PART 2

UNDERGROUND DESIGN AND EXCAVATION

Chapter 6

THE DESIGN O F UNDERGROUND EXCAVATIONS

by N. G. W. Cook

When an excavation is made underground the original rock stresses are removed from the surfaces of the excavation. These surfaces converge to partially close the excavation and the superincumbent rock mass moves down towards the excavation. If the convergence of the surfaces of the excavation were controlled, energy generated by their motion against the controlling forces could be extracted. Since the total forces in the rock mass are not significantly affected by the excavation, stress concentrations arise around the excavation, which compensate exactly for those stresses no longer transmitted through the excavation. These phe- nomena give rise to two general principles which provide the basis for analyzing the effects of making any underground excavation. The first of these is that the total force resulting from the stresses across any plane in the rock, including any plane through the excavation, must be the same before and after the excavation is made. The second of these is that energy must necessarily be released as a result of making an underground excavation.

It is instructive to consider the implications of these two principles for two extreme cases of material behavior between which the actual be- havior of rock must lie, namely, the behavior of a fluid and the behavior of an elastic solid. For the purpose of analysis in this paper, an underground excavation is defined as an excavation in which the smaller horizontal dimension of any open void a t a particular horizon is less than the depth of that horizon below surface.

I n the case of a fluid, any unsupported excavation would fill immedi- ately, as a fluid cannot sustain shear stresses. For the same reason, stress concentrations could not arise and the forces across any plane would necessarily be the same before and after excavation. If the filling of the excavation were controlled by forces applied to the surfaces of the exca- vation, an amount of energy equal to the product of the fluid pressure

N. G. W. Cook is Director, Mining Research Laboratory, Transvaal and Orange Free State Chamber of Mines, Johannesburg, South Africa.

167

a t the depth of the excavation and the volume of the excavation could be extracted. In the absence of control, this is the amount of energy released, W1,. Since no energy is stored, this is equal to the change in gravitational energy due to the downward movement of the fluid above the excavation, WG, SO that

W R Z W G = ~ ~ H V 111 where p = density of the fluid, H = depth of the excavation below surface, and V = volume of the excavation.

While the fluid case probably approximates the situation of an exca- vation after an infinitely long time, the immediate reaction of a rock mass to making an excavation is probably more nearly elastic. When an excavation is made in elastic rock, a change in gravitational potential, WG, and a change in the potential of the horizontal stresses, WH, occurs in the rock mass. The sum of these two changes must equal the sum of the energy stored in stress concentrations, Ws, the released energ?, WR, and the strain energy originally stored in the excavated rock, w,, so that

W G + W H = W ~ + W ~ + ~ W , r21 and

WRh+ (Wc, + WH - 2w,) 131 since WRhWs. The released energy is equal to the energy which could be extracted by controlling the convergence of the surfaces of the excavation from their initial positions. The amount of energy released is equal to the integral over the surface of the excavation of the product of the con- vergence and the average stress, during convergence, of elemental areas of the excavation surface. This provides a convenient means of illustrat- ing the differences between the energies released by making excavations in the following three circumstances: 1) an unsupported excavation in a fluid, 2) an unsupported excavation in an elastic solid, and 3) a supported excavation in an elastic solid which fails partially. Fig. 1 shows the rela- tionships between the mean surface stress during convergence and the volumetric convergence of an excavation, for these three cases. In each case the released energy is equal to the area beneath the volume-stress curve, which shows that for the same amount of volumetric convergence most energy is released by an excavation in a fluid and least by an open excavation in an elastic solid. For all three cases the total change in potential energy is the same; i t is equal to the energy released in the fluid case, so that the stored strain energy in the other three cases is equal to the difference between this and the releas6d energy.

When any excavation is made underground a change in potential energy occurs, strain energy is stored in the surrounding rock, and excess energy is released. The strain energy derives from the stress concentrations

Convergence volume

Fig. I-The energies released, WR, and Fig. %-The stored strain-energy, VS, stored, WS, as a result o f the same volu- ma metric convergence of ezcavations i n a and the energy ---- aR a fluid, an elastic solid without' support function o f the size, R, of ezcavations stresses, and a partially failed elastic with cross-sections o f a circle ( 6 ) and ( 7 ) ; solid. a slit ( 9 ) and (10 ) ; and a series o f slits

(14) and (16).

which arise to maintain force equilibrium and, by St. Venant's principle, significant stress concentrations are confined to the rock in the vicinity of the excavation. In general, the original stress differences existing in the rock before excavation are small in relation to the strength of the rock,~so that i t can be anticipated that most of the rock mass will respond elastically to the small stress changes brought about by making an excavation. Elastic displacements and strains in the rock mass are seldom of sufficient magnitude to present difficulties, especially if they can be predicted in advance and allowed for in the design of structures which

they may affect. The elastic displacements do, however, provide a valuable facility for calculating the energy release .and stored strain energy which result from an excavation. These two quantities provide the most objective measure of the hazards likely to arise from making an underground excavation.

