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2026-05-04

VE0085 | Thin-Walled Conical Vessel with Hydrostatic Pressure

Description

A thin-walled conical vessel of height h and peak angle 2φ is filled with water. Thus, it is loaded by hydrostatic pressure according to the following sketch. While neglecting self-weight, determine the stresses σ₁ and σ₂ at the test point at height h₀ = 1.000 m.

Material	Isotropic Linear Elastic	Modulus of Elasticity	E	210000.000	MPa
Material	Isotropic Linear Elastic	Poisson's Ratio	ν	0.296	-
Geometry	Conical Vessel	Vessel Height	h	2.000	m
		Shell Thickness	t	1.000	mm
		Vessel Angle	φ	π/6	rad
Load	Hydrostatic Pressure	Water Specific Weight	γ	9810.000	N/m³

Thin-Walled Conical Vessel with Hydrostatic Pressure

Analytical Solution

The analytical solution is based on the theory of thin-walled vessels. The stress state of the thin-walled vessel 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 surface line and circumferential direction, respectively, and R₁, R₂ are the radii in the corresponding directions. The mentioned stresses correspond to the principal stresses. The pressure p is, in this case, equal to the hydrostatic pressure:

p = h ρ g = γ h

Hydrostatic Pressure

The radius R₁ for the conical vessel is equal to R₁ ≈ ∞. The radius R₂ can be expressed, considering r = z tan φ:

R_{2} = \frac{r}{\cos φ} = z \frac{s i n φ}{\cos^{2} φ}

Radius Function for Conical Vessel

Sketch of Used Dimensions of Conical Vessel

The pressure in the depth h - z is equal to:

p (z) = γ (h - z)

Pressure Function

Substituting into the Laplace equation, circumferential stress σ₂ can be obtained:

σ_{2} = \frac{γ (h - z) z \sin φ}{t \cos^{2} φ}

Circumferential Stress

An additional equation has to be defined to obtain the remaining stress σ₁. The internal and external forces have to be equal. Furthermore, the external force Q due to the hydrostatic pressure is equal to the gravity force caused by the height of the water column:

Q = σ_{1} 2 π r t \cos φ

Q = γ [π r^{2} (h - z) + \frac{1}{3} π r^{2} z]

Equations for the Tangential Stress

The desired stress σ₁ can then be determined:

σ_{1} = \frac{γ z \sin φ}{2 t \cos^{2} φ} (h - \frac{2}{3} z)

Tangential Stress

For the test point at height z = 1.000 m, the above-mentioned quantities can be calculated:

σ_{1} \approx 9.249 MPa

σ_{2} \approx 13.873 MPa

Conical Vessel - Results

RFEM Settings

Modeled in RFEM 6.13 and RFEM 5.39
Element size l_FE = 0.025 m
Isotropic linear elastic material is used
Kirchhoff plate bending theory is used

Note: The hydrostatic pressure is modeled by means of Free Rectangular Load. The pressure at the top edge (z = 2.000 m) is p₁ = 0.000 N/m² and at the bottom (z = 0.000 m) it is p₂ = -19620.000 N/m².

Free Rectangular Load Definition in RFEM 6

Results

Von Mises stress distribution, Free Rectangular Load for the hydrostatic pressure and the test point location in RFEM 6

Quantity	Theory [MPa]	RFEM 6 [MPa]	Ratio [-]	RFEM 5 [MPa]	Ratio [-]
σ₁	9.249	9.265	1.002	9.264	1.002
σ₂	13.873	13.980	1.008	13.982	1.008

Remark: The stresses σ₁ and σ₂ are evaluated at the middle surface of the conical vessel. The corresponding stresses in RFEM are σ_2,m and σ_1,m, respectively.

Model 000183 | Verification Example 000085 | 1

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Verification Example 000085 | 1

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