Analogue modelling of stresses and displacements in bord and pillar workings of coal mines

Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 14. pp. 7-23. Pergamon Press 1977. Printed in Great Britain

Analogue Modelling of Stresses and Displacements in Bord and Pillar Workings of Coal Mines K. I. ORAVECZ*

In this paper a method is outlined for the determination of stress and displacement distributions in bord and pillar workings of coal mines. The method is based on elastic theory: a linear elastic medium and a simple linear relationship for the compression of the seam are assumed. The convergence distribution in the seam, from which stresses and displacements can be computed elsewhere in the medium, is obtained by modelling the mining geometry on an electrical resistance analogue computer. The validity of the proposed method is tested by comparing the predicted displacements with those measured in the field under a variety of conditions. From the evaluation of three mining geometries, it is deduced that by using the proposed meihod the displacement field can be predicted with sufficient accuracy for practical purposes. By extension it is accepted that the stress field and thus pillar load, as derived by using the method, can be used for the purposes of mine design and stability investigations.

1. INTRODUCTION

due to the shallowness of the workings. The compressibility of the seam can be taken into account by a modification of the electrical resistance analogue, which is used to obtain the analogue solution of seam convergence.t The influence of stress-free ground surface is not taken into account in the solution of convergence distribution on seam level: it is assumed that the seam is at an infinite depth. When displacements and stresses remote from the seam are calculated however, the presence of the ground surface is fully accounted for. Before a new method of stress or load computation can be used with confidence for design purposes, its validity must be ascertained. In the present case this was achieved by comparing theoretically-predicted displacements with those obtained from field measurements. As displacements around stable bord and pillar workings are quite small in magnitude (in the order of 5 mm) an elaborate measuring scheme was develo ~ d to ensure that accurate and reliable results were obtained. After the measuring scheme employed has been described, the measured displacements are presented together with those obtained on the basis of the method proposed. As the correlation proved to be satisfactory, the method was used to compute the load distribution across the mining panels to indicate its practical application.

The upper bound of average pillar stress or pillar load can be readily computed for the case when large areas are mined in horizontal or near-horizontal coal seams with uniform pillars at uniform spacing. In practical mining such conditions rarely exist; thus, generally the pillar load is not known accurately, so that the support capacity of pillars cannot be utilised fully when mine layouts are designed. Apart from the pillar load, the knowledge of detailed stress distribution in and around pillars is often required when dealing with pillar stability problems. Prompted by the success achieved by using the elastic theory for describing the response of hard rock to mining in deep-level mines, the development of a method for the determination of load on coal pillars has been sought also on the basis of this theory. From the point of view of rock mechanics however, important differences exist between deep-level hard-rock mining and bord and pillar coal mining. The method proposed in this paper takes into account these differences and allows for the special conditions that apply to most coal mining. These are essentially two-fold, when compared with deep-level hard-rock mining: first, the relatively high compressibility of the coal seam, and second, the influence of the stress-free ground surface

2. THE MODEL AND THE METHOD OF SOLUTION The concept of linear elasticity has been applied successfully to the solution of problems in rock mechanics

* Technical Assistant to the Research Adviser, Chamber of Mines of South Africa, P.O. Box 809, Johannesburg 2000, South Africa. "l'That is the convergence of the roof and the floor of the seam. irrespective of whether the seam is still present (pillars) or removed (bords).

7

8

K.I. Oravecz

assumptions are quite acceptable for the deep-level hard-rock mines of South Africa. In the case of coal mining, however, the assumptions do not hold: the compression of the seam is an appreciable proportion of the total displacements and, furthermore, the roof strata are more flexible than the floor strata, because of the stress-free ground surface in the vicinity. The compressibility of the seam can readily "1 j I I be taken into account in the analogue solution of the ! I convergence distribution as it is outlined in the next section. At this stage, however, that is, from the point of view of convergence distribution, the finiteness of depth is not taken into account. In the definition of function Fw, however, apart from H Sz ~ P satisfying the basic equations of the model, the boundary conditions that are applicable to near-surface excavations are also satisfied. This is achieved by means of superposition of suitably chosen infinite depth soluAA tions, as shown in some detail in Section 4. Section: a - ~ In the solution of an actual practical problem, the Fig. 1. Coordinate system for obtaining displacements, from known mining area is first modelled on the analogue computer convergence distribution lafter Salamon [1,2]). by subdividing the total area into face elements. The analogue solution gives the convergence at the centre (Salamon [1,2]); for example, the homogeneous elastic of each face element. The value of function Fw isthen model has been found to describe satisfactorily the dis- evaluated for the centre of each face element and mulplacements and stresses induced by tabular excavations tiplied by the corresponding convergence value. By in deep-level hard-rock mines (Ryder & Officer [3]; summing the contributions from each of the face eleOrtlepp & Nicolls [4]). It was shown that in the case ments the final result is obtained. of mining horizontal seams in elastic rocks, the induced stresses and displacements can be calculated from the 3. ANALOGUE SOLUTION O F CONVERGENCE convergence distribution of the roof and floor of the DISTRIBUTION FOR A COMPRESSIBLE SEAM seam. To obtain the vertical displacement (w) remote The general form of the solution for stresses and disfrom the seam, for example, it is necessary to evaluate placements for horizontal seams, when the plane of the the following type of integral: seam is at an infinite depth, has been given in terms w= s:(¢, q)Fw , ~ , ~1, U2, ~/3 . . . . ] Z2 , (1) of a single harmonic function, ~p, in both the homogeneous isotropie and the homogeneous transversely A where the notation of Fig. 1 is adopted, and A denotes isotropic elastic models [I].* In the present context the whole area of mining; Pz, /t2.... are dimensionless only the vertical displacement, w, and the vertical stress, a z, are of interest. Quoting from Salamon's results and material constants (Salamon [2]). The solution to a particular problem is thus reached using the coordinate system as shown in Fig. 2t: in three phases: /

