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Some aspects of rock pressure modelling on equivalent materials
Int. J. Rock Mech. Min. $cL Vol. 5, pp. 355-369.
P e r g a m o n Press 1968.
Printed in G r e a t Britain
SOME ASPECTS OF ROCK PRESSURE MODELLING ON
EQUIVALENT MATERIALS DEVENDRA NATH THAKUR M.B.M. Engineering College, Jodhpur, India
(Received 1 July 1967) Abstract--Fundamentals of the theory of rock pressure modelling on equivalent materials a r e discussed. Scale factors have been derived using the Theory of Dimensional Analysis, special emphasis having been laid on the time scale factor. A correction factor has been introduced in the modulus of elasticity of the equivalent materials of plane models to simulate triaxial loading conditions existing in the mine. For modelling deeper deposits a new technique with compensated load is suggested. 1. INTRODUCTION IN PRESENT-DAYmining practice accurate decisions are required to be taken on such intricate problems as the size and shape of workings, parameters of working faces, size of barrier pillars, parameters of supports etc. The increasing depth of workings, large-scale exploitation of reserves under adverse geological conditions and fast rate of advance of working faces aggravate problems of strata control underground, and thus a thorough knowledge of rock pressure phenomena is very essential. At present there are three methods commonly used for predicting rock pressure phenomena in mines; namely, the analytical method, in situ observations and observations on scale models. Due to the uncertain nature of the mining conditions, not only differing from one deposit to another, but 'also in one mining district, the analytical method has a limited application for practical problems, but a generalized picture may be obtained by this method. Of the remaining two methods, scale model observations lead to comparatively reduced costs, increased safety, reduced time, fine control over the parameters involved and ease and accuracy of measurements, and as such they are gaining wide popularity all over the world. There are three main modelling techniques applied for rock pressure studies, e.g. photoelastic modelling, centrifugal modelling and modelling on equivalent materials. Some of the basic aspects of the last method are discussed below. 2. THE SIMPLE MODEL To explain the principle of scale modelling let us first have a simple model of cubical shape (Fig. 1) and let it be made out of the same material as the prototype and work in the same gravitational field. Let the characteristics of the prototype and its model have the parameters as given in Table 1. It can be seen that o'M 7 • e L . L --
~
--
7 •L 355
~L.
(1)
356
DEVENDRANATH THAKUR
/
i
t_
_ lJ r I FIG.
Scheme illustrating the method of simple models.
Thus, the ratio of the stresses in a simple model, and its prototype is equal to the geometrical scale factor though they are of the same material. TABLE 1
Parameters (form) Linear measurement Geometrical scale factor Cross-sectional area Volume Mineral specific gravity (natural) Weight Stress at the base
Prototype (cubical) L
Model (cubical) l
L:L = 1
I:L = aL
L2 L3 7 y•L a
12 = aL2 . L~ 13 = aL 3 . L ~ y
crp
7.12 y . L 3
L~ --~,.L
=
7 • a/. a - La ~ • la
aM-- 12 --~'.aL.L
Under these circumstances, if a L < l , then c r , < c r p . From here it can be inferred that if the stress in the prototype exceeds its ultimate strength, it will fail, but the value of the corresponding stress developed in the simple model will be less and the model will not fail. The simple model, therefore, will not predict the correct phenomena occurring in the mine and may give erroneous results. For the simple model to represent the working of the prototype, we must either (i) increase the stress in the model in the same ratio as its linear measurement is related to its prototype, keeping the material of the model the same; or (ii) select the material of the model in such a way that its ultimate strength is less than that of the prototype and is reduced to a particular scale factor. The principle of the first method is used in the centrifugal method of rock pressure modelling and that of the second for modelling on equivalent materials. 3. SCALE FACTORS FOR ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS On first approximation, the convergence C in the working area of a mine may be expressed in the form C = f(H,
L , E , or, y , y ' , Q , t, to, 7, ~, I~) •
where C is the value of the convergence [L] H is the depth of the working from the surface [L]
(2)
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
357
L is the characteristic linear measurement [L] E is the modulus of elasticity of the rock [K/L ~] is the strength characteristic of the rock; e.g. ultimate strength of the rock in tension, compression or bending [K/L ~] ~, is the mineral specific gravity (natural) [K/L 3] ~,' is the mineral specific gravity (natural) of the mineral being worked [K/L 3] Q is the reaction of the support [K] t is the total time of deformation IT] to is the creep period of the rock [T] ~7 is the coefficient of dynamic viscosity of the rock [KT/L ~] is the angle of internal friction of the rock [1] tz is the Poisson's ratio of the rock [1] Equation (2) can be rewritten in the dimensionless form as:
C
H
~(H-L,Y ~'HyH
Q
0 ' --E" - ~ '
-
Q
t ~7
~7
-to,
) .
(3)
These dimensionless ratios must retain their forms in any system of measurement. Therefore, the dimensionless quantities of the function should he the same for the model as well as for the prototype. Under these conditions we obtain the following system of equations: -~Y
P
e
----- ~- " =
~'
u
or . . . . Hp
;
or
u;
or . . . . .
, =
M'
=
u"
P
He
or
QP
or
Q-;
top
Et'o
He
,.)/p
Hap
=
(10) te
or to - -e -- -Eu - • ~p --.
•
M
to e
(6) (7)
yP
or . . . .
P
e
)'p
(Tp
=
(5)
or . . . . . .
=
go
Y---~= Y--~,. yp yp
Ee
P
P
(4)
Lp
yu
(11)
HM ~e
/~P -----/,u.
(13)
~0, = ~ .
(14)
358
DEVENDRA NATH THAKUR
3.1 G e o m e t r i c a l s c a l e f a c t o r
From equation (4) we obtain HM LM --Hv Lv
--
(I 5)
aL -~ i d e m .
Equation (15) represents the condition of geometrical similarity of the model and the prototype, which states that the distance between two points in the model system must bear a constant ratio to the corresponding points in the prototype or the mine opening. The choice of the geometrical scale factor ( a z ) depends upon the nature of the investigation, laboratory facilities and also upon financial considerations. For investigations of a general nature on equivalent material models a geometrical scale factor of 1/100 is considered to be quite normal, but for detailed analysis of bed separation, load on face supports etc. higher geometrical scale factors of 1/50 or 1/20 are essential. 3.2 M i n e r a l s p e c i f i c g r a v i t y ( n a t u r a l ) From equation (5) we obtain
scale factor
yM __ )'M yv ~,v'
- - a~ =
(16)
idem.
