What is Tangential Stress

Pipe strength



Stress calculation of vessels and pipelines

The stress calculations are valid under the following conditions:
- Rotationally symmetrical body.
- The internal and external pressure is evenly distributed along the circumference.
- Tensions are in the elastic range.

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Tangential stress at internal pressure [1]


σ t = Tangential stress (N / mm²)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p i = Pressure inside wall (N / mm²)
σ t = Tangential stress (N / mm²)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p i = Pressure inside wall (N / mm²)
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Radial stress at internal pressure [1]


σ r = Radial stress (N / mm2)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p i = Pressure inside wall (N / mm²)
σ r = Radial stress (N / mm2)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p i = Pressure inside wall (N / mm²)
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Beware of external pressure: In the case of thin-walled pipes, the critical buckling stress must also be calculated when calculating the strength.

Tangential stress at external pressure [1]


σ t = Tangential stress (N / mm²)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p a = External wall pressure (N / mm²)
σ t = Tangential stress (N / mm²)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p a = External wall pressure (N / mm²)
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Radial stress at external pressure [1]


σ r = Radial stress (N / mm2)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p a = External wall pressure (N / mm²)
σ r = Radial stress (N / mm2)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p a = External wall pressure (N / mm²)
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Axial stress


σ a = Axial stress (N / mm²)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
p i = Pressure inside wall (N / mm²)
p a = External wall pressure (N / mm²)
σ a = Axial stress (N / mm²)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
p i = Pressure inside wall (N / mm²)
p a = External wall pressure (N / mm²)

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Equivalent stress according to the shape change energy hypothesis (GEH)


σv = Equivalent stress (N / mm²)
σt = Tangential stress (N / mm²)
σr = Radial stress (N / mm²)
σa = Axial stress (N / mm²)
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Equivalent stress in thin-walled pipe under internal pressure and torsion or bending



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Radial deformation for an infinitely long pipe [1]

Under internal pressure


Δr x = Radius expansion at radius x (mm)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p i = Pressure inside wall (N / mm²)
p a = External wall pressure (N / mm²)
E.   = Modulus of elasticity (N / mm2)
ν   = Poisson's ratio (-)
Δr x = Radius expansion at radius x (mm)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p i = Pressure inside wall (N / mm²)
p a = External wall pressure (N / mm²)
E.   = Modulus of elasticity (N / mm2)
ν   = Poisson's ratio (-)
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Under external pressure


Δr x = Radius expansion at radius x (mm)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p i = Pressure inside wall (N / mm²)
p a = External wall pressure (N / mm²)
E.   = Modulus of elasticity (N / mm2)
ν   = Poisson's ratio (-)
Δr x = Radius expansion at radius x (mm)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r x = Radius at any point x (mm)
p i = Pressure inside wall (N / mm²)
p a = External wall pressure (N / mm²)
E.   = Modulus of elasticity (N / mm2)
ν   = Poisson's ratio (-)

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Denting of thin-walled pipes when exposed to external pressure

In the case of thin-walled pipes with external pressure loads or internal negative pressure, the risk of buckling must be taken into account.
The theory of buckling in thin-walled pipes is very complex. Only the general formulas for bulging are listed here. Further formula can be found in AD Leaflets 2000 - B6.

Buckling stress in pipes

The theoretical buckling stress is calculated for a thin-walled pipe under uniform external pressure that is hinged at the ends as follows:


p a, kr = critical buckling pressure (N / mm²)
p a, perm = permissible external pressure (N / mm²)
E.   = Modulus of elasticity (N / mm²)
ν   = Poisson's ratio (-)
s   = Wall thickness (mm)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r   = mean Radius (mm)
S.   = Safety (-) elast. Stress state S = 3
p a, kr = critical buckling pressure (N / mm²)
p a, perm = permissible external pressure (N / mm²)
E.   = Modulus of elasticity (N / mm²)
ν   = Poisson's ratio (-)
s   = Wall thickness (mm)
r i = Inner wall radius (mm)
r a = Outer wall radius (mm)
r   = mean Radius (mm)
S.   = Safety (-) elast. Stress state S = 3

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Boiler formula - simplified stress calculation of thin-walled pressure vessels

The boiler formula is a simplified calculation of pressure vessels with internal overpressure.
The formula is only valid for thin-walled cylindrical containers with a diameter ratio of Da / Di <1.2.