Wherever the profile of an excavation changes sharply very high stress concentrations are likely. These are often sufficient to cause failure of the rock in the immediate vicinity of an excavation. IJocal effects of this nature are usually unimportant so that maximum stress, alone, does not adequately define t.he difficulties likely to be encountered in an excava- tion. However, strain energy which measures both the magnitude and extent of the stress concentrat,ions does provide such a measure. Further- more, damage to any structure, be it. a rock face, a concrete lining, or steel support, can only occur if energy is derived from some source to produce this damage. When an excavation is made underground, this source is the released energy which thus provides an objective measure of the potential damage.

In many circumstances, such as deep-level mining, failure of the rock in the immediate vicinity of the excavation cannot be avoided. I n this case difficulties arise when the failure proceeds in an uncontrolled fashion as i t does in the case of a rockburst.

This paper discusses methods of evaluating strain energy accumulation and excess energy release which result from making an underground excavation. These methods are then used to show how excavations can be planned to minimize these energies. The properties of failed rock and the interaction between failed rock and the stresses applied to i t by the rock mass and support media are examined, particularly with a view to controlling rock failure and avoiding violent failures.

EVALUATION OF STRAIN ENERGY AND RELEASED ENERGY

The shapes of underground excavations can be divided into two classes for the purposes of analysis. First, those in which the dimensions of a cross-section through the excavation are similar, such as horizontal tunnels and vertical shafts. Second, those in which one of the dimen- sions of a cross-section through the excavation is negligible compared with the other, such as narrow stopes associated with the extraction of thin tabular deposits. I n addition, excavations in both classes may have a dimension in the direction perpendicular to the cross-section which is so much greater than the other dimensions that the excavation can be treated as two-dimensional, or i t may be so similar to the cross-sectional dimensions that the problem must be considered in three dimensions. I n the former case i t is often possible to obtain analytical solutions to the problems of stress concentration, strain energy accumulation and energy

release resulting from excavation, whereas in the latter case analogue solutions of one form or another have to be used. Examples of both these approaches are considered in this section.

The energy released by excavation is equal to the work which could be extracted from the rock mass by controlling the convergence of the surfaces of the excavation from their original positions. If 5, g2, 5 are the average stress components on the surfaces of the excavation during controlled convergence, u,, u,, u, are the components of convergence of these surfaces, and d a,, d a,, d a, are the components of the surface area of the excavation, then the extracted work, or released energy is

/ WR=Saol UI dal +Jt1Z2 u2 da2+ JaZ3 U, da3. [4]

To evaluate these integrals it is necessary to know the stress and con- vergence components. In general, the original vertical component of rock stress is due to the weight of the superincumbent strata, and this can be regarded as a principal stress except in the vicinity of mountainous topog- raphy. As a result of creep, the horizontal components of the original rock stresses would tend to become equal to the vertical component. The hori- zontal components are disturbed by tectonic forces in the earth's crust and by stress concentrations due to mountainous topography. Measured values of the original horizontal components vary from a third to three times the vertical component (Hasty1 Jaeger and Cook12 and Leeman ,). For the purposes of specific calculations in this section it will be assumed that the vertical and horizontal stress components are both equal to the weight of the superincumbent strata and constant over the surface of the excavation. This facilitates calculation and, even where i t is not exact, the values of energy yielded by this approximation are often adequate.

As two-dimensional analytical examples the energy release and strain energy per unit length of a circle, a slit, and a series of co-planar slits are chosen because of their close similarity to many practical problems.

The convergence of the surfaces of a long circular excavation, far below surface in relation to its radius, R, can be found from the solution to the displacements in plane strain of a circular hole in an elastic material subject to equal bi-axial stresses, U. Muskhelishvili shows the radial displacement to be

where r=modulus of rigidity of the rock and v=Poissonls ratio. The surfaces of a circular excavation will not make contact with,one

another, so that in the absence of support the average stress during con-

vergence is half the original rock stress. In this case the stored strain energy is equal to the energy released.

and

aw, 2 ~ ( 1 - ~ ) -- aR - p

u2 R.