Equation (6) refers to the conditions of dynamic similarity of the model and the prototype. If the superincumbent strata over a coal seam is made up of rocks having different mineral specific gravities, and deformation is caused due to their own weight, then from equation (16) it can be concluded that to maintain dynamic similarity the following relationship must be maintained: s
~
n
~'M __ ~'M __ ' ~,~ Yv
__ ~M __ i d e m . ye
(17)
As the mineral specific gravities of sedimentary rocks do not differ appreciably, it has been recommended [2] that those of the equivalent materials should also be close to each other. In practice, while using wax, sand and mica dust as equivalent materials, their mineral specific gravity (natural) vary from 1.3 to 1- 5 g wt/cm 3. Thus, while modelling sedimentary beds, having average mineral specific gravity (natural) of 2- 5 g wt/cm 3, a mineral specific gravity (natural) scale factor of 0.6q3-7 is normally chosen. 3.3 Stress scale factor
Equations (6) and (7) can be rewritten as EM Ev
--
aM ~rv
--
yM X
HM -=
~v
Hv
aL.
%.
(18)
From this it can be seen that if the superincumbent strata consists of different beds having different strength characteristics then for each one of them equation (18) must be satisfied. The important strength characteristics to be taken into account are: the ultimate strength of the rock in uniaxial compression ( R c ) , in tension (RT), in bending (RB) and the cohesive strength of the rock (c). It will be found, however, that it is tedious to choose a material satisfying the similarity condition (18) for all these properties. KUZNETSOV [2]
SOME ASPECTSOF ROCK PRESSUREMODELLINGON EQUIVALENTMATERIALS
359
confines attention to only those properties which play significant roles in the process under investigation, and denotes these fundamental strength characteristics.
3.4 Similarity of caving spans of the roof in the model and in the mine If the criteria cited above have been satisfied, then the area and the form of the caved roof in the model and in the mine should be similar. However, it is very difficult to assess the exact process occurring in the goaf and as such it will be sufficient if similarities are maintained between the maximum caving spans of the main and immediate roofs in the mine and in the model. I f this does not occur, a method of trial and error has to be followed. First a few experimental models have to be made and observations taken for maximum caving spans of the roof [3]. For the first experimental model 2 SM = a L . Sr
(19)
where S,, is the average area of the caved roof on the model S~ is the average area of the caved roof in the mine, and aL is the geometrical scale factor, but b,~ - - m ~ a L
(20)
B,
where b~ is the average caving span of the roof on the model Bff is the average caving span of the roof in the mine m is the actual geometrical scale factor. If the immediate roof is supposed to work as a cantilever, then using equations (18) and (20) we shall get $
,
yu aL yF m s
(21)
where a',, is the ultimate strength of the equivalent material in bending ~ is the ultimate strength of the rock in bending in the mine fl = adm ~ = correction factor for the ultimate strength in bending of the equivalent material of the immediate roof on the first experimental model. The main advantage of this correction factor is that it excludes the effects of weakening due to cracks etc. on the model. However, it can be introduced only when investigations are carried out for a working mine.
3.5 Poisson's ratio and angle of internal friction The dimensionless parameters of rocks e.g. Poisson's ratio (~) and the angle of internal friction (4) must be the same for the rock and the equivalent materials. ~toeat 5p,~--l,
360
DEVENDRA NATH THAKUR
3.6 Force scale factor Reactions (tonnes weight) to be provided by face supports on the model may be calculated from equations (8) and (9): QM_y,., H~,_ %'a~ Q~ y~ H~ or
QM : a~. a~. Qp
(22)
and therefore, force scale factor ae =
a~,. a [ .
(22')
3.7 Time scale factor For modelling on equivalent materials, the time scale factor is one of the most important similarity criteria, and it refers to the kinetic conditions of similarity, being accomplished if similar points of the model and the prototype, moving on geometrically similar paths cover geometrically similar distances in time, differing from one another by a constant time scale factor [4]. From equation (10), time scale factor tM te
tOM toe
(23)
According to KUZNETSOV [2], the forces involved in the process of deformation around a mine opening are plastic in nature, therefore he suggests that e, =
~,~
1
--.
~
7}p
a L . a~
(24)
ILESTEIN [5] considers time scale factor a, =
vM h~, g v~ hp l~
. . . . .
(25)
where vM and v~ are the rates of sag of geometrically similar beams of the model and the rock h,, and h~ are the heights of the model and the rock beams lM and L~ are the lengths of the model and the rock beams. From equations (24) and (25) it can be seen that opinions differ over the form of deformation to be considered for determining the time scale factor. As with ILESTEIN [5], the rates of deformations shouM be taken in the settled portions of the time deformation curves of the model and the rock beams. But KUZNETSOV [2] takes into account the total deformation suffered by the two in a given time. Some authors prefer to determine the time scale factor by comparing times taken in accomplishing similar processes in the mine and on the model, e.g. amounts of yield suffered by similar supports at geometrically similar distances in the working faces of the mine and the model. However, the uncertain nature of the actual mining condition, unpredictable stoppages of the workings and other factors associated with underground mining, make it difficult to determine the time scale factor accurately by this method. Moreover for modelling virgin deposits this method has a limited application.
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
361
A simple analytical method for determining the time scale factor is suggested below. In an elastic medium the period of oscillation of a sound wave is determined by the equation t =
,/Y
(26)
where E is the modulus of elasticity of the medium o is the density of the medium. Taking this as a criterion, the time scale factor in an elastic medium
¢qe =
•
" Hp
"/n
as p = y/g; and acceleration due to gravity (g) is the same for the mine and its model on equivalent materials. For plastic deformations let us take equations (10) and (11) as criteria. Then we shall get tom
E~
to,,
E~ ~
~7,~
O-r/
(28)
0 , ~ • CtL
where ~. is the scale factor for coefficient of viscosity. From Reynold's equation pvL ~7
where p is the v is the L is the ~7 is the
idem
(29)
density [KT~L ~] velocity [ L T -1] linear measurement [L] coefficient of viscosity [KTL-~].
Therefore pu • vM • LM
pp •
"Ore
~)p * L p
"OP
or
r/__~__-- tZ . y__.. L_ ~Tv tM ~)/P L ~ or -Ctr/ - -
CL? . ¢2L _ _ CtZp
(30)
Putting the value of % into equation (28), we get 2 0 . ~ / . U, L
Q'L
CLip. CL~. CLL
(tip
OLtp - -
or O,tp =
~v,/CJ.L.
(31)
362
DEVENDRA NATH THAKUR
Thus, from equations (27) and (31), it can be seen that for modelling on equivalent materials the time scale factor is equal to the square root of the geometrical scale factor under both elastic and plastic conditions. The same conclusion can be reached by considering the equation of free motion of the geometrically similar particles of the model and the prototype in the same gravitational field as criterion [6]: 1
L
2 gt~"
As the acceleration due to gravity is the same for the model and the mine LM Lp or
t~ t~
~/aL.
~, =
4. BOUNDARY CONDITIONS O F PLANE EQUIVALENT MATERIAL M O D E L S
A three-dimensional model is the true representative of a mining panel. However, the technology of working out the coal seam in a three-dimensional model and the measurement of the deformation and rock pressure is so complicated that for investigating normal strata control problems a 'plane model' is considered quite suitable. A plane model is a thin, but long, beam in space. The working conditions of such a model beam are different from those of an imaginary strip vertically cut from the rock mass. For example, the model beam has free surfaces and can expand on the sides but the vertical rock strip cannot. There is no stress on the walls of the model beam, but stress exists on the vertical rock strip. In this way, the thin vertical rock strip is more stressed than the model beam. For the deformation in the model beam to be the same as that in the vertical rock strip, it is necessary that their modulus of elasticity should be different. Let us calculate the values of strains in a cubical rock piece under different stress conditions (Fig. 2).
