vutukuri katsuyama introduction to rock mechanics

Vutukuri Katsuyama

Editors

Industrial Publishing & Consulting, Inc.

Contents

Intrvdnction

Geological aspects

Lithology

Geological separations

Rock mass and rock material

Comparison to other materials

Characteristics of discontinuities

Orientation

Spacing

Persistence

Roughness

Wall strength

Aperture

Filling

Number of sets

Block size

Site investigations

Regional geological investigations

Mapping of exposed smctures

Diamond drilling for structural purposes

Rock mass classification

Rock quality designation (RQD)

Multi-parametric classification schemes

Exercises

Physical and mechanical properties of rock and rock mass

Physical propehes

Density

3.1.1.1 Grain density

3.1.1.2 Bulk density

3.1.2 Porosity

3.1.3 Water content

3.1.4 Swelling and slake-durability indices

3.2 Mechanical properties of intact rock

3.2.1 Strength properties

3.2. I. 1 Definitions

3.2.1.2 Stiff and servo-controlled testing machines

3.2.1.3 Common laboratory strength tests

3.2.1.4 Uniaxial compressive strength test

3.2.1.5 Point load strength test

3.2.1.6 Triaxial compressive strength test,

3.2.1.7 Tensile strength tests

3.2.1.8 Shear strength tests

3.2.2 Deformation properties

3.2.2.1 Definitions

3.2.2.2 Static elastic constants of rock

3.2.2.3 Dynamic elastic constants of rock

3.2.2.4 Comparison of static and dynamic elastic constants

3.2.3 'Iime-dependent properties

3.2.3.1 Creep

3.2.3.2 Time-dependent strength

3.2.3.3 Fatigue

3.2.3.4 Dynamic tensile strength

3.3 Mechanical properties of rock mass

3.3.1 Strength properties of discontinuities

3.3.1.1 Shear testing of discontinuities

3.3.2 Strength of rock (mass) with a single discontinuity of discontinuity set

3.3.3 Strength of rock mass with multiple intersecting discontinuities or discontinuity sets

3.3.4 Deformation propenies of discontinuities

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Contents

3 . 3 4 1 Normal deformation

3.3.4.2 Shear deformation

3.3.5 Deformation propenies of rock mass

3.3.6 Large-scale in situ tests

3.3.61 Compression tests

3.3.6.2 Shear tests

3.3.6.3 Deformability tests

3.3.6.4 Seismic tests

3.4 Fracture criteria

3.4.1 Fracture criteria for intact rock

3.4.1.1 Maximum tensile stress criterion

34.1.2 Coulomb's criterion

3.4.1.3 Mohr's criterion

3.4.1.4 Griffith's criterion

3.4.2 Empirical criteria for intact ~ o c k

3.4.2.1 Bieniawski's criterion

3.4.2.2 Hoek and Brown's criterion

3.4.2.3 Johnston'scriterion

3.4.2.4 Analysis by ~utukuri and Hossaini

3.4.3 Empirical criteria for rock mass

3.4.3.1 Analysis by Vutukuri and ~ossaini

3.5 Properties of soft and weak rocks

3.6 Exercises

4 Hydraulic properties of rock and rock mass

4.1 Definition of permeability

4.11 Flow location and model

4.1.2 Resistance law of flow

4.2 Permeability of intact rock

4.2.1 Coefficient of permeability of rock

4.2.2 Laboratory percolation tests of rock

4.2.2.1 Longitudinal percolation test

4.2.2.2 Radial percolation test

4.3 Permeability of rock mass

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4.3.1 Permeability of a single fracture

4.3.2 Permeability of rock mass

4.3.3 In situ test method for permeability of the rock mass

4.3.3.1 Lugeon Test

4.3.3.2 in situ test using E T (Johnston's Formation Tester)

4.4 Gas seepage in the rock mass

4.4. I Gas seepage in intact I-ock

4.4.2 Gas leakage in a rock mass

4.5 Exercises

5 Virgin rock stresses

5.1 Specification of the virgin state of stress

5.2 Compensation technique -Flat jack

5.3 Relief techniques - Undercoring or overcoring

5.31 General aspects

5.3.2 Measurements on surface

5.3.3 Measurements in borehole

5.3.3.1 Measurements in the borehole

5.3.3.2 Measurement at the back of borehole

5.4 Hydraulic fracturing technique

5.5 Methods using drill cores

55.1 Acoustic emission (AE) method

5.5.2 Deformation rate analysis (DRA)

5.5.3 Differelitial strain curve analysis (DSCA)

55.4 Anelastic strain recovery (ASR) method

5.6 Exercises

6 Methods of analysis for rock engineering

6. I Stresses around underground excavations

6.1.1 Calculation of stress fields

61.2 Closed-form solutions for simple excavation shapes

612.1 Circular excavation

6 1 . 2 2 Elliptical excavation

b 1.3 Coinplex profiles

61.4 Size of an excavation

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6.1.5 Preferred shapes for two different stress fields

61.6 Multiple excavations

6.2 Analysis of rigid blocks

6.2.1 Two-dimensional single plane sliding

6.3 Exercises

7 Rock reinforcement and support

7.1 Mechanisms of failure in rock engineering structures

7.1.1 Underground openings

7.1.1.1 Failure modes involving only intact rock

7.1.1.2 Failure modes involving discontinuities and intact rock

7.1.1.3 Failure modes involving only discontinuities (blocky medium only)

71.2 Slopes

7. I .2.1 Failure modes involving only intact rock


7.1.2.3 Failure modes involving only discontinuities

7.2 Rock loads

7.2.1 Empirical approaches:- Rock classifications

7.2.2 Semi-empirical approaches

7.2.3 Structural defect approach

7.2.4 Rock-support interaction approach

7.2.5 Unified approach

7.3 Supporting and reinforcement members

7.3.1 Steel liners

7.3.2 Steel ribs

7.3.3 Concrete liners

7.3.4 Shotcrete

7.3.5 Rockbolts

7.4 Design of support and reinforcement systems

74.1 Reinforcement of continuum by rockbo1ts:- Pattern rockbolting

7.4.1. I Contribution to the deformation moduli of the medium

7.4.1.2 Contribution to the strength ofthe medium

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7.4.2 Reinforcement of discontinuum by rockbolts

7.4.2.1 Increment of the tensile resistance of a discontinuity plane by a rockbolt

7.4.2.2 Increment of the shear resistance of a discontinuity plane by a rockbolt

7.4.3 Suspension effect of rockbolts

7.4.4 Beam building effect of rockbolts

7.4.5 Arch formation effect of rockbolts

7.4.6 Stabilisation against sliding

7.4.7 Support of a circular tunnel by shotcrete, rockbolts and steel ribs

7.4.7.1 Pure elastic behaviour of rock

7.4.7.2 Elasto-plastic behaviour of rock

7.4.7.3 Modelling of rockbolts

7.4.7.4 Modelling of shotcrete

7.4.7.5 Modelling of steel ribs

7.5 Exercises

8 Monitoring of structures

8 1 General features of monitoring systems

8.2 Monitoring systems

Convergence measurement

Multiple-point borehole extensometer

Hydraulic pressure cell

Stress change measurement

Microseismic activityiAcoustic emission monitorins

Examples of monitoring structures

Monitoring of rock mass behaviour around the caverns during excavation

Monitoring of excavation-induced microseismic activity

Exercises

References

index

Contents

Introduction

ock mechanics may be defined as the theoretical and applied science of the R achanical behaviour of rocks; it is that branch of mechanics concerned with the response of rocks to the force fields (natural and person made) of their physical environment. For civil and mining engineers, rock mechanics is just another engineering approach to solving problems that involve rock. It should guide the engineers wishing to build excavations and structures in or on rock such as tunnels for underground railways, water supply, drainage, etc., underground power houses, caverns for storage of oil, water, air, gasoline, etc. as well as disposal of nuclear waste, underground mines, quarries, open pit mines, deep cuts for spillways, etc.

Rock is quite different from other engineering materials. It is heterogeneous and anisotropic. The other materials have few or no intrinsic structurqs that are counterparts of discontinuities in rock masses except perhaps for the grain in wood and the layer boundaries in laminated or composite synthetic materials. Usually, there is considerably less choice in selecting material properties.

When dealing with rock in place there are many unknowns; the strengths and the stresses are both unknown and highly variable. The geological discontinuities play a vital role in the selection of the geometry of the structure.

The second Chapter deals with Geological aspects. This Chapter is very limited in scope covering only basic terminology and the definitions. The third Chapter covers Physical and mechanical properties of rock and rock mass. In any rock mechanics investigation, the knowledge of these properties is essential. This Chapter is quite comprehensive. The fourth Chapter covers Hydraulic properties of rock and rock mass. It should be pointed out that these properties determine pore water pressure which influences the strength of rock and rock mass. It should also be pointed out that the third and fourth Chapters cover not only the intact rock but also the rock mass. The fifth Chapter deals with Virgin rockstresses. These stresses are important in the design of excavations and structures. Various techniques developed to measure these stresses are described in brief. The sixth Chapter deals with Methods of analysis for rockengineering This Chapter is limited to stresses around simple excavation shapes and analysis of rigid blocks. However, numerical methods are not dealt with. The seventh Chapter deals with Rock reinforcement and support. Almost every exca-

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vation need reinforcement and support. This Chapter is quite comprehensive. The eight Chapter deals with Monitoring of structures. The principles used in various instruments are described in some detail.

Geological Aspects

ocks possess certain physical, chemical, mechanical and hydraulic properties R, epending upon their mode of origin and the subsequent geological processes that have acted upon them. These events lead to a particular lithology, a particular set of geological separations and a particular in situ state of stress. This Chapter is concerned with lithology and geological separations.

2.1 Lithology Lithology refers to the rock type, mineralog, texture and cementingmaterial. From

a genetic point of view, rocks are divided ;to three groups:- igneous, sedimentary and metamomhic. Rocktvoes eive valuableaid in the orediction ofcertainundesirable - ~~ ~ ~~- ~~ ~

engineering properties (Table 2.1). Figure 2.1 identifies water access in limestoneand swelling of the invert in bypsiferous stone in Belchen tunnel (Swiss jura mountain). Figure 2.2 shows the dependence of costs of excavation and support, and rate of progess on the lithology of a tunnel to the hydro power station in Austria.

2.2 Geological Separations Geological separations are classified intothree:- 1. microfissures, 2. micro fractures

and 3 , macro fractures. Micro fissures are the defects in the rock fabric. (Fabric refers to the spacial data about the grains constituting the rock i.e. orientation, mutual relationship to each other or packing..) They are 1 btm or less in width and about the lenyh of a crystal or two or three molecules of water.

Micro fractures are about 0.1 mm or less in width. Their extent is significant despite the fact that they are barely visible to the naked eye. They often depend on the schistosity of the material and have well-defined directions in space.

Macro fractures are wider than 0.1 mm. They may be up to several metres or more in length. The term "discontinuity" is widely used in the literature mean "macro fracture". It is a collective term for most types ofjoints, weak bedding planes, weak schistosity planes, weakness zones and faults. A discontinuity has zero or low tensile strength. There are many relationships between geological separations and the micro structure of rocks.

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2 Geolog~cal Aspects

Table 2.1 Undesirable engineering properties of some types of rock (after Bock, 1978).

~ -- -

Genetic group Name Undesirable engineering properties to be ~ =exoected ~ ~~~~ ~.~

Igneous rocks Granite Tends to decompose especially when coarse- grained

Basalt Weather to highly sensitive clay mineral montmorillonite

Sedimentaq Sandstone Poor drillability rocks

Shale Liable to swelling and shrinking

Limestone Karst:- subsidence; extremely high permeability

Gypsum Swelling when water access; aggressive ground water

Rock salt Tends to creep; leaching

Metamorphic Slate, schist High anisotropy; various platy minerals rocks (micas, graphite, talc) cause low shear strength

Gneiss Anisotropic

Quartzite Extremely ooor drillability

2.2.1 Rock m a s s and rock material

In order to clearly understand the engineering properties of rocks in various conditions, the distinction between rock mass and rock material must be understood. Rock mass (rock system, rock in situ) consists of rock material and discontinuities. (Rock mass does not mean "mass of rock".) Rock material (rocksubstance, intact rock, rock element) is the presumably continuous rock matter. The volume element of rock material is large with respect to grains, pores, and micro fissures: it is small with respect to the spacing between discontinuities. The frequently observed differences in enineerine, orooerties between rock mass and rock material iustify the distinction -. . . . between the two.

Micro fissures and micro fractures determine the engineering properties of rock material whereas the engineering properties of a rock mass depend far more on the system of macro fractures i.e. discontinuities within the mass than the properties of the rock material itself.

2.2.2 Comparison to other materials

What makes rock different from other materials such as concrete, ceramic, glass, metal. wood and synthetic is the extent of its heterogeneity and anisotropy, the range of size and the span of time that must be considered. and the fact that there is less choice i n selecting material properties. The other materials lhave few or no intrinsic

2.3 Characteristics of Discontinuities 5

structures that are counterparts of discontinuities in rock masses such as bedding except perhaps for the main in wood and the laver boundaries in laminated or . . - ~~~~ ~~

composite synthetics. The building blocks i.e. minerals constituting rocks are considerably different from the building blocks of most other materials.

Figure 2.1 Belchen tunnel (Swiss jura mountain); (a) Geological cross-section (b) Water access (preferably in limestone) (c) Swelling of the invert (preferably in gypsiferous stone) (after Bock, 1978).

B limestone EB gpsiferous strata 0 other lithological units

Distance. km

2.3 Characteristics of Discontinuities The characteristics of discontinuities that can be expected to affect the engineering

properties of the rock mass are orientation, spacing persistence, roughness, wall strength, aperture, filling, number of sets and block size. The particular contribution of geologists is to determine the geometrical properties of the discontinuities.

2.3.1 Orientation

Orientation is the attitude of discontinuity in space and is described by the dip direction (azimuth) and dip of the line of steepest declination in the plane of the discontinuity. The dip is measured from the horizontal and the dip direction is measured clockwise from the true north. Figure 2.3 indicates the strike direction. dip and dip direction of a plane.

Discontinuities are graphically represented by Schmidt's method (Figure 2.4). A discontinuity plane K is positioned at the centre of the hemisphere H. The line OP, normal to the planeK, will pierce the hemisphere at the pole 1'. this representing the orientation ofplane K. The poles ofail the discontinuities surveyed can be represented

6 2 Geological Aspects

on an equal-area projection ofthe hemisphere, producing a point diagram. The density of points indicates the number of discontinuities with approximately the same direction.

The orientations of discontinuities relative to the faces of excavations have a dominant effect on the potential for instability due to falls of blocks of rock or slip on the discontinuities. The mutual orientations of discontinuities determine the shapes of the blocks into which the rock mass is divided.

Figure 2.2 Tunnel to the hydro power station "Landl" in Austria, (a) Lithology met in 2.6 km long tunnel (b) Costs of excavation and support per metre (without scale) (c) Rate of tunnel progress (highly dependent on lithology) (after Bock, 1978).

evaporites unleached

P ~ Y leached limestone / '.,, hlly leached /

Figure 2.3 Strike direction, dip and dip direction of a plane

strike direcrion = a' dip = P" dip diredon = a' = a" + 90,'

2.3 Characteristics ofDiscontinuities 7

Figure 2.4 Representation of discontinuity planes; (a) Three-dimensional illustration of the method of W. Schmidt (b) HemisphereHin equal-area projection (after Jaeger, 1979).

line of strike N

2.3.2 Spacing

Spacing is the perpendicular distance between adjacent discontinuities belonging to the same set. It controls the size of individual blocks of intact rock. The classification for discontinuity spacing suggested by the International Society for Rock Mechanics (ISRM) in 1978 is given in Table 2.2.

Several closely spaced sets tend to give conditions of low mass cohesion whereas those that are widely spaced are much more likely to yield interlocking conditions. These effects depend upon the persistence of the individual discontinuities.

The ratio of discontinuity spacing to excavation size affects the mechanism of deformation and failure. The caveability, fragmentation characteristics and permeability of rock mass depend upon spacing.

2.3.3 Persistence

Persistence is the discontinuity trace length as observed in an exposure. It may give a crude measure of the areal extent or penetration length of a discontinuity Termina- tion in solid rock or against other discontinuities reduces the persistence. F i~vre 2.5

Y 2 Geological Aspects

gives simple sketches and block diagrams indicating the relative persistence of the various sets of discontinuities. ISRM uses the most common or modal trace lengths of each set of discontinuities measured on exposures to classify persistence according to Table 2.3. Persistence has a major influence on the shear strength developed in the plane of the discontinuity, on the fragmentation characteristics, caveability and permeability of the rock mass.

Table 2.2 Classification for discontinuity spacing (ISRM, 1978b)

. . ..Descn~~on_ ----- ~ S ~ a c i n & _ m L ~ . ~ . . ~ ~ ~ ~ . ~ . .- ... ..

Extremely close < 20

Very close 20 - 60

Close 60 - 200

Moderate 200 - 600

Wide 600 - 2000

Very wide 2000 - 6000

'6000 Eliem&z&de de==.-.__._-_ ~

Figure 2.5 Simple sketches and block diagrams indicating the relative persistence of the various sets of discontinuities (after ISRM, 1978b).

2.3 Characteristics of Discontinuities 9

Table 2.3 Classification of discontinuity persistence (after ISRM, 1978b). ----=-- > .=.- ~.*< .-.-=---= n- -----==*7&-...---.--- .:.-.-. -:?-

. . -P1-e~~cp.~_qn __==____;. Aodalf~a-~F~ _-=== Very low < 1

Low 1 - 3

Medium

High

2.3.4 Roughness

Roughness is a measure of the inherent surface unevenness and waviness of the discontinuity relative to its mean plane. Figure 2 6 shows small-scale surface irregularity or unevenness and large-scale undulations or waviness of the surface. Figure 2 7 gives typical roughness profiles and nomenclahue as suggested by I S M . The length of each profile is in the range of 1 to 10 m. The vertical and horizontal scales are equal.

Roughness influences shear strength especially in the case of undisplaced and interlocked features (e.g, unfilled discontinuities) Importance declines with increasing aperture, filling thickness or previous shear displacement

Figure 2.6 Different scales of discontinuity roughness are sampled by different scales of tests (after ISRM, 1978b). Waviness can be characterised by the angle (i).


Figure 2.7 Typical rou~hness profiles and suggested nomenclature (after ISRM, 1978b). Thelength of each protile is in the range 1 to 10 m. Thevertical and horizontal scales are equal.

rough I ZV-

slickensided 111

stepped

undulating

2 3 Characteristics of Discont~nuities 11

Aperture and its areal variation influence the shear strength of the discontinuity and the permeability or hydraulic conductivity of the discontinuity and ofthe rock mass.

Figure 2.8 Aperture of an open discontinutty (after ISRM, 1978b)

2.3.7 Filling

Filling is the material that separates the adjacentrock walls of a discontinuity and it is usually weaker than the parent rock. Typical filling matelials are sand, silt, clay, breccia, gouge, mylonite and quartz and calcite veins. The perpendicular distance between the adjacent rock walls is termed the width ofthe filled discontinuity (Figure 2.9). The filling materials have a major influence on the shear strengths of discontinuities.

much VII

Figure 2.9 W~dth of afilled discontinuity (after I S M , 1978b)

wldth

IX slickensided

2.3.5 Wall strength

Wall strenyh is the uniaxial compressive strength of the adjacent rock walls of a discontinuity. It may be lower than rock block strength due to weathering or alteration ofthe wails. It isan important component ofshear strength ifrock walls are in contact.

2.3.6 Aperture

Aperture is the perpendicular distance between adjacent rock walls of an open disconrinuity, i n which the intervenin~ space is air or water filled (Figure 2.8).

2.3.8 Number of sets

It is the number of discontinuity sets conlprising the intersecting discontinuity system. The rock inass may be further divided by individual discontinuities. Figre 2.10 gives examples that demonstrate the effect of the number of discontinuity sets on the mechanical behaviour and appearanceof a rock mass.


Figure 2.10 Examples that demonstrate the effect of the number of discontinuity sets on the mechanical behaviour and appearance of a rock mass (after ISRM, 1078b).

three discontinuity sets

2.3.9 Block size

It is the rock block dimensions resulting from the mutual orientation of intersecting discontinuity sets, and resulting from the spacing of the individual sets. Individual discontinuities may further influence the block size and shape.

Rock masses can be described by the following adjectives, to &ive an impression of block size and shape..

(i) Massive -few joints or very wide spacing. (ii) Blocky -approximately equidimensional. (iii) Tabular - one dimension considerably smaller than the other two. (iv) Columnar - one dimension considerably larger than the other two. (v) Irregular - wide variations of block size and shape. (vi) Crushed -heavily jointed

Figure 2. I 1 yives csamples of some of the above

2.4 Site Investigations 13

Figure 2.11 Sketches of rock masses illustrating (a) blocky, (b) irregular, (c) tabular, and (d) columnar block shapes (after ISRM, 1978b).

2.4 Site Investigations

241 Regional geological investigations

Structural discontinuities, upon which local failure can occur, are related to the regional smctural pattern of the area and it is therefore useful to start an investigation of the regional geology. Air photographs, topographic maps and regional geology maps should be consulted as early as possible in the investigation. Stereographic examination of adjacent pairs of air photographs can detect linear surface feahlres which usually indicate the presence of underlying geological structures. Full use should also be made of any exposures available on site, in adjacent mines or quarries, road cuts and exposures in river or stream beds.

2.4.2 Mapping of exposed structures

The most important tool for use in mapping is thegeological compass. It should read directly the dip and dip direction. Line sampling method is usually preferred for mapping. This involves stretching a 3 m tape at approximately waist height along a face or a tunnel wall and recording every stluctural feature which intersects the tape line. The features to be recorded for each discontinuity include:. distance along the scar~line to the point at whicll the discontinuity intersects the scanline, the length of the discontinuity measured above the scanline i e semi-trace length, nature of termination point. dip and dip direction, curvahire or waviness, roughness and comments such as nature of any infilling present, aperture, seepage and origin ortype of discontinuity. Stn~ctural mapping suffers from some fonn of bias since structures


parallel or nearly parallel to an exposed face will not daylight as frequently as those perpendicular to the face. To overcome this problem certain corrections are applied to structural data obtained from surface mapping. Photogrammetric techniques are alsoavailablealthough not yet in wideuse. Priest(1993) hasdealtthe subjectin detail.