Many longwall stopes approximate in cross-section a thin slit with . a large span. The convergence of the surfaces of such an excavation can be found from Muskhelishvili's solution for the displacements in plane strain of the surfaces of such a slit subject to an internal stress (Cook " ) .

The relative displacements normal to the plane of the slit are

where R = half span of the slit and x =distance from the center of the slit. Until the surfaces of such a slit make contact with one another, the

average stress during convergence, in the absence of support, is ~ / 2 . Therefore, the energy release and the stored strain energy are

and

aw, =(I-") --P

aR - p a2 R.

As soon as the surfaces of the slit make contact with one another, or in the presence of significant support loads, the average stress .during con- vergence exceeds half the original stress and the amount of energy released exceeds the amount stored as strain energy. In the extreme case, when the span is much greater than that a t which the surfaces make con- tact, the strain energy stored is negligible compared with the released energy. This becomes' equalto the total change in gravitational potential and is given by

WE =2u SR [ I l l

where S=thickness of the slit. The rate of energy release rapidly ap- proaches the value given by Eq. 12 after the surfaces of the slit make contact with one another (Cook, et al.G).

Several longwalls are often excavated in the same plane and it is

important to determine the effects of interaction between them. Salamon has studied the problem of an infinite number of co-planar slits on centers of 2L with half spans of R. He shows that the volumetric convergence of each slit is

Until the surfaces of these slits make contact with one another, the average stress during convergence is half u .so that the energy release and stored strain energy are equal.

WR=Ws= -4(1-v) log, ( cos - ;Jt) u2 L' "'P

The rates of energy release, Eqs. 7, 10, 12, and 15, and the total stored strain energies, Eqs. 6, 9, and 14, are as illustrated in Fig. 2 for L=1000. If Eqs. 9, 14 and 10, 15 are compared, it is seen that the effects of inter- action between excavations become noticeable a t 50 percent extrption and significant a t 75 percent extraction.

Even when the shape of an excavation cannot be approximated rea- sonably well in two dimensions, the two-dimensional calculation provides a useful and easy way of determining the maximum rates of energy release and strain energy storage, since truncation of the long axis can only reduce these quantities. Nevertheless, in practical problems it is some- times necessary to take cognizance of the three-dimensional nature of the excavation, especially where rates of energy release are of greater importance than the total energy release. In these cases, the geometrical complexity is usually such as to climinate the possibility of obtaining an analytical solution, and recoursc to analogue or numerical methods is necessary.

Excavations associated with the extraction of narrow tabular deposits often have a dimension normal to the planc of the deposit which is negli- gible compared with the dimensions in that plane. This allows certain simplifications in terms of the theory of elasticity, and Salamon has shown that the displacements and stress concentrations resulting from such excavations can be expressed in terms of a function which is a solution to Laplace's equation. Solutions to this function can be found by applying appropriate boundary conditions to analogues provided by electrical conduction. One form of the analogue (Salamon, Ortlepp, and Ryder %) makes use of the conductivity of a cubical volume of electrolyte contained between two electrodes. One electrode corresponds to the plane of the excavation, and the shape of the excavation, to a suitable scale,

is removed from this electrode. It is convenient that current densities on this electrode are proportional to rock stresses normal to the plane of the excavation and voltages within the removed shape are proportional to convergence within the excavation. Only the voltages can, in practice, be measured directly on this analogue and displacements in other regions, and stresses, must be found by numerical integration of these voltages. The other form of the analogue (Cook and Schiimann lo) consists of an orthogonal network of resistors which provide a finite-difference approxi- mation to the conductivity of t h e electrolyte, Fig. 3. The resistor nodes on one side of th'e block of resistors can be short-circuited or open-cir- cuited by means of plugs to simulate unit areas comprising the solid rock or the excavation, respectively. With this analogue it is possible to measure voltages (convergence) and currents (normal stresses) in the plane of the excavation directly. The product of the mean stress on an area of rock before excavation and the convergence after excavation

Fig. 3-Electrical resistance analogue for planning tabular mine excavations. Each plug represents an area of unmined rock and each hole an area of the excavation. Plug current and hole voltage are proportional to stress and convergence in the plane o f the excavation, respectively. .

provides a measure of the energy released by the extraction or complete failure of that area of rock.