O" I
I _
(a)
T
%
(b)
(d
Fio. 2. Scheme illustrating deformation of rock under different loading conditions. Let the vertical stress be equal to ol, and the horizontal stresses /L
(32)
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
363
where is the Poisson's ratio of the rock. Vertical strain [Fig. 2(a)] t E
--
ol
(33)
E where E is the modulus of elasticity of the rock under uniaxial compression. Under biaxial compression the vertical strain [Fig. 2(b)] 1
1--
04)
"
Therefore the modulus of elasticity of the rock in biaxial compression E"
--
E
_
/~
1 --
1
tz
.
-
/z2
.
E.
(35)
In triaxial compression [Fig. 2(c)], the vertical strain ,, ___ ~x _ t~ (o~ + o3) ---- ox -. 1--t~--2tz ~
(36)
Therefore, the modulus of elasticity of the rock in triaxial compression E " -----
1 -- #
. E.
(37)
1 - - / ~ - - 2/~ ~
Thus, the 'model beam' working in biaxial stress condition will represent the vertical rock strip working in triaxial stress condition, if the modulus of elasticity of the former is equal to =
E" --.
E -
E"
1 __ /~__ #2 1 --
tz --
E.
(38)
2tz ~
For example, if the Poisson's ratio of the rock is equal to 0-3, then the modulus of elasticity of the model beam must be increased by nearly 1.2 times. Substituting equation (38) in the stress scale factor (18), the modulus of elasticity of the material of the plane model Eu----aL.%.l
1 --tz--~
--t~--2/z~
E.
(39)
5. MODELLING DEEP DEPOSITS In laboratories having limited height, considerable difficulties might be experienced when investigating rock pressure phenomena around deposits lying at depths more than 200-300 m; one solution to the problem appears to be to construct models on low geometrical scale factors (less than 1 : 200), but on such models it is difficult to simulate rock structures because it is difficult to lay slices less than 0 . 4 - 0 . 5 cm thick. Apart from this, the working area on such models cannot be supported, and therefore the parameters of the support cannot be investigated there.
364
DEVENDRA NATH THAKUR
These difficulties can be easily overcome if only a certain portion of the superincumbent strata above the coal seam is laid on the model and the rest of the rock mass is represented by a compensated load. ILESTEIN[5] considers that the amount of convergence suffered by the roof in the working area of the truncated model with a compensated load must be equal to that on a full model. Using RUPENEIT'S [7] convergence formula, he recommends that the compensated load on the model
]
q"
= [Th~ ( S ~ + 4Uoh) ~ (S ~ + 2Uoh~) [ ' ~ - ~ +'4U--~O-~(S-~ +- 2. U---~ -- yh._ .aL. a~
(40)
where S is the width of the working area Uo is the amount of convergence of the roof in the working area at a distance S from the face edge h is the depth of working from the surface hi is the height above the coal seam modelled y is the average mineral specific gravity (natural) of the superincumbent strata a~ is the mineral specific gravity (natural) scale factor aL is the geometrical scale factor. However, the compensated loading problem must be solved depending upon the type of mine opening. If the opening is stationary or on the long wall face there has been no caving of the main roof, the calculation of the compensated load and technique of placing it over
D
(e)
C
Ii:l
Ap----~
FIG. 3. Illustrations for calculating compensated load.
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
365
the model is simple. The amount of compensated load qp is equal to the pressure due to the weight of the portion ABCD [Fig. 3(a)] and can be placed at a height sufficiently above the coal seam. However, on a longwaU face, after the first caving of the main roof, there is redistribution of the initial pressure due to bed separation in the strata above the coal seam. Some portion of the superincumbent strata in the goaf gets de-stressed, but rock pressure ahead of the coal seam qF increases 2-6 times the initial value and a zone of front abutment pressure is formed (Fig. 4). On the advance of the working face this zone also advances. Therefore, the system of compensated loading must ensure redistribution of load as shown in Fig. 4.
i~
S
._
I,,rl!/rT 1.,......~ ....... izo.e o,~o,.n,,
. .....
..
"
:...'"'"
s,ro,o m o v . m . n *
.
,
."
: .'.
-"~:~.~.~iiiiiiiiiiiiiii!!iiiil,_~.
FIG. 4. Scheme of placing compensated load. (Courtesy of Professor A. A. Borisov.)
Though the method shown in Fig. 4 is correct in principle, it is difficult to put it into practice due to the following: (i) The value of the front abutment pressure qF is not a constant quantity on the length of the panel and it also varies from one coal field to the other, depending upon the strength properties of surrounding rocks and coal, depth of workings, structural features, etc. (ii) In this system the width of the abutment zone S, the position of the maximum front abutment pressure qr, and change in their values at different position of the working face must be known before hand. It should be mentioned here that the true picture of the front abutment pressure on the whole working field and changes in its value with different rates of advance are problems still to be investigated. (iii) The character of the strata, bound between the planes A-B and C-D [Fig. 3(a)] are not taken into account here, but in actual mining practice they play a significant role in the redistribution of rock pressure in and around the working area. To overcome the above-mentioned difficulties the following method is suggested [8]. The load over the model should be imposed through a compensatory layer of hard equivalent material [Fig. 3(e)], whose strength should be equivalent to the average strength properties of the strata ABCD [Fig. 3(a)] in the mine, i.e. the bed abc'd' [Fig. 3(e)] should be representative of the superincumbent strata ABCD in the mine [Fig. 3(a)] or abcd of the full model [Fig. 3(c)]. The weight of the superincumbent strata above the surface A-B is an external load, and therefore, it can be replaced by an evenly distributed load q~ where qp
--
P
F or
--
7,.A2e.(H -
Hx)
°
100
1000. A~
q~ = 0" 1.7~. (H -- Hx), kg/cm ~
(41)
366
DEVENDRA NATH THAKUR
and yp is the mineral specific gravity (natural) of the rock (g wt/cm0 H is the depth of the coal seam from the surface (m) //1 is the height above the coal seam being modelled (m). In Fig. 3, scheme 'c' is the model of scheme 'a' and scheme 'd' is the model of scheme 'b'. Using equation (18) the value of the compensated load on the model (scheme 'd') q,~ = aL. a~. qp -----0" 1. aL. a~. 7p. ( H -- H1), kg/cm ~.