2.4.3 Diamond drilling for structural purposes

The important thiflg here is to obtain continuous cores which are as nearly undisturbed as possible so that the properties of discontinuities i.e. nature. infilling, inclination and orielltation are obtained. The drilling machine should have hydraulic feed so that a constant correct thrust is applied on the drill bit to obtain a good core recovery. A multi-tube core barrel is preferred. The inner hlbe or tubes are mounted pn a bearing so that they remain stationary while the outer barrel, which carries the diamond bit, rotates. The core, cut out by the bit, is transferred into the non-rotating inner barrel where it remains undisturbed until the barrel is removed from the hole. The most common size used at present is NX (56 mm).

The orientation of the core is important. Various techniques are available and details are given by Hoek and Brown (1980).

A borehole periscope, consisting of a rigid tube which supports a system of lenses and prisms, can be used for borehole examination. It is only effective to borehole depths of about 30 m. Borehole television cameras can also be used for borehole examination.

2.5 Rock Mass Classification The basic aim of a classification system is to be able to group certain things using

simple evaluatiodtesting techniques and using the results of such grouping to be able to predict the inherent nature of the material and how it would behave under different engineering situations. Classification of rockmass fonns the backbone of the empirical design approach and is widely employed in rock engineering. Geological classification system gives an idea to engineer about the typeof material that he has to deal with. This system based on mode of formation is very useful and often used by engineers. But as engineers are more concerned with mechanical and hydraulic characteristics and other geological features of rock masses in order to design and construct safe and efficient structures in rocks, this need prompted researchers to put forward classification systems taking into account the engineering aspects.

Since the engineering behaviour of a rock mass depends on the properties of intact rock and discontinuities. and ground water conditions, it is obvious that for classification of rock mass at least three parameters should be used; one parameter charac- tensing the intact rock properties, one parameter characterising the discontinuities and one parameter describing ground water conditions.

2.5.1 Rock quality designation (RQD)

A one-parametric classification of rock mass is possible by means of Roi l (Deere, 1968). It is a modified core recovery percentage in which all the pieces of sound core over I0 cm long are counted as recovery, and are expressed as a percentage of the length drilled. The smaller pieces resulting from closerjointing, faulting, or weather-

2 5 Rock Mass Classificat~on 15

ing are discounted. The core should be at least 50 mm in diameter and drilled with diamond drilling equipment with at least double-tube core barrel.

If the core is broken by handling or by the drilling process (i.e if the fractures are fresh breaks rather than natural surfaces). the fresh broken pieces should be fitted together and counted as one piece, provided they form the required length of 10 cm.

Material that is obviously weaker than the surrounding rock such as over-consolidated gouge is discounted. even if it appears as intact pieces that are 10 cm or more in length. (This type of material will normally only be recovered whenusing themost advanced drilling equipment and experienced or carefully supervised drilling crews.) The length of individual core pieces should be assessed along the centre line of the core.

RQD quantifies discontinuity spacing. An estimate of RQD can be obtained from discontinuity spacing measurements made on core or an exposure using the following equation developed by Priest and Hudson (1976):-

where h= the mean discontinuity frequency of a large discontinuity population; - 1 - -. -

X

where ir = the mean spacing.

Although RQD is a quick and inexpensive index, it bas limitation such as the disregarding of discontinuity orientation, tightness, and gouge (infilling) material. Consequently, while it is a practical parameter for core quality estimation, it is not sufficient on its own to provide an adequate description of a rock mass.

2.5.2 Multi-parametric classification schemes

Two schemes are currently widely used- 1. CSIR Geomechanics or Rock Mass Rating scheme developed by Bieniawski (1973) and 2. NGI Rock Mass Quality Index@) developed by Barton et al. (1974). In both schemes, the rock mass quality is deduced from six parameters.

Rock mass rating'(RMR) is determined from the following equation:-

where R = rating of uniaxial compressive strength for intact rock; RR~JII = rating for RQI)-index; Rd= rating for spacing of discontinuities;

rating for condition of discontinuities: Kw = rating for ground water conditions; and

RCXI= rating for orientation of discontinuities.

Ratings for the first five of these parameters are determined from Part (a) of Table 2.4. Rating adjustments for discontinuity orientation are summarised for tunnels.

Table 2.4 Rock mass rating scheme forjointed rock masses (after Bieniawski, 1976 and 1989). Part (a) Classification parameters and their ratings.

Part (b) Rating adjustment for discontinuity orientations. -- -- .- -:-. -

1 s r n k ~ ~ o d d ~ ~ ~ ~ ~ n _ 4 n ! ~ ~ ~ d i ~ ~ o n f i ~ ~ ~ j ~ ~ ~ ~ ~~. --= 1- Faroursble - - Fair Un&ouiable ..

, Krlinpr Tuliil~lr lo - -2 -5 -10 -12 --

i -.

Foundation3 -- : 0 -2 -7 - -15 -25 i -4

'Slopes 10 -5 -25 -50 -60 1 ; _ , ~ .-=----- i-- L . . . . ~

Part (c) Rock mass classes based onRMR

i llrlirle 81 to 100 , _ -- _=_-.--.. )61to80 141 to60 /2 l t040 I/czo -41 :, Cl!? "!!!he! . . ..-- ~~ ~ Ill 111 /IV v / O o o d mck I b n g o u d m c k . Pair mok !&>+!!?. ... . . Ven p o r rock

~ . - -

:

2 Geological Aspects 2.6 Exercises

foundations and slopes in Part (b) of Table 2.4. A summary of the five possible rock mass classifications based on the observed RA4R value is given in Part (c) of Table 2.4.

Rock mass quality (Q) is determined from the following equation:-

where ROD = RQD index; Jn = joint set number; Jr = joint roughness number; J , = joint alteration number; .Iw = joint water reduction factor; and

SRF = stress reduction factor.

The first term in the above equation is a crude measure of block size, the second term inter-block shear strength and the thirdtern active stress. The numerical ratings of the various parameters are given by Barton et al.

It is obvious that both systems emphasise the importance of properties of discontinuities (joints) (3 parameters in RMR system and 4 parameters in Q system). Both systems considertheRQD index in addition to ground water conditions. It is therefore not surprising that both systems can be related. Bieniawski (1976) has given the following equation relating the two ratings:-

RMR = 9 In (Q) + 44 (2.4)

2.6 Exercises 1 What is meant by "rock material" and "rock mass"? What makes them different

2 What are the characteristics of rock discontinuities that can affect the mechanical properties of the rock mass?

3 Write short notes on "Scanline survey in mapping surface and underground exposures''.

4 What is RQI)?

5 The mean spacing of all discontinuities in the direction of a scanline has been found to be 0.3 In. Estimate the RQII of the rock mass in the direction of the scanline.

6 In recent years, rock mass class~ficanon systems have become popular in the empir~cal d e ~ ~ g n of rock structures Expla~n why?

8 Outline various parameters used in the Rock mass quality (Q) system of the Nonueyan Geotechnical Institute.

9 A damp rock mass is characterised by the following parameters:-

Discontinuity water pressure = 0; Point-load strength index = 3 MPa; Discontinuity spacing = 0.5 m; and ROD = 55%.

Prepare a table of rock mass rating versus discontinuity condition

10 A tunnel is to be driven in a granitic rock mass. The following parameters have been determined for the rock mass:-

Strength of intact rock material - 150 MPa RQD - 70% Discontinuity spacing - 0.5 m Condition of discontinuities - slightly rough surfaces; separation < I mm; slightly weathered walls Ground water - wet The tunnel has been oriented such that the dominant discontinuity set becomes unfavourable. What is the rock mass rating for this application?

11 An underground crusher station is to be excavated in the limestone footwall of a lead-zincore body. From the geological investigations, the following parameters have been determined..

RQD = 80% Joint sets = 2 (This gives joint set number as 4) Joint roughness = rough (This gives joint roughness number as 3) Joint alteration = clay gouge (This givesjoint alteration number as 4) Joint water = large inflow (This gives joint water reduction factor as 0.33) Stress reduction = medium stress (This gives SRFas I )

What is the Rock mass quality (0 ) for this site?

7 Outline various parameters i n the Bieniawski's geomechanics classitication.

Physical and Mechanical Properties of Rock and Rock Mass

T he knowledge of physical and mechanical properties of rwks and rock masses is essential in any rock mechanics investigations connected with either civil or

mining engineeringprojects. Afterpredictingthe state of stress, strain or stored energy from the analysis of loads or forces being applied to the rocks and rock masses, the behaviour i.e. fracture, flow or simply deformation of rock and rock mass can be estimated from these physical and mechanical properties. This Chapter is concerned with these properties.

3.1 Physical Properties This Section deals with the general physical properties such as density, porosity,

water content, and swelling and slake-durability indices.

3.1.1 Density

Density, pis defined as mass per unit volume. If the weight (force), and not the mass, of a unit volume is measured, a unit weight, Wis obtained. The density is related to its unit weight as follows:-

where g = acceleration due to gravity

) The mass of a unit volume of rock in its natural state is different from the mass of

3 the same volume of rock containing only of its solid phase. Because ofthis, two terms "bulk den*" or simply "density" and "grain density" are in common use.

3.1.1.1 Gain density

Grain density. pgisthemass of aunit volurneofthegrains(i.e, solid phase ormineral skeleton) of a rock.

22 3 Properties of Rock and Rock Mass

where mg = mass of grains and V'= volume of grains.

Pg Grain specific gravity, F,? = - Pw

where pw = density of water at 4' C

3.1.1.2 Bulk density

Bulk density, p i s defined as the mass of a unit volume of rock. It depends upon the mineralogical composition, porosity and amount of water present in the pores. If bulk volume of the specimen is Vb (i e. pore volume, Vp + grain volume, V*) and bulk specimen mass is mh (mass of grains, mg + mass of water in the pores, mw), then

If the rock is completely dry, then dry density of rock, pdis obtained.

If the rock is saturated with water, then saturated density of rock, p , is calculated from the equation

Usually, the dry density of rock is determined and quoted as one of the rock parameters unless otherwise specified.

The density of rocks depends upon porosity, joints and other open spaces present. For the same rock type, the density increases as the depth increases due to decreased frequency of open cracks or closure of the cracks etc. under pressure of overlying rocks. Weathering of rocks near their outcrops decreases the density tirstly due to frachlring, secondly due to increase in volume of certain minerals as they come in contact with water (montmorillonites, anhydrites) and thirdly due to decomposition under the influence of atmospheric action (alteration of feldspars into clay minerals under the action of water and carbon dioxide).

3 I Phys~cal Properttes 23

Since most of the common rock-forming minerals have densities in the range 2650 to 2800 kgJm3, it could be argued that the presence of pore space in a rock would affect to a large extent the density of the rock.

3.1.2 Porosity

Porosity of a rock, n is defined as the ratio of the pore volume, Vp (i.e. volume of internal open spaces, interstices or voids) to the hulk volume of the rock, Vh.

The porosity can also be expressed in terms of grain density, pg and dry density of rock, pd as follows:-

The porosity of a rock depends upon its mode of formation and the following factors in general influence the porosity of rocks:-

1. Size distributio'n of grains. 2. Shape of grains. 3. Solidity of grains. 4. Orientation of grains. 5. Degree of compaction. 6. Amount of non-granular material (colloids or cement) in pores or coating thegrains.

During subsequent period., the rocks may have deformed developing cracks, fissures and joints or even certain minerals might have been dissolved away or chemically changed giving a decrease or an increase in porosity.

There are open pores (pores inter-connected with each other and linked to the external surface) and closed pores (pores that are locked up in the rock having no connection with the external surface or open pores) in a rock. Obviously, therefore, porosity is expressed as either total orapparent porosity. When all the pores are taken into account, then porosity value obtained is called total porosity. When open pores only (i.e, closed pores excluded) are considered, then porosity obtained is called apparent porosity.

For similar rocks, density and porosity are related. Tie nature of the relationships derived is linear In the case of sedimentary rocks (sandstone, limestone, dolomite, chalk, marble, shale, claystone, slate, sand. clay. gravel, alluvium and soils) the following equation with a correlation coefficient of -0.9648 has been derived:.

24 3 Properties of Rock and Rock Mass 3 1 Physical propemes 25

where t~ = porosity, %and 3 p = density, gicm

When only sandstoneisconsidered, thecorrelation coeficienthasimproved slightly to - 0.9674 with the following relationship:.

3.1.3 Water content e

The water content of a rock is the ratio of mass of water in the rock pores to the mass of a perfectly dry rock specimen expressed as a percentage. Saturated water content (when the pores are fully saturated) is the ratio of mass of water in the rock pores at saturation to the mass of a dry rock specimen as a percentage. It is also called void index. Void index is nothing but a measure of porosity of the rock and is extensively used as the p r imw characteristic of rock material in engineering. It should only he - - determined for rocks that do not appreciably disintegate uhen immersed in water

The mehod suggested by lntemdrional Society for Rock Mechanics (ISRM) in 1972 for determining void index is as follows:-

A representative sample is selected comprising at least ten pieces of rock, each having a mass of at least 50 g, to give a total sample mass of at least 500 g. The sample in an air-dry condition is packed into the container, each lump separated from the next and surrounded by crystals of dehydrated silica gel. The container is left to stand for a period of 24 hours. The container is emptied, the sample removed, blushed clean of lwse rock and silica gel crystals and its-mass A determined to 0.5 g. The sample is replaced in the container and water is added until the sample is fully immersed. The container is aeitated to remove bubbles of air and is left to stand for a oeriod of one - ~~~ - . hour. The sample is removed and surface-dried using a moist cloth, care being taken to remove only surface water and to ensure that no fragments are lost. The mass B of surface-dried sample is measured to 0.5 g.

The void index, 1" is calculated from the relationship

1" = - B-AxlOO% A (3.11)

Void index depends on the type and age of rock material. There seems to be a remarkable correlation between them.

3.1.4 Swelling and slakedurability indices

Many rocks particularly those containing clays are prone to swelling, cracking and disintegration when exposed toshortterm weatheringprocessesofwettinganddrying. Suppons placed in excavations made in such rocks experience cycles ofincreased and

decreased pressures depending upon the wetting and drying cycles. Special tests are required to estimate this aspect of mechanical behaviour These tests are commonly required for classification or characterisation of the softer rock materials. They may also be used, however, for the characterisation of harder rocks where the rock condition, its advanced state of weathering for example. indicates that they are appropriate.

Rocks that disintegrate during the tests may he further characterised using soil classification tests such as determination oftheliquid and plastic limits, the grain size distribution, or the content and type of clay minerals present.

The methods suggested by ISRM in 1972 are as follows:-

1. Swelling pressure index under conditions of zero volume change. 2. Swelling strain index for a radially confined specimen with axial pressure. 3. Swelling strain developed in an unconfined compression. 4. Slake-durability index.

lk%L This test is intended to measure the pressure necessary to constrain an undisturbed rock specimen at constant volume when it is immersed in water. W. This test is intended to measure the axial swelling strain developed against

a constant axial pressure, when a radially confined, undisturbed rock specimen is immersed in water. m. This test is intended to measure the swelling strain developed when an

unconfined, undisturbed rock specimen is immersed in water The test should only be applied to specimens that do not change their geometry appreciably on slaking; less durable rocks are better tested using a confined swelling test. lM.4. This test is intended to assess the resistance offered by a rock sample to

weakening and disintegration when subject to two standard cycles of drying and wetting.

In 1989, ISRM suggested methods for laboratory testing of argillaceous swelling rocks. The document consists of four parts:- I. Sampling, storage and preparation of test specimens; 2. Determining the maximum axial swelling stress; 3. Determining the axial and radial free swelling strain; and4. Determining axial swelling stress as a function of axial swelling strain. For details, reference should be made to the publication.

In the swelling index tests, peak values are usually obtained within 5 - 10 minutes, but for certain rocks. collapse mav take much longer time even up to a year or so. As - such, testsshould becarriedout foilonger ~e r iods i f an~ rock is suspected ofweakness or if it contains certain clayey constituents in the form of pockets and when the porosity and pore size are small.

Swelling aIso depends upon the degree of cementation. If the cementation bond is strong, swelling is small and the time required to achieve peak will be large. If the bond is weak, considerable swelling occurs and the peak arrives quickly.

The swelling characteristic of a rock is the funchon of its moisture content, grain size, nature of the bond between the grains and the chemical properties of the grain


material. The size of the pores governs the capillary suction pressures developed and perhaps also the osmotic suction and gravitational suction. The extent of cementation governs the ability of the individual gains to reorient themselvesunder the influence ofstressesdeveloped. Ifthebondisstrong, expansion will benegligible. A weak bond will show swelling and therefore swelling could be regarded as a measure of the bond strength and strength of the rock.

The relationship between swelling strain and void index for various rocks is given in Figure 3.1. It is predominantly linear. Similar linear relationship exists between bulk density and swelling strain index.

Figure 3.1 Swelling strain versus void index for various rocks (after Duncan et al., 1968).

shales and other compacted and indurared tine-gained rocks

sandstones and medium-gained cemented and indurated rocks

Void index

3.2 Mechanical Properties of Intact Rock Mechanical properties include strength and deformability properties. These proper-

ties will be discussed with respect to intact rock in this Section

3.2.1 Strength properb'es

3.2.1.1 Definitions

kkacnue is the formation of planes of separation in the rock. Strengh or peak strengh is the value of maximum stress attained before failure. It

corresponds to B i n Figure 3.2(a). Beyond peak strenyh, the rock may still have some

4 3 2 Mechanical Propert~es of Intact Rock 27

strength. The minimum or residual strengh is reached generally only after considerable post-peak deformation (point C in Figure 3.2(a)).

Brittle fr&ture is the process by which sudden loss of strength occurs across a plane following little or no permanent (plastic) deformation. It is usually associated with strain-softening behaviour of the rock as illustrated in Figure 3.2(a).

Ductile deformation occurs when the rock can sustain W h e r permanent deformation without losing load-canying capacity (Figure 3.2(h)).

Yield occurs when there is a departure from elastic behaviour i.e, when some of the deformation becomes irrecoverable as at A in Figure 32(a). The yield stress (qv in Figure 3.2) is the stress at which permanent deformation first appears.

1 Figure 3.2 (a) Strain-softening and (b) strain-hardening stress-strain curves (after Brady and Brown, 1993).

L V qv - yield stress

>

Strain

3.2.1.2 Stiff and se~o-~0rIt t '0l led testing machines

When compressing a rock specimen in a conventional hydraulic or screw driven loading machine, the specimen fails violently and uncontrollably after reaching the peak strength. Such afailure is not an intrinsic characteristic of the rock specimen but caused by the design characteristics of the loading system.

In a compression test, when a load I' is applied to the specimen, it is shortened by 6, and the testing machine is extended by 6,,, (Figure 3 3). Load-displacementcur~es for specimen and testing machine are shown in Fibare 3.4.


The elastic energy stored in the testing machine, Em is:.

Defining the slope of the load-deformation curve as stiffness, k (Figure 3.4):-

P k", = ~~ ~ (3.13) 6,))

This equation indicates that the stiffer the testing machine the lesser the elastic energy is stored in the testing machine. In the testing machine, this elastic energy is recoverable when the load is decreased. This may happen when the peak strength of the specimen is reached.

Figure 3.3 Deformation of the two system's components - specimen, Fs and testing frame, 6 , -when load P i s applied. (a) system:- specimen, jackandtestingframe; (b) specimen; (c) testing frame.

load and deformation

3.2 Mechanical Properties of Intact Rock 29

For controlled testing (non-violent failure):.

In the above equations, subscripts refers to specimen in post-failure state. In other words, the stiffness of the testing machine should exceed the stiffness of the rock specimen in the post-failure state to avoid violent failure. In a normal testingmachine, this is not so.

Today, the following technical means are available to avoid such violent failure and to get a loaddeformation curve in the post-failure range:-

1. Increase the stiffness of the testing machine. This can be achieved with very heavy columns and minimal amount of fluid in the hydraulic circuit. 2. Use of a servo-controlled testing machine.

Figure 3.4 Load-deformation curves for testing machine (left) and rock specimen (right). Dotted - Elastic energy stored in testing machine when peak strength of specimen is reached. Hatched - Elastic energy needed to deform specimen in post- failure region.

failure (yield) of testing machine

Load

port-failure curve

Testing machine Specimen

Deformation 6


The basic principle of servo-controlled testing machines is illustrated in Figure 3.5. A feedback signal (n, representing some typical experimental conditions (e.g. the axial deformation of the specimen, Ss), is generated by a transducer This electrical signal is compared with the program signal @)which represents the test conditions desired. If a difference exists between feedback and program signals, an error signal (e) is generated which acts on the servo-valve. If the feedback signal indicates eg . a higher deformation than desired (just the beginning of an uncontrolled failure), the servo-valve is opened leading to a rapid reduction of the effective fluid pressure acting in the hydraulic jack ofthe load frame. The continuous operation of such a closed-loop system ensures that the experiment is automatically controlled and follows the course set by the program.

Actual servo-controlled systems do not, of course, respond instantaneously. A finite time of the order of milliseconds will elapse between detection of incorrect actual conditions and the adjustment of the applied force, but testing practice has shown that this response time of a servo-system is small enough to prevent an uncontrolled failure.

Figure 3.5 Principle of a servo-controlled testing machine (after Hudson et a]., 1971).

elecwonic somparimn offeedbacksignal (n and pragam r i g a l (P)

I

displacement

I I . Y..

load frame. hydrnvlic jack and rpecimen

/' hydraulic

Y N I ) - V ~ V ~ circuit (10 Pump)

3.2 Mechanical Properties of Intact Rock 3 1

3.2.1.3 Common laboratory strength tests

To characterise the strength of intact rocks, uniaxial (unconfined) and triaxial (confined) compression tests, direct and indirect tension tests, and shear tests are widely used. The important features of these tests are reviewed here.

Equipment for preparing test specimens for laboratory tests include a drill press, a diamond saw and a surface grinder.

3.2.1.4 Uniaxial compressive strength test

The uniaxial compressive strength test is the most frequently used strength test for rocks, yet it is not simple to perform properly and results canvary by afactor of more than 2 as procedures are varied. The uniaxial compressive strength value is often used for comparison, and it serves as a good index value.

The main features of the test are as follows:-

I . Shape and size of specimens - Cylinders havingadiameter ofnot less than NX core size (approximately 54 mm). 2. Height-to-diameter ratio of specimens - 2.5 to 3.0. 3. Tolerances on dimensions of specimens - (a) The ends of the specimen flat to 0.02 mm. (b) The ends of the specimen perpendicular to the axis of the specimen within 0.001 radian or 0.05 mm in 50 mm. 4. Rate of loading - 0.5 to 1.0 MPa/s. 5. Environmentalconditions - Specimens, as far as possible, with the natural water content. 6. Specimens carefully centred in the loading machine and placed on steel platens. No capping material used. 7. The uniaxial compressive strength of the specimen calculated by dividing the maximum load carried by the specimen duringthe test, by the original cross-sectional area.