Where the dimensions of an excavation are approximately equal but so irregular as to make an analytical solution impossibly difficult, two- or three-dimensional photoelasticity can be used to study stress concen- trations around the excavation (Hoek ll). These methods are too well known to need description here (Frocht 1 2 ) . The results obtained from them do not lead directly to a measure of the energy release or strain energy accumulation, but can be of considerable value in studying regions of rock failure (Hoek l3! 14). When the dimensions of an excavation are similar, contact between the surfaces does not occur as a result of con- vergence, so that the energy release and strain energy are both equal to half the product of the original rock stress and the volumetric convergence of the excavation. This suggests that i t would be easy to find these quantities by modeling thc excavation in an elastic material such as metal or plastic and measuring the volumetric convergence when the model is subjected to externally or internally applied stresses. The volume could easily be determined by measuring the displacement of a fluid filling the model excavation.

DESIGN OF EXCAVATIONS TO MINIMIZE ENERGY CHANGES

From the preceding sections it is apparent that the energy release and the stored strain energy depend upon the volumetric convergence of the excavation. This is different from the volume of the excavation to the extent that the rigidity of the rock prevents complete closure. If the surfaces of the excavation are stress free, the released energy and stored strain energy are equal. If they make contact with one another, or with support loads, then the surfaces of the excal-ation are not free of stress and the released energy is grcater than the stored strain energy.

The magnitude of the problems arising from making an excavation underground can be reduced by limiting the volumetric closure of an excavation. This may be done by minimizing the volume of rock ex- tracted, by the introduction of artificial support, by filling the excavation with sand or rock, or by lcaving pillars within the excavation.

If thc surfaces of an excavation are supported by evenly distributed loads generating a mean stress, u,,,, then the volumetric convergence of these surfaces will be less than that of an unsupported excavation, Vf. However, the mean stress during convergence will increase to (o+u,,)/2, as is illustrated in Fig. 4. This shows that differcnces between the energy released by supforted and unsupported excavations are

Fig. 4-The effects of a' constant .support stress, om, on the released, WR, and stored, Ws, energ.ks.

released by supported and unsupported excavations are

If the support stress is applied during convergence, a fraction of the released energy will be used in compression of the support. The rate of energy release by a supported excavation is

aw, T ( I - V )

aR - (a2-am2) R. P

These equations show that om has to be a large fraction of u before support has an appreciable effect on the energy release. I n practice, am

is less than 100 psi, so that support is only effective in reducing the energy release and strain energy a t depths of less than 1000 ft below surface.

The released energy and stored strain energy can be reduced by filling an excavation with sand or rock only if the fill reduces the volumetric convergence of the excavation. To do this the volume of fill, after com- paction, must exceed the volume of the void which would be left by an unfilled and unsupported excavation. Provided that this condition is satisfied, the effects of filling can be studied in termsof Fig. 5.

I V Vf

Convergence volume

10 I

I v ConvDrgence volume

VJ

l b )

0 50 100 Parc8ntog8 artroction.

too SO . 0 Percantag8 fill.

Fig. 5-The ef fects o f compressible fill Fig. 6-(a) T h e percentage reduction o n the released, Wn, and stored, Ws, en- in energy release achieved b y partial ex- ergies. T h e fill i n 5 ( a ) is less compressible traction or filling using strip pillars. ( b ) than that in 5 ( b ) , due to the relatively T h e dependence o f parameter k o n rela- large total volume of the latter excava- tive stress and geometry for Poisson's tion. ratio o f 0 2 .

In this figure the effect of different ratios between the volumetric con- vergence and the total volume of the excavation is illustrated. For excava- tions with a relatively small total volume, little convergence is required to compress the fill (Fig. 5a) and filling is very effective. Excavations with relatively large total volumes require more convergence to compress a larger volume of fill and filling is less effective (Fig. 5b). The difference in the amount of energy released by a filled excavation and an excavation with stress-free surfaces is

where Vf=volumetric convergence of the stress-free excavation and V = volumetric convergence of the filled excavation.

Some of the energy release, WA, is absorbed in compaction of the fill. The difference in the stored strain energies is

Eqs. 19 and. 20 show that filling reduces t.he amount of stored strain energy more than it reduces the amount of energy released.

Filling is most effective where the volumetric convergence is a large fraction of the total volume of an excavation. It is, therefore, of most significance in connection with extraction of extensive, thin tabular de- posits, particularly a t depth. I n this type 1 of excavation, significant reductions in energy release and stored energy can be achieved by partial extraction, or partial filling, where a regular system of strip pillars is left, or installed, (Cook and Salamon 15) as support.