(42)
In Fig. 3(e) the external compensated load q,~ has been replaced by the compensated load q'M and by the compensating layer having thickness hc (cm) and specific gravity (natural) ~, (g wt/cm0. Therefore , qM =qM
~ . hc kg/cm2" 1-0~'
(43)
The imaginary specific gravity (natural) of this system of compensated loading ~lma$
--
1000. qM h~
100. aL. yM. ( H -- H0, g wt/cm 8. h,
(44)
Let the characteristic strength of the system be oc (kg/cm2). In the full model [Fig. 3(c)], which may be taken as the prototype for the model shown in Fig. 3(e), the specific gravity (natural) is equal to ~,,~ and the characteristic strength oM. The geometrical scale factor of this modelling system aL is unity. Therefore, using equation (18) we shall get
or
O'¢
--
~',=,2 . ~,M
o'c
=
100. a~. a~. (H -- //1) h,
(7"M
z
100. aL. ( H -- //1) hc • O'e~
.
tTm
kg/cm2"
(45)
5.1 Problem The depth of the deposit from the surface is 500 m. The first 115 m of the strata above the coal seam is modelled on a geometrical scale factor of 1 : 100. The specific gravity (natural) scale factor for the equivalent material is 0.6. The problem is to find the suitable ultimate strength of the compensating layer in bending if the corresponding value for the rocks of the superincumbent strata is 75 kg/cm 2. 5.2 Solution
1 Here H = 500 m; Hx = 115 m; eL -- 100'-ay = 0 . 6 ap = 75 kg/cm 2. Putting the above values in equation (45), we obtain the strength of the compensating layer in bending 100 × 1 × 0.6 (500-115) 173.25 crc---. 75---kg/cm 2 1002 • hc hc
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
367
where h, is the thickness of the compensating layer in centimetres. It will be clear from the above that the strength of the compensating layer is inversely proportional to its thickness. Inserting different values of h, we obtain different values of the strength of the compensating layer in bending, which is given in Table 2. TABLE 2 he(era)
10
20
40
60
80
100
120
140
180
200
17.32
8"66
4"33
2-88
2.17
1"73
1.44
1"24
0"96
0"87
h¢(crn)
220
250
300
385
o~(kg/cm2)
0.79
0" 693
0- 578
0" 450
ac(kg/cm s)
In the above example when the thickness of the compensating layer is maximum, i.e. 385 cm i.e. equal to H-H1; or in other words the model represents the full superincumbent strata, the strength of the equivalent material of the compensating layer in bending is 0.45 kg/cm ~. The average strength of the equivalent material of the superincumbent strata of the full model in bending as calculated in equation (18) I ~,~------ x 0 " 6 x 7 5 = 0 . 4 5 k g / c m 2. 100
This conclusion, therefore, justifies the validity of the formula (45). In laboratories the thickness of the compensatory layer below the external load depends upon the working height available, the thickness of the seam being worked, etc. While working thicker seams non-systematic caving of the roof occurs up to an appreciable height above the coal seam, and therefore hi should be as large as possible. In such cases, the space available will be limited and the thickness of the compensatory layer less. Our experiences show that for inducing bed separation in the compensatory layer and thereby redistributing the rock pressure at this height, its thickness should not be less than 25 cm. Now if the thickness of the compensatory layer in the above example is 25 cm, the strength of the equivalent material for the compensatory layer will be 6.93 kg/cm'. With hc = 25 cm and ~,c = 1-38 (g wt/cm3) (to be determined); , 0.1z 1x0.6x2-4x385 qM = 100
1.38x25 -- 0.52 kg/cm~. 1000
If the width of the model is 20 cm and its length 500 cm, then the total amount of load to be placed over the model will be Q = 20 x 500 x 0.52 kg = 5"2 tonnes. For putting this load over the model, in some laboratories mechanical levers connected to blocks are being used. In some cases gunny bags filled with sand or lead pebbles have also been used. The main defects associated with these techniques are their unsafe nature, stamping effect and lack of uniformity over the surface. Moreover, the amount of rock pressure coming over the worked-out area is the same as that ahead of the face. But, as explained earlier the value of the vertical pressure in the goaf should be less than the normal i.e. yH.
368
DEVENDRA NATH THAKUR
T o a v o i d these difficulties a p n e u m a t i c l o a d i n g device is suggested below (Fig. 5). Here, the c o m p e n s a t e d l o a d q'M is given to the m o d e l t h r o u g h p n e u m a t i c b l a d d e r s or pillows 3, each 20-30 c m long a n d connected individually t h r o u g h valves 5 to a c o m m o n metallic air collector 4, which in t u r n is connected via a pressure gauge to a h a n d p u m p . P n e u m a t i c b l a d d e r s are held between two side channels 7 a n d a thick w o o d e n p l a n k 6 f r o m the top. D u e to the subsidence o f the c o m p e n s a t o r y layer, while w o r k i n g the coal seam below, the v o l u m e o f the air space a b o v e the c o m p e n s a t i n g layer 2 increases a n d the pressure in
I
Fro. 5. Side view of the arrangement for pneumatic loading over the model; 1--model; 2---~ompensatory layer; 3--pneumatic pillow or bladder; 4--air collector tube; 5--valve; 6--wooden plank; 7--side channels. the b l a d d e r decreases. Separate connexion o f the b l a d d e r s to the air collector ensures pressure d r o p only in the w o r k e d - o u t area. D u e to the bed s e p a r a t i o n occurring in the c o m p e n s a t i n g layer a b o v e the w o r k e d - o u t a r e a there is a pressure rise ahead o f the w o r k i n g face a n d r e d i s t r i b u t i o n o f pressure is in the m a n n e r shown in Fig. 4. Acknowledgements--The author gratefully acknowledges the help and guidance rendered by Docents I. D.
VOSTROV and I. M. PANIN of the Peoples' Friendship University, Moscow. He is indebted to Professor A. K. GHOSEof Indian School of Mines, Dhanbad for his valuable suggestions while preparing this paper. Thanks are due to Dr. K. N. SINHA,Director, Central Mining Research Station, Dhanbad for encouragement and kind consent for publication. REFERENCES 1. BUCKINGHAMJ. On physically similar systems, illustrations of the use of dimensional equations. Phys. Rev. 4, 4 (1914). 2. K.13ZNETSOVG. N. Study of the Behaviour of Rock Pressure on Models (in Russian), Ugletexizdat, Moscow (1959). 3. BoRISOVA. A. Calculation of Rock Pressure on the Faces of Moderately Inclined Coal Seams (in Russian), Nedra, Moscow (1964). 4. I(3RrUCrmVM. B. Theory of Dimensional Analysis (in Russian), Academy of Sciences, U.S.S.R. (1953). 5. ILESTEINA. M. Laws of the Manifestation of Rock Pressure (in Russian), Ugletexizdat, Moscow (1958).
SOME ASPECTS OF ROCK PRESSURE MODELLINGON EQUIVALENTMATERIALS
369
6. PANn~ I. M. Rock Mechanics (in Russian), Peoples' Friendship University Press, Moscow (1965). 7. RUPENErrK. V. Rock Pressure and Convergence on Working Faces of Moderately Inclined Coal Seams
(in Russian), Ugletexizdat, Moscow (1957). 8. THAKURD. N. Investigations into the Behaviour of Rock Pressure While Working Thick and Moderately Inclined Coal Seams with the Shield KTU Under Conditions of Jharia Coal Field [India] (in Russian).
