Pipes and Shells Calculators

Toroidal Shell Under Pressure

In this calculation, a toroidal shell with a mean diameter D, tube diameter d, and wall thickness t is considered. The shell is under the influence of a hydrostatic pressure P. For the calculation, the elastic modulus E and Poisson's ratio ν of the shell should be specified.

As a result of calculation, the maximum meridional σ₁ and circumferential σ₂ stresses in the shell are determined. At that, the maximum stresses are concentrated along the inner diameter of the torus. The tube deformation Δd in the radial direction is also calculated.

Calculation of toroidal shell

Toroidal shell under pressure

INITIAL DATA

D - Toroidal shell diameter;

d - Tube diameter;

t - Wall thickness;

P - Internal pressure;

E - Young's modulus;

ν - Poisson's ratio.

RESULTS DATA

σ₁ - Maximum meridional stress;

σ₂ - Circumferential stress;

Δd - Deformation of the tube in the radial direction.

Metric Imperial

Shell diameter (D)

Tube diameter (d)

Wall thickness (t)

Pressure (P)

Young's modulus (E)

Poisson's ratio (ν)

Meridional stress (σ₁)

Circumferential stress (σ₂)

Tube deformation (Δd)

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BASIC FORMULAS

Maximum meridional stress:

σ₁ = P*d*(2D - d) / 2t*(D - d);

Circumferential stress:

σ₂ = P*d / 2t;

Tube deformation:

Δd = 0.25*[P*d / 2E*t]*[D+d - ν(2D + d)] - 0.25*[P*d / 2E*t]*[D-d - ν(2D - d)].

INITIAL DATA

D - Toroidal shell diameter;

d - Tube diameter;

t - Wall thickness;

P - Internal pressure;

E - Young's modulus;

ν - Poisson's ratio.

RESULTS DATA

σ₁ - Maximum meridional stress;

σ₂ - Circumferential stress;

Δd - Deformation of the tube in the radial direction.

MATERIALS PROPERTIES

Material	Young’s modulus Pa (psi)	Poisson’s ratio
Steel	1.86÷2.1×10¹¹ (2.7÷3.05×10⁷)	0.25÷0.33
Cast iron	0.78÷1.47×10¹¹ (1.1÷2.1×10⁷)	0.23÷0.27
Copper	1.0÷1.3×10¹¹ (1.45÷1.9×10⁷)	0.34
Tin bronze	0.74÷1.22×10¹¹ (1.1÷1.8×10⁷)	0.32÷0.35
Brass	0.98÷1.08×10¹¹ (1.4÷1.6×10⁷)	0.32÷0.34
Aluminum alloy	0.7×10¹¹ (1.0×10⁷)	0.33
Magnesium alloy	0.4÷0.44×10¹¹ (5.8÷6.4×10⁶)	0.34
Nickel	2.5×10¹¹ (3.6×10⁷)	0.33
Titanium	1.16×10¹¹ (1.7×10⁷)	0.32
Lead	0.15÷0.2×10¹¹ (2.2÷2.9×10⁶)	0.42
Zinc	0.78×10¹¹ (1.1×10⁷)	0.27
Glass	4.9÷5.9×10¹⁰ (7.1÷8.5×10⁶)	0.24÷0.27
Concrete	1.48÷2.25×10¹⁰ (2.1÷3.3×10⁶)	0.16÷0.18
Wood (along the grain)	8.8÷15.7×10¹⁰ (12.8÷22.8×10⁶)	-
Wood (across the grain)	3.9÷9.8×10¹⁰ (5.7÷14.2×10⁶)	-
Nylon	1.03×10¹⁰ (1.5×10⁶)	-

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