The factors affecting the test results are as follows:-

I . Internal factors - mineralogy, density, porosity, ga in size, void index and anisot- ronv ~-r,.

2. External factors - specimen geometry i.e. height-to-diameter ratio and size, friction between platens and end surfaces i.e. end effects, rate of loading and environment i.e. moisture, liquids and temperature

Internal factors

The rocks containing quartz as the binding material are the strongest followed by calcite. ferrous minerals: rocks with clayey binding material are the weakest. In general, the higher the quartz content, the greater is the strength.

The uniaxial compressive strength increases with increase in density. Figure 3.6 gives the relationship between the ratio (densityluniaxial compressive strength) and


uniaxial compressive strength. The following equation may be used to estimate the uniaxial compressive strength of rocks, cc:-

3 where p = density, glcm

In the case of carbonate rocks, the following equation may be used:-

crc = 0.0863 e2.85 MPa (3.18)

Figure 3.6 The ratio (densityiuniaxii compressive strength) versusuniaxial compressive swength for rocks (after Imazu, 1986).

Uniaxial compressive suenslh, MPa

The uniaxial compressive strength decreases with increase in porosity. The relationships between uniaxial compressive strength, GC in MPa and porosity, n in % for carbonate rocks and quartz rocks are given in Figures 3.7 and 3.8 respectively.

For carbonate rocks:.

- 254 r-" "9 ,, c - MPa

For quartz rocks:.

3.2 Mechanical Properties of Intact Rock

,: !j Dc, = 343 e- 0.108 n MPa

Figure 3.7 Uniaxial compressive strength versus porosity for carbonate rocks (after Smorodinov et al . 1970)

0 5 10 15 20 25 30 35 40

Porosity. %

Figure 3.8 Uniaxial compressive strength versus porosity for quartz rocks (after Smorodinov et al., 1970).

i

0 , 2 3 4 5 6 1

Porosity. %


Finer grain size leads to higher uniaxial compressive strength. The uniaxial compressive strength decreases with increase in void index. The

relationship for granite is given in Figure 3.9.

Figure 3.9 Uniaxial compressive strength versus void index for granite (after Serafim and Lopes, 1962).

0 2 4 . 6 8 1 0 Void index. %

Anisotropy It is seldom found that rock contains mineral grains, cracks and pores of equal size

and random orientation. Consequently, rock specimens are anisohopic and tested under different orientations to the direction of applied load. Figure 3.10 gives the relationship between the uniaxial compressive strength, GC and the angle between the minimum principal stress direction and the plane of lamination, a fo r three specimens of two kinds of schist. Theinfluence is maximum f o r a = 60". the decrease in strength being 75 lo 90% of that for a = 0"

External factors

End effects and the influence of height-to-diameter ratio When a specimen is compressed between the steel platens of a testing machine, it

tends to expand laterally as it shortens in axial direction (Poisson's effect). On the other hand, friction between platens and end surfaces of the specimen tends toprevent expansion. As a result, the specimenisnot in abomogeneous stressstate(Figure3.11). A high stress concentration exists at the circumferential contact and failure usually initiates at it ziving rise to conical or wedge-shaped frabments based on each platen, which are commonly observed in uniaxial compression tests (Figure 3.12). If end effects can be reduced or eliminated, a totally different failure mechanism occurs -


Instead of shear fractures with conical framents, tensile fractures with axial splitting are now observed.

Figure 3.10 Uniaxial compressive strength versus inclination angle of lamination plane to minimum principal stress direction for schists (after Akai et al., 1970).

- A-specimen 0 . . . . . . . . . . B - specimen

0 .-.- C -specimen

I I I I 0 30 60 90

Inclination angle of lamination plane,

As a consequence of these end effects, the stress distribution varies throughout the specimen as a function of specimen height-to-diameter (hid) ratio. As the hid ratio increases, a greater proportion of the specimen volume is subjected to an approximately uniform state of uniaxial stress. It is for this essential reason that a hld ratio of at least 2.0 should be used in laboratory uniaxial compression testing of rock. Figure 3.13 shows some experimental dataillustrating this effect. Effect of size

Rocks are composed of crystals and grains in a fabric that includes cracks and fissures; understandably, rather large specimens are required to obtain statistically complete collections of all the components that influence uniaxial compressive strength. When the size of aspecimen is so small that relatively few cracks are present, failure is forced to involve new crack growth, whereas a large specimen may present Pre-existing cracks in critical locations. Thus uniaxial compressive strength of rock is size dependent. It could also be expected that with sufficiently large specimens so that distribution of cracks and fissures would not be affected by size. uniaxial compressive strengh would not be affected with further increase in size.

36 3 Propelties of Rock and Rock Mass

Figure 3.11 Influence of end constraint on stresses and displacements induced in a uniaxial compression test. (a) Desired uniform deformation of the specimen; (b) defonnation with complete radial restraint at the specimen end-platen contact; (c) non-uniform normal suess, D,, and shear stress, r induced at the specimen end as a result ofend restraint (after Brady and Brown, 1993).

Often the uniaxial compressive strengh, or. is given by the following relationship:-

where K= a constant; a = a characteristic dimension of the specimen and x = a constant. For coals, various values of x have been suggested, ranging

from 0.2 to 0.5

The uniaxial compressive strenah test results of granite, sandstone and limestone are given in Figure 3.14 indicating a decrease in uniaxial compressive strength with increase in size. Rate of loading

High rates of loading tend to increase the uniaxial compressive strength of rocks. Figure 3.15 shows the results on various rocks. However, normal rates of loading, such as 0.5 MPaJs to 3 MPds, show no significant change in uniaxial compressive strenM. Moisture

Moisture in rock could alter the uniaxial compressive strength by interacting with ~nineral surfaces and altering their surface properties and the nature of bonding. Reduction in uniaxial compressive strength due to moisture has been reported by numerous investigators. Figure 3.16 shows theeffects ofmoisture content on uniaxial compressive stten& of sandstone. Since the extent of reduction depends on the rock type and the test conditions, it is best to determine the uniaxial compressive strength of rockunder the moisture conditions expected to be encountered in the structure in the field.

3.2 Mechanical Propelties of intact Rock 37

Figure3.12Typical conical failurein auniaxial compression test specimen(afterPau1 and Gangal. 1966)

Figure 3.13 Uniaxial compressive strength versus heighu'diameter ratio for granite (I), dolomite (2). trachyte (3). sandstones and siltstones (4 - 8) and saturated granite (9) from various sources (after Hawkes and Mellor, 1970)

300 r \

0 1 2 3 4

Heightidiameter ratio

38 3 Properties ofRock and Rock Mass

Figure 3.14 Uniaxial compressive strength versus volume for cubical specimens of various rocks (log-log scale) (after Lundborg, 1968).

50 1 1 I I I I

0 lo 100 I 000 10000

Volume, cm'

$ a 10

5 200 - - D > .- * * e ? O, LCQ-

Liquids Figure 3.17 shows the effect of various liquids on the uniaxial compressive strength

of quartzitic sandstone. The uniaxial compressive strength is inversely proportional to the surface tension of different liquids with which the specimen is saturated.

Figure 3.18 shows the effect of different liquids on the compressive strength of sandstone. The figure shows the importance ofpH on the compressive strength ofthis silica rock.

Tivo principal theories have been proposed to explain the mechanism by which the liquids affect the compressive strength of various rocks. The first, advanced by Rehbinder and co-workers in 1944, proposes that the strength is altered by a change in the surface-free energy ofthe developing crack or fracture due to adsorption of the

- sandstone

\ limesrone

l ~ q u ~ d The se~.ood theory or'hesruood dnd co-workers (Westwood. 1971) proposes thsr thellquld altersd~slocauondena~tv and mob~lltv around thecrack noasa funcuon

~ >~ ~~~ ~ ~ - - . ~ ~-.~. of the zeta potential of the liquid-rock system a& thus inhibits or enhances crack propagation. %ere is still a controversy about the merits ofthe two mechanisms. Temperature

Simpson and Fergus reported results of their study on the effect of temperature on the uniaxial compressive strength of diabase in 1968. The specimens were air-dried at 27" C. 1 IOU C, 190" C and 345" C. Theresults show a pattern of increased strength with an increaseindrying temperature. They explained the result sin terms ofmoisture content.

3 2 Mechanical Propelbes of Intact Rock 39 i

i Figure 3.15 Uniaxial compressive strength versusloading rate forvarious rocks (aftel

6 sano et a1 , 1981) i

s ' 40 . F -.---- Sera sandstone

2 0 ~ " " ' " " l ~ 1

lo4 lo-' lov 1 0 10' to6 tan Smas rate. ma/$

I . I I 8 I I I I . / . I

10.' 1 ~ 3 I@ 10.' lo1 10'

Sbain me. r"

The effect of cryogenic temperatures on the uniaxial compressive strength of two limestones and iwo sandstones was studied by Brighenti and results were reported in 1970. His general conclusions are that (1) the rocks tested donot show notable strength decreases on account of aging at cryogenic temperatures; and (2) strength of dry and saturated rocks increases, generally speaking, when the temperature decreases

Normally, tests are conducted at room temperatures. If the i n situ conditions are different, the tests should be conductedin simulated atmosphere. Very little work has been done in this area and more investigations are needed.

3.2.1.5 Point load strength test

The uniaxial compressive strength test, as described above, requires careful specimen preparation and is time consuming. Sometimes this is not justified and approxi-


mate strength values are adequate. In that case, a point load strength test may be performed by means of portable equipment either in the laboratory or in the field.

In this test, rock specimens in the form of either core (the diametral and axial tests), cut blocks (the block test), or il~egular lumps (the irregular lump test) are broken by application of concentrated load through a pair of spherically truncated, conical platens (Figure 3.19).

Figure 3.16 Uniaxial compressive strength versus moisture content for quartzitic sandstone (after Colhack and Wiid, 1965).

dried overp20s 1. ~ i ~ u r e 3 . 1 7 Uniaxial compressive strength versus surface tension ofimmersion fluids for quartzitic sandstone (after Colback and Wiid, 1965).

I

0 0.001 0.002 O.W3 0 . 0 ~ 0.005 0.006 O W 7

Surface tension of immersion fluids. Nlm

where 1'= the load required to break the specimen and [Ic = the equivalent core diameter

= I ) for dia~netral tests and 4 W I )

= F f o r axial. blockandlump tests. Pi

WandD aredefinedinFigure3.19.

I~varieswith O,,sothatasizecorrection must beappliedtoobtain auniquepoint loadstren~thvaluefortherockspecimen.


The size-corrected point load strength index i,(jm of a rock speclmen is defined as the value of /, that would have been measured on a specimen with De = 50 mm.

When testing specimens of De other than 50 mm, size correction may be accom- plished by the use of the following formula:.

where F= the size correction fact01

- - Theoretical consideration of point load test shows that it gives a measure of tensile

strength. The results are, however, also sufficiently related to uniaxial compressive strength. /,(.mi is approximately 0.80 times the uniaxial tensile or Brazilian tensile strength. On average, the uniaxial compressive strength is 20 - 25 times the point load strength. However, in tests on many different rock types the ratio can vary between 15 and 50, so that errors of up to 100 % are possible in using an arbitrary ratio value to predict uniaxial compressive strength from point load strength.

Figure 3.18 Uniaxial compressive strength versus pH value of immersion fluids for sandstone. Closed circles - dodecylammonium chloride; small circles - sodium carbonate; large open circles -aluminium chloride (after Street and Wan& 1966).

3 5 7 9 1 1

pH value

3.2.1.6 Triaxial compressive strengm test

Rocks in the earth's crust generally exist in a confined state i.e. surrounded by other rock which exerts a stress from all sides on the element under consideration. Thus,


the element is under a triaxial state of stress. Hence, to obtain a more realistic idea of i how the rock will behave during protolype loading it is oRen important to measure

the triaxial compressive strength of rocks.

1 Figure 3.19 Specimen shape requirements for (a) the diametral test: (b) the axial test: 1 i

(c) the block test; and (d) the irregular lump test (after ISRM, 1985). I I i (4 (b) 1 I L. 0 5 D

A true triaxial (polyaxial) test is carried out by applying different normal stresses to three pairs of faces of a cube, plate or rectangular prism of rock. The great difficulty with such a test is that the end effects exert a substantial influence on the stress distribution within the specimen introducing marked errors. Owing to experimental difticulties, afew investigations of failure under true triaxial stress systems, in which


all three principal stresses are different, have been made. Although the value of the intermediate principal stress increases the triaxial compressive strenyh of the rocks somewhat, it can be neglected.

In a conventional ttiaxial tesf axial stress, 01 > lateral stress, 0 2 = 0 3 , The triaxial compressive strength of rock is measured on cylindrical specimens prepared in a similar manner to those for uniaxial compression tests and of similar dimensions and tolerances. The specimen is encased in a flexible, non-porous sleeve and placed in a suitably designed cell. It is subjected to a confining pressure via a hydraulic fluid around the sleeve and axial pressure by means of the platens of a loading machine. The axial load is generally applied at a selected deformation rate and under a predetermined, constant confining pressure. Failure is achieved when either the load reduces, or becomes constant, or a predetermined deformation is reached. Instead of a single value, different compressive strengths are obtained with various confining pressures.

The most commonly used arrangements are:.

1 . Longitudinal compression and confiningpressure. 2. Longitudinal compression with confining pressure and interstitial pore pressure.

For normal engineering applications, equipmentsimilarto Hoek andFranklin (1968) apparatus (Figure 3.20) can be recommended.

The factors affecting the test results include:-

1. Confining pressure. 2. Pore pressure. 3. Temperamre. 4. Strain rate.

Confining pressure

The effects of confining pressure on the stress-strain curve are shown in Figures 3.21 and 3.22. From these figures, it can be concluded that theconfiningpressureincreases the compressive strengih of the rock. It is also observed that rock tends to flow with higher confining pressures i e , behaves in a more ductile manner (Figure 3.21): For confiningpressures of up to about 50 MPa, there is a well defined peak strength with a decreasing strength in the post-failure range. It is said that the rock is brittle (or strain-softening) under the conditions the test is performed. The curves for confining pressure greaterthan about 68.5 MPaare completely different, since the rock can now undergo strains ofover 7% with noloss in strength. This is generally known as ductile (or strain-hardening) behaviour The conclusion from Figure 3.21 is that at a rather ill-defined value of the confining pressure there is a transition from brittle to ductile behaviour This is called the brittle-ductile transition.

I 3 2 Mechanical Propenies of Intact Rock 45

8

I Figure 320 Hoek and Frankhn tnaxial cell (after Hoek and Franklin, 1968) E

hardened and ground steel spherical seas

clearance gap

mild steel cell body

rwk specimen

oil inlet

main Bauges

rubber sealing sleeve

j In connection with engineering problems most rocks can be regarded to be in a brittle $ state although some rocks (e.g. water saturated shales) tend to a more ductile i behaviour.

Pore pressure

An increase in pore pressure decreases the compressive strength. The effect of pore pressure on the compressive strength is commonly described by the "law of effective stress". Hence, the effect of the confining pressure is negated by the introduction of pore pressure (Figure 3.23). However, it is probable that in many rocks the effects of water adsorption on the grain boundaries cause departure from this relationship.

Temperature

Generally an increase in temperature results in a decrease in compressive strenyh and an increase in ductility. F iq re 3.24 shows deviator stress (01 - 03) - axial strain curves for granite at a confining pressure of 500 MPa and different temperarures At room temperaturethebehaviouris brittle, but at 800°C the rock is almost fully ductile. The effect of temperature on the deviator stress at failure for dityerent types of rocks is different (Figure 3.25).


Strain rate

Tne compressive strength usually increases with an increase in strain rate (Figure 3.26).

Figure 3.21 Effects of confining pressure on the axial stress - axial strain curve for Carrara marble (the numbers on the curves are confining pressures in MPa) (after von Karrnan, 191 1)

Axial mais %

Brittleductile transition

Rock behaviour is affected by the surrounding pressure, pore pressure, temperature. strain rate and the presence of water or other substances that may affect it chemically. The last parameter is notwell understood and requiresfurther investigation. The other factors have been briefly discussed above. In general, the ductiliv of the rock increases with pressure and temperature. and decreases with higher pore pressures or strain rates. This situation is schematically depicted in Figure 3.27.


Figure 3.22 Effects of confining pressure on axial stress - axial strain curve for Tennessee marble (the numbers on the curves are confining pressures in MPa) (after Wawersik and Fairhurst, 1970)

ZOO k I

0 10 20 30 40 50 60 70

Axial main. 10~~emlnn

Figure 3.23 Effects of pore pressure on the axial stress-axial strain curve of a limestone tested at a confining pressure of 69 MPa (the numbers on the curves are pore pressures in MPa) (after Robinson. 1959).

r

1 .o 2.0

Axial strain. %

7 : 3 Properties of Rock and Rock Mass . < : 3.2 Mechanical Properties of Intact Rock

Figure 3.24 Effects of temperature on the deviator stress (DI - D?) - axial strain curve for granite at a confining pressure of 500 MPa (the numbers on the curves are temperatures in degree C) (atter Griggs et al., 1960)

Axial strain. %

3.2.1.7 Tensile strength tests

Rocks are weak in tension, hence if tensile stresses exist in a rock structure, there is little doubt that these will contribute towards its failure. Some of the tests developed for determining the tensile strength of rocks are described here.

Direct pull test

The best way to measure the tensile strength of rock is in direct pull. However, the design of grips for the ends without damaging them and the accurate axial and centric loading of the specimen need attention. Several specimen shapes and methods of attaching specimens to the uullin~ svstems were develooed. but most of these - . , ,~~

techniquesdo not suit the engineering requirements of simplicity and reliability. Right cylindrical NX - cores (54 mm diameter). with a height-to-diameter ratio of 2.0 to 2.5 bonded with end-caps by epoxy cement are used. The tension is introduced through chains that prevent bending or torsional forces on the specimen (Figure 3.28).

Brazilian test

The direct pull test discussed above gives the most reliable tensile strength values for rocks but it is time-consuming (and thus expensive). For most construction or design purposes. an approximation of the tensile strength is adequate. Hence, the Brazilian test is used.

Figure3.25 Deviator stress (01 - 03) at failure versus temperature for different types of rocks at 500 MPa confining pressure (after Griggs et al . . 1960).

dunite

PYroxenite

Solenhofen .. limestone ~..

Test specimens (discs) with a diameter of about 50 mm and a thickness-to-diameter ratio of about 0.5 to 1.0 are cut from a core. These then are compressed to failure across a diameter (Figure 3.29) and the failure load P noted. The tensile strength, o, is given by

where I = thickness of the disc and < I = disc diameter.

A preferred method of testing is to use machined steel loading jaws (Figure 3.30) designed so as to contact the specimen surfaces over an arc of approximately 10" to Prevent flattening of the specimen at the contact points.


The Brazilian test sives proper results only when failure starts with the development of avertical tensile crack in the inner parts of the specimen. Iffailureis initiatedunder theloading platens, theBrazilian test has to be regardedasinvalid forthe measurement of tensile strength.

Figure 3.26 Effects of strain rate on compressive strength-confining pressure curve for Westerly granite (the numbers on the curves are strain rates per second) (after Logan and Handin, 1970).

Confining pressure. hCa

3 2 Mechan~cal Properties of Intact Rock 5 1 E

Figure 3.27 Schematic d~agrarn showing the brittle-ductile transition in rock (after Gnggs and Handin, 1960)

C increasing pmsure - increasing rernprature - increasing pre Ouid p m r e - increasing main rare -

relative duciiliy -

brinle Umi$onal d u d e

Figure 3.28 Loading arrangement for direct pull test

t chain


Figure 3.29 Schematic of the Brazilian test.

load

induced tensile stress

Figure 3.30 Loading jaws for the Brazilian test (after ISRM, 1978a).

hole with clearance on dowel

i i w k j a w

test s~ecimen

Bending test

When a specimen (prismatic or cylindrical) is strained by bending, tensile, compressive and shear stresses are developed. In cases where a portion of the specimen is placed in pure bending, only tensile stresses are developed on the convex side ofthe specimen and compressive stresses on the concave side. The maximum tensile stress at theeatreme most fibre at failurecan betaken as thetensile strength (it is alsotermed


the modulus of rupture). This property is particularly useful in evaluating the tensile strength of rock spans such as are found in tunnel and mine roofs.

The four-point loading (Figure 3.31) is preferred to three-point loading because it gives more uniform bending stress contiguration between the two line-load points. With three-point loading i.e, centre loading, the maximum stress occurs directly under the load and causes the fracture to occur at or near the point of applied load regardless of the relative strength of the surrounding material. The tensile strength measured by the four-point loading, therefore, is slightly less than that given with three-point loading because the developing fracture can seek out the weakest material between the load application lines.

Figure 3.31 Four-point loading fixture for bending test (after Lewis and Tandanand. 1974)

The stress, a at any section of the beam is given by

= 4 9 I

whereM= bending moment; y = distance from the neutral axis; and I = cross-sectional area moment of inertia.

For the four-point loading, Mis uniform in the central portion of the beam:-

where P = load and I. = span length between support loads.

54 3 Properties ofRock and Rock Mass

Also for a rectangular beam,

where b = width and t = thickness.

The modulus of mpture, oh is therefore defined as the tensile stress at the lower I surface of the specimen wherey = 2.

where Pm = the applied load at failure.

Since the exact location of the neutral axis is unknown and the above equation is also based on the assumption thatthematerial islinearly elastic up to failure, therefore, the test result is only related to, but is not, the tensile strength

Since cylindrical specimens are occasionally used, the above equation becomes for these specimens

where d = core diameter.

Factors affecting the test results

The factors affecting the results are the same as discussed under uniaxial compressive strength test.

Comparison of results obtained by different methods


but to a combination of stresses. Applicability of results will hence differ from case to case.

Table 3.1 Comparison of tensile strength values obtained by different methods Vutukuri et al.. 1974).

Various investigators have used more than one method in their tests. The results obtained by some of them are given in Table 3.1 for comparison. The results vary i

li within wide limits. Strength values determined by direct test are the lowest in most I

cases, because the volume of material under the tensile stresses is the highest. In other arc of contact - 7 degrees 2 methods. not only is the volume under maximum tensile stress too small, but there is

a combination of stresses also and the combined effect of these stresses has not been 3.2.1.8 Shear strength tests evaluated as yet. Obviously, the results obtained from these methods are questionable.

