4.12.2024

VE0067 | Nestlačitelný materiál - Silnostěnná nádoba

Popis

Tento verifikační příklad je modifikací VE0064 - Silnostěnná nádoba, kde je rozdíl že materiál nádoby je nestlačitelný. A thick-walled vessel is loaded by inner and outer pressure. The vessel is open-ended, thus there is no axial stress. The problem is modeled as a quarter model and described by the following set of parameters. While neglecting self-weight, determine the radial deflection of the inner and outer radius u_r(r₁), u_r(r₂).

Materiál	Elastic Incompressible	Modul pružnosti	E	1.000	MPa
Materiál	Elastic Incompressible	Poisson´s Ratio	ν	0.499	-
Geometrie		Inner Radius	r₁	200.000	mm
Geometrie		Outer Radius	r₂	300.000	mm
Load		Inner Pressure	p₁	60.000	kPa
Load		Outer Pressure	p₂	0.000	kPa

Nestlačitelný materiál - silnostěnná nádoba

The aim of this verification example is to show the phenomenon of volumetric locking of finite elements. This problem can occur in incompressible material cases when the Poisson ratio ν is approaching 0.5 (rubber materials, materials in plastic state). In this case, the finite element displacements tend to zero. The volumetric locking cannot be avoided by the mesh refinement. All fully integrated elements will lock when the incompressible material is used. The simplest way to avoid locking is using reduced integration. In this case the number of integration points is reduced. The integration scheme is one order less accurate than the standard integration scheme. In RFEM the reduced integration is used so the result will be correct, as seen bellow.

Analytické řešení

The analytical solution is the same as in the case of VE0064 - Thick-Walled Vessel. Desired radial deflection of the inner and outer radius of the open-ended vessel u_r(r₁), u_r(r₂) can be determined using the following equations:

u_{r} (r_{1}) = \frac{r_{1}}{E} [σ_{t} (r_{1}) - ν σ_{r} (r_{1})],

u_{r} (r_{2}) = \frac{r_{2}}{E} [σ_{t} (r_{2}) - ν σ_{r} (r_{2})]

Radiální průhyb vnitřního a vnějšího poloměru otevřené nádoby

Nastavení programu RFEM

Modeled in RFEM 5.06 and RFEM 6.06
The element size is l_FE = 2.000 mm
Isotropic linear elastic material model is used

Výsledky

RFEM 6 Výsledky – celková deformace

Quantity	Analytické řešení	RFEM 6	Poměrná hodnota	RFEM 5	Poměrná hodnota
u_r(r₁) [mm]	29.988	29.986	1.000	29.987	1.000
u_r(r₂) [mm]	22.497	22.495	1.000	22.495	1.000

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Verifikační příklad 000067 | 1

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