/

~//

-(I%

s=0,75m

/

0.5m

o.,,o

,_t_

w =

(a) finding the convergence distribution, s.(~,r/); (b) definition of function Fw; and (c) integration.

2(1

-

v)~p

o'. = 1 + v \ ~ z

- ~¢

(2)

z t~z2],

(3)

The convergence distribution can be obtained either where E and v are the modulus of elasticity and the experimentally or derived on the basis of the elastic. Poisson's ratio for the medium, respectively. Function model. Salamon [2] showed that the convergence dismust satisfy both the Laplace equation: tribution satisfies the Laplace equation: this in turn V2~ = 0 (4) led to the development of analogue techniques (the electrolytic and electrical resistance analogues) [5]. The and the boundary conditions as applicable to the minanalogue solution of convergence was based on the ing situation. At seam level the boundary conditions assumption that the reef or the seam are at infinite must be defined separately for the mined-out (bord) depth and that the unmined parts of the seam are in- and the unmined, solid regions (pillars), denoted colleccompressible. It was found in practice that these tively with B and P, respectively. In the mined-out region the induced vertical stress is equal in magnitude * In this paper only the homogeneous isotropic model is conto the vertical primitive stress or cover load, q., but sidcred. opposite in sense, mathematically: + Note that for the infinite depth solution in this sequel, the origin -=0; a . = -q_- for region B. (5) of the coordinate system is placed at the seam level.

Bord and Pillar Workings of Coal Mines

9

The ratio of proportionality, ,;., can be expressed as

It

~. = _E+

(8)

h' where E, is the elastic modulus of the seam, and h is the height of the pillars. At great distance from the seam the effect of mining vanishes, thus: : = m;

Fig. 2. Coordinate system for the infinite-depth solution of convergence distribution.

The cover load is customarily taken to be proportional to the depth of cover, H: q: = 7H, (6) where 7' is the pressure exerted by a column of rock of unit height. Over the solid region P the conditions are more complex. In the present approach Winkler's hypothesis is used: the reaction of elastic support is proportional to the deflection. Translated into the mining situation, it is assumed that the induced vertical stress in region P is proportional to the pillar convergence, s.: : = 0;

o: = ,;.s. for region P.

(7)

cr_=w=0.

(9)

All these boundary conditions can be modelled on the analogue. The boundary condition that cannot be represented on the analogue is that the ground surface is stress-free. The convergence distribution obtained by means of the analogue can thus be considered as only an approximation of the finite depth convergence. It will be shown, however, that this approximation is acceptable in the present context. The theory of the analogy was presented in detail elsewhere (Salamon & Oravecz [-6]); here only the description and operation of the analogue and essential relationships are given. In the electrical resistance analogue (Fig. 3) the continuous conducting medium is represented by an orthogonal resistance network, forming a prism of 120 x 120 x 51 unit nodes. One of the 120 x 120node faces of the network is connected to a voltage supply {current source). Each of the central 60 x 60 nodes on the opposite face are connected to individual sockets; these nodes form the addressable 'window' of the analogue computer. Groups of the boundary nodes surrounding the central window are connected to sockets also.

Fig. 3. The electrical resistance analogue.

10

K . I . Ora\rec/ V=O

T ~°T iii V

r

Vz

•

V

Vz

I Rd ERN

_~

ERN

r

I_-± °...-E v R a Z,

Rd

I Z, V~ V~ Fig. 4. t'ndisturbcd ,,late nct~ork. I!RN. electrical resistance network: R. unit resistance of network; d. depth: r, resistance representing the seam: V~. applied voltage: V z, voltage on undisturbed network: V. voltage: I, current: z~, vertical coordinate.

of

Nodes in the window and in the boundary region can be either earthed by inserting shorting plugs or 'pins'. or connected to earth through a series resistor: the plug in this instance has a small built-in potentiometer. To model the pre-mining situation corresponding to a compressible seam, all the addressable and boundary nodes are earthed through series resistors, r (Fig. 4).* As boundary nodes are paralleled in groups, the effective series resistance that is ph, gged into the boundary sockets will have a value r/n, where n is the number of nodes connected in parallel. The uniform potential. I~ that is measurable at the junction of the nodes and series resistors, in fact, corresponds to the primitive vertical stress or cover load on the seam. 'Mining' on the analogue is represented by the removal of the resistors corresponding to the areas extracted (Fig. 5). The resulting change in potential ( L - 1~) is proportional to the convergence, s.. The following relationships can be used for the modelling, convergence: s. - hq: V - Ii2 ;# E~ V2