Tangential stress in the container wall


σ t = Tangential stress (N / mm²)
p   = Internal pressure (N / mm²)
D.   = Mean diameter (mm)
s   = Wall thickness (mm)
D. a = Outer diameter (mm)
D. i = Inner diameter (mm)
σ t = Tangential stress (N / mm²)
p   = Internal pressure (N / mm²)
D.   = Mean diameter (mm)
s   = Wall thickness (mm)
D. a = Outer diameter (mm)
D. i = Inner diameter (mm)
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Axial stress in the container wall


σ a = Axial stress (N / mm²)
p   = Internal pressure (N / mm²)
D.   = Mean diameter (mm)
s   = Wall thickness (mm)
D. a = Outer diameter (mm)
D. i = Inner diameter (mm)
σ a = Axial stress (N / mm²)
p   = Internal pressure (N / mm²)
D.   = Mean diameter (mm)
s   = Wall thickness (mm)
D. a = Outer diameter (mm)
D. i = Inner diameter (mm)
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Wall thickness calculation of cylindrical tanks


s min = minimum wall thickness (mm)
p   = Internal pressure (N / mm²)
D.   = Mean diameter (mm)
σ   = permissible stress (N / mm²)
s 1 = Surcharge for tolerance errors (mm) (1
s 2 = Surcharge for corrosion or erosion (mm) (2
D. a = Outer diameter (mm)
D. i = Inner diameter (mm)
s min = minimum wall thickness (mm)
p   = Internal pressure (N / mm²)
D.   = Mean diameter (mm)
σ   = permissible stress (N / mm²)
s 1 = Surcharge for tolerance errors (mm) (1
s 2 = Surcharge for corrosion or erosion (mm) (2
D. a = Outer diameter (mm)
D. i = Inner diameter (mm)
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Wall thickness calculation of spherical containers

With spherical containers there is no tangential stress, which is why the wall thickness is halved.



s min = minimum wall thickness (mm)
p   = Internal pressure (N / mm²)
D.   = Mean diameter (mm)
σ   = permissible stress (N / mm²)
s 1 = Surcharge for tolerance errors (mm) (1
s 2 = Surcharge for corrosion or erosion (mm) (2
D. a = Outer diameter (mm)
D. i = Inner diameter (mm)
s min = minimum wall thickness (mm)
p   = Internal pressure (N / mm²)
D.   = Mean diameter (mm)
σ   = permissible stress (N / mm²)
s 1 = Surcharge for tolerance errors (mm) (1
s 2 = Surcharge for corrosion or erosion (mm) (2
D. a = Outer diameter (mm)
D. i = Inner diameter (mm)

(1 See manufacturer information - reference values: s ≤ 10 mm - s1 = 0.35 mm and for s> 10 mm - s1 = 0.5 mm
(2 Ferritic steels s2 approx. 1 mm - stainless steels or with corrosion protection see p2 = 0 mm

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Wall thickness calculation of cylindrical containers according to AD 2000 Merkblatt B1 [2]


s min = Wall thickness (mm)
D. a = Outer diameter (mm)
p   = Calculation pressure (bar)
K   = Strength value at calculation temperature (N / mm2)
S.   = Safety value (-)
ν   = Utilization factor for voltage (-)
c 1 = Surcharge for under-wall thickness (mm)
c 2 = Wear allowance (mm)
s min = Wall thickness (mm)
D. a = Outer diameter (mm)
p   = Calculation pressure (bar)
K   = Strength value at calculation temperature (N / mm2)
S.   = Safety value (-)
ν   = Utilization factor for voltage (-)
c 1 = Surcharge for under-wall thickness (mm)
c 2 = Wear allowance (mm)
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Safety values ​​of pressure vessels according to AD 2000 data sheet B0 [3]


Safety values ​​against yield strength, yield strength or creep strength

materialSecurity p
at calculation temperature
Security S '
at the test print
Rolled and forged steels1,51,05
Cast steel2,01,4
Ductile iron cast iron
EN-GJS-700-2 / 2U
EN-GJS-600-3 / 3U
5,02,5
EN-GJS-500-7 / 7U4,02,0
EN-GJS-400-15 / 15U3,51,7
EN-GJS-400-18 / 18U-LT
EN-GJS-350-22 / 22U-LT
2,41,2
Aluminum and aluminum alloys
Kneading materials
1,51,05

Safety values ​​against tensile strength

materialSecurity p
at calculation temperature
Security S '
at the test print
Cast iron with lamellar graphite (gray cast iron)
- not annealed9,03,5
- annealed or enamelled7,03,5
Copper and copper alloys including rolled and cast bronze
- for seamless and welded tanks3,52,5
- with soldered containers4,02,5

Literature:
[1] Holzmann / Meyer / Schumpich: Technical mechanics strength theory
[2] AD 2000-Merkblatt B 1 - Cylinder and spherical shells under internal overpressure
[3] AD 2000-Merkblatt B 0 - Calculation of pressure vessels


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