Ph.D. Thesis, Peoples' Friendship University, Moscow (1966).
P e r g a m o n Press 1968.
Printed in G r e a t Britain
SOME ASPECTS OF ROCK PRESSURE MODELLING ON
EQUIVALENT MATERIALS DEVENDRA NATH THAKUR M.B.M. Engineering College, Jodhpur, India
(Received 1 July 1967) Abstract--Fundamentals of the theory of rock pressure modelling on equivalent materials a r e discussed. Scale factors have been derived using the Theory of Dimensional Analysis, special emphasis having been laid on the time scale factor. A correction factor has been introduced in the modulus of elasticity of the equivalent materials of plane models to simulate triaxial loading conditions existing in the mine. For modelling deeper deposits a new technique with compensated load is suggested. 1. INTRODUCTION IN PRESENT-DAYmining practice accurate decisions are required to be taken on such intricate problems as the size and shape of workings, parameters of working faces, size of barrier pillars, parameters of supports etc. The increasing depth of workings, large-scale exploitation of reserves under adverse geological conditions and fast rate of advance of working faces aggravate problems of strata control underground, and thus a thorough knowledge of rock pressure phenomena is very essential. At present there are three methods commonly used for predicting rock pressure phenomena in mines; namely, the analytical method, in situ observations and observations on scale models. Due to the uncertain nature of the mining conditions, not only differing from one deposit to another, but 'also in one mining district, the analytical method has a limited application for practical problems, but a generalized picture may be obtained by this method. Of the remaining two methods, scale model observations lead to comparatively reduced costs, increased safety, reduced time, fine control over the parameters involved and ease and accuracy of measurements, and as such they are gaining wide popularity all over the world. There are three main modelling techniques applied for rock pressure studies, e.g. photoelastic modelling, centrifugal modelling and modelling on equivalent materials. Some of the basic aspects of the last method are discussed below. 2. THE SIMPLE MODEL To explain the principle of scale modelling let us first have a simple model of cubical shape (Fig. 1) and let it be made out of the same material as the prototype and work in the same gravitational field. Let the characteristics of the prototype and its model have the parameters as given in Table 1. It can be seen that o'M 7 • e L . L --
~
--
7 •L 355
~L.
(1)
356
DEVENDRANATH THAKUR
/
i
t_
_ lJ r I FIG.
Scheme illustrating the method of simple models.
Thus, the ratio of the stresses in a simple model, and its prototype is equal to the geometrical scale factor though they are of the same material. TABLE 1
Parameters (form) Linear measurement Geometrical scale factor Cross-sectional area Volume Mineral specific gravity (natural) Weight Stress at the base
Prototype (cubical) L
Model (cubical) l
L:L = 1
I:L = aL
L2 L3 7 y•L a
12 = aL2 . L~ 13 = aL 3 . L ~ y
crp
7.12 y . L 3
L~ --~,.L
=
7 • a/. a - La ~ • la
aM-- 12 --~'.aL.L
Under these circumstances, if a L < l , then c r , < c r p . From here it can be inferred that if the stress in the prototype exceeds its ultimate strength, it will fail, but the value of the corresponding stress developed in the simple model will be less and the model will not fail. The simple model, therefore, will not predict the correct phenomena occurring in the mine and may give erroneous results. For the simple model to represent the working of the prototype, we must either (i) increase the stress in the model in the same ratio as its linear measurement is related to its prototype, keeping the material of the model the same; or (ii) select the material of the model in such a way that its ultimate strength is less than that of the prototype and is reduced to a particular scale factor. The principle of the first method is used in the centrifugal method of rock pressure modelling and that of the second for modelling on equivalent materials. 3. SCALE FACTORS FOR ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS On first approximation, the convergence C in the working area of a mine may be expressed in the form C = f(H,
L , E , or, y , y ' , Q , t, to, 7, ~, I~) •
where C is the value of the convergence [L] H is the depth of the working from the surface [L]
(2)
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
357
L is the characteristic linear measurement [L] E is the modulus of elasticity of the rock [K/L ~] is the strength characteristic of the rock; e.g. ultimate strength of the rock in tension, compression or bending [K/L ~] ~, is the mineral specific gravity (natural) [K/L 3] ~,' is the mineral specific gravity (natural) of the mineral being worked [K/L 3] Q is the reaction of the support [K] t is the total time of deformation IT] to is the creep period of the rock [T] ~7 is the coefficient of dynamic viscosity of the rock [KT/L ~] is the angle of internal friction of the rock [1] tz is the Poisson's ratio of the rock [1] Equation (2) can be rewritten in the dimensionless form as:
C
H
~(H-L,Y ~'HyH
Q
0 ' --E" - ~ '
-
Q
t ~7
~7
-to,
) .
(3)
These dimensionless ratios must retain their forms in any system of measurement. Therefore, the dimensionless quantities of the function should he the same for the model as well as for the prototype. Under these conditions we obtain the following system of equations: -~Y
P
e
----- ~- " =
~'
u
or . . . . Hp
;
or
u;
or . . . . .
, =
M'
=
u"
P
He
or
QP
or
Q-;
top
Et'o
He
,.)/p
Hap
=
(10) te
or to - -e -- -Eu - • ~p --.
•
M
to e
(6) (7)
yP
or . . . .
P
e
)'p
(Tp
=
(5)
or . . . . . .
=
go
Y---~= Y--~,. yp yp
Ee
P
P
(4)
Lp
yu
(11)
HM ~e
/~P -----/,u.
(13)
~0, = ~ .
(14)
358
DEVENDRA NATH THAKUR
3.1 G e o m e t r i c a l s c a l e f a c t o r
From equation (4) we obtain HM LM --Hv Lv
--
(I 5)
aL -~ i d e m .
Equation (15) represents the condition of geometrical similarity of the model and the prototype, which states that the distance between two points in the model system must bear a constant ratio to the corresponding points in the prototype or the mine opening. The choice of the geometrical scale factor ( a z ) depends upon the nature of the investigation, laboratory facilities and also upon financial considerations. For investigations of a general nature on equivalent material models a geometrical scale factor of 1/100 is considered to be quite normal, but for detailed analysis of bed separation, load on face supports etc. higher geometrical scale factors of 1/50 or 1/20 are essential. 3.2 M i n e r a l s p e c i f i c g r a v i t y ( n a t u r a l ) From equation (5) we obtain
scale factor
yM __ )'M yv ~,v'
- - a~ =
(16)
idem.
Equation (6) refers to the conditions of dynamic similarity of the model and the prototype. If the superincumbent strata over a coal seam is made up of rocks having different mineral specific gravities, and deformation is caused due to their own weight, then from equation (16) it can be concluded that to maintain dynamic similarity the following relationship must be maintained: s
~
n
~'M __ ~'M __ ' ~,~ Yv
__ ~M __ i d e m . ye
(17)
As the mineral specific gravities of sedimentary rocks do not differ appreciably, it has been recommended [2] that those of the equivalent materials should also be close to each other. In practice, while using wax, sand and mica dust as equivalent materials, their mineral specific gravity (natural) vary from 1.3 to 1- 5 g wt/cm 3. Thus, while modelling sedimentary beds, having average mineral specific gravity (natural) of 2- 5 g wt/cm 3, a mineral specific gravity (natural) scale factor of 0.6q3-7 is normally chosen. 3.3 Stress scale factor
Equations (6) and (7) can be rewritten as EM Ev
--
aM ~rv
--
yM X
HM -=
~v
Hv
aL.
