Strona 2_redak - Instytut Agrofizyki im. Bohdana Dobrzańskiego ...

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10. PRESSURE RATIO
The pressure ratio is one of the three most important physical properties of
bulk solids, commonly used for calculation of pressure in a silo. Almost all design
codes use a Janssen-type [73] pressure distribution to predict silo pressures [170].
The well known Janssen formula uses the equilibrium of a horizontal slice of the
granular material to estimate pressures in deep silos. The fundamental assumption
of Janssen’s method involves a relationship between the average stresses acting
on the finite dimension of a slice, and stresses that act at the walls of a silo.
Janssen assumed that the ratio between the average vertical stress 1 z and the stress
normal to the wall, 1 x is a constant for a given bulk material stored in a silo:
(10.1)
and k is to be determined from measurements. Other Janssen’s assumptions are:
fully mobilized friction at the interface of the bulk material and the walls of the
silo, and constant bulk density [45].
Since the work of Janssen, several attempts aiming at an expression for k,
based on postulating a mechanical model for bulk solids, have been proposed [38,
45, 115]. A majority of the estimations are based on the assumption that the bulk
material stored in or discharged from a silo is at a limiting state of stress. Another
important assumption concerning the location of a region inside the slice of
material where the yielding conditions occur involves relations between local
stresses and the stresses averaged over the area or the perimeter of a slice [45].
In the case of a deep silo the following two stress cases are commonly
considered: active for filling and storage mode, and passive for discharging mode.
In the active case the vertical stress is higher than the lateral stress, while in the
passive case the lateral stress is higher than the vertical one.
10.1. Yielding at the silo centre
Considering yielding at the silo centre, the stress ratio k can be easily
obtained from Mohr’s circle construction [82, 115] for the active case (fig. 10.1a):
and for the passive case (fig. 10.1b):
1
1
x
z
= k = const.
1− sinϕ
k = , 1 + sinϕ
(10.2)