Generally, around a rathole or a mining pit, the missing stresses on the boundary of such bridge or rathole are compensated by tangential pressure. Mohr developed a model of the stress distribution in an elastic body with a hole in the middle.
The Mohr equations for radial and tangential stresses for the case of elastic deformation are:
radial stress: σre = σx ( 1 – ri2/r2 ) (1)
tangential stresses: σte = σx ( 1 + ri2/r2 ) (2)
where:
ri – the radius of bridge
r - the radius
σx – stress before the bridge occur
σre – radial stress in elastic body after bridge occur
σte – tangential stress in elastic body after bridge occur
The average vertical stress in vertical cross section of silos is.
σx = pv
The graphical presentation of stresses as function of the radius is shown in Fig.2 However, from mining technology is known, that the material around an pit deforms plastically and on this way reduces the tangential and radial stresses near to zero. We can postulate:
For r=ri is σt~σr~0
The relation between the major stresses can not be bigger than relation obtained by shear test. For internal angle of friction φ= 30o follows the relation of the principal stresses σ1/σ3 <= 3. This, is valid for each point in the bulk mass, thus for the maximum relation of the principal stresses to. In our case we obtain for the transition between elastic and plastic region:
ptp/prp<=3
From the equations 1 and 2 and this value we can calculate the radius of the plastic region.
rp = 1.41 ri (3)
At the inner radius of the bridge the radial pressure must be equal to zero. The maximal tangential pressure at the inner radius must be equal or smaller than the unconfined compressive strength.
Thus, for r=ri is:
pr=0 (4)
0 <σt< σd (5)
Within the possible tangential stresses on the boundary of the bridge according (3), the most stable bridge is when
σt = 0 (6)
In mining technology this allow to secure the pit with a small contra-pressure in form of wooden supports. On this way relatively weak wooden supports can resist the pressure of hundreds of atmospheres given by the depth above the pit. In our case, because of cohesion the tangential stress can decrease in the plastic region much faster then for free flowing materials and that means that the loading of the bridge can decrease to zero.
Because o the described phenomena, we can assume:
1. that the loading of the bridges is only their own weight.
2. that the material in the bridge is consolidated by a consolidation stress, equal to the vertical pressure (lit 8,14).
Schematic bridge in a hopper:
d – diameter of the bridge
P – Load of the bridge
R – Radius of the bridge
α – Angle of the hopper
σn- Stress in the bridge support
The critical diameter of bridging is, is dependent from the angle of the cone, from the coefficient of wall friction and from unconfined compressive strength, corresponding with the silo pressure.
The angle of the cone is for the mass flow silos mostly between 15 and 25 degree with the vertical. For further calculations we will limit the angle of the cone:
15°<α<25°
For mass flow silos we can assume, that the coefficient of friction, especially in the cone, change during the use. For a new silo, the coefficient of wall friction can be high, but after a certain time, the polishing effect lead to a very small coefficient of wall friction, in which case a value of 5 degree can be reached. For silos from stainless steel or aluminum, the wall friction can be very low by a new silo too.
The angle β of the stress in the bridge support is:
0<β<α+ρ
A stable arch must fulfill two conditions: the stresses mast be smaller then the unconfined compressive strength and the support has to be stable. Independent of all other conditions, a bridge have a stable support when the angle of the arch in the support has an angle smaller than the angle of the hopper. In this case sliding or rolling of particles in the support is not possible. From the measurement of the arch geometry in a model bin (lit 15) it follows, that the preferable angle in the support of a bridge is between zero and α. The observations of bridges in the practice confirms this too. Therefor, we assume that possible bridges can occur under an angle between zero and α. The bridges with an angle bigger than the angle of the hopper, can easily partially slide because of locally small coefficient of friction or break as result local irregularities.
For calculation of the critical bridge we will take the limit
α=β
The critical bridge is the bridge with a smallest possible radius. All other bridges with a bigger radius would have a bigger stress then the critical bridge and would break. All bridges with a smaller radius would slide down along the hopper wall as shown in Fig. 4. Consequently, the critical bridge have a radius:
R= d/(2sinα). (7)
where d – diameter of the bridge, R - radius of the of the bridge, α – angle of the hopper
If we assume that the critical bridge just can resist the load of their own weight. Then, all bridges with bigger radius should break under this load. The maximum total force loading the bridge is:
P=Α*γ (8)
where: A - the cross section of the bridge
The forces in the bridge support are:
P=L*σn*sinα (9)
Where L– the length of the bridge support σn- the stress in the bridge support
From equations 8 and 9 we obtain the relation between the bridge geometry and the loading forces of the bridge.
A/L= σn *sinα/γ
The mayor pressure in the silo is the vertical pressure. This pressure is responsible for the consolidation of the powder in the silo. The pressure in the silo is dependent of the silo geometry the density of the powder and the wall friction The pressure in the silo can be calculated according the theory of Janssen (lit 16). According Janssen, for the infinite depth of the silo is valid:
ph=γD/4µ (11)
pv=ph/λ (12)
After the filling of the silo, the relation of horizontal to the vertical pressure is
λ=ph/pv=0.5 (13)
This corresponds with experimental results on real installations and models (lit 15), especially for mass flow silos with smooth walls, polished trough the flow of material.