(10)

the induced vertical stress over the intact region of the seam:

V - V2. ~.-i = q : -

(II)

Fig. 5. Disturbed state of net~ork. ERN. electrical resistancc network: R. unit resistance of net~ork: r. resistance representing the seam; V~ applied ~oltagc: V, ,,okage on undisturbed nets~ork: V. ~oltage: I, current: z~ vertical coordinate: VG, ~oltage generator: VM, voltmeter: A, ammeter: DVM, digital ~oltmcter.

tion ip can also be obtained directly from the expression: 0 =

hq:

l-

4(1 -- t')E~

I,

t',

(14)

The voltage distribution over the whole or a portion of the addressable window of the analogue can be scanned and the results printed atttomatically. In the actual analogue computer used. the network can be switched to one of the two available patchboards, that is, the mining geometry can be set up or modified on one board while the voltage distribution is printed out from the other one. 4. DISPLACEMENTS AND STRESSES REMOTE FROM THE SEAM To obtain stresses and displacements remote from the seam requires the definition of kernel, as for example function F,, in the integrand of equation (1). The infinite-depth kernels have been given by Salamon [1], and to satisfy the additional boundary conditions introduced by the presence of stress-flee ground surface, the principle of superposition as suggested by Berry I-7] may be used. Using the notation of Fig. 6

and the total stress: /"

a. = q: ~22"

(12)

In order to obtain correct modelling, resistor r should be such that r tt N E R - 4(1 - v2) L E~' (13)

where the ratio N/L is the scaling factor of the minin~ layout on the analogue: a linear distance, L, in the mine is represented by N nodes on the analogue. The value of resistor R, that is, the unit resistance of the network, is determined at the construction of the analogue. From the voltage distribution on the analogue, func* In the case of an incompressible seam the nodes would be shorted to earth, that is, the series resistor is equal to zero. to correspond to no convergence in the solid. + The expression hq=:'E~ is actually the primitive convergence, that is, a fictious convergence which would be caused by the action of the primitive stress on an originally unstrained seam.

mirrorimageseam 2 t,.,'F,.A-Z.~)~ Z "~-.:t"Z F2,,,.3ZZ L~.I-_-

.-Z

///~\\\\'~= ~///

H ZI=Z-H actuo[

seam

z,

Fig. 6. Principle of superposition to obtain the finite-depth solution (after Salamon [1] t.

Bord and Pillar Workings of Coal Mines the additional boundary conditions to be satisfied can be expressed as follows: :t =

-

H(:=0);

a:=

%:=

r,._-=0.

(15)

ll

similar manner to that of the displacement kernel:

q), :]

F , [ ( x - ~), (y -

r:i-

-

r-')

-

L (-#-' + =r) (,: + :?) " If. as shown in Fig. 6, the stress field of a mirror image seam, positioned at a distance equivalent to 8z~13r" - z~) - 3r 'v] - 3Hz (r: -T- _~7,~_] • (20) twice the mining depth above the actual seam, is superz-2) imposed on the field of the actual seam, the plane of ground surface will be free of shear stresses. To eliminSimilar expressions can be derived for the other ate the normal stresses from the ground surface an ad- stress and displacement components, thus once the conditional stress component must be introduced which vergence distribution at seam level is obtained, all stress cancels the normal stresses introduced by the first two and displacement components can be calculated remote components, without introducing further shear stresses from the seam anywherc in the medium. on the plane of the seam. Thus the final form of function F will have three components. Actually. at seam 5. FIELD MEASUREMENT OF level the boundary conditions will not be satisfied preDISPLACEMENTS cisely: some residual stresses appear in the mined-out areas (region B) from the field of the mirror image seam 5.1. The needJbr and means o f displacement nwasurement and from the third component, which balances the norThe elastic model and analogue approach for the mal stresses at ground level. These spurious comdetermination of displacements and stresses in bord ponents could be eliminated by iterative methods, but and pillar workings in coal mines can be validated only as their magnitude is small in comparison with th.e through the comparison of calculated or predicted beother stress components, they can usually be neglected. haviour with that observed in the field. To this end Using superposition as outlined above, the finitea comprehensive programme of field measurement of depth kernel of vertical displacement will be of the foldisplacements was carried out at two collieries, and lowing form: results were obtained from three mining geometries. F.,[(x - ~), (y - r/), :] As the magnitude of displacements around stable bord and pillar workings is quite small, special measur_ 1 f [2111- v)z~ + (1 -- 2v)r2]zl ing systems were developed for the observation of these 4rt(1 -- v) 2(r 2 + zi) small displacements accurately and reliably over long periods of time. In the case of horizontal or near-hori[2(l - v)z~ + (1 2v)r2]22 + 2(1 - v)z2 zontal seams and horizontal ground surface, the vertical 2(r 2 + z~) s2 (r + z2) 3j2 displacements are the most important ones: displacements in the horizontal direction are very small and 2z~ - r2 difficult to observe. In the system used therefore, + [:2 + (1 - 2,')H] 02 ; :~)5'2 measurements were concentrated on the determination of vertical displacements only. -_ _,r-_(. (16) To obtain the vertical displacement field around a + 3Hzz,_ 02 + :~_).;, _,j full panel, vertical boreholes were drilled from the surface into the floor of the seam which were positioned where in the centres of future pillars and in one case in the r-'=(x-~)2+(y-r/)2 zl = z - H . centre of bord intersection. The vertical displacements at various horizons in the borehole, as a result of minand z 2 = z + H. (17) ing, was measured relative to the collar of the boreThe vertical displacement can be obtained directly by holes. substituting the voltage distribution from the analogue: On the surface a network of stations was laid out, the collars of boreholes forming part of the network. hq 1'(~.q) - rt~ F,,,E(x - ¢ ) , ( y - q ) , - ] d ~ d , 1. The elevation of the stations was then determined relaw = E, 3 3 ~tive either to an arbitrary surface reference station or .4 (18) to a point within a vertical borehole. The displacements of all measuring points were thus obtained relative to The expression for the induced vertical stress as an arbitrary point. The arbitrary reference station was obtained from the convergence distribution is: usually so chosen that its displacement induced by mining was negligibly small. E_ -