The difference between the maximum amount of energy released over a span of 2L by a slit-like excavation of thickness S, and that which would be released if partial extraction were used is, according to Eqs. 11 and 14

A W , ~ , = ~ U L S - , - 1 - ( ;;)rL] [ TP log, cos-

Eq. 14 can also be used to study the effects of partial filling where insufficient material is available for a complete fill. If the thickness of a complete fill, after compaction, is AS, then the reduction in energy release over a span of 21, due to complete filling is

Awn,, = 2~ L AS. [22I Assuming that the compacted thickness of partial filling in the form of regular strips is also AS, then the reduction in energy release due to partial ,

filling is

The effectiveness of partial extraction can be gauged by comparing the ratio between the reduction in energy release due to partial extraction, Eq. 21, with the maximum possible reduction of energy release obtained by reducing the excavation thickness to zero, Eq. 11. This yields

where k is related to the pillar spacing and thickness, and the rock stresses

and elastic properties. A similar equation provides a comparison between the. effectiveness of complete and partial filling. Eq. 24 is shown as a function of percentage extraction or fill and k in Fig. 6a; k, for a Poisson's ratio of 0.2, is shown as a function of the geometry and relative stress in Fig. 6b. This shows that for high values of k, given by moderate stresses or small pillar spacing, partial extraction or filling provides about 80 per- cent of the maximum possible reduction in energy release with pillars occupying about 10 percent of the excavation.

I n partial extraction it is unlikely that the surfaces of the excavation will make contact with one another so that the released energy and stored strain energy will be equal. The rates of energy release and storage de- pend upon the direction of extraction in relation to theIpillar axes. If the pillars are formed by faces advancing perpendicular to the axes, the rate of energy release will increase as mining progresses, reaching a maximum value a t the minimum pillar dimension. If the pillars are formed by faces advancing parallel to their axes, the maximum rate of energy release will be less. The ratio between the maximum perpendicular and parallel rates of energy release, derived from Eqs. 14 and 15, is

nR TR tan 2L p=-- [25 I 2L l o g , ( c o s ~ ) '

It should be stressed that, the total energy release and the rate of energy release are of similar importance. Large differences in the rates of energy release can result from different ways of extracting the same volume of rock. This can be very significant in the extraction of highly stressed remnants where excessively high rates of energy release are known to give rise to severe problems of rock failure (Cook, et al.G). Examples of the change in energy release rate with different sequences of extraction are shown for some idealized cases in Fig. 7. These results were derived using the analogue computer, and the detailed variations in rate of energy release reflect the finite size of the elemental areas represented by each plug. I n general, the most uniform rate of energy release is obtained when extraction comlnences a t the most highly stressed point and pro- ceeds towards the most intact area of rock.

Damage to ancilliary excavations, such as access ways to the main excavation, is usually caused by stress increases associated with the stor- age of strain energy around the main excavation. The techniques used to minimize the energy changes often have a greater effect on the reduc- tion of strain energy than on the reduction of energy release, but the effects are equal in the case of partial extraction. The stress concentra- tions ahead of, and above, an extensive tabular excavation with partial

10 20 30 LO 50 60 70 00 90 Extraction (Plugs1

Fig. 7-The rates 01 energy release due to different sequences o f remnant extraction.

extraction on regular strip pillars is shown in Fig. 8. The extent of the stress concentrations, and hence the stored strain energy, are determined by the geometry, and decrease as the pillar spacing 'is reduced. This is important in two respects. First, i t means that the reduction in stored energy resulting from partial extraction allows ancilliary excavations to be placed within half the pillar spacing of the plane of the excavation

Strain

Fig. 8-Stress concentrations ahead of Pig. 9--A complete strain-stress curve and above a n extensive tabular excava- for rock showing ( 1 ) the elastic region, tion with regular strip pillars. ( 2 ) the region o f stable crack growth, (3)

the region o f potentially unstable faihrre, and applied stress charactelistics result- ing i n stable or unstable failure i n region (3) .I

without ever being subject to stresses significantly greater than the virgin rock stress. Secondly, i t means that with this system of extraction, excavations on different horizons can be regarded as independent, provided that they are separated by a distance greater than the pillar spacing.

ROCK FAILURE

The strength of most rocks can be described in terms of the modified Griffith's theory. Even where the techniques described in the previous section are used to alleviate the problems arising from energy release and strain energy storage, i t is usually not possible, particularly a t depth, to avoid stresses sufficient to cause rock failure. It is a matter of common experience that rock in the vicinity of the surfaces of most underg~ound excavations is both failed and subject to stress. In general, the failed rock is in stable equilibrium with the stresses applied to i t ; difficulties arise when this equilibrium breaks down, resulting in a violent and uncon- trolled release of energy. It is, therefore, of great importance to study the behavior and stability of failed rock.