From equation 11, 12, and 13 we obtain the consolidation pressure for a circular silo:
pcons=pv=2*γD/4µ
The stability of an bridge is highly dependent of the unconfined compressive strength of the powder. The relation between the consolidation pressure and the unconfined compressive strength can only be found trough a measurement with a shear tester. From the shear test of the powder, we obtain the yield loci for various pressure. From this yield loci we can determine the unconfined compressive strength for different consolidation pressure, which corresponds with the silo pressure.
The critical unconfined compressive strength σd can be expressed as function of consolidation pressure – equal to the vertical silo pressure - we call this the flow function – as 
σ dcrit=f(pv)=f(σcons) (15)
In many cases the flow function is linear and can be expressed as function of the angle of flow function Ψ and the cross section trough σd-axes at the value σdo, as shown in Fig. 7.
σdcrit= σdo+tan (ψ) * pv
The stresses in the bridge and the bridge support are can limited to the value of unconfined compressive strength, valid for the consolidation pressure in the silo. The equilibrium of the load of the bridge and the stresses in the support depends of the geometry of the silo outlet as can be seen in equation 10.
For further calculations we can postulate σn=σd
From the equilibrium of forces we obtain in case of plain symmetry the forces in the support:
For plane symmetry is valid:
A/L=d/2 (17)
From equations 8,10 and 17 we obtain the critical diameter of arching for plane symmetry
dcrpl=2*σd*sinα / γ (18)
For axial symmetry, in case of an circular outlet, it is valid
A/L=d/4 (19)
From equations 10, and 13 we obtain the critical diameter for arching for the axial symmetry :
dcrax=4*σd*sinα / γ (20)
The σd in equations 18 and 20 has to be found from the flow-function corresponding with the pressure in the silo.
The critical σd for bridging can be calculated using the data from the flow-function as expressed in the equation 16. Then we obtain:
FOR PLANE SYMMETRY
dcrpl =2sin(α)*(σdo+tan(ψ)*pv) / γ (21)
FOR AXIAL SYMMETRY
dcrax =4sin(α)*(σdo+tan(ψ)*pv) /γ (22)
For dimensioning of the silo outlet, the calculated critical outlet has to be multiplied by a safety factor C: Doutlet = σcrit *C (23)
For dimensioning of a safe silo-outlet, a number of safety ( un-knowledge ) coefficients should be taken as recommended below:
for inaccuracy of measurements C1= 1.1
for inaccuracy of theory C2= 1.5
for fluctuation of material properties C3= 1.2 – 2
engineering safety C4= 1.5 – 2
REMARK
The theoretic approach contain a number of conservative assumptions.
1. we assumed that the tangential pressure in the bridge is zero. In reality this is true only for the boundary of the bridge.
2. we calculate the loading area of the bridge as cross section of the outlet which is smaller than the real area.
3. we assumed that the wall friction is zero, but in reality it is higher than zero.
Because of the conservative assumptions, the inaccuracy of measurements, and the theory as well as engineering safety may be already covered and we ca use the theoretic value for calculation. However, be careful to neglect the safety factors.
After a rathole occur, the radial and tangential stresses has to be calculated according the horizontal stress given in equation (16). As discussed in chapter 3-loading of a bridge, the stresses are very small. In case of rathole the average pressure is px = ph. The loading of a rathole is the vertical pressure according Janssen. The vertical pressure is also the consolidation pressure
The vertical pressure is growing from the top of the silo downwards until the pressure corresponding the infinite depth of the silo.
By a cohesive powder a rathole can exist until the depth where the unconfined compressive strength equals the vertical pressure. By growing of the depth of the rathole above this level, the rathole must collapse.
A stable rathole over the whole height of the silo can occur when the unconfined compressive strength is bigger then the vertical pressure at the bottom of the silo.
The diameter of the rathole is practically independent of the pressure because of the reduced tangential and radial stresses. The diameter is predict only by the size of the outlet.
For infinite depth of the silo, the rathole can be stable when
σd > pv=λγDsilo/4µ (25)
More detailed calculation we can obtain by calculating the vertical pressure for the upper part of the silo. There is the pressure dependent from the depth below the surface.
Any cohesive material can form a rathole, only it is the question by which depth will the pressure be higher than the unconfined compressive strength. In this case the rathole can be stable if:
σd>Dγλ(1-expK)/4µ (26)
where: K=-2λµH/D H – depth of the rathole
CONCLUSIONS
Authors calculation method, based on soil mechanics, is used for similar problems in mining an civil engineering, contain all important data from geometry of the silo as well from the measured powder characteristics. The results of calculation are many times checked in build or redesigned silos. From experience with new installations we can conclude that the outlet opening was sufficient oversized. From the redesigned silos we could see that, the silo opening was under-dimensioned before redesign and after redesign was de outlet opening over-dimensioned. Both situations can be calculate and give an answer according the practice. In this case we can conclude that the theoretic approach, by the rules of soil mechanic, is accurate enough for application in the practice.
Seminar – MIT Boston 14 May 99