%.
(18)
From this it can be seen that if the superincumbent strata consists of different beds having different strength characteristics then for each one of them equation (18) must be satisfied. The important strength characteristics to be taken into account are: the ultimate strength of the rock in uniaxial compression ( R c ) , in tension (RT), in bending (RB) and the cohesive strength of the rock (c). It will be found, however, that it is tedious to choose a material satisfying the similarity condition (18) for all these properties. KUZNETSOV [2]
SOME ASPECTSOF ROCK PRESSUREMODELLINGON EQUIVALENTMATERIALS
359
confines attention to only those properties which play significant roles in the process under investigation, and denotes these fundamental strength characteristics.
3.4 Similarity of caving spans of the roof in the model and in the mine If the criteria cited above have been satisfied, then the area and the form of the caved roof in the model and in the mine should be similar. However, it is very difficult to assess the exact process occurring in the goaf and as such it will be sufficient if similarities are maintained between the maximum caving spans of the main and immediate roofs in the mine and in the model. I f this does not occur, a method of trial and error has to be followed. First a few experimental models have to be made and observations taken for maximum caving spans of the roof [3]. For the first experimental model 2 SM = a L . Sr
(19)
where S,, is the average area of the caved roof on the model S~ is the average area of the caved roof in the mine, and aL is the geometrical scale factor, but b,~ - - m ~ a L
(20)
B,
where b~ is the average caving span of the roof on the model Bff is the average caving span of the roof in the mine m is the actual geometrical scale factor. If the immediate roof is supposed to work as a cantilever, then using equations (18) and (20) we shall get $
,
yu aL yF m s
(21)
where a',, is the ultimate strength of the equivalent material in bending ~ is the ultimate strength of the rock in bending in the mine fl = adm ~ = correction factor for the ultimate strength in bending of the equivalent material of the immediate roof on the first experimental model. The main advantage of this correction factor is that it excludes the effects of weakening due to cracks etc. on the model. However, it can be introduced only when investigations are carried out for a working mine.
3.5 Poisson's ratio and angle of internal friction The dimensionless parameters of rocks e.g. Poisson's ratio (~) and the angle of internal friction (4) must be the same for the rock and the equivalent materials. ~toeat 5p,~--l,
360
DEVENDRA NATH THAKUR
3.6 Force scale factor Reactions (tonnes weight) to be provided by face supports on the model may be calculated from equations (8) and (9): QM_y,., H~,_ %'a~ Q~ y~ H~ or
QM : a~. a~. Qp
(22)
and therefore, force scale factor ae =
a~,. a [ .
(22')
3.7 Time scale factor For modelling on equivalent materials, the time scale factor is one of the most important similarity criteria, and it refers to the kinetic conditions of similarity, being accomplished if similar points of the model and the prototype, moving on geometrically similar paths cover geometrically similar distances in time, differing from one another by a constant time scale factor [4]. From equation (10), time scale factor tM te
tOM toe
(23)
According to KUZNETSOV [2], the forces involved in the process of deformation around a mine opening are plastic in nature, therefore he suggests that e, =
~,~
1
--.
~
7}p
a L . a~
(24)
ILESTEIN [5] considers time scale factor a, =
vM h~, g v~ hp l~
. . . . .
(25)
where vM and v~ are the rates of sag of geometrically similar beams of the model and the rock h,, and h~ are the heights of the model and the rock beams lM and L~ are the lengths of the model and the rock beams. From equations (24) and (25) it can be seen that opinions differ over the form of deformation to be considered for determining the time scale factor. As with ILESTEIN [5], the rates of deformations shouM be taken in the settled portions of the time deformation curves of the model and the rock beams. But KUZNETSOV [2] takes into account the total deformation suffered by the two in a given time. Some authors prefer to determine the time scale factor by comparing times taken in accomplishing similar processes in the mine and on the model, e.g. amounts of yield suffered by similar supports at geometrically similar distances in the working faces of the mine and the model. However, the uncertain nature of the actual mining condition, unpredictable stoppages of the workings and other factors associated with underground mining, make it difficult to determine the time scale factor accurately by this method. Moreover for modelling virgin deposits this method has a limited application.
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
361
A simple analytical method for determining the time scale factor is suggested below. In an elastic medium the period of oscillation of a sound wave is determined by the equation t =
,/Y
(26)
where E is the modulus of elasticity of the medium o is the density of the medium. Taking this as a criterion, the time scale factor in an elastic medium
¢qe =
•
" Hp
"/n
as p = y/g; and acceleration due to gravity (g) is the same for the mine and its model on equivalent materials. For plastic deformations let us take equations (10) and (11) as criteria. Then we shall get tom
E~
to,,
E~ ~
~7,~
O-r/
(28)
0 , ~ • CtL
where ~. is the scale factor for coefficient of viscosity. From Reynold's equation pvL ~7
where p is the v is the L is the ~7 is the
idem
(29)
density [KT~L ~] velocity [ L T -1] linear measurement [L] coefficient of viscosity [KTL-~].
Therefore pu • vM • LM
pp •
"Ore
~)p * L p
"OP
or
r/__~__-- tZ . y__.. L_ ~Tv tM ~)/P L ~ or -Ctr/ - -
CL? . ¢2L _ _ CtZp
(30)
Putting the value of % into equation (28), we get 2 0 . ~ / . U, L
Q'L
CLip. CL~. CLL
(tip
OLtp - -
or O,tp =
~v,/CJ.L.
(31)
362
DEVENDRA NATH THAKUR
Thus, from equations (27) and (31), it can be seen that for modelling on equivalent materials the time scale factor is equal to the square root of the geometrical scale factor under both elastic and plastic conditions. The same conclusion can be reached by considering the equation of free motion of the geometrically similar particles of the model and the prototype in the same gravitational field as criterion [6]: 1
L
2 gt~"
As the acceleration due to gravity is the same for the model and the mine LM Lp or
t~ t~
~/aL.
~, =
4. BOUNDARY CONDITIONS O F PLANE EQUIVALENT MATERIAL M O D E L S
A three-dimensional model is the true representative of a mining panel. However, the technology of working out the coal seam in a three-dimensional model and the measurement of the deformation and rock pressure is so complicated that for investigating normal strata control problems a 'plane model' is considered quite suitable. A plane model is a thin, but long, beam in space. The working conditions of such a model beam are different from those of an imaginary strip vertically cut from the rock mass. For example, the model beam has free surfaces and can expand on the sides but the vertical rock strip cannot. There is no stress on the walls of the model beam, but stress exists on the vertical rock strip. In this way, the thin vertical rock strip is more stressed than the model beam. For the deformation in the model beam to be the same as that in the vertical rock strip, it is necessary that their modulus of elasticity should be different. Let us calculate the values of strains in a cubical rock piece under different stress conditions (Fig. 2).
