignificant. This is not exactly true representation and accounts in large part to the deviation from the presented numerical results.

# 8 Mesh Convergence Study

A mesh convergence study was also conducted to evaluate the impact of meshing on the FEA results.

MeshingVerticesCells\sigma_{\text{hoop}}~(\text{MPa})\Delta R~(\mu\text{m})\Delta y~(\mu\text{m})
Very Coarse38291239.237.6-1.11
Coarse781232539.237.6-1.11
Medium2027743839.237.6-1.11
Fine40531561039.137.7-1.11
Very Fine204158850139.137.7-1.11

Table 4: Meshing comparison for different tiers of OnScale Solve’s automated meshing.

The outcomes for the problem are highly consistent across the different mesh densities with very little change between the most dense and least dense mesh. The values converge close to the value obtained for the analytical approach showing that the second order elements used with OnScale Solve are capable of recovering most of the physics of the problem even with a few elements across the cylinder thickness.

# References:

[1] W. C. Young, R. G. Budynas, and A. M. Sadegh, Roark’s formulas for stress and strain. McGraw-Hill Education, 2012.
[2] A. Ibrahim, Y. Ryu, M. Saidpour, et al., “Stress analysis of thin-walled pressure vessels,” Modern Mechanical Engineering, vol. 5, no. 1, p. 1, 2015.