f[

t~: i

4n(l

v) 2 f f s : ( 5 , q)F~

5.2. Instrumentation

A

[(x - ~), (y - q), :] d~ d,1.

(19)

where the finite depth stress kernel is obtained in a

For the measurement of displacements in boreholes the well-established method as developed in the Department of Mining Engineering. University of New-

12

K.I. Oravecz MEASURtNG WI# -

~W[RE PULLEY ......- ~ ~ - ~

PERMANENT LOAD

' Itt~/-ISOmm

I __LI lll SEP

DIAMETER STEEL PIPE

E

Fig. 7. Borehole instrument station.

castle-upon-Tyne under Professor E. L. J. Potts was adapted. Measuring wires were clamped in the boreholes at selected horizons: the wires passed through installed anchors subsequently and emerged at the collar of the holes. After all the wires had been installed, an instrument station was affixed to the collar of the borehole, to provide mountings for the borehole extensometer and permanent load to each wire. The wire clamp or borehole anchor served to clamp the measuring wire to the wall of the borehole. It is a spring-loaded device, but the anchorage depends on the permanent load through the wire. The displacement of the selected horizon relative to the collar of the borehole was measured by a borehole extensometer. Usually readings were taken at several values of load. In this way the extension characteristics of the wires could be monitored and any irregular behaviour detected. An instrumented borehole is shown in Figs. 7 and 8. The change of elevation of the stations in the network was determined by means of precise levelling, using self-compensating automatic instruments and double-graduated invar staves. Standard surveying methods were used, as recommended for precise levelling. 5.3. Adjustment of observations When the number of observations is just sufficient to determine the change in the coordinates of a stationas a result of displacements induced by mining, no precise estimate of the accuracy of calculated displacements is possible. Although the inherent accuracies of basic instruments such as precise levels and borehole extensometers are known, or can be determined experimentally, the accuracy of the measured displacements remains unknown. The only way to determine the accuracy of the observations is to arrange the instrumentation and layout of stations so that 'redundant' measurements can be made, that is, more than the minimum number of observations required to define displacements uniquely. In this way. by using the

Fig. 8. Fully instrumented borcholc with exlensometer mounted on

method of 'least squares', appropriate adjustments can be made to the field results. In the process of adjustment the accuracy of any set of observations can be determined readily. In surveying practice, the procedures for leastsquares adjustments are well established. The introduction of these procedures to the measurement of mininginduced displacements is relatively recent and was necessitated by the stringent requirements brought about by the small magnitude of displacements which occur in bord and pillar operations. To illustrate the principle of adjustment consider for example a line of levelling stations 1, 3, 5 and 7 (Fig. 9). Suppose that the elevations of stations 3-7 along the straight line must be determined relative to the elevation of station 1. Normally, this would require twoway levelling between adjacent stations 1-3, 3-5. and 5-7. Even if precise levelling procedures were used, the accuracy of computed elevation difference, say between stations 7 and 1, could not be determined• 1

e 2

2

3

e4

..e6

z.

S

._%

e8

6

7

Fig. 9. Triangular network of JcvelJin~ stations.

