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2026-09-24

VE0089 | Bourdon Effect

Description

A pipe with the tubular cross-section is loaded by means of internal pressure, see the following figure. The internal pressure causes axial deformation of the pipe, which is called the Bourdon effect. Determine the axial deformation u_x of the pipe endpoint. The problem is described by the following parameters.

Material	Steel	Modulus of Elasticity	E	210000.000	MPa
Material	Steel	Poisson's Ratio	ν	0.300	-
Geometry	Pipe	Length	L	10.000	m
		Outer Diameter	D_e	200.000	mm
		Inner Diameter	D_i	196.000	mm
Load		Pipe Internal Pressure	p	1.000	MPa

Bourdon Effect

Analytical Solution

The pipe is considered to be a thick-walled closed-end vessel. A detailed description of the thick-walled vessel calculation can be found in VE0064 | Thick–Walled Vessel . The stress state of the pipe is generally spatial due to the radial stress σ_r, tangential stress σ_t and axial stress σ_x. The axial deformation u_x of the pipe endpoint is defined by means of Hooke's law:

u_{x} = \frac{L}{E} (σ_{x} - ν (σ_{r} + σ_{t}))

Axial deformation

The axial stress σ_x when considering zero outer pressure is:

σ_{x} = \frac{p r_{i}^{2}}{r_{e}^{2} - r_{i}^{2}}

r_e	Outer radius
r_i	Inner radius

Axial stress

The radial and tangential stresses are defined as follows:

σ_{r} = σ_{x} - \frac{C}{r^{2}},

σ_{t} = σ_{x} + \frac{C}{r^{2}}

r	Radial coordinate
C	Real constant

Radial and tangential stress

The real constant C is in this case equal to:

C = p \frac{r_{i}^{2} r_{e}^{2}}{r_{e}^{2} - r_{i}^{2}}

Real constant

Due to the agreement with RFEM / RSTAB analysis, further calculations are carried out for the middle radius r_m = (r_e + r_i) / 2. Using the formula for the axial deformation above, the axial deformation u_x of the pipe endpoint results:

u_{x} \approx 0.462 mm

Result axial deformation of the pipe

RFEM and RSTAB Settings

Modeled in RFEM 6.15, RSTAB 9.15 and RFEM 5.16, RSTAB 8.16
Element size l_FE = 0.500 m
Isotropic linear elastic material is used
Member load 'Pipe internal pressure' is used
Displacements due to the member loads of type 'Pipe internal pressure' (Bourdon effect) are enabled

Results

Quantity	Analytical Solution [mm]	RFEM 6 [mm]	Ratio [-]	RSTAB 9 [mm]	Ratio [-]
u_x	0.462	0.462	1.000	0.462	1.000

Quantity	Analytical Solution [mm]	RFEM 5 [mm]	Ratio [-]	RSTAB 8 [mm]	Ratio [-]
u_x	0.462	0.462	1.000	0.462	1.000

Results - axial deformation

Verification Example 000089 | 1

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