20x

2024-11-04

VE0065 | Two-Layered Thick-Walled Vessel

Description

A two-layered thick-walled vessel is loaded by inner and outer pressure. The vessel is open, thus there is no axial stress. The problem is modeled as a quarter model. Determine the radial deflection of the inner and outer radius u_r(r₁), u_r(r₂) and the pressure (radial stress) in the middle radius p_m. The self-weight is neglected.

Material	Inner Vessel	Modulus of Elasticity	E₁	1.000	MPa
	Inner Vessel	Poisson´s Ratio	ν	0.250	-
	Outer Vessel	Modulus of Elasticity	E₂	0.500	MPa
	Outer Vessel	Poisson´s Ratio	ν	0.250	-
Geometry		Inner Radius	r₁	200.000	mm
		Middle Radius	r_m	250.000	mm
		Outer Radius	r₂	300.000	mm
Load		Inner Pressure	p₁	60.000	kPa
Load		Outer Pressure	p₂	10.000	kPa

Two-Layered Thick-Walled Vessel

Analytical Solution

The analytical solution of the given problem is analogous to the analytical solution of VE0064 - Thick-Walled Vessel. The radial deflection of the middle radius of both the inner and outer vessel can be calculated using following equations.

u_{r} (r_{m}) = \frac{r_{m}}{E_{1}} [K_{1} + \frac{C_{1}}{r_{m}^{2}} - ν (K_{1} - \frac{C_{1}}{r_{m}^{2}})]

u_{r} (r_{m}) = \frac{r_{m}}{E_{2}} [K_{2} + \frac{C_{2}}{r_{m}^{2}} - ν (K_{2} - \frac{C_{2}}{r_{m}^{2}})]

Radial Deflection of Middle Radius of Both Inner and Outer Vessel

Constants K₁, C₁, K₂ and C₂ are calculated subsequently for each vessel from the corresponding radii and boundary pressures. Using these equations, the pressure in the interface p_m can be determined.

p_{m} = \frac{2 (E_{1} p_{2} r_{2}^{2} (r_{1}^{2} - r_{m}^{2}) + E_{2} p_{1} r_{1}^{2} (r_{m}^{2} - r_{2}^{2}))}{E_{2} (r_{2}^{2} - r_{m}^{2}) ((1 + ν) r_{1}^{2} + (1 - ν) r_{m}^{2}) + E_{1} (r_{m}^{2} - r_{1}^{2}) ((1 + ν) r_{2}^{2} + (1 - ν) r_{m}^{2})}

Pressure in Interface

In turn, the radial displacements u_r(r₁), u_r(r₂) can be calculated.

u_{r} (r_{1}) = \frac{r_{1}}{E_{1}} (K_{1} + \frac{C_{1}}{r_{1}^{2}} - ν (K_{1} - \frac{C_{1}}{r_{1}^{2}}))

u_{r} (r_{2}) = \frac{r_{2}}{E_{2}} (K_{2} + \frac{C_{2}}{r_{2}^{2}} - ν (K_{2} - \frac{C_{2}}{r_{2}^{2}}))

Radial Displacement of Inner and Outer Radius

RFEM Settings

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

Results

RFEM 6 Results – Total Deflection

Quantity	Analytical Solution	RFEM 6	Ratio	RFEM 5	Ratio
p_m [kPa]	21.655	21.663	1.000	21.648	1.000
u_r(r₁) [mm]	33.605	33.602	1.000	33.605	1.000
u_r(r₂) [mm]	27.287	27.283	1.000	27.287	1.000

444x

Verification Example 000065 | 1

;