The concept of a complete strain-stress curve for rock has been ad- vanced by Fairhurst and Cook lo to facilitate an understanding of this problem. This distinguishes three regions of rock behavior, see Fig. 9. First, an elastic region where the strain increases almost linearly with t.he stress. Second, a region of stable crack growth within the rock, where the slope of the strain-stress curve decreases with stress but always remains positive. Third, a region where the slope of the strain-stress curve be- comes negative, due to the onset of some potentially unstable mode of failure. The stresses applied to failed rock by a testing machine, or by the rock mass surrounding an excavation, decrease if the failed rock yields. If the applied stress decreases more rapidly than the slope of the strain-stress curve, the rock is in stable equilibrium with the stress. If the applied stress decreases less rapidly than the slope of the strain-stress curve, violent failure will occur. Thus, regions (1) and (2) of the com- plete strain-stress curve are intrinsically stable with respect to an applied stress, but region (3) is potentially unstable. The behavior of rock in regions (1) and (2) is regarded as a fundamental property of the rock. Violent failure can only occur in region (3) if the applied stress decreases less rapidly than the slope of the strain-stress curve, with further com- pression. The point a t which potentially unstable failure starts, and the particular form that this mode takes, are regarded as functions of the geometry of the system and the properties of any support, rather than as fundamental properties of the rock.

CONTROL- OF ROCK FAILURE UNDERGROUND

There are several types of rock failure which give rise to problems in underground excavations. One type of failure occurs in the vicinity of a

surface as a result of an excessive compressive stress parallel to this surface. This type of failure is most often observed on the sidewalls of horizontal tunnels or chambers, and a t stope faces. Another type of failure occurs in the hanging above excavations, where the rock pos'sesses inadequate cohesion to resist vertical tensile stresses. This type of fail- ure most often manifests itself as falls of hanging. Inadequate cohesive strength is often due to a combination of geological weakness and failures induced by compression a t some earlier stage of mining.

The first type of failure, Fig. 10, probably occurs in the form of buckling of incipient slabs generated by stable crack extension parallel to the sur- face as a result of compressive stresses in excess of the strength of the rock (Fairhurst and Cook I"). This type of failure can be controlled by increasing the resistance to buckling of the incipient slabs surrounding the excavation. This can be achieved by changing the profile of the excavation, by support, or by a combination of these techniques. It is obvious that resistance to buckling can be achieved by providing the most highly stressed surfaces with curvature towards the solid, and by decreasing the radius of curvature as the stress increases. Where this is inadequate or inconvenient, resistance to buckling can be achieved by rockbolting the surfaces. The effectiveness of rockbolting in this respect depends upon the modulus of foundation (Timoshenko and Gere I T ) with which the bolts inhibit buckling of the rock, rather than upon the stresses they apply to the rock. It is, therefore, the resistance of the rockbolt to extension (that is its modulus, rather than its strength) which is of im- portance; it is fortunate that the modulus exceeds the strength by almost two orders of magnitude. The rockbolts must be sufficiently long to be anchored in solid rock beyond the region of incipient slabbing, so as to resist the movement of the slabs towards the excavation. Crack exten- sion giving rise to slabs is strongly inhibited by confining stress and, in general, will occur no further into the rock than a distance equal to the radius of curvature of the free surface, or in the case of profiles with flat surfaces, a distance equal to half the dimension of the flat surface. The reason for this is illustrated in Fig. 11. This shows the maximum and minimum principal stresses on the horizontal center line of a square and circular excavation subject to a uniaxial vertical stress. The principal stresses become comparable a t this distance and cannot, therefore, lead to crack extension.

Stable tunnel profiles for different magnitudes of vertical stress, ex- pressed as a fraction of the uniaxial compressive strength of the rock are as illustrated in Fig. 12. Strengthening an excavation in this manner is very effective in inhibiting the onset of a potentially unstable mode of failure. It is far more difficult to control the progress of a potentially unstable mode of failure once it has started. The strength of an exposed rock surface decreases with time, and the stresses around an excavation often increase as a result of nearby mining. It is important that

Maximum

Minimum

s t r e s s .

s t r e s s .

Fig. 11-Maximum and minimum principal stresses ahead of a flat and curved tunnel surface in uniaxial compression. (From Timoshenko and Goodier,'l and Savin.)

strengthening should be done as soon as possible, before these factors lead to the onset of a potentially unstable mode of failure.

Gravity introduces a degree of asymmetry into the problems of under- ground support.

