Sign up for the Dlubal Extranet to get most of the software and have exclusive access to your personal data.

Create Account

Verification Example

Back to Examples

19x

009064

2024-10-11

VE0064 | Thick–Walled Vessel

Description

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. Determine the radial deflection of the inner and outer radius u_r(r₁), u_r(r₂). The self-weight is neglected.

Material	Elastic	Modulus of Elasticity	E	1.000	MPa
Material	Elastic	Poisson´s Ratio	ν	0.250	-
Geometry		Inner Radius	r₁	200.000	mm
Geometry		Outer Radius	r₂	300.000	mm
Load		Inner Pressure	p₁	60.000	kPa
Load		Outer Pressure	p₂	10.000	kPa

Thick-Walled Vessel

Thick-Walled Vessel

Analytical Solution

The stress state of the thick-walled vessel is described by the equation of equilibrium

\frac{d σ_{r}}{d r} + \frac{σ_{r} - σ_{t}}{r} = 0

σ_r	Radial stress
σ_t	Tangential stress

Equilibrium Equation for Thick-Walled Vessel

Using strain-deflection equations and Hooke´s Law, second-order differential equation is obtained

\frac{d^{2} u_{r} (r)}{d r^{2}} + \frac{d u_{r} (r)}{d r} - \frac{u_{r} (r)}{r} = 0

Second-Order Differential Equation for Thick-Walled Vessel

The solution leads to the radial stress σ_r and tangential stress σ_t.

σ_{t} (r) = K + \frac{C}{r^{2}},

σ_{r} (r) = K - \frac{C}{r^{2}}

K, C	Real constants

Radial and Tangential Stress of Thick-Walled Vessel

Constants K and C are obtained using boundary conditions.

K = \frac{p_{1} r_{1}^{2} - p_{2} r_{2}^{2}}{r_{2}^{2} - r_{1}^{2}},

C = (p_{1} - p_{2}) \frac{r_{1}^{2} r_{2}^{2}}{r_{2}^{2} - r_{1}^{2}}

Constants K and C

The 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})]

Radial Deflection of Inner and Outer Radius of Open-Ended Vessel

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

RFEM 6 Results – Total Deflection

Quantity	Analytical Solution	RFEM 6	Ratio	RFEM 5	Ratio
u_r(r₁) [mm]	27.000	26.998	1.000	27.000	1.000
u_r(r₂) [mm]	21.750	21.747	1.000	21.750	1.000

Verification Example 000064 | 1

434x

13x

Verification Example 000064 | 1

;