Bord and Pillar Workings of Coal Mines TABLE

1. VARIANCE AND STANDARD DEVIATION OF OBSERVATIONS

13

INITIAL

used for modelling a particular layout, but it permits the finding of the final value in fewer 'analogue runs'. Standard The modulus ratio, the mining dimensions and the Type of No. Variance deviation geometric scaling used for the layout, determine the observation of sets (mm 2) (mmJ value of r, that is, the resistance representing the comLevelling 13 0.0149 0.12 pressibility of the seam namely, equation (13). Boreholes i3 0.0067 0.08 After the potentiometers have been set to the selected (extensometer) value, theoretically at first, the undisturbed state of Relative weight of observations: levelling to boreholes --- 0.45. strata, that is, the undisturbed state of network, has to be modelled by inserting the set values of resistors at every addressable node. The area constituting the Subsidiary stations were thus laid out with the aim boundary region should also be modelled by insertion of determining the elevation differences of all adjacent of the appropriate values of resistors. As boundary stations using precise levelling procedures, thus obtaining elevation differences et, e2. . . . etc. If the observa- regions are represented by blocks equivalent to nodes tions were without random errors, the algebraic sum from 4 x 5 to 20 x 20, the single plugs representing the of the elevation differences between the apexes of a region would have resistors one-twentieth to one-fourtriangle would obviously vanish after it had been hundredth of the resistance representing the seam. ensured that no systematic errors had been introduced After a fixed voltage has been applied to the network, by the observational procedures. In practice, because the voltage on the undisturbed network can be of random errors of observation, this condition is measured. This voltage can also be computed readily usually not satisfied. The field observations, therefore, on the basis of Fig. 4. An 'analogue run' on the undishave to be adjusted to meet the geometric condition. turbed network is thus used only to check the basic As the number of observations exceeds the number of design parameters of the network. imposed geometrical conditions, a unique set of adjustA given mining layout is then represented by the ments can be derived by imposing a further restriction removal of appropriate resistors. If a certain percentage that the weighted sum of the squares of the adjustments extraction is allowed in some part of the boundary should be a minimum. The adjustments thus derived regions, the value of an equivalent resistor has to be are the well known 'least-squares' adjustments, which calculated on the basis of the pillar layout. It is, howgive the most probable corrections if the random errors ever, important to limit extraction in the boundary of the observations are distributed normally. areas such that, firstly, the overall extraction in the Once the adjustments are calculated, the precision whole 120 x 120 node area of the analogue is limited of a set of observations can be derived readily; thus to a maximum of 25%, and secondly, the voltage at a measure of reliability can be assigned to the particu- the outermost boundary nodes should reach the value lar set of observations. More specifically, the precision corresponding to the undisturbed state of the network. of any combination of the adjusted results can be com- These conditions have to be satisfied to ensure that puted (for example, the precision of elevation difference the influence of errors brought about by the finite size between selected points). of the network is limited. After the layout has been Examples of the degree of precision achieved in one modelled, the analogue computer gives an automatic of the field measurement projects (described in Section listing of the voltages at the 60 x 60 addressable nodes. 6.2.) are presented in Tables 1 and 2. In Table 1 the This voltage distribution can then be converted to the precision of pre-mining observations are given separ- convergence distribution on the basis of equation (10). ately for levelling and borehole observations (extensThe fixed value of modulus ratio still allows an enometer), taken over a period of 3 months. In Table tirely arbitrary value to be assigned to the modulus 2 the precision of combined levelling and borehole for the seam E,. This implies that at a fixed ratio of observations for unit weight of observation is shown moduli, the convergence derived from the analogue is covering a period of one and three-quarter years. inversely proportional to the modulus of the seam. As the field experiments were designed with the aim of obtaining comprehensive patterns of displacements, 6. VALIDATION OF ELASTIC MODEL AND ANALOGUE SOLUTION 6.1. Derivation of elastic solution corresponding to given mining geometries To represent the elastic response of the strata when a compressible seam is extracted, the electric resistance analogue must be set up to model the correct ratio of elastic moduli between the strata and the coal seam. Values determined by small-scale laboratory tests can be used as a first approximation; the initial choice of modulus ratio, however, does not affect the final value

TABLE 2. VARIANCE AND STANDARD DEVIATION FROM ADJUSTED OBSERVATIONS For unit weight of observation

Variance (mm 2)

Standard deviation (mm)

Lower limit Upper limit Mean value

0.0112 0.0704 0.0293

0.11 0.26 0.17

Number of observations = 86. Weight of levelling = 1. Weight of boreholes = I/0.45