Tensile stresses are induced in the rock surrounding an excavation and the extent of this induced tension is determined by the dimensions of the surfaces of the excavation. The resultant stress is equal.to the sum of the induced and virgin rock stresses. The vertical component of the virgin rock stress is generated by the weight of the superincumbent strata and

, increases linearly with depth. Where the vertical component of the induced tension exceeds the vertical component of the virgin compres- sion, the resultant vertical stress is tensile. This fact is of major sig- nificance in problems of hanging control. The induced tension and virgin compression above the centre of excavations with circular and slit-like profiles a t different depths below surface are as illustrated in Fig., 13. This shows that in the immediate hanging of an excavation the resultant

Fig. 12-Stable tunnel profiles for different vertical stresses, u, in relation to the uniazial strength of the rock, u,, where the vertical stress is the mazimum principal stress.

I 1. Induced vert ical tension.

I a. Circle. hydrastatic stress. 1 h Circle. uniaxial stress. stress.

Fig. 13-The vertical components of the induced tenswn and virgin compresswn above the center o f circular and slit-like excavations for hydrostatic and uniaxial stresses.

stress can be tensile. The extent of the region of tensile stresses depends upon the ratio between the depth of the excavation below surface and its horizontal dimensions, and increases as this ratio decreases.

Even where the horizontal dimension is small, as is often the case with tunnels, the design of hanging support should be such that failure does not result ,in falls of hanging. For this reason, hanging support should be independent of sidewalls (Fig. 12e) or, if rockbolts are used for hanging support, they should be grouted so that failure near the nut or washer of the bolt does not destroy the support provided by the bolt. ' I n stratified rock significant weaknesses in a near vertical direction

often exist a t bedding planes or contacts between different rock types. Where such a weakness exists within the region of tension, s t ra ta . sepa- ration is likely to occur unless adequate support is provided. If such sepa- ration occurs and the detached slabs so formed are divided vertically by weaknesses such as faults or slips, or by fractures induced by mining, severe problems of strata control can be anticipated. Two solutions to these problems can be found for excavations with large horizontal dimensions.

First, if the horizontal dimension is a large fraction of the depth below surface, so that the tensile region extends far into the hanging, and, if the hanging is homogeneously weak in tension so that it will break up

uniformly, caving with temporary face support can be practised. In these circumstances, the failure of the hanging may proceed through to surface where the excavation is shallow, or the bulking of the caved rock may provide a form of fill where the excavation is deeper. Caving will not be effective if the hanging is strong in tension except a t a few,par- ticular horizons. In this case attempts a t caving merely transfer the void of the excavation to these weak horizons by bulk movement of the inter- vening hanging towards the excavation.

Secondly, if the depth below surface is great, so that the tensile region is of limited extent and, if the rock is generally strong in tension, per- manent support adequate to hold the weight of the detached slabs must be used. Fig. 14 shows the maximum vertical extent of the tensile region above a slit-like excavation. This defines the distance within which a

c. 0

5 L . - I" 0 10 20 30 LO

jhpt h below surf= Half .span.

Fig. 14-The mazimum extent o f the region o f vertical tension above a slit-like excavation.

weakness must occur to cause separation and hence the maximum loads required of the support. Where the support loads which can be obtained are limited, the extent of the tension must be similarly limited by limiting the effective span with a regular system of pillars, or by limiting the, thickness of extraction. The support must not only be capable of carry- ing the weight of the detached hanging but it should reduce the displace- ment of the hanging to a minimum. This minimum is virtually the displacements forced upon the hanging by the motion of ' the rock mass. If they are greater than this, voids will be created in the hanging, resulting in disruption and loss of hanging cohesion. Therefore, support should either be installed with the required load or should generate this load with a minimum of compression (Cook I s ) .

CONCLUSION

The basic mechanical principles governing the changes brought about in the rock by underground excavation have been considered. These show that strain energy is stored in the rock around an excavation and that excess energy must necessarily be released. The magnitudes of these energy changes are determined by the volumet,ric convergence of the excavation, and the virgin rock stresses a t the position of the excava- tion. These energies provide an objective measure of the problems likely to arise from failure of the rock around an excavation. The amounts of energy which are stored and released can be reduced by limiting the volumetric convergence of the excavation in the following ways:

1) Underground support a t depths of less than 1000 ft below surface. 2) Partial extraction, where it is possible to leave adequately strong

permanent pillars. 3) Complete or partial filling where the volumetric convergence is an

appreciable fraction of the volume of the excavation, as is the case a t great depth.

4) Minimizing the volume of rock extracted, where the volumetric con- vergence is a large fraction of the volume of rock extracted.

The rates of energy release and the total energy release are of equal significance. The former can be controlled by planning the layout and sequence of extraction of an exca~at~ion.

