Janssen equation

The stresses in a silo can be described with the Janssen equation (after HA Janssen, 1895) . It describes the equilibrium of forces in the vertical direction and is a special case of the general equation of hydrostatic equilibrium , taking into account the silo wall.

For a disk-shaped volume element (radius r) of height  dz , which spans the entire cross-section of  the silo with circumference  , applies ${\displaystyle A=\pi \cdot r^{2}}$ ${\displaystyle U=2\cdot \pi \cdot r}$

${\displaystyle A\cdot \sigma _{v}+\rho _{b}\cdot g\cdot A\cdot dz=A\left(\sigma _{v}+d\sigma _{v}\right)+\tau _{w}\cdot U\cdot dz}$,

in which

• ${\displaystyle \sigma _{v}}$ the vertical tension
• ${\displaystyle \rho _{\rm {b}}}$the mean density of the bulk material
• g is the gravitational acceleration
• ${\displaystyle \tau _{\rm {w}}}$the wall shear stress
• z denotes the depth from the upper edge of the bulk material.

The weight of the bulk material acts in the disk ${\displaystyle \rho _{b}\cdot g\cdot A\cdot dz}$

• the difference in stresses (pressure gradient ) on the top and bottom of the disc (the first terms on both sides) as well
• against the thrust exerted by the silo wall . ${\displaystyle \tau _{w}\cdot U\cdot dz}$

The Janssen equation can be transformed into an ordinary differential equation :

${\displaystyle {\begin{alignedat}{2}\Leftrightarrow \rho _{b}\cdot g\cdot A\cdot dz&=A\cdot d\sigma _{v}&&+\tau _{w}\cdot U\cdot dz\\\Leftrightarrow \rho _{b}\cdot g\cdot dz&=d\sigma _{v}&&+\tau _{w}\cdot {\frac {U}{A}}\cdot dz\\\Leftrightarrow d\sigma _{v}&=(\rho _{b}\cdot g&&-\tau _{w}\cdot {\frac {U}{A}})\cdot dz\\\Leftrightarrow {\frac {d\sigma _{v}}{dz}}&=\rho _{b}\cdot g&&-\tau _{w}\cdot {\frac {2}{r}}\end{alignedat}}}$

The solution of the like. can be found by choosing suitable boundary and initial conditions . ${\displaystyle \sigma _{v}(z)}$

literature

1. ↑ Janssen, HA: Experiments on grain pressure in silo cells . In: Z. Ver. German Ing. Band 39 , 1895, p. 1045-1049 ( duke.edu [PDF] scan of the article (memento in the web archive from October 30, 2011)). Experiments using grain pressure in silo cells ( memento of the original from March 3, 2011 in the Internet Archive ) Info: The archive link was automatically inserted and not yet checked. Please check the original and archive link according to the instructions and then remove this notice.  

Web links

• Calculation of stresses in silos ( Janssen equation )