The best method seems to be the direct pull test. Strength values obtained by other In the field most often racks fail in shear. Hence, the shear strength of a rock is an methods may be useful where failure takes place not due to uniaxial tensile stresses important desibn criterion. Two kinds of tests are available to establish the shear

strength parameters, nameiy. direct shear test and tnaxial test.

5s 3 Properties of Rock and Rock Mass

Triaxial test

The triaxial test (Figure 3.35) can be used to establish the shear strengths at, and in the close vicinity of, their peak values The limited displacement available to this method is not usually enough however to establish the residual values of the shear strengths.

i 3 2 Mechanical Properties of Intact Rock 59

3 Figure 3.36 Coulomb's strength envelopes in terms of(a) shear and normal stresses 1 . Mohr's circle with strai&t line envelope - and (b) principal stresses. f

Figure 3.35 Principle of determining shear strength from a triaxial test

I t i i Figure 3.37 Curved line Mohr's envelope showing the variation of the shear strength

1 parameters.

The specimen is enclosed in an airtight flexible membrane; confining pressure, a3 is applied and held constant during the test by means of a cell fluid. Axial stress, a, is then applied and continuously increased until failure occurs. The test is repeated at various confining pressures of interest.

The resultant shear stren@h envelope, drawn tangent to the Mohr's circles, is either a straight line as shown in Figure 3.36(a), or is more often a curved line as shown in Figure 3.37. In the case of a straight line envelope (also called Coulomb's strength or failure or

fracture envelope), evaluation of the test results is simple because the same shear strength parameters i e . cohesion, c and friction angle, (p apply over the total range of the normal stress, on. The shear strength is then evaluated from the following equation:.

7 : (: + G,, tan ip (3.30)

. * ;,,L'* ,_....I ?a ....'

03 01 a,, a' a3

This equation is also called Coulomb's shear strength criterion. Applying the stress transformation equations ;o the case shown in Figure 3 35 gives

I a n d ~ = ~ ( o ~ -o3) s in2a (3.32)

From the equations (3.30). (3.3 I ) and (332). the limiting stress condition on any plane defined by a can be obtained as:-


There will be a critical plane on which the available shear strength will be first reached as ol is increased. The Mohr's circle construction of Figure 336(a) gives the orientation of this critical plane as

For the critical plane,

sin 2 a = cos (p cos 2a = - sin cp

Substitution of these equations (3.35) and (3.36) into equation (3.33) gives the following equation:-

2ccoscp+o3(1 +sin(p) S, =

1 - sin v

The linear relationship between o3 and the peak value of 01 is shown in Figure 3.36(b). The slope of this envelope is !I, i.e.

y is related to (p by the equation

I + s i n 9 tan {v = I - sin (p

and GC is related to c and cp by

2 c c o s 9 oc = I - sin Q

If the Coulomb's envelope shown in Figure 3.36(b) is extrapolated to 01 = 0, it will intersect the o? axis at an apparent value of uniaxial tensile strength of the material given by

2 c cos (p "I = i ;sin-<; The calculated tensile strength values are usually higher than measured values. I t is somewhat more difficult to evaluate the test results if the strength envelope is

a curved line, because in this case each pair of strength parameters, such as C I and


rpl, cz and cp2, etc., applies only to its particular portion of the curved envelope that can be safely replaced by a tangent without excessive error

The triaxial test seems to be the best method of obtaining shear strength of rocks, but it requires expensive equipment andthe testis time-consuming. Besides, it is very difficult to vary independently the shear stress and normal stress on apredetermined plane of failure.

3.2.2 Deformation propetties

3.2.2.1 Definitions

A material iscalled elastic when it recovers toits original state after being subjected to a loading-unloading cycle. The relationships between stresses and strains are represented by constants called the elastic constants of the material. The total load acting on an area divided by the area is called the stress. The displacement between two measurement points on application of force is called deformation. This deformation divided by the original length is called the strain.

YOWP'S modulus (often referred to as Modulus of elasticiiy), E - The ratio of axial stress to corresponding axial strain below the proportional limitunderuniaxial loading (Figure 3.38(a)).

Poisson's ratio, v - The absolute value of the ratio of transverse strain to the corresponding axial strain resultingfromuniaxial loading below the proportional limit of the material Figure 3.38(a)).

&ear modulus (often referred to as Modulus of riediiy)), G - The ratio of shear stress to corresponding shear strain (Figure 3.38(b)). Bulk (often referred to as . . .

), K - The ratio of hydrostatic pressure to the volumettic strain it produces (Figure 338(c)).

The above properties are constants for rocks which behave in a linear elastic manner according toHooke's law If the material is also isotropic, these parameters are related to each other as follows:-

3.2.2.2 Static elastic constants of rock Many methods of measurement of strain and loading techniques are employed to

determine the stress-strain relationships for rocks. The techniques usually involve the measurement of load and corresponding strain under some stress state, usually


uniaxial, triaxial with two principal stress components being equal, or hydrostatic. Some details regarding measurements under uniaxial loading are given here.

Figure 3.38 Loading geometries to define elastic moduli. (a) Young's modulus and Poisson's ratio; (b) Shear modulus; (c) Bulk modulus.


Uniaxial loading

Most ofthe available data is for the uniaxial compressive loading condition; very little data exist for tensile loading. Uniaxial compression tests are usually conducted on cylindrical specimens with a height-tediameter ratio of about 2. Axial deformations are usually measured over some gauge length by either electrical resistance strain gauges (Figure 3.39). transducers with a deformation jacket which attaches to the specimen, or by a transducer which measuresthe crosshead displacement ofthetesting machine. Two or three axial sensors equally spaced around the specimen are used so that more accurate average strains may be recorded. Lateral strains are usually measured by two or three electrical resistance strain gauges equally spaced around the circumference of the specimen (Figure 3.39), or by two or three transducers to measure radial dispkacement~, or a single transducer utilising a chain which wraps around the specimen to sense the change in circumference.

Figure 3.39 Use of electrical resistance strain sauges for measuring axial and lateral strains in uniaxial compression.

axial load

axial sWin gauge

Fibxre 3.40 shows the shape of atypical stress-strain curve for a rock behaving in a non-linear manner The upward concavity of the axial stress-axial strain curve in the lower part is due to the closing of pre-existing cracks. The linear part of the curve involves predominantly the elastic behaviour of the rock. The downward concavity in the upper part is due to the initiation and goivth of micro cracks in the rock as failure is approached. The volumetric curve departs from linearity. The relative prominence ofthe three regions described depends upon the type of rock.

3 Properties of Rock and Rock Mass 64 I 3.2 Mechanical Propenies of Intact Rock 65

Figure 3.40 Typical stress-strain curves (after Thill, 1984).

o Poison's ratio r axial strain n vdumenicsuain

- 8 - 6 - 4 - 2 0 2 4 6 8 10

The post-failure portion of the curve can only be obtained in a stiff or servo-controlled testing machine. In this region, the overall mechanical behaviour of the specimen can only be given as a load-displacement curve. The load decreases with increasing deformation i.e. the load-deformation curve has a negative slope and the rock is now in a fractured state. This implies that even cracked, fractured rock offers resistance to loads applied to it. An excavation may be such that it will not collapse although the rock material surrounding it has failed by exceeding its material strength. Actually, fractured rock may even be desirable since it will not fail suddenly and violently.

Figure 3.41 shows methods for calculating Young's modulus from axial stress-axial strain curve. Since most rocks do not show an elastic behaviour it is more appropriate to use the term modulus of deformation instead of Young's modulus.

Factors affecting the stress-strain curve

1 . Specimen geometry 2. Platen conditions. 3. Rate of loading. 4. Temperature 5. Confinement. 6. Stress level. 7. Cyclic loading. 8. Water content.

Figure 3-41 Methods for calculating Young's modulus from axial saess-axial strain curve. (a) Tangent modulus measured at a fixed percentage of ultimate strengrh, 0"; (b) Average modulus of linear portion of axial stress-axial strain curve; (c) Secant modulus measured up to a fixed percentage of ultimate strength (after ISRM, 1979).

The following is a summaly from the work of Lama and Vutukuri (1978):

Modulus values of homogeneous rocks are not influenced by specimen size, but for non-homogeneous specimens, the heightidiameter (hid) ratio and specimen size play


important roles. The drop in value with increase in (hid) ratio may be considerable (a factor of 2 - 3).

Platen conditions, rate of loading and confinement affect the stress-strain curve, but not to ageat extent. ThemodulusvaluescalculatedwiII vary only by 5 - 10% between extreme conditions.

The influence oftemperature on the stress-straincurve becomes marked particularly when a higher confining pressure is in existence. The most notable effect is the large decrease in the maximum stress at which the rock yields as the temperature increases. The ductility of rocks increases with increase in temperature and pressure and othenvise brittle rocks show ductile behaviour.

Thevariation in modulusvaluewith stressis due totemporary or permanent changes occurring in the structure ofthe specimens and hence the loading and unloading paths followed by the specimens may be quite different. The modulus value obtained on unloading represents to a greater measure themodulus ofthe intact material. The ratio between the unloading and loading modulus is a measure of the degree of fracturing of specimens.

Modulus anisotropy is the result of micro- and macro-fabric of the rock. Rocks containing spherical grains or randomly distributed grains have low or no modulus anisotropy Elongated shapes result in modulus anisotropy in rocks and the ratio may vary between 1 - 2.5.

Presence ofjoints and bedding planes geatly influence modulus values. Modulus of deformation for rocks stressed parallel to the bedding planes is greater than when stressed at right angle to their bedding planes.

Poisson's ratio of rocks is ve j m u c h dependent upon the stress level and is greatly influenced by theopening orclosing ofcracks. Thecalculatedvaluesat direrentstress levels may vary from 0.1 to 1.0 or even more. Under such circumstances, specimens should preferably be loaded to a stress level lower than that at which any permanent changes occur in rocks.

Modulus decreases as water content increases. Poisson's ratio increases as water content increases.

3.2.2.3 Dynamic elastic constants of rock

So far, the elastic constants of rock have been considered from the point of view of its reaction to static stresses. However. a rock may be subject to transient dynamic !oading and the way it reacts to the dynamic stresses is important.

'Two methods are usually used to determine the dynamic elastic constants, namely, the resonance method and the ultrasonic pulse method. However, only the ultrasonic pulse method is described here in some detail. The compression (longitudinal) and shear wave velocities determined in this method are used to calculate the dynamic elastic constants.

Ultrasonic pulse method

The ultrasonic pulse method best suited for determining elastic wave velocities in rock is the pulse transmission method that employs separate driving and receiving


transducers (Figure 3.42). Transducer elements commonly usedfor longtudinal wave propagation are lead-zirconate titanate, or barium titanate ceramic discs operating in a thickness expansion mode. For shear wave transmission, ac cut quartz or specially cut ceramic elements are frequently used or a longitudinal wave is mode converted with suitably cut prisms.

Figure 3.42 Ultrasonic pulse measurement system (after Lewis and Tandanand, 1974).

In order to measure longitudinal "infinite medium" velocity, I;, specimen dinien- sionsnormal to the direction of propagation, dshould be large in comparison with the

d acoustic wavelength, h employed i.e. - > 2. The longitudinal bar of intinitely thin rod

h d .

velocity, I; , can be obtained by the pulse methad if the . I S made small (0.6) by A

reducins dor increasing ?,. To propagate the longitudinal "infinite plate" velocity. 1 jii wavelength of the pulse must be long compared with the thickness of the plate but short in comparison with the planar dimensions of ihe plate. In all cases, to avoid excessive attenuation of the transmitted elastic wave. acoustic wavelength should be long i n con~parison wit11 specimen grain size. Obviouslv. either specimen dimension or acoustic wavelength, orboth, nlust beadjusted to conform with these lsquirenlents. Ofen it is convenient to adjust the acoustic wavelengh by changing the transdi~cer element for another of a dilferent resonant frequency. For bar and plate velocity measurements, it is usually Inore convenient to control specimen shape to that of a long, th in bar or a thin plate.


In the pulse transmission technique, a rock specimen is placed between, and in physical contact with, two piezoelectric transducer elements; one acts as a driver, and theotheracts as areceiver Apulse generator supplies ashort-duration electrical pulse to the driver transducer. The electrical pulse is converted into a mechanical wave or impact by the driver transducer, and this wave is transmitted to the specimen. After travelling through the test specimen, the mechanical pulse, or elastic wave, is picked up by the receiving transducer, reconverted to an electrical signal, and displayed on the screen of a cathode ray oscilloscope (CRO) (Figure 3.42). The transit time required for the mechanical pulse to pass through the specimen is used to determine elastic wave velocity. Transit times are displayed on a timer and recorded by a printer. Amplitude of the pulse used is typically in the range of 50 to 700 V. Pulse rise time i.e. the time the pulse takes to rise from 10 to 90% of its final amplitude, is typically about 0.2 11s or less in order to excite the pulse frequencies commonly employed in testing rock i.e, those from about 100 WIz to 1 MHz. Pulse length is set in the range from about 1.5 to 10 ps, and the pulse recurrence frequency usually is set at about 60 Hz. The couplant found most satisfactory for the transfer of longihldinal wave energy between transducer and specimen is one or two thicknesses of plastic electrician's tape placed over the cover plates of the transducers. Couplants used for shear wave propagation include Salol, Canada balsam, Lakeside 70 cement and Nonaq stopcock grease.

Elastic wave velocities are calculated by the equation

where s = distance travelled by the wave in the rock and t = pulse transit time in the rock.

Measured transit times are corrected for instmmentation delays, operator errors, system errors, etc.

Given the appropriate wave velocities and density, the modulus of elasticity, E Can be computed from the following equations:-

where p = density; V,, =longitudinal bar velocity; Vv = longitudinal infinite medium velocity; and

C< = shear velocitv.


The modulus of rigidity, (;can be calculated from the following equation:-

Factors affecting propagation velocity of waves in rocks (hence dynamic elastic constants)

I . Rock type. 2. Texture. 3. Density. 4. Porosity. 5. Anisotropy. 6. Stress level. 7. Water content. 8. Temperature.

The following is a summary from the work of Lama & Vutukuri (1978):-

Generally it can be said that velocities are higher for more dense and compact rocks, lower for less dense and compact rocks.

The velocity in a rock may be related with the velocities in rock's various mineral components. Such relationship does not take into account factors such as grain size, orpreferred orientation of crystals. Some results indicate an increase in velocity value with increasing hornblende percentage and a decrease in velocity with increasing quartz content.

The velocity of waves is influenced by the size of the grains constituting the rock. The velocity is greater as a rule in fine-grained rocks than in coarse-grained rocks.

In general, the velocity increases as the density of rock increases. The relationship may be linear or curvilinear.

The propagation velocity decreases as the porosity increases. This is true for both dry and saturated rocks.

In layered rocks, the velocities of elastic waves differ along and across the layers, and the velocity parallel to the layers is always greaterthan the velocity perpendicular to the layers, Increase in porosity increases anisotropy Anisotropy usually corresponds with the micro-structural sub-fabrics, namely. crystallographic orientation of constihlent anisotropic minerals and shape andorientation ofpores or cracks. Wetting may change the symmetry.

Velocities generally increase with increasing stress. There is a rapid increase at low stresses dueto a decrease in porosity, theclosing of cracks and defects, and an increase in the mechanical contact between the grains. The velocity increase at a higher stress resultsfrom changesin theintrinsic properties ofthe rocks, such as finite compression of the crystals. In certain cases, the velocity is observed to decrease when the stress exceeds a certain value.

70 3 Propelties of Rock and Rock Mass

The degreeofanisotropy decreases appreciably asthe hydrostatic confining pressure is increased.

The wetting of rocks usually leads to a rise in the elastic wavevelocities. The wave velocity inmore porous rocks completely saturated with water, however,is lowerthan in slightly porous rocks, because the elastic wave velocity in water is less than the elastic wave velocity in the mineral skeleton.

Transverse waves can only pass through the mineral skeleton; consequently V, remains almost constant in rocks whatever the degree of wetting.

Usually, the velocities and dynamic elastic constants fall with temperature. As the rock cools, only a part of the decrease is reversible because as the temperature rises, the unequal expansion of the crystals cause some internal cracking and the crystals would be loosely joined. Because of hysteresis, measurements of velocity versus temperature are recommended at elevated pressures, usually gseater than 100 MPa.

In wet rocks, an abrupt rise in velocity is observed when their temperature falls to below the freezing point of water

3.2.2.4 Comparison of static and dynamic elastic constants

Static methods give rise to a large scatter of results, but can be extended to the high strains 10.' occurring in mining processes. In dynamic methods, low strains of are involved with high rates of loading and the scatter is comparatively small.

Since dynamic methods usually involve low stresses, a compiuison of static and dynamic values of modulus of elasticity is meaninghl only if the values of the static modulus of elasticity are taken at comparable stress levels i.e. using initial or zero stress tangent modulus.

Usually, values obtained by dynamic methods are higher than those obtained by static techniques. The greater the degree of compactness, the more nearly dynamic and static elastic constants may agree.

3.2.3 Timedependent properties

Rocks also exhibit time-dependent behaviour, such as can be seen in the slow deterioration and closure of coal mine workings The understanding and prediction of time-dependent behaviour is also necessary for engineering works in those rocks showing time-dependent deformation.

3.2.3.1 Creep

Usually, the creep deformations (or strains) are plotted against the time as indicated in Figure 3.43. A schematicgraph isgiveninFigure3.44. Ifaconstant stress is applied to the rock material, an instantaneous elastic strain. se first appears. This is followed by aregion I in which thestrain-timecurveis concavedownwards; creep in thisregion is called primary or transient creep. The second region is characterised by a curve with approximately constant slope(secondary orsteady-state creep). Finally thecurve becomes convex (tertiary or accelerating creep) leading rapidly to failure.


Figure 3.43 Creep (strain-time) diagram for marble (after Lama, 1974).

rlgure 3.44 Schematic creep diagram indicating the instantaneous elastic strain se, primary (I), secondary (11) and tertiary creep (Ill) (after Jaeger and Cook, 1979).

If the applied stress is suddenly reduced to zero in region I, the s - 1 curve takes the form PQR. Figure 3.44, in which PQ =sr and QR tends asymptotically to zero. There is thus no permanent deformation. If the applied stress is suddenly reduced to zero in


region11 a curve T W i s followed which leads asymptotically to permanent deformation.

The mechanism of creep is not well understood and several hypotheses have been suggested. However, it may be stated that micrwfracturing in the rock plays a significant role in the phenomenon.

Creep laws

There are no universally accepted creep laws that apply to rocks. Most of the equations have been derived empirically. The following are examples:-

Primary creep may be represented by:-

where E = strain; a = a constant; and t = time.

Secondary creep may be governed by:.

where p is a constant

Tertiary creep may be represented by:-

where y and n are constants.

Phenomenological rheological models of creep

The creep behaviour of rocks and other substances has been explained in terms of mechanical models consisting of simple elements such as springs, dashpots and sliding blocks. This permits the development of mathematical relationships for these materials, but some of the models are so complex that using them becomes cumbersome, and they are not popular Some of these models are given in Figure 3.45.

Factors influencing creep

The time-dependent deformation of rocks is dependent upon a number of factors. Some imponant ones are:.

1 . Nature of stress. 2. Level of stress.


3 . Confining pressure. 4. Temperature. 5. Cyclic loading. 6 . Moisture and humidity. 7. Structural factors.

For details, refer to the publication of Lama and Vutukuri (1978)

Figure 3.45 Rheological models of creep.

Kelvin o u = I q = + k a

'1

Maxwell rnEfanmBnSO"s

u = k ~ e - ~ $ ( e l m c m n and ~eeondq

h inmtanews strain. primary

+ z r and mondary ". 3 '12 h creep

3.2.3.2 Time-dependent strength

Time-dependent strength of a rock can be defined as the maximum stress sustained by the rock at which the failure will just not occur or just occur no matter how long the force has been applied. This strength has been described by various terms such as fundamental strength, true strength, time safe saess, long-term shengh and sustained load stren~th. There are many methods for the determination of this strength. They may be grouped under two main headings: direct methods and indirect methods.

In the direct method, specimens are subjected to different sustained loads and the highest value of the load at which no failure takes place with time is determined. The

74 3 Propert~es of Rock and Rock Mass 3 3 Mechan~cal Properties of Rock Mass 75

method is very cumbersome and lnvolves much labour and time. As such, many indirect methods have been developed. These indirect methods are dependent upon the understanding of the phenomenon of creep, microseismicity, volumetric strain, etc. and are named after the phenomenon utilised. For details, refer to the publication of Lama and Vutukuri (1978).

3.2.3.3 Fatigue

Very little attention has been paid to rock fatigue under cyclic loads although rock structures are often subjected to. Repetitive loading generally weakens the rock by a significant amount. Fatigue characteristics are usually presented in the form of S - N curves (stresslevel versusthenumber ofcyclesrequired to bring aboutfailure)(Figure 3.46). This curveindicates clearly that at a high maximum stress thenumber of cycles required to cause failure is small and increases as the maximum stress is lowered. The maximum applied stress level at which the material can stand an infinite number of cycles is called fatigue strength. The fatigue strength is about 65 to 75 % of the quasi-static strength. The primaly mechanism for rock failure in fatigue is probably micro-fracturing.

Figure 3.46 General form of a typical S - Ncurve

high cycle , fafigve regian

i 2 2 ti FI .- - a a i z .- 3

j

Number of cycles, N

3.2.3.4 Dynamic tensile strength

Dynamic tensile strength of rock is important in the breakage ofrock by explosives. The principle underlying the determination of this propenv is to subject the specimen

to a known intensity of transient stress waves. As the compressive stress wave moves and reaches the free end of the specimen, it is reflected as a tension wave which on being superimposed upon the tail of the incident compressive wave develops an increasing amount of tension in the material (AB in Figure 3.47). The tensile stress at failure is the dynamic tensile strength of the material. Experiments are repeated over and over again at slightly increasing or decreasing stress intensities to determine the threshold stress at which slabbing just develops.

The dynamic tensile strength is one to thirteen times the static tensile strength. The probable reason for very high dynamic tensile strength is that with the increase in the rate of loading, the weakest link in the rock may not necessarily have an opportunity to participate in the fracturing process. The situation which causes the spall is highly localised and the strength measured is that of the rock lying in this highly localised region and hence the volume of the rocks subjected to the maximum tensile stress is vely small. The dynamic tensile strength is not constant and increases with straining rate

Figure 3.47 Mechanism of development of tensile stress by the reflection of the incident (compressive) wave.