14

K.I. Oravecz

the method based on the face element principle as shown in Section 3 was utilised to obtain theoreticallydeduced displacements at positions corresponding to the surface stations and boreholes anchors. In particular. the integral in equation (1) using the kernel for finite-depth as expressed in equation (16), provides the vertical displacements in the medium away from the plane of the seam. The convergence distribution as obtained from the voltage distribution on the analogue is substituted into the equation and the limits of integration extend over the addressable area of the analogue. see equation (18). As will be observed, the modulus for the seam is outside the integral sign; the displacements as computed will thus also be inversely proportional to the selected value of E~. Considering the form of kernel Fw in equation (16). it will be realised that as the plane of the seam is approached, zl--,0, singularity is introduced in the solution of the displacements. A good approximation of the displacement at the mid-seam position is obtained by neglecting the term in the kernel containing the vertical co-ordinate z . To obtain the displacement in the immediate roof and floor of the sefim, the convergence at the centre of the pillars is first determined. As each nodal point gives the average voltage corresponding to the average convergence over the area represented by it. the convergence at the centre of the pillar can be found only by extrapolation. One half of the convergence is then added to the mid-seam displacement to obtain the roof displacement, and one half of the convergence subtracted to give the displacement of the floor. The procedure followed is thus as follows. After the voltage distribution, corresponding to the initially selected value of modulus ratio, is obtained from the analogue, an arbitrary value is assigned to the modulus of the coal seam. Then. using the 'integrator', the displacements of the points, corresponding to those where displacements were measured, are computed. As the measured displacements are normally derived relative to a reference point, the theoretically-deduced displacements are also re-computed relative to the same reference point. In comparing the measured and theoreticallydeduced displacements, the difference between them at each point (Awi) can be expressed in a general form A w i = w,,i - ewai, (21 ) where w,,i, w~ are the measured, and theoreticallydeduced displacements of the i-th station, and c is a factor of proportionality. It is, in fact, the ratio of the arbitrarily chosen modulus for the seam to the actual in-situ modulus c=

E*(arbitrary) Es(in-situ)

(22)

To determine the optimum value of E~, the customary 'least-squares' approach is used. Let the sum of the squares of deviation be ['~ = E ( A w I ) 2.

(23)

This function will have a minimum value, where ?c

0;

(24)

whence after substitution c-

~- WmiWai

~;w'i

(25)

After the value of c is determined, the variance of the fit between the measured and theoretical displacements will be given by f~ sz = -, (26) n

where n is the number of points used for the comparison excluding the reference point. After the values of variance and S.D. (s) have been computed, the initially selected modulus ratio can be considered to be either too high, adequate, or too low for satisfactory modelling. If modelling is not satisfactory, a new analogue run can be prepared by using an improved value for the ratio of elastic moduli. In this task the graphical presentation of displacements w,,i and wai will be very helpful. Normally, a satisfactory value for the ratio of moduli can be found after 2--4 analogue runs. At each run the new value of s-' is monitored to determine whether the required improvement in the variance of the fit between measured and deduced displacements had been achieved. The above described process is, in fact, finding numerically the minimum of the two-variable function f~ = ~(r,c).

6.2. Modelling the No. 5 seam workings, Colliery A The geological section across the experimental area and the position of measuring anchors are shown in Fig. 10, while the plan of workings and of the surface boreholes are shown in Fig. I1. The details of surface levelling network, comprising of the borehole collars and levelling stations are shown in Fig. 12. For the comparison, the mean values of displacements measured after the completion of the panel were used. The starting value for the resistor, r, representing the seam was chosen as 1000 f2. Altogether, the vertical displacements of 51 points were calculated from the analogue runs: 9 surface points and 42 borehole anchor positions. Borehole A was not included in the computation as measured results there were affected severely by non-mining-induced displacements. The first analogue run with r = 1000 ~, gave an unsatisfactory result, with a variance of fit of 1.30 mm 2 at a modulus value of E~ = 1,08 GPa. The second analogue run, using a resistance of 400 f~ improved the variance of fit to 0.55 mm z, at E~ = 1.54 GPa, and it was accepted as satisfactory. The computed and measured surface subsidence and pillar convergence are shown in Fig. 13 for comparison. The results reveal that particularly good representation of the surface subsidence was achieved, and more vari-

15

Bord and Pillar Workings of Coal Mines 0

18

20

~ I

BH. F

BH. H

BH. E

BH, O

BH. C

i I

,, _

,, _

-

-

_

-

-

.....

,, _--, .

__

- .

I. I

....

-'1--

--,:

-,_

..

I ~

_-

_

_--

.

.

.

-

,.. .....

-

i

-

.

----

:....

-,,-

ii

-

BH. A

I

I

_

~,

S.A,t

-

- -',---_,: ....-..

BH. B

-

-

,,

"

I Z

_- - ----

-

.,......-

.

~

...-

- -- I1-- -

_--'--_---,

-_1,------2---',,-_.

.....

--

........-

............

~.

'....,,-:...~..:.-.,.. ..:...,•..:.........,...,.....,...... .... II '.":. .:""

".

"

3

•" -

".".S

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ation in the representation of pillar convergence. The overall representation of displacement profiles in the vertical boreholes, Figs. 1 4 ( a ) a n d ( b ) , is considered highly satisfactory•

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From the average seam compression in boreholes B-H the average pillar stress or pillar load was computed, using the value of modulus for the seam that gave the best fit or lowest variance (Fig. 15). The data