O" I
I _
(a)
T
%
(b)
(d
Fio. 2. Scheme illustrating deformation of rock under different loading conditions. Let the vertical stress be equal to ol, and the horizontal stresses /L
(32)
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
363
where is the Poisson's ratio of the rock. Vertical strain [Fig. 2(a)] t E
--
ol
(33)
E where E is the modulus of elasticity of the rock under uniaxial compression. Under biaxial compression the vertical strain [Fig. 2(b)] 1
1--
04)
"
Therefore the modulus of elasticity of the rock in biaxial compression E"
--
E
_
/~
1 --
1
tz
.
-
/z2
.
E.
(35)
In triaxial compression [Fig. 2(c)], the vertical strain ,, ___ ~x _ t~ (o~ + o3) ---- ox -. 1--t~--2tz ~
(36)
Therefore, the modulus of elasticity of the rock in triaxial compression E " -----
1 -- #
. E.
(37)
1 - - / ~ - - 2/~ ~
Thus, the 'model beam' working in biaxial stress condition will represent the vertical rock strip working in triaxial stress condition, if the modulus of elasticity of the former is equal to =
E" --.
E -
E"
1 __ /~__ #2 1 --
tz --
E.
(38)
2tz ~
For example, if the Poisson's ratio of the rock is equal to 0-3, then the modulus of elasticity of the model beam must be increased by nearly 1.2 times. Substituting equation (38) in the stress scale factor (18), the modulus of elasticity of the material of the plane model Eu----aL.%.l
1 --tz--~
--t~--2/z~
E.
(39)
5. MODELLING DEEP DEPOSITS In laboratories having limited height, considerable difficulties might be experienced when investigating rock pressure phenomena around deposits lying at depths more than 200-300 m; one solution to the problem appears to be to construct models on low geometrical scale factors (less than 1 : 200), but on such models it is difficult to simulate rock structures because it is difficult to lay slices less than 0 . 4 - 0 . 5 cm thick. Apart from this, the working area on such models cannot be supported, and therefore the parameters of the support cannot be investigated there.
364
DEVENDRA NATH THAKUR
These difficulties can be easily overcome if only a certain portion of the superincumbent strata above the coal seam is laid on the model and the rest of the rock mass is represented by a compensated load. ILESTEIN[5] considers that the amount of convergence suffered by the roof in the working area of the truncated model with a compensated load must be equal to that on a full model. Using RUPENEIT'S [7] convergence formula, he recommends that the compensated load on the model
]
q"
= [Th~ ( S ~ + 4Uoh) ~ (S ~ + 2Uoh~) [ ' ~ - ~ +'4U--~O-~(S-~ +- 2. U---~ -- yh._ .aL. a~
(40)
where S is the width of the working area Uo is the amount of convergence of the roof in the working area at a distance S from the face edge h is the depth of working from the surface hi is the height above the coal seam modelled y is the average mineral specific gravity (natural) of the superincumbent strata a~ is the mineral specific gravity (natural) scale factor aL is the geometrical scale factor. However, the compensated loading problem must be solved depending upon the type of mine opening. If the opening is stationary or on the long wall face there has been no caving of the main roof, the calculation of the compensated load and technique of placing it over
D
(e)
C
Ii:l
Ap----~
FIG. 3. Illustrations for calculating compensated load.
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
365
the model is simple. The amount of compensated load qp is equal to the pressure due to the weight of the portion ABCD [Fig. 3(a)] and can be placed at a height sufficiently above the coal seam. However, on a longwaU face, after the first caving of the main roof, there is redistribution of the initial pressure due to bed separation in the strata above the coal seam. Some portion of the superincumbent strata in the goaf gets de-stressed, but rock pressure ahead of the coal seam qF increases 2-6 times the initial value and a zone of front abutment pressure is formed (Fig. 4). On the advance of the working face this zone also advances. Therefore, the system of compensated loading must ensure redistribution of load as shown in Fig. 4.
i~
S
._
I,,rl!/rT 1.,......~ ....... izo.e o,~o,.n,,
. .....
..
"
:...'"'"
s,ro,o m o v . m . n *
.
,
."
: .'.
-"~:~.~.~iiiiiiiiiiiiiii!!iiiil,_~.
FIG. 4. Scheme of placing compensated load. (Courtesy of Professor A. A. Borisov.)
Though the method shown in Fig. 4 is correct in principle, it is difficult to put it into practice due to the following: (i) The value of the front abutment pressure qF is not a constant quantity on the length of the panel and it also varies from one coal field to the other, depending upon the strength properties of surrounding rocks and coal, depth of workings, structural features, etc. (ii) In this system the width of the abutment zone S, the position of the maximum front abutment pressure qr, and change in their values at different position of the working face must be known before hand. It should be mentioned here that the true picture of the front abutment pressure on the whole working field and changes in its value with different rates of advance are problems still to be investigated. (iii) The character of the strata, bound between the planes A-B and C-D [Fig. 3(a)] are not taken into account here, but in actual mining practice they play a significant role in the redistribution of rock pressure in and around the working area. To overcome the above-mentioned difficulties the following method is suggested [8]. The load over the model should be imposed through a compensatory layer of hard equivalent material [Fig. 3(e)], whose strength should be equivalent to the average strength properties of the strata ABCD [Fig. 3(a)] in the mine, i.e. the bed abc'd' [Fig. 3(e)] should be representative of the superincumbent strata ABCD in the mine [Fig. 3(a)] or abcd of the full model [Fig. 3(c)]. The weight of the superincumbent strata above the surface A-B is an external load, and therefore, it can be replaced by an evenly distributed load q~ where qp
--
P
F or
--
7,.A2e.(H -
Hx)
°
100
1000. A~
q~ = 0" 1.7~. (H -- Hx), kg/cm ~
(41)
366
DEVENDRA NATH THAKUR
and yp is the mineral specific gravity (natural) of the rock (g wt/cm0 H is the depth of the coal seam from the surface (m) //1 is the height above the coal seam being modelled (m). In Fig. 3, scheme 'c' is the model of scheme 'a' and scheme 'd' is the model of scheme 'b'. Using equation (18) the value of the compensated load on the model (scheme 'd') q,~ = aL. a~. qp -----0" 1. aL. a~. 7p. ( H -- H1), kg/cm ~.