3.3 Mechanical Properties of Rock Mass Many rock masses consist of relatively intact rock layers or blocks separated by

more or less planar discontinuities, such as joints, bedding planes, faults, etc. There are two methods for the determination of properties of rock mass:-

I . Synthesising the properties of mass from separate determination of the intact rock and the discontinuities. 2. Large-scale in situ experiments to test a large enough .volume to determine the representative properties ofthe mass.


3.3.1 Strength properties of discontinuities

This Sub-section deals with the shear strength of discontinuities. The main factors controlling the shear strength of a discontinuity are:-

1. Friction angle of the discontinuity s u ~ a c e . 2. Water pressure. 3. Strength of intact rock in the discontinuity walls. 4. Surface roughness. 5. Infill material. 6 . Degree of continuity.

Friction angle of the discontinuity surface

Figure 3.48 (a) shows the simplest type of discontinuity that could be encountered; a hlockrestine on abase separated by aclean, planar discontinuity The resultant force, ~~ ~ ~ - Ftransmitted across the discontinuitv is inclined at an angle Q to the normal to the . . . . . - diicantinuir) if O is incrulsed 10 the iriction angle. .,the bloik will r l ~ d t i e. fail b) shcarins3lonl?rhe lnvriace Theshear beh3\iwrofrhcdiscontinui~ I S bestdescribcd - - in terms of the normal and tangential i.e. shear components of stress acting on the interfaceas shown in Figure3.48@). Figure 3.48(c) shows thevariation ofthe applied shear stress with the shearing of the surfaces at a constant rate of displacement. The peak shear strength develops after afinite amount of displacement. The shear strength reduces to a residual value at large displacements.

The peak shear strength, T~ is given by the following equation:-

where an = normal stress on the interface and 9 =friction angle.

tan IP is called the coefficient of friction.

Water pressure

The shear strength of the discontinuity shown in Figure 3.48 depends on the forces transmitted by the inter-particle contact across the discontinuity so that, if part ofthe load is transmitted by pressure of water in the discontinuity, the shear strength is reduced. If the water pressure in the discontinuity is "a" then:-

3.3 Mechanical Properties of Rock Mass 77

Figure 3.48 Clean planar discontinuity; (a) geometry; @)analytical formulation; (c) shear stress-shear displacement curve (after Bock. 1978).

Shear displacement

Strength of intact rock

This parameter becomes important when shearing is considered along rough, wavy surfaces. The peak shear strength, rp of the intact rock is given by the following equation:.

rp = c; + u,, tan (o, (3.55)

where ci = cohesion of intact rock and Q; = angle of friction of intact rock.


Once the shear displacement of the previously intact rock is increased beyond the peak strength there is a large decrease in strength to a residual shear strength, r r and it is given by the following equation:-

TI. = CI- + a,, tan mr (3.56)

where subscript r refers to residual portion of the curve.

Surface roughness (or waviness)

As indicated in Figure 3.49(a), a discontinuity may have a larger scale irregularity than that forwhich cp has been estimated. The presence of this waviness wouldincrease shear strength above that described by cp. A simple model shown in Figures 349(b) and (c) was proposed by Panon in 1966 to describe this phenomenon.

For sliding to occur in the arrangement shown in Figure 3.49(h), the shear stress, rm on the plane of serration inclined at angle, i to the general plane ofthe discontinuity is:-

T,, = am tan cp (3.57)

Resolving along the plane inclined at angle, i:-

Resolving perpendicular to this plane:.

a,,, = on cos i + rp sin I (3.59)

Combination of these equations (3.57). (3.58) and (3.59) gives:.

This equation would be expected to apply up to the level of normal stress at which the intact rock is sheared through at the base of the serrations (Figure 3.49(cn. This will occur when:-

T p = Ci + a,, tan pi

Figure 349(d) shows the complete envelope of peak shear strength of Patton's simple model.

In reality, roughness features do not conform to the model shown in Figure 3.49. in most cases, discontinuity surfaces exhibit asperities that are variable in their angle,


amplitude and wavelength. The simple roughness angle model is, therefore, inade- quate to describe the shear behaviour of real discontinuities.

Figure 3.49 Rough discontinuity; (a) geometry; (b) and (c) failure mechanisms; (d) failure envelope (after Bock, 1978).

-...-. average slope (a)

__.- - ofirregularifies

failure ~ '

0"

Infill material

Where the thickness of the infill material is greater than the maximum height of surface irregularities. the shear behaviour of the discontinuity corresponds to that of the infill material. Where the thickness of the infill is less than the height of


irregularities, shear behaviour is complicated. At small displacements, the shear strength will be dominated by the infill, but interlocking of surface irregularities becomes increasingly important as displacement increases.

Degree of continuity

At low normal stresses, the peak shear strength of the intact rockwill be at least an order of magnitude higherthan that ofthe rest of the discontinuity. The shear stiffness of the intact rock may be considerably higher than that of the discontinuity so that the peak strengths cannot be compounded, the intact rock failing at smallerdisplacements than that required to develop the full strength of the discontinuity. The behaviour of such discontinuities is complex and it is recommended that laboratory or large scale in situ tests are conducted for evaluation.

3.3.1.1 Sheartesting of discontinuities

The shear behaviour of discontinuities may be evaluated either by laboratory testing or in situ testing. In the laboratory, either hiaxial testing of cores or shear testing of specimens is conduded. The procedures are as described for intact rock testing except that the specimens contain discontinuity In the case of hiaxial testing, the discontinuity to be tested is oriented roughly at 45" to the vertical. In the case of shear testing, the discontinuity is oriented so that displacement of the movable piece causes shear movement.

Large scale in situ tests can be conducted for determining the properties of discontinuities. The method in principle is very simple. A block containing the discontinuity to be investigated is prepared and is subjected to the normal and shear loads (Figure 3.50). The test involves making a block of 70 cm x 70 cm. The block is surrounded by a reinforced concrete or steel frame and the normal and the shear loads are applied though cylinders. Very often the shear force is applied at an angle 0 and in such a case, the normal and shear stresses can be given by the following relationships:-

S cos 0 T =--- A

where S = inclined shear force; N= normal force; A = area of cross-section of the specimen at the base; and 0 = inclination of the shear force with the base.

The value of 0 may vary from test to test, but in case it is in the range from 30" to 40". it is very usual to make N = 0 and thus only one hydraulic jack is necessary. When twojacks are applied, thegeometry of the test should be such that the axes ofthejacks pass through the centre ofthe base of the specimen.

3.3 Mechanical Properties ofRock Mass 81

Figure 3.50 In situ testing of a discontinuity plane.

The purposes of application of the shear force at an angle are:

1. To limit the amount of excavation required for placing the jack and 2. To avoid development of tensile stresses due to bending.

To avoid development of tensile stresses, the value of d plays an important role (Figure 3.50). According to Rocha (1964), thevalue of d should be so chosen from:-

In this case the tension due to bending will equal compression, If the value of S is above this limit tensile stresses mav occur and it is advisable to check the occurrence

of tens~le cracks in the test The usual value of dmay be taken as f t o t , depending . -

upon the angle of friction of the rock, provided N is not very small. There is another factor which is very important and influences the results greatly.

There may be concentration of the shear stresses which are higher the lower the value ofd. (It is in general noted that the shear strenb!h decreases with increase in the value of d.) To avoid this, the shear force frame should be as rigid as possible. The best arrangement naturally would be to apply the shear force with the plane to be investigated placed in the centre of the specimen. But this arrangement is more costly both in terms of equipment and the conduct of the tests.


3.3.2 Strength of rock (mass) with a single discontinuity or discontinuity set

The failure of a rock (mass) can take place either by sliding along the discontinuities or failure of the intact rock. A fault plane, a thin seam of shale or a thin weathered zone can be treated as a single discontinuity. Bedding or cleavage planes can be considered as a discontinuity set.

The magnitude and orientation (with respect to the discontinuity) of the principal stresses together with the shear strength parameters of the discontinuity determine whether sliding along the discontinuity can take place. Assume that the discontinuity plane has shear strength defined by Coulomb's shear strength criterion as follows:-

where cd = cohesion and qd= angle of friction of the discontinuity.

Slip on the discontinuity plane (ah) (Fiw're 3.51) will become incipient when the shear stress on the plane becomes equal to or greater than the shear strength. The stress transformation equaiions give thenormal and shear stresses on (ab) as

where a = angle between the discontinuity and the direction ofthe minimum principal stress.

From these equations (364). (3.65) and (3.66), the criterion for slip on the plane of discontinuity can be obtained:-

The principal stress difference required to produce sliding failure tends towards infinity as a approaches 90" and as a approaches q,r Between these values of a, sliding failure on the discontinuity plane is possible, and the stress at which sliding failure occurs varies with a according to the above equation.

The minimum stress required for failure occurs when


Figure 3.51 A sinde discontinuity plane, ab.

Ifthe discontinuity is unfavourably inclined i.e. a approaching 90" and in the range 0" to q~i, failure can take place along another plane which passes through the intact rock. (This plane is not controlled by the presence of planes ofdiscontinuiry.) Again, Coulomb's shear strength criterion can be applied, hut the intact rock constants, ci and qi are used. The variation of ol with the angle a predicted by these criteria for two confining pressurespl andp2 is given in Figure 3.52. The horizontal lines in Figure 3.52, which represent the failure criterion for the intact rock, intersect the respective parabolas at two points. For those a values which fall between these points failure takes place by sliding along the discontinuity, for inclinations outside these points the failure surface passes through the intact rock.

Although the above results have been derived for a single discontinuity only, they are equally valid for an entire set of parallel discontinuities.

Figure 3.53 shows somemeasuredvariations in peak principal stress difference with the angle of inclination of the minor principal stress to the plane of discontinuity. Although the peak principal stressdifference curvesvary with a and show pronounced minima, they do not take the same shape as Figure 3.52. In particular, the constant strength region at low values o f a predicted by the theory is not always present in the experimental results. This indicates that the two-strength model provides an oversim- plified representation of strength variation in anisotropic racks.


Figure 3.52 Maximum principal stress, a1 versus angle of inclination of the minor principal stress to the plane of discontinuity, a for the different coniining pressures,

3.3.3 Strength of rock mass with multiple intersecting discontinuities or discontinuity sets

The theory for a single discontinuity can be extended and applied to a rock mass with several intersecting discontinuities. Figure 3.54 shows the composite failure criterion for two symmetrically oriented discontinuities. This has been obtained by superimposition. From this figure, it can be seen that the strength of the rock mass now is almost wholly determined by the properties of the discontinuities; the strength of the intact rock contributes only a small part to the failure curve.

For a rock mass, it is possible to establish a wide band, with definiteupper and lower hounds, within which thestrength envelope ofa rock mass [with discontinuities) must lie. The lower bound is given by the failure envelope for sliding along a smooth discontinuity and the upper bound by the failure envelope for the intact rock (Figure 3.55). In general, it can be said that at relatively high normal stresses, the strength of the rock mass will be closer to that of the intact rock, whereas at low normal stresses, it will be determined mostly by the properties of the discontinuities. The more regular and continuous the discontinuities, the planarand smootherthediscontinuity surfaces and the smaller the degee of interlocking between the individual blocks, the nearer will be the failure envelope to the lower hound. If a rock mass contains randomly oriented discontinuities. then its failure characteristics would be in the upper region of the defined failure band.

3 .3 Mechanical Properties of Rock Mass 85

Figure 3.53 Principal stress difference, ol -0; versus angle of inclination of the minor principal stress to the plane of discontinuity, a for various confining pressures indicated; (a) a phyllite (after Donath. 1972): (b - d) a slate and two shales (after McLamore and Gray, 1967).

3.3.4 Deformation properties of discontinuities

Deformation propehes of discontinuities can be considered with respect to movements in normal direction of the discontinuity surface (normal deformation) and movements parallel to the plane of discontinuity [shear deformation).

3.3.41 Normal deformation

The maximum possible closure. has to be less than the thickness of the discontinuity, e (Figure 3.56(a)). A typical normal stress, a,~ 1 normal deforma-


tion, 6" relationship is shown in Figure 3.56@). Asimple equationgiven by Goodman in 1976 is as follows:-

Figure 3.54 Composite failure curve for two symmetrically oriented discontinuity planes.

Figure 3.55 Bounds for rock mass strength envelope.

intact rock

3.3 Mechanical Propelties of Rock Mass 87

where 6 = seating pressure, defining the initial condition for measuring the normal deformation Sn;

A and t = dimensionless empirical factors.

Figure 3.56 Normal deformation of discontinuities; (a) Discontinuity with its thickness "e" (b) ldealised hehaviour of a discontinuity in compression (< = seating pressure) (after Goodman, 1976).

3.3.4.2 Shear deformation

Typical relationships between shear stresses and shear deformations for discontinuities are given in Figure 3.57. It is customary to subdivide these curves into three regions, namely, elastic (pre failure) region, peak region and plastic (post failure)

i region In the pre failure region, - curve is usually idealised as a straight line (linear 6. -..

elastic case). The slope of the curve, k , is defined as the "shear stiffness" of the discontinuity.

3.3.5 Defmation properties of rock mass

Two models can be used to determine the deformation properties of rock mass containing several discontinuities. In the first model. the deformation properties of


discontinuities can be combined with the deformation properties of the intact rockto produce a constitutive relationship for a continuum that is equivalent to the discon. tinuous mass. In the second model, the specification of each discontinuity geometv, the deformation properties of the intact rock and the deformation properties of each discontinuity are usedin numerical methods. For details, refer to publication by Priest (1993).

Figure 3.57 Shear stress, ? versus shear deformation, 6 , curves; (a) Two types of curves (A) and (B) at constant normal stress, an (b) Idealised curve with parameters of shear deformation. Subscript p refers to peak and r to residual respectively (after Goodman, 1976).

3.3.6 Large-scale in situ tests

The need for in situ testing arises when large-scale erects are anticipated which cannot be ascertained from small scale laboratory tests. Large-scale in situ tests have


the advantage that the rock (mass) is tested under the environmental conditions prevailing in the rock mass but are time-consuming and expensive. These tests can be divided into four main categories:.

1. Compression tests. 2. Shear tests. 3. Deformability tests. 4. Seismic tests.

3.3.6.1 Compression tests

Compression tests are conducted to determine the strength and deformation characteristics of rock masses or of specific rock structures such as mine pillars. The test should reproduce as faithfully as possible the state of stress that shall be occurring during and after completion of the structure. If the object of the test is to determine the strength anddeformation characteristics ofmine pillar sthen obviously theloading conditions must be the same as that to which an actual mine pillar is subjected. The more important conditions are as follows:-

1. The width-to-height ratio of the in situ specimens should be the same as that of actual pillars (usually between 2 and 5). 2. The lateral constraint conditions at the interfaces between pillar top and roof layer as well as between pillarbottom and floor layer should be simulated for the specimen. 3 Actual mine pillars are loaded by the convergence between the roof and the floor. To simulate this condition either a stiff loading date must be introduced between the pillar and roof or theloading must ensure that fue deformation isuniformly distributed over the pillar cross-section. The latter method i e . displacement controlled loading, is recommended

Since fractured rock can still support some load it is also useful to investigate the deformation behaviour of rock after failure. For this purpose the complete stress-strain or load-deformation curve ofthe specimen should be determined.

3.3.6.2 Shear tests

In situ shear tests are usually conducted on a prepared test block to determine the shear strength and deformation properties, in particular, along discontinuities. In underground openings, the roof and sidewalls are used to carry the reaction of the applied normal and shear loads whereas on surface, cable anchors are used for this ~. purpose.

Some details about the shear test have already been given earlier in Sub-section 33.1 . I Shear testing of discontinuities. During specimen preparation, the block and Panicularly the shear plane should be retained as close as possible to its natural i n situ water condition. For determining deformation properties, provision is made for the measurement of shear and normal displacements by means of dial gauges fixed to a reference frame which is anchored in rock some distance from the test block. The


normal force applied should be through rollers or a similar low friction device to ensure low resistance to shear displacement.

3.3.6.3 Deformability tests

The deformability of rock in situ has been determinedusing different methods. The tests can be divided into the following three types:-

1. Plate bearing test. 2. Pressure tunnel test 3. Borehole tests.

Plate bearing test

This is one of the simplest and most commonly used tests for determination of the deformability of rock mass in situ. The test consists of applying normal load to a specially prepa~ed flat surface through arigid or flexible finite plate and the deformation is measured at the centre of the plate or at its edge or at any other convenient point within the rock mass. The d e h a t i b n modulus can be calculated using the relationships developed depending upon the shape of the loading plate and the nature of the rock (assuming isotropic or anisotropic behaviour).

The layout of a typical test is shown in Figure 3.58. The load is applied by means of hydraulic jacks while the flat jack serves as a load transfer plate resulting in a uniform stress distribution over the loaded area. Loaded areas with a diameter of 0.5 to 1.0 m are common. The size of the flat rock surface to which load is applied should belarge compared tothe sizeofthe loaded area. Othenvisecorrections must be applied to the results to take this into consideration.

The displacements on the rock surface are measured by a reference frame with dial gauges as shown in Fibwre 3.58. The displacements inside the rock are measured by borehole extensometers installed in a borehole drilled at the centre of, and normal to, the loaded circular area, as also shown in Figure 3.58. Modifications of plate bearing test include (i) Compressionin narrow slits. (ii) Cable jacking method and (iii) Goffl's method. For details, refer to publication by Lama and Vutukuri (1978).

Pressure tunnel test

The test consists of applying uniformly distributed radial load to the surface of the tunnel while the diametral deformation of the tunnel b o u n d q is measured. The calculations are based upon treating the tunnel as a thick-walled cylinder and assuming the external radius to be infinite. Two types of tests come under this category. namely, hydraulic pressure chamber test and radial jacking test.

Hydraulic pressure chamber test In this test, a uniformly distributed hydraulic (water) pressure is applied to a full size section of a tunnel (Figure 3.59). This is achieved by sealing off a section of the tunnel by means of concrete plugs and by pumping water into the sealed off section until the required pressure is reached.


Displacement and other measurements are made at selected pressure intervals. For details, refer to publicahon by Lama and Vutukuri (1978).

Figure 3.58 Layout of a plate bearing test (after van Heerden, 1976)

U! concrete pad

hydraul~c lacks

reference frame wlh d18I mugs lo-

Radial jacking test This test is a modification of the hydraulic pressure chamber test where pressure is applied through a series ofjacks placed close to each other The testis carried out in asection of atunnel preparedsothatthediameter with the concrete lining in position is about 2.5 m. After preparing the test section. a number of steel rings are placed next to each other in the test section. The loading equipment consists offlatjackswhich areinserted between wooden blocks, placedbetween the steel rings and the concrete linine of the test section. A typical arrangement is shown in Figure - . . 3.60.

Pressure is applied to the flatjacks by a pump and hydraulic manifold. The pressure is transmitted from the jacks to the concrete and the rock surfaces. Deformation is measured by means of dial gauges between the fixed reference frame (pipe) and the rock surface. It is also ~ossible to use multi-point extensometers installed into the rock.

Advantages of radial jacking test in comparison to hydraulic pressure chamber test include:-

I . Less expensive and easier to carry out. 2. No fluid to come into contact with the rock. This feature permits testing of rocks which would otherwise not be suitable because of leakage through discontinuities.

92 3 Properties of Rock and Rock bfass 1 3.3 Mechanical properties of ~ o c k Mass 9 3

3 . Accessibility to instruments duringthe test. This allows repairs, if necessary, during the test without intempting the test for too long. In a hydraulic pressure chambertest it would be necessary to empty the chamber if something goes wrong or to proceed without taking some of the measurements. 4. Measuring instruments simple because they are not required to work under water at high pressure.

For details, refer to publication by Lama and Vutukuri (1978).

Figure 3.59 Details of hydraulic pressure chamber test (after Rocha et al., 1955).

2 - prersure gaug Nbe 9 - P- wse 3 - w m i n l d I 0 - vibnting metm 4 -air wder I I -air preswe equaliring chamber 5 -cable admirim rube 12 - invar red! 6-cablerubewal I3 - air p-re quaiiring rube 7 -water meter 14 -cable

Borehole tests

Many instruments have been developed for measurement of rock deformability in small diameter boreholes. These instruments depend upon the application of pressure to the walls ofa boreholeand measurement oftheradial response ofthe borehole wall. The instruments that apply aunifonn internal pressureto the borehole walls arecalled dilatometers or pressure meters. Those which apply force along a limited portion of the circumference of the borehole by forcing apai circular plates are called borehole jacks.

Borehole dilatometers These instruments are essentially borehole pressure cells which apply pressure radially in all directions to the wall that encloses them. The devices are cylindrical, and incorporate some means of measuring changes of diame-

teror volume with variation in pressure. The change in diameter may be deduced and ,yeraged from measurement of the volume change or it may be measured across

directions. Typical of the latter, more elaborate, device is theLNEC (Lisbon) borehole dilatometer (Figure 3.61). It can operate submerged in water to depths exceeding 100 m, and it applies radial pressures to the borehole wall up to 15 MPa. Diametral deformations are measured by linear variable differential transformers aligned in four directions 45" apart.

Figure 3.60 Layout of a radial jacking test (after van Heerden, 1966)

m m e flat jacks

easuranent device

woodm b l a k supported by steel rings

flat jack

Pressure is applied by pumping water into the annulus between the inner steel cylinder and the rubberjacket. Thisjacket is deflated to allow the device to be raised or lowered by cable and winch to any desired measurement location along the length


of the borehole. Observations on four diameters enable the deformation ellipses and coefficients of anisotropy to be deduced.

Figure 3.61 LNEC borehole dilatometer (after Rocha et al., 1966).

Borehole jacks Instead of applying a uniform pressure to the full cross-section of the borehole, high pressure can be applied to a part of the borehole surface by driving plates against the borehole walls using hydraulic pistons, wedges or flat jacks. The load can be directed in any desired direction and this offers an advantage over the dilatometers. The test can be oriented to study special geological features e.g, it can be used to measure the force necessary to open a discontinuity plane cutting a holeor determine deformation in any one p d c u l a r direction. Facilities for measuring diametral deformation are incorporated in the insimment. in general, the moduius observed when using borehole jacks is likely to be 30 or 40 % low, due to fractures caused by the jack in the rock at the contact areas.