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relevant to the modelling of the No. 5 seam workings at Colliery A are summarized in Table 3. 6.3. Modelling the No. 2 seam workings, Colliery B According to the geological section (Fig. 16) the seam under consideration is a composite seam, formed by the merging of the No. I and 2 seams. As a result of increased seam thickness, the final mining height was reached in 2 steps. In the development stage the pillars were formed only to a limited height (2.5-3.0 m) in the lower section of the seam under a coal roof (primary extraction). The final pillar height was reached in a secondary operation 'topcoaling', that is, the mining height was increased by extracting coal from the roof of the bord areas (secondary extraction). In this way, two mining geometries became available for modelling on the analogue. It is apparent that the plan of the mining geometry (Fig. 17) remains the same for both stages of operation, only the mining height changes appreciably, and to a small extent the cover load. The detailed layout of the boreholes and levelling 2

BH.A 20

BH.B 40

BH£ 60

stations relative to the planned positions of the pillars is shown in Fig. 18. The modelling on the analogue was carried out with a few values of r, until sufficiently accurate fitness was achieved between measured and analogue results. The best fit for the first stage of extraction was achieved at a resistance value of r = 450 f2 and seam modulus of Es = 2.97 GPa, and the final stage was modelled by using the resistance value of r = 679 fL at a seam modulus of 3.92 GPa. The theoretical and measured subsidences of the two mining stages are shown in Fig. 19 and the theoretical and measured vertical displacements in the boreholes are shown in Figs. 20 (a) and (b). The data relevant to the two mining stages are summarized in Table 4. The corresponding variances of fit amounted to 0.26 mm z and 0.23 mm-', which were considered highly satisfactory. The successful representation of the displacements is even more apparent from the study of Figs. 19 and 20. From the analogue results the distribution of average pillar stress or pillar load across the half panel and the stress distribution in the solid ribside, are shown in Fig. 21 for the final stage of mining. BH.D

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Fig. 13. Comparison of theoretical and measured subsidence and pillar convergence, Colliery A.

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Fig. 14(a). Comparison of theoretical and measured vertical displacements, boreholes B, C and D, Colliery Ao Ib) Comparison of theoretical and measured vertical displacements, boreholes E, F. G and H, Colliery A.

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Fig. 15. Derived load distribution across the panel, Colliery A.

6.4. Evaluation of the results

Before the elastic model is accepted as being generally suitable for the representation of stress and displacement distributions around bord and pillar workings, some of the more important salient points, evident when the derived and measured displacements are studied together and relevance to the analogue method of solution must be emphasized. (1) The analogue solution is based on the assumption that the medium is homogeneous; that is, in practice, only the average material properties can be represented. Variations from the average conditions would thus appear as discrepancies between measured and predicted displacements. TABLE 3. SUMMARYOF DATA--No.

Mining geometry Average depth to mid-seam Seam height Bord centre distance Pillar width Bord width Extraction No. of pillars in panel Panel width Analogue parameters Resistance representing seam Unit resistance of analogue No. of nodes per pillar No. of nodes per bord Extraction modelled Elastic modulus of seam Elastic modulus of strata Modulus ratio Poisson's ratio Variance of fit

(2) As with the properties of the material, the geometry of workings has to be also represented by average dimensions, as the available size of the analogue does not allow exact reproduction of irregularities in the plane of workings. In addition, variations in other dimensions such as depth of cover, height of workings, and of the seam and changes in floor elevations characterise normal mining operations. Although the effects of these factors are small within a relatively small test site, they certainly affect the final quality of fit between theory and the real situation. (3) When only a limited portion of a seam is extracted the proposed model does not represent fully the actual situation. In the model, it is assumed that 5 SEAMWORKINGS,COLLIERYA H h C = 2L w B = 21 e n S

40.3m 1.5m 10.7m 5.2m 5.5m 76.4% 16 176.2

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Derived load distribution

Gradient of primitive stress Cover load Modified cover load Average load on pillar F Average load on pillar E Average load on pillar D Average load on pillar C Average load on pillar B Load reduction at centre of panel

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SUMMARY OF DATA--No.

Mining geometry Average depth to mid-seam Height of workings Bord centre distance Pillar width Bord width Areal extraction Number of pillars in panel Panel width An alggue parameters Resistance representing seam Unit resistance of analogue Number of nodes per pillar Number of nodes per bord Extraction modelled Elastic modulus of seam Elastic modulus of strata Modulus ratio Poisson's ratio Variance of fit Derived load distribution Gradient of primitive stress Cover load Modified cover load Average load on pillar A Average load on pillar B & H Average load on pillar C Average load on pillar D Load reduction at centre of panel

2 SEAM W O R K I N G S , C O L L I E R Y

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Secondary

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r,bslae

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B C O PIN POSITIONS ON ANALOGUE ( • UNMINED, :o MINED OUT)

I o oio

•

O0

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•

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0 0 0 0

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Fig. 21. Derived load distribution across the panel. Colliery B.

