Boiler formula

Cylinder segment with sectional stresses due to internal pressure

The boiler formula is a calculation formula from technical mechanics . It is of elementary importance in the calculation and design of steam boilers , pressure vessels and pipelines . The formula for pipes can be found in DIN EN 13480, Part 3.

application

The boiler formula indicates the mechanical stresses in rotationally symmetrical bodies loaded by internal pressure , such as those found in pipes or pressure vessels , for example . As membrane stress, it is based on an equilibrium of forces , so neither deformation assumptions nor elasticity quantities are necessary to calculate the stresses .

The boiler formula only applies to thin-walled and curved pressure vessels. The boiler formula does not apply to boilers made from flat metal sheets or plates , or to thick-walled containers .

A pressure vessel can be considered thin-walled if its dimensions (diameter) are much larger than its wall thickness (i.e. outer diameter / inner diameter = D / d ≤ 1.2). The greatest stress in cylindrical bodies is the tangential stress , which is why pipes that are too weakly designed and similarly shaped containers always burst in the longitudinal direction. ${\displaystyle \sigma _{\rm {t}}}$

calculation

The tangential stress and axial stress in a cylinder loaded by internal pressure , which is closed at the ends, are:

${\displaystyle \sigma _{\rm {t}}={\frac {p\cdot d_{m}}{2\cdot s}}}$

${\displaystyle \sigma _{\rm {a}}={\frac {p\cdot d_{m}}{4\cdot s}}}$

Cylinder segment with dimensions

• p = internal pressure
• s = wall thickness
• ${\displaystyle d_{m}}$ = Mean diameter ${\displaystyle d_{m}=d+s={\frac {D+d}{2}}}$
• ${\displaystyle \sigma _{\rm {t}}}$ = Tangential stress in the wall
• ${\displaystyle \sigma _{\rm {a}}}$ = axial tension (longitudinal direction) in the wall

In this form, the kettle formula is also known as the Bockwurst formula . The designation serves as a donkey bridge to remember which of the two tensions is the greater. The circumferential tension is twice as great as the tension in the longitudinal direction, therefore sausages always burst in the longitudinal direction if improperly heated.

From the shear stress hypothesis which ultimately follows as boiler formula called equivalent stress with ${\displaystyle \sigma _{\rm {v}}}$

${\displaystyle \sigma _{\rm {v}}=\sigma _{\rm {max}}-\sigma _{\rm {min}}=\sigma _{\rm {t}}-\sigma _{\rm {r}}={\frac {p\cdot d}{2\cdot s}}+{\frac {p}{2}}}$

• ${\displaystyle \sigma _{\rm {r}}}$= Radial stress; on the inside of the tank , on the outside (unloaded surface) , in the middle of the wall the arithmetic mean is used ${\displaystyle \sigma _{\rm {r}}=-p}$ ${\displaystyle \sigma _{\rm {r}}=0}$ ${\displaystyle (\sigma _{\rm {r}}=-p/2)}$

or.

${\displaystyle \sigma _{\rm {v}}={\frac {p\cdot (d+s)}{2\cdot s}}={\frac {p\cdot d_{\rm {m}}}{2\cdot s}}}$

Including wall thickness surcharges, the minimum wall thickness is calculated using the following formula:

${\displaystyle s_{\rm {min}}={\frac {p\cdot d_{\rm {m}}}{2\cdot \sigma _{\rm {zul}}}}+s_{\rm {1}}+s_{\rm {2}}}$

• ${\displaystyle s_{\rm {1}}}$ Surcharge for corrosion
• ${\displaystyle s_{\rm {2}}}$ Surcharge for tolerance errors

With spherical vessels there are no tangential stresses; the axial stresses correspond to those of the cylinder. Therefore the minimum wall thickness is halved:

${\displaystyle s_{\rm {min}}={\frac {p\cdot d_{\rm {m}}}{4\cdot \sigma _{\rm {zul}}}}+s_{\rm {1}}+s_{\rm {2}}}$

Additional note: Since a tank is closed at both ends, the end surfaces must be taken into account when calculating the force and area and thus the voltage. It follows that the tangential and radial stress depends on the ratio of the diameter d to the length l of the boiler:

${\displaystyle {\frac {\sigma _{t}}{\sigma _{r}}}={\frac {2}{1+{\frac {d}{l}}}}}$

• d = l: ${\displaystyle \sigma _{t}=\sigma _{r}}$
• d> l: ${\displaystyle \sigma _{t}<\sigma _{r}}$
• d <l: ${\displaystyle \sigma _{t}>\sigma _{r}}$

Individual evidence

1. ↑ DIN EN 13480-3, December 2017 edition: Metallic industrial pipelines - Part 3: Design and calculation; German version EN 13480-3: 2017.
   For unfired pressure vessels, the equivalent formula can be found in DIN EN 13445 Part 3, Section 7.4: Cylindrical and spherical shells .
2. ^ Statics, especially the cutting principle : Gerhard Knappstein, page 243, Verlag Harri Deutsch, ISBN 978-3-8171-1803-8