Borehole tests have certain advantages. The devices allow measurements at points far removed from the excavation and hence at places which are not affected by drivage of access galleries and disturbances caused due to site preparation. The test is simple requiring less site preparation and permitting a large number of tests to be conducted in a sholt time at low costs. The main disadvantage ofthe test is the small volume of rock involved i n each test so that the results obtained may not be representative ofthe deformability of the rock mass. These tests are very useful for first assessment of the deformability and subsequent zoning of the rock mass because of the ease of access to deeper regions through boreholes from sulface or galleries.

3.3.6.4 Seismic tests

Seismic tests are aimed at determining the dynamic moduli of rock masses. The method is based upon the measurement of wave propagation velocity through the rock medium and Figure 3.62 shows a typical experimental set up.

To measure longitudinal and shear velocities of seismic waves in rock masses exposed on the surface. an explosive is detonated in a shallow drill hole. The wave motions generated on the rock surface are measured by velocity gauges at several distances from the source and recorded with a tape recorder, oscillograph or an oscilloscope. Zero shot time is determined by an ionisation probe attached to the charge. Velocities are computed from the time intervals between the detonation ofthe


explosive and the arrival of a longirudinal or shear wave at each gauge. Shear wave arrivals are most frequently recorded by transversely oriented gauges. Longitudinal and shear wavevelocities are computed by dividing the shot-to-gauge distance by the time interval between the detonation of the charge and the arrival of the appropriate signal at the gauge position. The velocities can also be computed from travel time plots asshowninFigure3.63. Figure3.63 showssometypical panicle velocity records and thecomputation of longitudinal and shear wave velocitiesfrom a travel time plot.

Using rock density and velocities determined in tests, all the elastic constants can be calculated from the standard elastic equations given earlier in Sus-section 3.2.2.3 Dynamic elastic constants of rock.

While this technique is an attractive one, it is not without problems. Often it is quite ditficult to generate shear waves and even more difficult to measure arrival rates. Since the velocity of shear waves is in the neighbourhood of half of the longitudinal velocities for rock, both shear and longitudinal waves normally generated by the original longitudinal wave being reflected from a free face may mask the original shearwaves. Also, the number of paths to bemeasured results in tedious calculations.

The averagedynamicelasticproperIies of a rockmass can also be evaluatedthrough velocity measurements between pairs of holes. Generally, the distance between receivers in the same hole and the distance between two subsequent positions ofthe emitter are chosen to be of the same order of magnitude (or at least equal to half to one-third) as the distance between the holes, in order not to have equivalent paths. The receivers, the subsequent positions of the emitter at different depths &d the hypothetical rectilinear paths between emitter and receivers are shown in Figure 3.64 for a pair of holes.

Figure3.62 Experimental setup for measuringlongitudinal and shearwave velocities (after Lewis and Tandanand, 1974).


Figure 3.63 Velocity records and computation of longitudinal and shear wave veloci. ties from travel time plot.

11 and I3 arrival t i m e for longimdinal wave zero time mark R and a arrival t i m a for shear wave

- , 4 . 4 I -

lime

wvel time plot = -

0 0 1 0 2

shot-tc-gauge dimnee

Thelocation of holes is chosen according totheshape ofthemassunderexamination. An example of a hole arranaement relative to the investigation of arock mass is eiven

~ - in Figure 3.65. The test begins by placing geophones inholes 2 and 3 and shooting at various depths in hole 1 . The geophones are taken out of holes 2 and 3 and put in holes 4 and 5 and shots are made in hole 2. hole 3 and so on. Each rock prism so formed (e.g. FI - F2 - Fz; F2 - F3 - F4; etc.) is analysed separately.

The speed values along the various emitter - receiver paths are normally scattered, and the statistical methods are employed to obtain average parameters characteristic of the rock.

3 4 Fracture Criteria 97

Figure 3.64 Positions of receivers Rand sources S for a pair of holes (after Bernabini and Borelli, 1974).

Figure 3.65 Hole arrangement used for a survey (after Bernabini and Borelli, 1974).

3.4 Fracture Criteria Fracture is the dominant mechanism of rock failure at the relatively low pressures

and temperatures at shallow depths in the earth's crust. Fracturing may be defined as the processes which involve at least momentary loss of cohesion, ability to resist differential stress, separation into two or more parts. and release of stored elastic strain energy (Griggs and Handin. 1960). In general there are two basic types of fractures, namely. extension fractures and shear fractures. Extension fractures occur normal to the least principal compressive stress. 03; these are termed tensile fractures if 03 is


tensile i.e, negative in sign. In the shear fracture, particle motion is parallel to the fracture surface that may be inclined from 45" to a few degrees to the direction of the maximum principal compressive stress, 01. Fracture criteria express relationships between stress components which will cause fracture.

3.4.1 Fracture criteria for intact rock

The state of stress at any point within a solid can be specified in terms of three principal stresses, ol , o2 , 03 , where 01 > a2 > 03. The set of o~ ,a2 , oi values at which fracture occurs in an element of a solid can be represented by a point in a1 .02 , a! space and the totality of these points describes the fracture surface i.e.

Such a relationship is called a criterion of fracture. Essentially, experimental measurements under different conditions should provide the form of this surface. Discussion will be confined to the region a1 2 0. Figure 3.66 shows the information available about this surface, namely, the uniaxial tensile strength 03 =-or, 01 =a? = 0; the uniaxial compressive strength 01 = oc , az = a3 = 0; and values obtained in the conventional triaxial test 01 > 02 = o3 > 0 which lie on a curve oc T.

Figure 3.66 Fracture surface.

3.4 Fracture Criteria 99

3.4.1.1 Maximum tensile s t r e s s criterion

The material is assumed to fracture in tension if the minimum principal stress a3 is equal to minus the uniaxial tensile strength i.e.

This is adequate under certain restricted conditions

3.4.1.2 Coulomb's criterion

This criterion has already been introduced while discussing the triaxial test for the determination of shear strength. For a given rock type, the angle q? may vary considerably with the applied load. This criterion indicates that the ol , a3 fraclure curve should be a straight line. This condition is reasonably well satisfied for most igneous and other hard crystalline rocks. However, for the evaporite minerals, shales and carbonates, the slope of the 01, a3 relationship usually decreases as 03 increases.

This criterion of fracture can be expressed in terms of at and ac, the uniaxial tensile and uniaxial compressive strengths of the material as follows:-

This criterion predicts that the compressive strength is greater than the tensile strength, but the ratio is not as large (10 to 50) as found in practice. For a nominal

oc value oftan q = 1, the ratio - = 5.8. Pi, -.

According to this criterion, the angle of fracture is the same for tension and compression fracture. In compression the angle of fracture is relatively constant for most rock types, but in tension the fracture sulface is usually normal to the direction of the tensile stress. This criterion assumes shear fracture, so that at should not be the actual (brittle) tensile strength but the value at which shear fracture in tension would takeplace ifin fact brirtlefracture didnot occur in practice beforethisvalueis reached. The difference in the appearance of the fracture surfaces created in tension and shear also indicates the mechanism of fracture is not the same in the two cases.

If the normal stress across the fracture plane is negative, the concept of an internal friction becomes meaningless. as this would tend to separate the fracture surfaces; hence. the Coulomb's criterion should not be expected to hold for fracture in tension.

Thevalue of tan ~p varies from 1.0 to 2.5. In that case the shear strength (cohesion) would be about 0.2 to 0. I times the compressive strength of rocks.

3.4.1.3 Mohr's criterion

A material may fracture when either the shear stress T in the plane of fracture has increased to a certain value which in general will depend also on the normal stress


a,, active acrossthesame plane orwhen thenumerically largest tensileprincipal stress has reached a limiting value at dependent on the properties of the material. Thus at fracture either

The functional relationship T = f ( a,, ) must be determined experimentally and is represented by a typical curve Aur c B (Figure 3.67). As this curve is the envelope to theMohr3s circles for the values of a 3 and GI at fracture, its physical significance is as follows:- For any state of stress represented by aMohr's circle lying completely within the envelope, the material will not fracture, whereas, if any part ofthe circle lies outside ofthe envelope, thecritical stresses will be exceeded. Forthecircletangent to the envelope, thematerial will fracture on the planemaking the angle a with respect to the minor principal stress. Mohr's criterion further implies that the intermediate principal stress, a2 has no influence on fracture.

Figure 3.67 Representation of Mohr's criterion of fracture

This criterion offracture not only specifies the state of stress at fracture but predicts the direction of the fracture plane. For specimens tested in triaxial compression, r increases monotonically with a,,. Hence Mohr's criterion implies that a material will not fracture in hydrostatic compression, a consequence that is consistent with experi-


mental fact. The envelope curves, if projected into the - a quadrant, do not predict the correct mawitude or angle offractureintension. In thesequadrants. themaximum tensile stress criterion is assumed; i.e. fracture will occur when a3 reaches a critical value - or, and the fracture plane will be normal to the direction of fracture stress.

3.4.1.4 Grifith's criterion

This criterion postulates the presence of microscopic cracks within the material. When the material is stressed, large tensile stress concentrations occur around the tips of the so-called Griffith cracks. When the tensile stress at or near the tip attains a certain critical value, the crack starts extending, ultimately. contributing to fracture. The theory has been substantiated by experimental work on glass. According to this criterion, the stress necessary to cause brittle fracture varies inversely with the length of the existing cracks.

an I If -- - -, fracture will occur when the minor principal stress equals the uniaxial

a1 3 tensile strength i.e. when a3 = - m~ and, in this case, the angle of fracture a = 90'.

a3 t G I - G ~ ) ~ If - > - 3' fracture will occur when = 8 rrt and at an angle given by

01 (a1+03)

~ - If a 3 = 0 and 01 = a , it follows that ae = 8 or. Thus, Griftith's criterion of fracture

predicts that the magnitude of the un~axial compressive strength should he exactly eight times the uniaxial tensile strength, a condition that is not consistent with observation, as the uniaxial compressive strength of most rocks varies from 10 to 50 times the uniaxial tensile strength

Murrell(1962) showed that this fracturecriterion corresponds to aMohr's envelope at fracture given by

2 r + 4 a, a,, = 4 a? (3.76)

Thus, Grifith's comparatively simple model of a brittle material containing micro cracks of a specified length leads to a fracture criterion represented by a parabolic Mohr'senvelope. Although someof thesedimentary rocks(limestone, sandstone, and carbonaceous rocks) have a non linear Mohr's envelope, it is common for the more brittle rocks such as yanite and quartzite to have a linear Mohr's envelope in compression. Moreover, as the stress concentration around a GriGth crack is calculated on the basis of elastic theory, this mechanism of fracture should be time-independent and hence would not account for variation in strength with stress or strain rate. McClintock and Walsh (1 962) extended the Griffith's criterion for the case of high

biaxial conditions. where the compression forces are sufficient to close the crack and thereby allow the action of friction forces on the crack surfaces. This modified

102 3 Properties of Rock and Rock Mass I 3 4 Fracture Criteria 103

Griffith's criterion includes two critical quantities, namely, the critical tensile stress at the tip, expressed by the value of uniaxial tensile strength of the material (as in the original Grifith's criterion) and the coefficient of friction between crack surfaces, Fracture occurs when

best not to extrapolate the equation far beyond the range of the data from which the equation has been fitted.

Asthetheoretical criteriado notfit the experimental results properly, many empirical have been formulated for rocks. The fracture cnteria can be written in terms

of either

acr 0.5 ' 3 1 [ ( 1 + p 2 ) 0 ~ 5 - ~ l - 0 1 [ ( ~ + ~ 2 ) U 5 + p ] = 4 ' 3 , ( 1 + ) - 2 1 1 0 ~ I ( I ) principal stresses, '31 and '33 at fracture such as at

(3.77) where p= the coefficient of friction for the crack surfaces and

cr,= the stress normal to the crack required to close it.

It should be noted that the coefficient of friction of the crack surface is not the same as the coefficient of internal friction that appears in Mohr's criterion of fracture, although these quantities may be related.

Brace (1963) pointed out that ' 3 ~ ~ is small and can be neglected. Hence the above equation (3.77) becomes

or (2) normalised principal stresses at fracture obtained by dividing the principal stresses, GI and '31 at fracture by the relevant uniaxial compressive strength, crc such as

In the above equations, a, b, B and a are empirical curve-fining parameters

The relationship between a1 and '33 is linear as in the Coulomb's criterion. If q = oc and 0 3 = 0 are the conditions for simple compression, the ratio of uniaxial compressive to uniaxial tensile strengths is

For p = I, the ratio of the uniaxial compressive to uniaxial tensile strengths is approximately 10, which is an improvement over the Coulomb's predicted ratio of 5.8; but lower than generally observed values.

The treatment of Griffith's criterion given hereis based on a two-dimensional model i.e, on a crack in a thin plate. Sack (1946) extended the criterion to three dimensions considering a penny-shaped crack and determined that the maximum and minimum boundary stresses and the surface energy differ from the two-dimensional case by only a few percent. Fracture occurs by a crack growing in a plane normal to the principal stress and parallel to the intermediate stress. This implies that crack growth and strength are not affected by the magnitude of the intermediate stress.

The fracture criteria can also be written in terms of shear and normal stresses or normalised shear and normal stresses with respect to uniaxial compressive strength Here, only three criteria in terms of principal stresses ornormalised principal stresses are described in some detail.

(3.79)

3.4.2 Empirical criteria for intact rock

3.4.2.1 Bieniawski's criterion

The criterion proposed by Bieniawski (1974a) is as follows:-

An empirical criterion is one where an arbitrary equation is statistically fitted to a real set of experimental data. This gives a sufficiently accurate prediction of fracture for most practical purposes although the criterion has no theoretical justification. It is

where B = 3.0 for siltstone and mudstone: = 4.0 for sandstone; = 4.5 for quartzite; and = 5.0 for norite and

a = 0.75 for all rock types.

3.4.22 Hoek and Brown's criterion

The criterion proposed by Hoek and Brown (1980) is as follows:-

where n7 = a constant

104 3 Properties of Rock and Rock bfass

The values ofconstantm derived or estimated by Hoek et al. (1992) for various intact rocks are given below:-

Amphibolite - 31.2 Andesite - 18.9 Anhydrite - 13.2 Basalt - 17 Chalk - 7.2 Chert - 19.3 Claystone - 3.4 Conglomerate - 20 Dolerite - 15.2 Dolomite - 10.1 Gabbro - 25.8 Gneiss - 29.2 Granite - 32.7 Gypstone - 15.5 Limestone - 8.4 Marble - 9.3 Norite-21.7 Quartzite - 23.7 Rhyolite - 20 Sandstone - 9.6 Slate - 11.4

Substihltion of 61 = 0 in this criterion, and solution of the resulting quadratic equation for 03, gives the uniaxial tensile strength of a rock, 01 as follows:-

3.4.2.3 Johnston's criterion

The criterion proposed by Johnston (1985) is as follows:-

where Mand Hare constants. These constants depend upon a< as follows:-

M=2.065 + k ( log oc )'

H = l -0.0172(loSo,)'

j q Fracture Criteria 105

where k = 0.170 for dolomite, limestone and marble: = 0.23 1 for mudstone, shale, slate and clay; = 0.270 for sandstone and quartzite; = 0.659 for amphibolite, gabbro, gneiss, granite, norite and grano-diorite; and

= 0.276 for all rock types combined (overall) and oC = uniaxial compressive strength in kPa.

When 01 = 0, 0 3 becomes tensile strength, or and

This is the only criterion which suggests that the values of the parameters are not only dependent on rock type but also on uniaxial compressive strength of the rock. This equation has also been proposed independently by Sheorey et al. in 1989.

3.4.2.4 Analysis by Vutukuri and Hossaini

For non linear regression analysis of triaxial test data, the selection of appropriate software is cmcial in the results obtained. Earlier, Hoek and Brown (1980) recommended a particular transformation to make their criterion a simple linear model to determine the appropriate values for the parameters by the use of a calculator. It must be mentioned here that linearisation of a non linear model does not produce an equivalent model. Shah and Hoek (1992) now use the simplex reflection technique which has definite advantage over linear regression analysis for fitting anon linear fracture criterion to laboratory data. They concluded that although linear regression analysis may, at best, give similar estimates of the parameters of any non linear criterion, the use of the simplex technique will always give better estimates of the parameters of any non linear criterion. Vutukuri and Hossaini (1992) also reported similar conclusions. As a matter of fact, non linear regression analysis through the minimisation of the sum of the squares of the relative errors fit the results better particularly in the lower range of independent variable i.e. confining pressure. Table 3.2 gives values determined from the analysis of published triaxial test data

on various rocks. It must be mentioned here that the values of parameters not only vary with rock type but also depend upon uniaxial compressive strength. The criteria serve an important practical purpose. By using the appropriate values of parameters, it is possible to carty out preliminary design calculations on the basis of the uniaxial compressive strength which can be estimated from a simple test such as the point load test.


Table 3.2 (a) Parameters for Bieniawski's criterion (after Hossaini, 1993).

Coal 0.60 10152-4.709 (logo,) 0.8889

r2= 0.9164

Granite 0.65 3.4452 i-21.617 logo^)-^^^' ? = 0.68

Granodiorite 0.70 2.56 + 85819 (log oc)-12.43

? = 0.998

Limestone 0.76 - 3.3538 + 10.883 logo^)^^^'^' rz = 0.8949

Sandstone

Shale

Table 3.2 (b) Parameters for Hoek and Brown's criterion (after Hossaini, 1993). -=-.-_-- =.--

m __--A__;-=--=--= .Rc!%k=& -- <-=*-- - - Claystone & liparite - 460.74 + 478.88 (log oe ) 4 . 0 2 4

?=0.51

Coal 62.903 - 34.213 (log oc)09772 2 = 0.8514

Granite - 971 + 1055.3 ( l ~ g o ~ ) - ~ " ~ ~ ?=0.70

Granodiorite

Limestone

Sandstone

7.86 + 8.493 e7 ( logo, )-1836 rZ = 0.941

Shale 16.1 l l - 41.543 (log a, )-2.43XX

. . . . . . . ~ ~ ~ _~~ ~.~ L:.=~Ea .-.- . , .

q, is in MPa. r- is coefficient of determination.


3.4.3 Empirical criteria for rock mass

Amongst the various criteria developed for rock mass, only Hoek and Brown's criterion will be discussed here in some detail. The orignal Hoek and Brown's criterion proposed in 1980 is as follows:-

where s and m , are rock mass constants

The uniaxial compressive strength of the rock mass, oc,,, can be obtained by substituting o3 = 0 into the above equation, namely,

In 1988, they suggested the following revised set of approximate relationships between the constants mm and s and the rock mass rating (M) developed by Bieniawski (1974b):-

Disturbed rock masses e.g. rock mass adjacent to rock slopes exposed to blast vibrations:.

mm =exp( RMR- 100.

m 14 1 RMR - 100

s=exp(--- 6 )

Undisturbed rock masses e g , rock mass adjacent to machine bored tunnels:-

nbrz RMR- 100 - m = e q ( - 2 8 )

RMR- 100 ,s= exp ) (3.93) 9

where mnl and s are the rock mass constants and nt is the constant for theintact rock.

They recommended the application of the criterion only to isotropic rock mass i.e. when the volume of rock under consideration contains four or more closely spaced discontinuity sets and where none of these discontinuity sets is significantly weaker than any of the others. In this original criterion, the uniaxial compressive strenah is used both in the RMR system and the criterion.

According to Hoek et al. (1992) the criterion predicts too high an axial strength for low values of a3 and also a finite tensile strength. They believe that the rock mass

108 3 Properties of Rock and Rock Mass I should have zero tensile strength and in view of this deficiency in the original Hoek and Brown's criterion, they presented the following modified criterion:-

where mn, and a are constants for rock mass. I m," Table 3.3 can be used to estimate values of and a. Table 3.4 gives approximate m

block sizes and discontinuity spacings in discontinuous rock masses. The values for m have already been given in Hoek and Brown's criterion under Sub-section 3.4.2.2 Hwk and Brown's criterion. This modified criterion gives uniaxial strength for any class of rock mass the value of zero.

3.4.3.1 Analysis by Vutukuri and Hossaini

After analysing triaxial test data on discontinuous plaster and sandstone models, the following modifications have been found necessarv with resvect to Bieniawski's and Hwk an i Brown's criteria:.

If Equation (3.96) is used instead of Equation (338) in the case ofHoek andBrown's criterion, it is required to estimate om andmnr. The following relationships have been determined from Hoek and Brown's equations:-

For disturbed rock mass:- I From Equations (3.89) and (3.91):- I Gem - RMR- 100 0.5 - rexp( ) ] D c 6

110 3 properties of Rock and Rock Mass

Table 3.4 Approximate block sizes and discontinuity spacings for discontinuous rock masses (after Hoek et al., 1992).

- ~ ~ . ~ ~.. --

Term Block size Equivalent discontinuity ---- ~ -.--- ... - ~- .spaccnps

Very large (> 2 m)' Extremely wide

Large (600 mm - 2 m) 3 Vely wide

Medium (200 - 600 mm)j Wide

Small (60 - 200 r n n ~ ) ~ Moderately wide 3

Verv small ~~ ~ ! ? ~ ! - . = . < Moderatelv -. wide

For undisturbed rock mass:

From Equations (3.89) and (3.92):.

For discontinuous models of plaster:.

For discontinuous models of sandstone:-

a,,, = 0.7

3 5 properties of Soft and Weak Rocks 111

3.5 Properties of Soft and Weak Rocks The soft and weak rocks are defined as geomaterials constituting semi-consolidated

ground in which solidification or decomposition is in progress. They can be classified according to geological age, geological process (solidification or decomposition) they are subjected to and mineral composition as follows:-

I . Sedimentary soft rock produced by the consolidation of deposited earth by a pressure from overlying sediments or chemical cementation. This kind of rock is widely distributed. 2. Weathered or altered soft and weak rock produced from hard rock by weathering or alteration resulting from repetitive dry and wet seasons or a chemical action such as oxidation, leaching etc. Although this kind of rock is limited in distribution, hydrothermally altered rock, fracture zones of faults, etc. become an issue. In many places, the thickness of weathered zones reaches about 30 m. hut in some places, the thickness may reach as large as 100 m. 3 . Volcanic soft and weak rock resulting from less welded pyroclastic flows from volcanoes. This kind of sot? and weak rock shows a non-uniform physical property distribution and may be argillised by hydrothermal alteration.

