# 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.

### Tangential stress at internal pressure [1]

_{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)

_{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)

### Radial stress at internal pressure [1]

_{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)

_{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)

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]

_{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)

_{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)

### Radial stress at external pressure [1]

_{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)

_{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)

### Axial stress

_{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)

_{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)

_{v}= Equivalent stress (N / mm²)

σ

_{t}= Tangential stress (N / mm²)

σ

_{r}= Radial stress (N / mm²)

σ

_{a}= Axial stress (N / mm²)

### 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

_{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 (-)

_{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 (-)

### Under external pressure

_{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 (-)

_{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:

_{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

_{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

_{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)

_{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)

### Axial stress in the container wall

_{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)

_{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)

### Wall thickness calculation of cylindrical tanks

_{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)

_{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)

### Wall thickness calculation of spherical containers

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

_{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)

_{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 - s_{1} = 0.35 mm and for s> 10 mm - s_{1} = 0.5 mm

(^{2} Ferritic steels s_{2} approx. 1 mm - stainless steels or with corrosion protection see p_{2} = 0 mm

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

_{min}= Wall thickness (mm)

D.

_{a}= Outer diameter (mm)

p

_{ }= Calculation pressure (bar)

K

_{ }= Strength value at calculation temperature (N / mm

^{2})

S.

_{ }= Safety value (-)

ν

_{ }= Utilization factor for voltage (-)

c

_{1}= Surcharge for under-wall thickness (mm)

c

_{2}= Wear allowance (mm)

_{min}= Wall thickness (mm)

D.

_{a}= Outer diameter (mm)

p

_{ }= Calculation pressure (bar)

K

_{ }= Strength value at calculation temperature (N / mm

^{2})

S.

_{ }= Safety value (-)

ν

_{ }= Utilization factor for voltage (-)

c

_{1}= Surcharge for under-wall thickness (mm)

c

_{2}= Wear allowance (mm)

### Safety values of pressure vessels according to AD 2000 data sheet B0 [3]

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 / 2U EN-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-LT EN-GJS-350-22 / 22U-LT | 2,4 | 1,2 |

Aluminum and aluminum alloys Kneading 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:

[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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