the seam, represented by modulus E,, is extracted completely and that the rest of the strata are composed of an elastic medium of modulus E. With a limited extraction of the full height of the seam the remaining portion of the seam forms the roof and floor of the workings, clearly of the same modulus as that of the pillars. The medium with the different modulus is not immediately in contact with the pillars. The modulus value of the strata that is obtained by the method described in Section 5.1 can thus reflect only the averaoe combined property of the main strata and part of the seam. However, it appears that the model is quite capable of describing even this particular situation, but the modulus values derived have to be considered in this light. (4) It is established, both experimentally and theoretically, that the highest gradient of displacement occurs within, and in the near vicinity of the seam. Unfortunately, in the immediate vicinity of the seam the integration process becomes singular (cf. equation (16), and Section 6.1). The theoretical displacement of points in the immediate roof and floor can thus be obtained only by linear interpolation. This will also have an effect on the 'goodness of fit' achieved. The examination of the 3 cases of analogue modelling as presented in the light of the foregoing discussion, leads to the favourable conclusion that the pro-posed model is suitable for the description of displacement distributions around bord and pillar geometries in coal mines. By extension, the model is also accepted as being suitable for the realistic representation of stress and load distributions around such workings. 7. SUMMARY The theory of elasticity has been used to develop a method for predicting pillar loads in bord and pillar workings of coal mines in which the stratification is horizontal.

In using the elastic theory the following basic assumptions were made: (1) the surrounding rocks behave as homogeneous is•tropic elastic media; (2) the induced vertical stress on the coal seam is related linearly to the seam compression; and (3) the convergence distribution as obtained by assuming the seam to be at an infinite depth, is an acceptable approximation of the true convergence distribution. For practical mining geometries, a solution could be obtained only by means of an analogue technique. The theory of the electrical resistance analogue, on the basis of which rock mechanics problems in deep-level hardrock mines could be solved, had first to be extended to permit the simulation of the compressibility of the coal seam. In practice, this was achieved by connecting a resistance, the value of which was determined by the scaling factor of modelling and the modulus ratio of the strata and the seam, in series with every node that represented unmined portions of the seam. To prove the validity of the analogue method of s01ution, field data, representative of whole mining panels, were required. Strata displacements caused by mining are the quantities that can be measured most conveniently and used for comparison with theoretically predicted values. In stable bord and pillar workings, displacements are normally quite small, being of the order of a few millimetres. Special techniques and measuring systems were thus developed in order to obtain reliable and accurate results on the basis of which meaningful comparison could be made with theory. Neglecting the obviously non-mining-induced displacements, the accuracies of measured displacements were within the requirements. Field observations were carried out at two collieries in the Transvaal, and results were obtained for three mining geometries. The displacement patterns corresponding to completed mining operations were com-

Bord and Pillar Workings of Coal Mines pared with those obtained by analogue modelling of the same mining situations. In the process of comparison, an empirical in-situ value for the modulus ratio of the strata and the coal seam was obtained. Also, a definite measure of the 'goodness of fit' was calculated for each case examined. The comparison led to the favourable conclusion that the analogue technique, when used in conjunction with the appropriate integration procedure, can be used to describe the stress and displacement distributions around practical bord and pillar mining geometries. Acknowledgements--The research work described in this paper was carried out under the sponsorship of the Coal Mining Research Controlling Council, within the Research Organisation of the Chamber of Mines of South Africa, under the supervision and directorship of Dr. M. D. G. Salamon. The paper is an extract from a thesis submitted by the author to the University of the Witwaters and, Johannesburg. The writer is deeply grateful to Dr. Salamon for the guidance received during the numerous research projects carried out over the years, and for the valuable advice given in the preparation of this paper. The writer is also indebted to Prof. R. P. Plewman for his active interest, and for placing various computing facilities at the writer's disposal. The Councirs and the Chamber's permission to publish this paper is gratefully acknowledged.

23

Receired 26 July 1976.

REFERENCES 1. Salamon M. D. G. Elastic analysis of displacements and stresses induced by the mining of seam or reef deposits--I. Fundamental principles and basic solutions as derived from idealised models. Jl. S. Afr. Inst, Min, Metall. 64. 128-149 (1964). 2. Salamon M. D. G. Elastic analysis of displacements and stresses induced by the mining of seam or reef deposits--II. Practical methods of determining displacement, strain and stress components from a given mining geometry. Jl. S. Aft. Inst. Min. Metall. 64. 197-218 (1964). 3. Ryder J. A. & Officer N. C. An elastic analysis of strata movement observed in the vicinity of inclined excavations. JL S. Aft. Inst. Min. Metall. 64. 219-244 (1964). 4. Ortlepp W. D. & NicoUs A. The elastic analysis of observed strata movement by means of an electrical analogue. JI. S. Aft. Inst. Min. Metall. 65. 214-235 (1964). 5. Salamon M. D. G., Ryder J. A. & Ortlepp W. D. An analogue solution for determining the elastic response of strata surrounding tabular mining excavations. JI. S. Aft. Inst. Min, Metall. 65. 115-137 (1964). 6. Salamon M. D. G. & Oravecz K. I. The electrical resistance analogue as an aid to the design of pillar workings. Proc. 2nd Congr. Int. Soc. Rock Mech. Beograd, 1970. Paper 4-18. 7. Berry D, S. An elastic treatment of ground movements due to mining--I. Isotropic ground. J. Mech. Phys. Solids 8. 280-292 (1960).