Unlike hard rock properties which are mainly controlled by discontinuities, a considerably close relationship is observed between the properties determined for rock and those determined for rock masses and, accordingly, the behaviour of the ground composed of such soft and weak rock largely depends upon the physical and mechanical properties of rock elements. Therefore, property tests on the elements of soft and weak rock become important. The rock has the following common features in spite of its different origins:-

1 The rock is rich in voids and largely affected by its water content. 2. Its basic physical property varies depending upon the consolidated state of its constituent particles. The diagenesis or alteration taking place in the rock produces a difference in the interlocked or contacting state or degree of cementation between each constituent particle and, thus, the rock has various kinds of physical and mechanical properties.

3. The property change from soil to hard rock through the soft and weak rock is essentially continuous and, therefore, the soft and weak rock has both properties the soil and hard rock have.

I Recently, construction work has been carried out against soft or weak rock such as (3 105) the Se~kan tunnel cut through soft sedimentary rock and Honshu-Shikoku bridge

I

112 3 properties of Rock and Rock Mass

constructed on weathered granite. A number of nuclear power plants is constructed on soft and weak rock in Japan.

A soft and weak rock has a low uniaxial compressive strengh of 20 MPa or below, mostly 10 MPa or below, and a high deformability, with its failure strain being about 1 % or higher The rock has a void ratio of about 0.5 to 1.5, with its water content reaching 50 % in the case of saturated mudstone. In recent years, Core Laboratoly of OYO Corporation. Japan carried out investiga-

tions of the ground characteristics at various construction sites of nuclear powerplants in Japan. The properties investigated include:-

Swelling and slaking. Uniaxial compressive stren@ Wave velocities. Permeability Triaxial strength. Creep. Deformation.

The effects of water and confining pressure on various properties were studied in detail. For details, refer to publication by Tanaka in 1993

3.6 Exercises 1 Distinguish between strain-sofiening and strain-hardening stress-strain curves. How do you obtain strain-softening stress-strain culves?

2 Write short notes on "Servo-controlled testing machine".

3 For determining the uniaxial compressive strength of rock material, the test specimens should have a height-to-diameter ratio of 2.5 to 3.0. Why? Explain in terms of end effects and the influence of height-to-diameter ratio on the uniaxial compressive strength of rock material.

4 Discuss the factors which can influence the determination of the uniaxial COmPreS-

\ sive svenyh of rock specimens tested in the laboratoty.

5 What is "Point load strengh index"? How is it related to compressive and tensile strenghs?

6 Using the generally accepted notation 01 . 02 a? for the principal stress cOmPo- nents, define the multiaxial test conditions referred toas

(a) Triaxial. (b) Polyaxial

3.6 Exercises 113

7 Describe briefly the effects of increasing confining pressure on the mechanical behaviour of rock in triaxial compression..

g Explain the effects of increasing pore pressure on the mechanical behaviour of rock at a constant confining pressure in triaxial compression.

9~escribe the variouslaborato~metbods which may be used to determine thetensile of a rock material.

10 Write short notes on "Brazilian test"

11 When carrying out a Brazilian test on a granite disc. 3.8 cm diameter and 2.5 cm long, the load at failure was found to be 35.6 kN. Calculate the tensile strength of the granite.

12 Give a graphical representation of the relationship between shear stress and shear displacement for rocks in a direct shear test.

13 Describe, with the aid of sketches, two methods of measuring the confined shear strength of intact rock. Explain how the strength envelope graph is obtained foreach.

14 Describe the procedures and the interpretation of uiaxial compressive strength testing for rock.

15Resgts of triaxial compression tests can beplottedin terms of(i) shear and normal stresses, and (ii) principal stresses. The second method is usually preferred. Explain why? How do you determine the cohesion and the internal friction angle from this plot? Assume the plot is linear

16 (a) A series of triaxial compression tests on intact specimens of a sandstone gave the following results:-

using the plot in terms <,f principal stresses, determine the Coulomb's shear strength that best tits the data. Assume the plot is linear.

(b) The initial state of'stress at a point in the sandstone is as follows:-

I , 6 ~ ~ e r c i s e s 115 3 Properties of Rock and Rock Mass

,9 Suppose rock at considerable depth is subjected to a triaxial, compressive'stress tield, which is not sufficient to cause failure. Use a Mohr's stress diagram to show what effect pore water pressure has on the stresses applied.

The pore water pressure, 11 will raise by the consuuction of a reservoir. What valueof u will cause fracture of the sandstone in situ. 1 20 (a) Draw a sketch ofthe failure envelope ~ostulated by the Coulomb's criterion,

with the stress circle indicating the result of a triaxial strength test. If the measured

17 A series of Viaxial compression tests on specimens of marble gave the followinp I test data consist of

Establish the Coulomb's shear strength criterion for this marble that best fits the data. Assume the plot is linear

- ~- results:-

-----.~--.wa==. - .-.= n~

m3, MPa . . - - ~ E L ~ L . ~ ~. . . ~ ~ .

IS Aseriesoftriaxial compression tests on specimens of sandstonegave the following results:-

(i) confining pressure; (ii) axial stress at failure; and (iii) angle between the failure plane and the minimum principal stress direction.

Establish the Coulomb's shear strength criterion for this sandstone that best tits the data. Assume the plot is linear

derive from these data, and the trigonometry ofyour sketch, expressionsforthevalues of-

(I) Angle of friction. q. (2) Cohesion, c ofthe material tested.

@)Given values of data in part (a) of

(i) l MPa; (ii) 9 MPa; and (iii) 60'.

calculate the values of rp and c and verify your results with a plot on graph paper.

21 (a) Describe with the aid of sketches the Coulomb's criterion of failure of rocks. (b) A cylindrical rock specimen is subjected to an uniaxial compression test and develops a shear fracture, running at 22.5' to the core axis. when the axial stress reaches 20 MPa. Derive an expression for the cohesion of the rock, using trigonomet- ilc principles and. assuming that the Coulomb's failure criterion is applicable, calculate the cohesion.

22 A sandstone has cohesion. c of 15 MPa and angle of friction. of45" What axial

Stress, 01 would be required to break a specimen subjected to a confining pressure. o ~ o f 15MPa'

23 The following are results tiom triaxial compression testson core specimens oftwo rock types. In the case of shale. theobserved values of a (the angle between the failure plane and the direction ofthe minimum ~ r i n c i ~ a l stress, 0 3 ) are also given. Determine the values of angle of friction, ,p and cohesion, c for both rocks. In the case of shale. how well does the observed values of acompare with that predicted by theCoulomb's failure criterion? Comment on any differences.

116 3 Properties of Rock and Rock Mass =- ..---

Rock ty .~ A*.-z=

Shale 125.7 4.4 53

176.3 110 55

150.3 6.6 53

21 1.0 26.3 54

Granite 330.0 11.0

387.4 21.9

487.4 32.9

526.0 43.9

54.9 ~ . 6 4 9 ? 4 - ~~ .. ~- .. ... ..--A=-k

24 Draw a stress-strain curve which may be obtained, in a stifftesting machine, on a cylindrical rock specimen in uniaxial compression. Describe briefly the important points on and sections of the curve.

25 Give a concise definition of I (a) Secant modulus of elasticity. @) Tangent modulus of elasticity

26 Given below are the results of a compressive test of a granite specimen, cylindrical in shape.

Gauge length = 5.1 cm Diameter of specimen = 1.8 cm ._""__.,;_______-__~_______;~-~~.~,___I_.*_ ___.__I --- .Load, kN ~ a=xa ~ ov-7>.u.G.m ~ ---**...- % ! ~ E $ ~ ~ A ~ ~ ~ = ~ m-7w ~----- ~-=.==

0 0

4.45 6 1 0 x 10"

8.90 1.63 x 10'

13.34 2.62 x 10.'

17.79 3.71 x 10.'

22.24 . 508x . 10.'

26.69 6.60 x 10.'

3114 8.64 n 10"

35 58 ~ . . 1 . 1 4 ~ 10.' . .

3.6 Exercises 117

1" addition to the above, at a total 4 load of 17 79 kN, the total transverse expansion was determined to be 3.45 x 10- cm.

(,) Tangent modulus Ea t zero load. (b) Secant modulus E at 17.79 kN load. (c) Secant modulus Ea t 35.58 kN load: (d) Poisson's ratio.

27 Describe in detail the ultrasonic pulse method for the measurement of dynamic elastic constants of a rock material.

28 (a) Define "creep" and indicate in a sketch the different stages of creep strain progressing towards failure. On the sketch show what would happen if the stresswere instantaneously removed at some time during (i) the primary creep stage and (ii) the secondary creep stage. Label all components of the diagram. (b) Show on the same sketch the general form of another creep curve for the same material if a significantly larger constant stress were applied.

29 What are the essential components of the rheologjcal models which may be used to simulate time-dependent swain behaviour of rocks under constant stress? Derive equations fortotal strain E experienced by thevarious models at constant stress a over time 1. Show strain-time curves for these models subjected to a constant stress.

30Draw theBurger's rheological model whose time-strain curve under constant stress is closely analogous to the time-dependent strain behaviour of many rock types under constant stress. Name each element and its constant. Also give the equation for the total strain E experienced by this model at constant stress o over time 1. Derive expressions for the recoverable and unrecoverable strains upon unloading the model during secondary creep stage.

31 What do you understand by the term "Time-dependent strength of rock"

32 What is the most commonly used method for the shear testing of discontinuities i n rock? Describe the test briefly.

33 Explain why strongly bedded rocks may sometimes fail along the bedding planes in a triaxial compression test, and at other times may fail across the rock material, depending on the orientation of the applied stresses. Quantify the answer as much as Possible.

118 3 Properties of Rock and Rock Mass I 34 A series of triaxial compression tests on specimens of a slate gave the following results:- .-=.. :-- .--- --- ..&----- ~- ---=---.-:=- ------ 03. MPa 01. MPa Angle between cleavage

planes and u3 direction,

In each test, failure occurred by shear along the cleavage. Determine the shear strength criterion for the cleavage planes. ! 35 Triaxial compression tests on specimens of a distinctly bedded shale cause failure of the rock material when tested normal to the bedding plane, such that cohesion is 18.5 MPa and the angle of friction is 3 lo. When tested at an angle to the bedding such that failure occurs along the bedding planes, cohesion is found to be 4.8 MPa, and the angle of friction is 25'. (i) What is the minimum axial stress which will cause failure, and at what bedding plane orientation, when a 3 = 8 MPa? (ii) A shale of this type, dipping at Is0, is subjected to an all-round horizontal stress of 10 MPa. What verfical stress is likely to cause failure?

36 A rock mass is intersected by a series of parallel discontinuities, oriented at angle a to the minimum principal stress direction.

The rock substance has c = 20 MPa; rp = 44" The discontinuities have c = 2 MPa; rp = 27"

(a) Investigate the stability of the following cases:- I ( i ) a 1 = 1 1 8 M P a ; u ~ = 2 M P a ; a = 1 5 " (ii) 0 1 = I05 MP~; 03 = 15 MPa; a = 82" ( i i i ) 01 = 90 MPa; u? = 30 MPa, a= 58"

(b) Determine the range of orientations of failure paths along the discontinuities, relative to the minimum principal stress direction, if

01 = 26 MPa; 03 = 2 MPa

3.6 Exercises 119

(c) If the discontinuities are oriented at 40'' to the minimum principal stress direction, determine

(j) the maximum sustainable GI, if cr3 = 2 MPa. (ii) if U I = 34 MPa, whatmagnitudeof~3 would havetobeimposed to ensure stability.

37 A sandstone block contains a planar discontinuity The discontinuity has cohesion, c', of 0.35 MPa and friction angle, (pd of 30' and the rock material has cohesion, cng of I MPa and friction angle, qh of 40". Construct a graph showing the expected variation of axial stress at failure at a confining pressure, 03 of 2.5 MPa, as the angle between the discontinuity and the minor principal stress direction, avaries from 0" to 90" in specimens cored from the block.

38 A triaxial compression test is to be camed out on agranite specimen with thejoint plane inclined at 55' to the minimum principal stress direction. A confining pressure, cr1 of I .5 MPa and an axial stress, ul of 3.3 MPa are to be applied. Then ajoint water pressure will be intraduced and gradually increased with U I and a held constant. At what joint water pressure is slip on the joint expected to occur? Repeat the calculation for a similartest in which 01 = 4.7 MPa and 03 = 1.5 MPa. The shear strength criterion for the joint is as follows:-

39 Give a summary of common in situ rock mass property tests. Describe any one test in detail.

40 (a) State Mohr's failure criterion applicable to rock. @)The strength characteristic of a sedimentary rock is given by the equation

Determine the compressive and tensile strenb$hs, cohesion and angle of friction (as a point on theculve when u = 0) ofthis rock. Will this rock withstand the state of stress. when 01 = 11.03 MPa, u2 = 4.83 MPa and 0 3 = - 4.62 MPa ?

41 In a series of triaxial and uniaxial tests, camed out on a sedimentary rock the test results given in the table were obtained. Construct the Mohr circles for these results and the corresponding Mohr's envelope. If the following states of stress are expected to be set up at various points in the same rock:-

120 3 Propenies of Rock and Rock Mass

indicate which of the three stress conditions will induce failure. -- ... ----____ ~3 ma-.-. (TI, MPa =z4=s==-_------- - 0 - 2.4

0 30.1

10.3 93.1

20.7 121.5

42 If the data are available on the uniaxial compressive and tensile strengths of rocks, show how an estimate can be obtained of the magnitude of cohesion of rocks.

43 What are the essential differences between the Coulomb's criterion of failure and the Mohr's criterion of failure.

44 A series of laboratory tests on intact specimens of quartzite gave the following results:-

Determine the constants B and a in the Bieniawski's criterion.

Hydraulic Properties of Rock and Rock Mass

T he permeability of rock mass is an important property that influences the pore water pressure, ground water level, flow rate, and flow velocity of sloping

surfaces of rock mass or the rock mass surtounding dams and underground cavems. To begin with, pore water pressure influences assessments of the mechanical stability of rock mass, while the ground water level affects assessments of ground settlement, even though this depends on the soil at the surface of the ground The flow rate, on the other hand, influences assessments of water inflow, etc, in underground cavems, while the flow velocity affects assessments of piping, permeation damage, etc. on sloping rock masses and dam foundation rock masses.

The primary content ofthis Chapterincludes adefinition of rock mass permeability, the coefficient ofpermeability, and the basic properties of rock and rock mass, as well as a description of laboratory tests and field tests conductedtomeasure the coefficient of permeability Because underground openings have been used in recent years to store petroleum, liquefied gas, compressed air, etc., this Chapter also discusses the gas permeability of rock mass.

4.1 Definition of Permeability

4.1.1 Flow location and model

Macroscopically, rock mass can be viewed as a porous medium, however, when it is considered in terms of permeability, the flow field of it can be divided into two pans:- primary porosity and secondary one (Barenblatt, et al., 1960). Figure4. I shows adoubleporosity rockmodel to outline rockmass ofthis type (Satoand lizawa, 1982). The flow location consistsofthe primary porosity h~ ofalarge number of microscopic pores and small cracks in rocks as well as its secondary porosity hz which is many fractures. Here, the primary porosity ?LI is usually geaterthan the secondary porosity k2. Therefore, this primary porosity 11 is believed to store large quantities of ground water, and is called a storage porosity. The fractures which form the secondary porosity hl on the other hand, are called seepage porosity because water can pass

122 4 Hydraulic Properties of Rock and Rock M~~~ 1 4.1 ~efinition of Permeability

through them. The geometrical properties of pore and fracture are based on geological and weathering conditions in natural state

where y = specific weight of the fluid; Figure 4.1 Double porosity model of a fractured rock mass (after Sato and lizawa, 1982). p = density of the fluid;

w = acceleration of rrravity; and I

- . z = height above a reference datum line . -. -

rock k , 1, ,,, (secondary porosity)

contact points

4.1.2 Resistance law of flow

When ground water permeates saturated porous media, the flow rate Q is propor- A h tional to the sectional area A, and the hydraulic gradient i = - This law, originally

I deduced by Henry Darcy in 1856. is called Darcy's Law. As shown in Figure 4.2, when the hydraulic gradient i is applied to a cylindrical rock sample, the flow rate Q is represented as follows:-

where I, = velocity; A = sectional area of the sample; k = coefficient of permeability; i =hydraulic gradient;

A h difference of piezometric head; 1 = length of the sample;

K = intrinsic permeability; ~i=viscosity of the fluid; and

A (1 =pressure difference.

-

Figure 4.2 Seepase through an inclined rock sample (after Bear, 1979).

U

d a m levd

The coefficient of proportionality k (#~mensions LIT) is known as the coeficient of permeability, and is generally used to represent the permeability of ground. When, however, fluid bodies other than water such as oil and gas infilh-ate the mass, are considered, the hydraulic conductivity K (dimensions L' or Darcy unit) is used. K is called the inbinsic permeability. I Darcy is defined as follows:-

0 AP 2 1 Darcy = ( - ) I/( ) = (1 cm31seclcm ) x (I centipoise) / ( I atmospherelcm)

A I (4.3)

The relationship between k and Kis as follows:-

I ! where v = kinematic viscosity ofthe fluid.

'1 On the basisofthis. the relationship between the units forwater at 20 degrees Celsius Here. piezometric head h is provided by the sum ofthe pressure head and elevation

can he represented as follows:- head. and is represented as follows:-

124 4 Hydraulic Properties of Rock and Rock Mass

1 rn/s=1.024x10~7m2=1.038x105~arcy 1 Darcy = 9.869 x 10-13 m2 = 9.635 x m/s

Darcy's Law meansthat a linear law of resistance is established when a flow through a permeable media is laminar. However, anon- linear law of resistance is established when the water molecules have absorption resistance due to extremely low flow velocity, or when a turbulent flow occurs due to a large flow.

Here it is possible to make a categorisation based on Reynold's number, Re.

where v = velocity of the fluid and 6 = effective diameter of the permeable matrix

The flow may take one of three forms; micro seepage, laminar flow, and transition flowiturbulent flow, as shown by the relationship between the hydraulic gradient and thevelocity showninFigure4.3. TheDarcy flow is establishedinthis internal laminar

-6 flow space, its minimum value is considered to be Red= 10 to (Sato, 1971). while its maximum value is taken to be about Rec = 1 to 10 (Bear, 1979).

Generally the hydraulic gradient of the ground water is small and the permeation resistance of rock mass is large, so most ground water flows occur at Re well within the laminar flow range, when the linear Darcy's Law is applicable. When, however, the coefficient of permeability or the hydraulic gradient is extremely small, or a high velocity converging flow appears near a well, the flow is a non-Darcy one.

Figure 4.3 Velocity of fluid, v versus hydraulic gradient, i

0 k -

i flow

i, id IC i

4.2 Permeability of Intact Rock 125

4.2 Permeability of Intact Rock The coefficient of permeability of rock is, as shown by Equation (4.1). defined on

the basis ofDarcy's Law, and is measured by laboratory percolationtests. This Section discusses the coefficient of permeability of rock and laboratoty testing methods.

4.2.1 Coefficient of permeability of rock

Table 4.1 gives representative values of coefficient of permeability, k for various rocks and rock masses (Vumkuri and Lama, 1986; Ito, 1989). The coefficient of permeability and the porosity are interdependent, and an example of granite is shown in Figure 4.4 (Watanabe, 1983 ).

Figure 4.4 Permeability versus porosity for granites (after Watanabe, 1983)

4.2.2 Laboratoly percolation t e s t s of rock

lod -

It is difficult to measure the coetlicient of permeability of intact rock at its original site because it is impossible to separate it from fractures. Therefore, it is measured by means of the laboratory percolation tests. The laboratory methods of measuring the coefficient of permeability of rock described in this Sub-section are the longitudinal percolation test and the radial percolation test.

lo4

-z E 5 .-

4.2.2.1 Longitudinal percolation test

This test is pelformed by first coating the side ofthe sample with epoxy resin or the like or applying lateral pressure to a rubber sleeve so that it adheres completely to the

-+ - + o weathered granite

a heated m i t e - a +. S*Ongly weathered

sail (Masado) a. a hydrathemally altered granite -

- * .-

lo" - E

p" -

4.2 permeability of Intact Rock 127

sample surface, thereby causing a vertical one-dimensional flow inside the sample and measuring the flow rate. Figure 4.5 is an outline diagram of the test (Sato and Sasaki, 1984). Many core samples have a diameter between 5 and 6 cm and a length ranging from 10 to 20 cm. This test method can be used on soft rock samples or rock samples that have many micro-cracks. The coefficient of permeability of rock, however. is generally very small and a low flow rate is produced by high-pressure

which limits coeEcient of permeability measurements. As a result, the approximate marginal precision is considered to be about k - lo-'" mls (Lama and Vutukuri. 1978).

Figure 4.5 Schematic view of longitudinal percolation test

pressure regulating valve m

water tank

JJ porous plate sample Nbber L I

This test is usually conducted by applying the constant head method, although the falling head method may alsobe applied. The permeability formulas aregiven below:.

I (Constant head method)

~ (Falling head method) I


where Q = the measured flow rate; A =sectional area of the sample; I = lengb of the sample;

Ah = difference in head between inlet and outlet; a = sectional area of the stand pipe;

12 - fl =measurement time; and hl, hz = water head in the stand pipe at time t l , t 2

Another method that has been recently employed is the transient pulse method proposed by Brace et al. (1968). This method is avariation of the falling head method, which uses a triaxial cell. This method is considered to be useful with low-permeability rock For this test, the sample is set between two vessels of known volume, as shown in Figure 4.6 (Sugimoto et a]., 1985). At first, the pressure inside the two pressure vessels is equal, but later a pressure pulse A P is added to the pressure inside one of thevessels. As time passes, the hydraulic gradient within the sample changes, and ultimately, the pressure within the two vessels converges to a fixed pressure. The coefficient of permeability is determined based on the relationship between the pressure and time during this interval using the following formula:.

Figure 4.6 Schematic view of transient pulse test and changes of pressures during an experiment.

whereP~, FI = pressure and volume of the vessel subjected to a pressure pulse; P2, Ji = pressure and volume of the vessel not subjecred to a pressure pulse;

4.2 Permeability of Intact Rock 129

I?= pressure convergence value; A P = pressure pulse;

t = time. K =intrinsic permeability; A = sectional area of the sample; p =viscosity of the fluid; q = wmpressibility of the fluid; and I= length of the sample.

To perform these tests, the sample has to be fully saturated and when a rubber sleeve is selected, its water absorption properties and permeability must be carefully considered. With soft rock and rock that has many fractures, the permeability also fluctuates according to the confining pressure. Another troublesome factor is the time dependence of rock seepage. In other words, if a percolation test is conducted over a long period of time, the flow rate gradually falls. Figure 4.7 presents an example of the results of along-period percolation test conducted on therock (Satoand Sasaki, 1987). This time dependence is believed to be a result of clogging caused by a suspension and bubbles in the ground water or the reaction of dissolved material in the injected water and pore water (Sato and Ito, 1988). This phenomenon is generally taken into consideration, but it is necessary to pay careful attention to this problem when performing a percolation test.

