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2026-04-10

VE0084 | Thin-Walled Spherical Vessel

Description

A thin-walled spherical vessel is loaded by inner pressure p. While neglecting self-weight, the goal is to determine the von Mises stress σ_Mises and the radial deflection u_R of the vessel.

Material	Isotropic Linear Elastic	Modulus of Elasticity	E	210000.0	MPa
Material	Isotropic Linear Elastic	Poisson's Ratio	ν	0.296	-
Geometry	Spherical Shell	Radius	R	0.500	m
Geometry	Spherical Shell	Shell Thickness	t	5.000	mm
Load	Pressure	Internal Pressure	p	5.000	MPa

Thin-Walled Spherical Vessel

Analytical Solution

The analytical solution is based on the theory of thin-walled vessels. This theory assumes the membrane stress state of the shell; thus, the following conditions have to be met:

The thickness of the shell can not change discontinuously.
The distributed loading can not change discontinuously.
The curvature radii and positions of centers cannot change discontinuously.
The outer forces including the reaction forces have to be tangential to the shell surface.

The stress state is described by the Laplace equation:

\frac{σ_{1}}{R_{1}} + \frac{σ_{2}}{R_{2}} = \frac{p}{t}

Laplace Equation

Where σ₁, σ₂ are stresses in meridian and parallel direction respectively and R₁, R₁ are the radii in the appropriate directions. For the spherical vessel, this can be simplified due to symmetry (σ₁ = σ₂ = σ, R₁ = R₂ = R) into the form:

σ = \frac{p R}{2 t}

Laplace Equation for the Spherical Vessel

The von Mises stress σ_Mises can be determined from the principal stresses:

σ_{Mises} = \frac{1}{\sqrt{2}} \sqrt{(σ_{1} - σ_{2})^{2} + (σ_{2} - σ_{3})^{2} + (σ_{3} - σ_{1})^{2}} = σ \approx 250.000 {Nmm}^{- 2}

Spherical Vessel - von Mises Stress

The radial deflection u_R of the vessel follows from Hooke's law:

u_{R} = \frac{R}{E} (σ - μ σ) \approx 0.419 mm

Spherical Vessel - Radial Deflection

RFEM Settings

Modeled in RFEM 5.39 and RFEM 6.13
Element size l_FE = 0.010 m
Isotropic linear elastic material is used
Full and eighth computational models are used

Results

RFEM Results - Eighth Model of the Spherical Vessel, von Mises Stress

Computational Model	Theory σ_Mises [MPa]	RFEM 6 σ_Mises [MPa]	Ratio [-]	RFEM 5 σ_Mises [MPa]	Ratio [-]
Full Model	250.000	249.984	1.000	249.987	1.000
Eighth Model	250.000	249.984	1.000	249.984	1.000

The von Mises stress is probed from the test point A for both computational models. There are minor stress deviations on the surface due to the mesh topology.

Computational Model	Theory u_R [mm]	RFEM 6 u_R [mm]	Ratio [-]	RFEM 5 u_R [mm]	Ratio [-]
Full Model	0.419	0.419	1.000	0.419	1.000
Eighth Model	0.419	0.419	1.000	0.419	1.000

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Verification Example 000084 | 1

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