# 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 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 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 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 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)  up

### 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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### Under internal pressure 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 (-)
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 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 (-)
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 (-) up

### 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)
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)
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 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 

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

 material Security p at calculation temperature Security S ' at the test print Rolled and forged steels 1,5 1,05 Cast steel 2,0 1,4 Ductile iron cast iron EN-GJS-700-2 / 2UEN-GJS-600-3 / 3U 5,0 2,5 EN-GJS-500-7 / 7U 4,0 2,0 EN-GJS-400-15 / 15U 3,5 1,7 EN-GJS-400-18 / 18U-LTEN-GJS-350-22 / 22U-LT 2,4 1,2 Aluminum and aluminum alloysKneading materials 1,5 1,05

Safety values ​​against tensile strength

 material Security p at calculation temperature Security S ' at the test print Cast iron with lamellar graphite (gray cast iron) - not annealed 9,0 3,5 - annealed or enamelled 7,0 3,5 Copper and copper alloys including rolled and cast bronze - for seamless and welded tanks 3,5 2,5 - with soldered containers 4,0 2,5

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

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