While these techniques can be effective in reducing the amount of energy released and the amount of stored strain energy, they may not be adequate to avoid the failure of rock immediately around excavations. Rock failure does not necessarily constitute a major problem provided that its occurrence can be controlled. Where failure is caused by high stresses the design and support of excavation must be planned so as to prevent unstable modes of failure. Where failure is caused directly by

gravitational forces support must be designed to carry the weight of the detached rock and prevent its disruption.

Though the arguments have been presented deductively in this paper they are, in fact, based upon the results obtained from the interaction of observation, theory, and experiment by many persons over several years (Cook, et and can be paved experimentally (Hodgson and Joughin 1 9 ) .

An understanding of the problems concerning the shbili ty of rock failure is only now beginning to emerge. When these studies have come to fruition the knowledge of rock mechanics will be adequate to define and solve most of the problems of strata control likely to be encountered in mining. When mining is pursued a t even greater depths than those a t present, the problems of strata control may be so severe as to dictate changes in mining techniques before they can be solved.

ACKNOWLEDGMENT

This paper forms part of the rock mechanics research done by the Mining Research Laboratory of the Transvaal and Orange Free State Chamber of RIines.

APPENDIX

Energy Changes Resulting from Underground Excavation

Thc changes resulting from making an excavation in an elastic solid subject to gravity are readily studied with the aid of the reciprocal theorem which states that for an elastic body Zcted upon by two states of stress, the work that would be done by the first state of stress acting , through the displacements produced by the second state is equal to the work that would be done by the second state of stress acting through the displacements produced by the first state ' (Timoshenko and Goodier 21).

Consider a large rectangular prism of rock surrounding an excavation and bounded by the surface of the earth and planes perpendicular to the axes of the principal virgin rock stresses, 1, 2, 3. Let the first state oi stress with second subscripts 1 be stresses applied to the surface of the excavation only and let thesc stresses be equal in magnitude but opposite in sign to those existing on these surfaces prior to excavation. The first state of stress will produce:

1 ) Displacements of the excavation surface having components u,,, u,,, and u,, in the directions of the principal stresses.

2) Displacements v,, and v,, of the vertical surfaces of the prism. 3) A displacement v,, of the surface of the earth. (The bottom of the

prism will not displace as it cannot move towards the center of the earth). These displacements are identical with those that result from making

Fig. 1-A-(a) T h e first state of st.ress applied t o the prism containing the ezcavat ion. ( b ) T h e second state o f stress applied t o the prism containing the ezcavat ion.

the excavation, which is equivalent to the superposition of these stresses on the virgin rock stresses.

Let the second state of stress with second subscripts 2 within the exca- vation be the same as the first state, but, in addition, let stresses equal in magnitude but opposite in sign to the principal virgin rock stresses a t the center of the excavation be applied uniformly to the corresponding surfaces of the prism. The second state of stress will produce:

l j Displacements of the excavation surface u,,, u,?, and u,?, equal and opposite to those corresponding to the compression of a volume of rock of the shape of the excavation by the virgin stresses.

2) Displacements of the prism surfaces which will be of no concern since in the first state there was no stress on these surfaces.

Let da,, da,, and da, be components of the excavation surface normal to the principal stress axes, and dA,, dA,, and dA, be the components of the prism surfaces. The work done by the first state of stress on the displacements of the second state is equal to the work done by the second state of stress on the displacements of the first. The former and latter amounts of work are given by the left- and right-hand sides, respectively, of the following equation

- J a ~ l ~ 1 2 d a 1 -J,~,u,,da? - J , , ~ ~ u ~ ~ d a ~ = J,alulldal + S , ~ , u ~ ~ d a , +Jnaau3lda:<

-2J4 ulvlldA1 - 2Ju,~,~dA, - 2Ja,v3,dA,. PA1

The sum of the terms on the left-hand side of Eq. 1A is -x twice the strain energy originally existing in the volume of rock removed by excavation, w.

The sum of the first three terms on the right-hand side of Eq. 1A is equal to twice the energy which would be released if the surfaces of the excavations were held in their original positions and then released, W,. It is also equal to twice the strain energy stored around the excavation, WS, by the first system of stresses.

The last three terms on the right-hand side comprise, respectively, the change in potential of the horizontal stresses, WH, and gravity, WG.

Thus, Eq. 1A can be expressed as excavation by these stresses.

-2wS= 2WR-WH-WG WG+WH= WR+WS+2ws PA]

wG= ~3 v3

where V,=volumetric subsidence of the surface due to the excavation

where V, and V2 are the volumetric displacement's of the vertical surfaces due to the excavation.

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