(42)
In Fig. 3(e) the external compensated load q,~ has been replaced by the compensated load q'M and by the compensating layer having thickness hc (cm) and specific gravity (natural) ~, (g wt/cm0. Therefore , qM =qM
~ . hc kg/cm2" 1-0~'
(43)
The imaginary specific gravity (natural) of this system of compensated loading ~lma$
--
1000. qM h~
100. aL. yM. ( H -- H0, g wt/cm 8. h,
(44)
Let the characteristic strength of the system be oc (kg/cm2). In the full model [Fig. 3(c)], which may be taken as the prototype for the model shown in Fig. 3(e), the specific gravity (natural) is equal to ~,,~ and the characteristic strength oM. The geometrical scale factor of this modelling system aL is unity. Therefore, using equation (18) we shall get
or
O'¢
--
~',=,2 . ~,M
o'c
=
100. a~. a~. (H -- //1) h,
(7"M
z
100. aL. ( H -- //1) hc • O'e~
.
tTm
kg/cm2"
(45)
5.1 Problem The depth of the deposit from the surface is 500 m. The first 115 m of the strata above the coal seam is modelled on a geometrical scale factor of 1 : 100. The specific gravity (natural) scale factor for the equivalent material is 0.6. The problem is to find the suitable ultimate strength of the compensating layer in bending if the corresponding value for the rocks of the superincumbent strata is 75 kg/cm 2. 5.2 Solution
1 Here H = 500 m; Hx = 115 m; eL -- 100'-ay = 0 . 6 ap = 75 kg/cm 2. Putting the above values in equation (45), we obtain the strength of the compensating layer in bending 100 × 1 × 0.6 (500-115) 173.25 crc---. 75---kg/cm 2 1002 • hc hc
SOME ASPECTS OF ROCK PRESSURE MODELLING ON EQUIVALENT MATERIALS
367
where h, is the thickness of the compensating layer in centimetres. It will be clear from the above that the strength of the compensating layer is inversely proportional to its thickness. Inserting different values of h, we obtain different values of the strength of the compensating layer in bending, which is given in Table 2. TABLE 2 he(era)
10
20
40
60
80
100
120
140
180
200
17.32
8"66
4"33
2-88
2.17
1"73
1.44
1"24
0"96
0"87
h¢(crn)
220
250
300
385
o~(kg/cm2)
0.79
0" 693
0- 578
0" 450
ac(kg/cm s)
In the above example when the thickness of the compensating layer is maximum, i.e. 385 cm i.e. equal to H-H1; or in other words the model represents the full superincumbent strata, the strength of the equivalent material of the compensating layer in bending is 0.45 kg/cm ~. The average strength of the equivalent material of the superincumbent strata of the full model in bending as calculated in equation (18) I ~,~------ x 0 " 6 x 7 5 = 0 . 4 5 k g / c m 2. 100
This conclusion, therefore, justifies the validity of the formula (45). In laboratories the thickness of the compensatory layer below the external load depends upon the working height available, the thickness of the seam being worked, etc. While working thicker seams non-systematic caving of the roof occurs up to an appreciable height above the coal seam, and therefore hi should be as large as possible. In such cases, the space available will be limited and the thickness of the compensatory layer less. Our experiences show that for inducing bed separation in the compensatory layer and thereby redistributing the rock pressure at this height, its thickness should not be less than 25 cm. Now if the thickness of the compensatory layer in the above example is 25 cm, the strength of the equivalent material for the compensatory layer will be 6.93 kg/cm'. With hc = 25 cm and ~,c = 1-38 (g wt/cm3) (to be determined); , 0.1z 1x0.6x2-4x385 qM = 100
1.38x25 -- 0.52 kg/cm~. 1000
If the width of the model is 20 cm and its length 500 cm, then the total amount of load to be placed over the model will be Q = 20 x 500 x 0.52 kg = 5"2 tonnes. For putting this load over the model, in some laboratories mechanical levers connected to blocks are being used. In some cases gunny bags filled with sand or lead pebbles have also been used. The main defects associated with these techniques are their unsafe nature, stamping effect and lack of uniformity over the surface. Moreover, the amount of rock pressure coming over the worked-out area is the same as that ahead of the face. But, as explained earlier the value of the vertical pressure in the goaf should be less than the normal i.e. yH.
368
DEVENDRA NATH THAKUR
T o a v o i d these difficulties a p n e u m a t i c l o a d i n g device is suggested below (Fig. 5). Here, the c o m p e n s a t e d l o a d q'M is given to the m o d e l t h r o u g h p n e u m a t i c b l a d d e r s or pillows 3, each 20-30 c m long a n d connected individually t h r o u g h valves 5 to a c o m m o n metallic air collector 4, which in t u r n is connected via a pressure gauge to a h a n d p u m p . P n e u m a t i c b l a d d e r s are held between two side channels 7 a n d a thick w o o d e n p l a n k 6 f r o m the top. D u e to the subsidence o f the c o m p e n s a t o r y layer, while w o r k i n g the coal seam below, the v o l u m e o f the air space a b o v e the c o m p e n s a t i n g layer 2 increases a n d the pressure in
I
Fro. 5. Side view of the arrangement for pneumatic loading over the model; 1--model; 2---~ompensatory layer; 3--pneumatic pillow or bladder; 4--air collector tube; 5--valve; 6--wooden plank; 7--side channels. the b l a d d e r decreases. Separate connexion o f the b l a d d e r s to the air collector ensures pressure d r o p only in the w o r k e d - o u t area. D u e to the bed s e p a r a t i o n occurring in the c o m p e n s a t i n g layer a b o v e the w o r k e d - o u t a r e a there is a pressure rise ahead o f the w o r k i n g face a n d r e d i s t r i b u t i o n o f pressure is in the m a n n e r shown in Fig. 4. Acknowledgements--The author gratefully acknowledges the help and guidance rendered by Docents I. D.
VOSTROV and I. M. PANIN of the Peoples' Friendship University, Moscow. He is indebted to Professor A. K. GHOSEof Indian School of Mines, Dhanbad for his valuable suggestions while preparing this paper. Thanks are due to Dr. K. N. SINHA,Director, Central Mining Research Station, Dhanbad for encouragement and kind consent for publication. REFERENCES 1. BUCKINGHAMJ. On physically similar systems, illustrations of the use of dimensional equations. Phys. Rev. 4, 4 (1914). 2. K.13ZNETSOVG. N. Study of the Behaviour of Rock Pressure on Models (in Russian), Ugletexizdat, Moscow (1959). 3. BoRISOVA. A. Calculation of Rock Pressure on the Faces of Moderately Inclined Coal Seams (in Russian), Nedra, Moscow (1964). 4. I(3RrUCrmVM. B. Theory of Dimensional Analysis (in Russian), Academy of Sciences, U.S.S.R. (1953). 5. ILESTEINA. M. Laws of the Manifestation of Rock Pressure (in Russian), Ugletexizdat, Moscow (1958).
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6. PANn~ I. M. Rock Mechanics (in Russian), Peoples' Friendship University Press, Moscow (1965). 7. RUPENErrK. V. Rock Pressure and Convergence on Working Faces of Moderately Inclined Coal Seams
(in Russian), Ugletexizdat, Moscow (1957). 8. THAKURD. N. Investigations into the Behaviour of Rock Pressure While Working Thick and Moderately Inclined Coal Seams with the Shield KTU Under Conditions of Jharia Coal Field [India] (in Russian).
Ph.D. Thesis, Peoples' Friendship University, Moscow (1966).