Figure 4.7 Permeability change with time in a rock (after Sato and Sasaki, 1987).

q, = flow rate qO= initial flow rate qe= flow rate at timer = m

q, - 0 5 q"

0 0 10 20

time, day

42.2.2 Radial percolation test

r h e r e ~ ~ a rock percolation ir'stmerhod tnar takcsadvanra~eoirad~al tlow 4sshoir n in Fiuure4 81Rishardand I'anchanarllam. 1380)- thistcsl isconduaed b) first dnlllny - a central hol;in the middle of the sample, then either causing a converient flow from the sides of the sample or a radial flow from the central hole, and measuring the resulting tlow rate. In either case, this test method employsthe constant head method.


When this test method is employed. the results are affected by the compression or tension of the sample. Therefore, it can be used to test homogeneous hard rock, but not soft or fractured rock.

Figure 4.8 Schematic view of radial percolation test

water rank

kplrm When this method is employed, a radial flow is postulated, and the coefficient of

permeability k can be determined using the following formula:-

where r2 =external diameter of the sample; rl =radius of the central hole in the sample; and H = height of the sample

When this test is performed, it is essential to be careful about the same factors. as in the case of a longitudinal percolation test.

4.3 Permeability of Rock Mass The permeability of rock mass includes the double porosity discussed with respect

to the rock mass model described above; however. when the principal issue is permeability, the fracture flow is the core issue. Table 4.1 presents typical coefficients of permeability of rock mass measured by i n situ tests, and these results are larger than those obtained for intact rock.

4.3 Permeability of Rock Mass 131

4.3.1 Permeability of a single fracture

To begin with, the coefficient of permeability kf for the flow in the single fracture shown in Figure 4.9, is represented as follows by considering the viscous flow in parallel smooth fractures (Poiseuille's Law):-

.- .

where w =width ofthe fracture; g = acceleration of gavity; and v = kinematic viscosity of the fluid.

Figure 4.9 Simple model of an open fracture

But because an actual fracture is influenced by the roughness offracture wall surface, tortuosity, filler, etc., its coefficient of permeability differs from that obtained with Equation (4.10). It is difficult to quantitatively assess the filler. But the roughness and tortuosity of the fractures are represented as follows (Louis, 1969; Watanabe, 1983) (Figure 4.10 shows a model of the fracture):-

1 where E = tomosity (usually E = 7);

I = horizontal length; I' = pass length of flow; f - coefficient of roughness: u = coefficient of relative roughness (approx. 4.5 to 125);

Ilir = hydraulic mean radius; hf= mean roughness height of fracture wall; and m = coefficient (approx. 1.5 to 3.0).

132 4 Hydraulic Properties of Rock and Rock Mass I 4 3 permeability of Rock Mass 133

Figure 4.10 Open fracture models used in experimental studies of resistance laws of 1 the case of fresh rock with few fractures, however. the permeability of the rock

flow contributes to the coefficient of permeability of the rock mass

(a) I Figure 4.11 Permeability versus axial stress of a fracture (after Gale. 1982). I I

It is possible to compute the ideal coefficient of permeability of a fracture using Equation (4.11), but the width of the fracture is, in the end, the factor that determines the coefficient of permeability of a fracture. It is therefore possible to estimate the coefficient of permeability of a fracture with some degree of accuracy in cases where the width of it can be accurately estimated.

On the other hand, the width of a fracture varies according to stresses in the surrounding rock mass. Figure 4.1 1 presents the relationship between axial stress and the permeability of a fracture, as obtained through laboratory percolation tests (Gale, 1982). This figure reveals that as the stress increases, the degree of permeability decreases sharply. This indicates that as the stress increases, the fracture becomes narrower When a structure has been constructed on top of or inside a rock mass, it is necessary to assess the coefficient of permeability by paying careful attention to the fact that thecoefficient ofpermeability of fractures such as this depends on thedegree of stress.

Moreover, although this suggests thatjust asin the caseof rock, rockmass ismarked by the time dependence of the coefficient of permeability of a fracture. there is no clear data on the matter.

4.3.2 Permeability of rock mass

The permeability of rock mass is generally determined by factors such as the density and continuity of fractures since the fracture system is the principal flow channel in

I 0.1 I 10 I

40 Axial stress. oz MPa

Therefore, ifthe situation shown in Figure 4.12 is considered, in which the fractures are ideal, the coefficient of permeability of the rock mass can be represented as follows:-

where h =spacing between fractures

Figure 4.13, in which kr = 0 is assumed, shows an example of the relationship between fracture width and the coefficient of ~emieabilitv (Hoek and Brav. 1981) . . As can be seen, as the fracture width and density increase, the coefficient of permeability also increases. In many cases, the fracture width is not isotropic and homoge-


neous, but rather anisotropic. Figure 4 14 shows a simple model of two orthogonal fracture systems.

Figure 4.12 Simple model of a rock mass.

Figure 4.13 Permeability versus width of fracture

Fracmre width, w, mm

Next, the relationship between coefficient of permeability and stress in a rock mass is considered. Figure 4.15 shows the permeability distribution with depth measured in a granite mountain using the Lugeon Test (Gale et d., 1982).

Theresults show thatthe coefficient of permeability decreases as the depth increases. This fall in the coefficient of permeability caused by the increase in depth is summarised by the rock block model shown on the right side of the figure. In a word. as the depth increases. the width of horizontal fractures narrows under the efect of the mountain rock's dead weight. In many cases, the coetticient of permeability in horizontal direction varies as much as 100 times between two points when the depth varies by 100 m. Vertical fractures, on the other hand, vary under the effect of lateral stress.

4.3 permeability of Rock Mass

Figure 4.14 Two-dimensional fracture model

Figure 4.15 Depth versus permeability in agranite mountain (after Gde et al., 1982).

1 rack mass

Permeability, m/s

136 4 Hydraulic Propemes of Rock and Rock Mass

4.3.3 In situ test method for permeability of the rock mass

It is not suEcient to measure the coefficient of permeability of a rock mass in a laboratory because it is necessary to account for a number of fracture systems. Therefore, on the whole, in situ tests are conducted. The most widely used methods of testing rock mass are the Lugeon Test and the JFT (Johnston's Formation Tester). In situ test methods used to test shallow strata of sandy ground include pumping tests, permeability tests in a single borehole, etc. Testing methods of this kind are applied to rock masses when the permeability is large in a weathered rock mass near the surface. More recently, a test called the crosshole sinusoidal pressure test has been developed as a way to measure the permeation anisotropy of rock masses.

Below, the Lugeon Test and JFT, the most widely used rock mass percolation test methods are explained.

43.3.1 Lugeon Test

The Lugeon Test is a type of injection test using a borehole. In principle, a packer is inserted into a borehole, which has a diameter between 46 mm and 66 mm, and then water is injected under a fixed pressure (0.98 to 1.47 m a ) into the test cavity. As shown in Figure 4.16 (ISSMFE, 1989), there are two methods:- the single packer method and the dwble packer method. In many casesthe test cavity is about 5 m. The measurement results are then obtained by defining and organising the Lugeon value as shown below:-

where Lu = Lugeon value; Q = water quantity injected, ilmin; p = effective injection pressure, MPa; L = length of test cavity, m;

po = water pressure in enhance of borehole, MPa; h~ =difference in altitude between the pressure gauge and the middle ofthe

test cavity, cm h2 =head from the ground water depth to the middle of the test cavity, cm; h3 =head loss caused by pipe resistance, cm; and yw = weight per unit volume of the water, MPaIcm (i 9.8 x 10.' MPaIcm,

zoo C).

One Lugeon represents the permeability when one lihe per minute of water is injected into a rock mass under a pressure of 0.98 MPa through a one metre borehole

The coefficient ofpermeability obtained by the Lugeon Test canbedetermined using the following formula in which the radial flow around the borehole is estimated, and the effective radius is considered to beR'=.L.


Figure 4.16 Schematic view of Lugeon Test.

PRssure gauge water supply flow late

single packer dwble packer

where ~p = borehole diameter.

According to this, when the diameter of the borehole, 9 - 40 to 80 mm, one Lugeon generally corresponds to k k 10.' mls.

When the Lugeon Test is it is important to make sure that the test cavity is not too short, because if it is, the flow will be a spherical radial flow, and it will be impossible to assess the coefficient of permeability using Equation (4.14). Further- more, if the Lugeon Test is performed in the non-saturated zone on top of the ground water surface, theinjected water will flow downward because of gravity. One possible approach in such cases calls for the injection of air instead of water An air permeability test is described later in Section 4 .4.

The following are other important points to keep in mind when pefoming aLugeon Test:.

(1) Hydraulic fracturing and fracture deformation of the rock mass around the borehole caused by high pressure water injection.


(2) Water tightness (cut-off properties) of the packer, (3) Lenb*h of the test (time dependence of the coeficient of permeability). (4) Assessment difficult if the rock mass is marked by strong anisotropy.

4.3.3.2 In situ test using JFT (Johnston's Formation Tester)

This test, which is normally called the JFT, is atype of water-table recovery test that uses a borehole. It is, in principle, identical to the in situ permeability in a single horehole test:- a percolation test method for sandy ground. Like the Lugeon Test, a single borehole is formed, and as shown in Figure 4.17 (JSSMFE, 1989). the length of the test cavity is established with a packer, and the coefficient of permeability of the rockmassis determined by measuring thevolumeandvelocity of theground water flowing into the measurement tube from the section being measured, and at the end of the test, also measuring the stabilised ground water level. The equipment used to perform this test includes a packer unit. a water level measurement tube, atrip valve that controls the flow of the ground water into the measurement tube from the area tested, and a water-level measuring apparatus. The measurement begins by opening thetripvalveafterall theequipment is set up. Themeasurementresults are categorised as confined aquifer and unconfined aquifer, and the coefficient of permeability is computed using the following formulas:-

(Confined Aquifer)

where d= effective internal diameter determined by subtracting the sectional area of the water-level measuring cable from the sectional area of the water-level measuring tube;

I. =length of the test cavity; D = diameter of the test cavity; s = difference between the mean water level and the water level measured

in the tube (Fisure 4.18 shows the relationship between logs and I.); and

1 =time.

(Unconfined Aquifer)

0.66 L? log ( 2 ) log ("' ) k .. '2 ~. ~. 'L8

1. ' D - (4.1 6) 12-11

Here. unconiined aquifer refers to a water-bearing stratum with an unconfined ?round watersorface. while the confinedaquiferisa water bearing stratum lyingunder


an impermeable ground layer, which stores ground water although it has no unconfined ground water surface.

When the JFT method is used, it is essential to keep the following points in mind:-

(I) The test is performed below the ground water surface. (2) The water level inside the hole must not be lowered a lot. If it is, the hydraulic gradient could become too large, which would result in an undervalued coefficient of permeability. (3) Water tightness ofthe packer. (4) Assessments are difficult if the rock mass is marked by strong anisotropy.

Figure 4.17 Schematic view of Johnston's Formation Tester.

@stank d recorder

n1F-U ...................... ..................... ..,.,.,.,.,.. . ..,., . ..,.,.,.,.,., , ....,. wire

level meanuremcnl tube

Figure 4.18 Log sversus t .

standerd point

): initial groundwaterlwel

lime, I. s

140 4 Hydraulic Properties of Rock and Rock M~~~ 1 4.4 Gas seepage in the Rock Mass 141

4.4 Gas seepage in the Rock Mass Problems of gas percolation and air leakage in the rock mass are encountered when

an underground space is used for the underground storage of petroleum or liquefied gas, or for the storage of compressed air. Therefore, measurements similar to those used to determine the coefficient of permeability are conducted to deal with this problem. The problems of gas seepage and air leakage in rock masses will become increasingly critical with progress in this use of underground space. Gas seepage and gas leakage actually encountered in a rock mass, however, vary in complex ways depending on the water content and the form of the voids in the rock. As a result, so the problem has to be handled in a different manner than that employed to deal with the coefficient of permeability.

Here, a basic approach is presented to handle the gas permeability of rock, and air leakage through fractures.

44.1 Gas seepage in intact rock

When considering theissueof gas seepage, it isimportantto keep in mind that while water is an incompressible fluid with a constant fluid density, gas is a compressible fluid with a variable density. Consequently, it is necessary to introduce equations of state to represent therelationship bet&een themass of agasand the pressure affecting it, even though it is considered possible to establish Darcy's Law for gases just as has been done for water.

First, when defining the flow rate Qg of gas for use in Equation (4.1) it is necessary to generalise and consider Darcy's Law. Next, the percolation mass flow velocity is defined based on Darcy's Law, and when it is represented by a one-dimensional flow in the z-direction, it is as shown in the following formula:-

where kg = coefficient of permeability of gas

Next the equation of state for gas:-

where p =,constant; n = adiabatic index (the process of expansion or contraction of gas);

cp = specific heat at a constant temperature; and cil = specific heat at a constant volume

When Equation (4.18) is substituted into Equation (4.17),

This shows that if the gas flow rate at an atmospheric pressure Qg i s in the case of air, in theisothermal state, rr= 1, andit can be written as follows (Aravin et al., 1965):-

where pa = atmospheric pressure; PO, p~ = boundary pressures of gas at z = 0 and z = I;

I = length ofthe sample; and z = ordinate.

In this way, the gas permeation volume differs from that obtained with Equation (4.1) in the case of water. The gas permeation volume is proportional to the difference of the square of the pressure, while the gas permeation resistance is represented by the intrinsic permeability This is a consequence of the fact that the gas peneaation depends on the pressure in the case of a gas.

The gas permeation of rock can therefore be measured using the same methods as the laboratory test methods used to find the coefficient of permeability. Figure 4.19 summarises the longitudinal percolation test for a gas (Sakaguchi et al. , 1992). The intrinsic permeability K is determined from the following equation:-

Basically, a gas percolation test should be performed on rock, when the sample is completely dry, the reverse of what must be done during a water permeability test. The presence of pore water in the rock sample lowers the characteristic coefficient of percolation. Moreover, if it is saturated, gas will either not permeate the sample or it will permeate it by forcing out the pore water

Attempts have been made to test the gas permeability of rock using laboratory gas percolation tests based on radial flow, and methods similar to the above-described Lugeon Test.

In these cases, the intrinsic permeability can be determined by the following equation:-

1 42 4 Hydraulic Properties of Rock and Rock Mass

where r? = external radius of the sample or the influence circle of in situ test and rl =radius of the interior space of the sample or the interior diameter of

the borehole.

Table 4.2 gives an example of the properties of rock samples measured by using a laboratory gas percolation test.

Figure 4.19 Schematic view of longitudinal percolation test for gas seepage

(I) Gas tank (4) Graduated cylinder (2) Valve (5) Rack sample (3) Resmregauge (6) Water

Table 4.2 Intrinsic permeability coefficients of typical rocks

I

granite 0

4.4 Gas seepage in the Rock Mass 143

4.4.2 Gas leakage in a rock mass

An outline of gas leakage in a rock mass is shown as a model of underground space in Figure 4.20. Gas which is either stored or generated in an underground space leaks easily through fractures. Since the undergound water pressure affects their leakage, it is necessary to determine whether ornot gasis leaking. Figure 4.21 shows a system of experimental apparatus for gas leakage test through a fracture (Sakaguchi et al., 1992). The pressure balance at an inlet of fracture is given by the underground water pressure PW . the gas pressure Pg and the capillary pressure PC . A safe water pressure without a gas leakage depends on the ground water depth aboverock cavern ofstorage. Thus, the balance equation for realising a critical pressure of gas penetration can be written as:-

where (.' = constant; o = sutface tension of intetface between gas and water, and w =width of the fracture.

Figure 4.20 Concept of gas leakage mechanism around a cavern.

Figure 4.22 shows the relationship between PPiP,,. and w. According to the same figure, it is possible to obtain the equation for PgiPP. values for the air leakage from the water pressure level Plr and the fracture width.

Equation (4.23) can also be called the critical pressure of penetration in rock masses. On the other hand, if gas leakage occurs and gas invades the inside of the fracture, it will form bubbles and rise in the fracture (Miyashita and Sato, 1984). It is said that if the vertical hydraulic gradient of the cavern wall is over 1 at this time, bubbles will not rise, and no gas leakage will occur (Aberg, 1977). After the bubbles have formed and risen inside the fracture, the gas will travel to the alluvial ground and weathered layers where it will be scattered in the atmosphere.


Figure 4.21 System of experimental apparatus with gas seepage in open fracture.

(1) Gas tank (5) Pressure gauge (2) Water tank (6) Transparent cylinder (3) Cushion lank (7) Rubber balloon (4) Sample

ps Figure 4.22 Effect of Pw on the - - fracture vidth, w curve P,"

1107-0.426 logw

0

0.01 0.05 0.1 0.2 0.5 1.0

Fracture width. w , mm

4.5 Exercises I The Lugeon test was performed on the rock mass of a mountain. The test conditions and test results are shown below. Find the Lugeon value and coefficient of permeability

Water quantity injected :- C) = 0.4 l i s Water pressure at the entrance of borehole:-p = 0.98 MPa

4.5 Exercises 145

Difference in altitude between the pressure gauge and the middle of the test cavity:- h l = 3 6 m Head from the ground water depth to the middle of the test cavity:- h2 = 30 m Head loss caused by the pipe resistance:- h3 = 1.0 m Length of the test cavity:-L = 5 m Diameter of the borehole:. 9 = 66 mm

2 A one-dimensional gas percolation test was performed on dry rock, and the following results were obtained. Find the intrinsic permeability Also, find the coefftcient of permeability in the same rock when it is in a saturated condition.

3 3 Flow rate of gas:- Qg=2.0 x 10- cm is Injection pressure:-po = 395.525 kPa (Gauge pressure = 294.200 kPa) Outflow-side pressure:-pi = 101.325 kPa (atmospheric pressure) Diameter of the sample:- I$ = 5 cm Length of the sample:- I= 10 cm Test temperature:. T= 20" C

3 Derive the coefficient of permeability computation formula for a falling head permeability test based on Darcy's Law. Also determine the coefftcient of permeability when the test conditions and results are as shown below:- Sectional area of the stand pipe:- a = 2 cm2

Sectional area of the sample:- A = 78.54 cm2 (= n x 5') Head at r ~ : - h ~ = I00 cm Head at tz:- h2 = 50 cm Sample length:- 1 = 10 cm Measurement time:. t2 - ti = 10 hours

4 The following are the geometric properties of a horizontal fracture. Determine the coefficient of oermeabilitv of this fracture. Also. determine the coefficient of oerme- ability of the rock mass i n k e horizontal direction when the interval between frectures is, h = 0.1 m, and the coefficient of permeability of the rock is k, = 4.0 x 10 .~ mls.

Width of the fracture:- M, = 0.05 mm Tortuosity:- E = 0.6 Constant:. a= 17.0 Mean roughness height of the fracture wall:- hr= 0.01 mm Coefficient:- m = 1.5 Temperature = 20" C

5 An underground cavern with a radius, r of 3 rn has been excavated at a point where the surface of the ground water, H, is 5 m above the interface between the alluvial ground and the rock ground as shown in Figure 4.23 The distance between the

146 4 Hydraul~c Propert~es of Rock and Rock Mass

interface and the middle of the cavern, H, is 10 m. A JFT test was performed on this rock mass, and the following data was obtained. Compute the coefficient of permeability, and then compute the discharge rate into the cavern per unit depth.

Test Results

Effective diameter of the water level measurement tube:- d = 30 mm L e n d of the test cavitv:- L = 2 m Internal diameter of the test cavity:- D = 100 mm Mean water level and water level in the test tube (Linear ponion)

Difference in water lev&- sl= 2.0 m ( t ~ = 5 minutes) sz = 1.15 m (22 = 35 minutes) .

Computation formula of discharge into the cavern

Figure 4.23 Rlustration for Exercise 5

n - - - T

alluvial graund H , = S r n

discharge, q

Virgin Rock Stresses

P rior to excavation of an underground structure in rock, the rock medium is subject tovirgin stresses . Since induced stresses are directly related to thevirgin

stresses, it is clear that determination of these stresses is a necessary precursor to any design analysis. The need for reliable estimates of the virgin state of stress has resulted in the

expenditure of considerable effort in the development of stress measurement devices and procedures. Methods developed to date exploit three separate and distinct principles in the measurement, although most methods use a borehole to gain access to the measurement site. Themost common set of procedures are based on determination of strains at the bottom or wall of a borehole, or other deformations of borehole, induced by overcoring that part of the hole containing the measurement device. This method is commonly called as '"overcoring method". If sufficient number of strain and deformation measurements are made during this stress-relief operation, the six components of the field stress tensor can be obtained directly from the experimental observations using solution procedures developed from elastic theory. The second type of procedure, represented by hydraulic fracturing and flat jack measurements. determines a circumferential normal stress component at a particular location in the wall of a horehole or an excavation. At each location, the normal stress component is obtained by the pressure. The third procedure involves the estimation from rock core taken from the underground space using drilling. Kaiser effect method, deformation rate analysis (DRA), differential strain curve analysis (DSCA) and anelastic strain recovery (ASR) are included in this type. These methods are less reliable than overcoring method and hydraulic fracturing method. However, this procedure is simple and not expensive. So, recently, much research has been done to develop these Procedures.

5.1 Specification of the Virgin State of Stress In general, both vertical and horizontal stresses increase with depth of overburden.

However. the magnitude and direction of the horizontal stress can be affected by additional factors. These factors may include the nature ofthe geological structures, tectonic forces existing within the earth's clust, residual stress, and thermal stress.

148 5 Virgin Rock Stresses

Since the ground surface is always traction-free, simple statics requires that the vertical normal stress component,p, at a sub-surface point be given by

where y = the rock unit weight and Z = the depth below ground surface

A common but unjustified assumption in the estimation of the in situ state of stress is a condition of uniaxial strain during development of gravitational loading of a formation by superincumbent rock. For elastic rock mass behaviour, horizontal normal stress components,p, and pp are then given by

v pu:=pYV = (---- )pa l - v

(5.2)

where v =Poisson's ratio for the rock mass

If it is also assumed that the shear stress components pry, py., pzx are zero, the normal stresses defined by Equations (5.1) and (5.2) are principal stresses.

5.2 Compensation Technique - Flat Jack The flat jack technique involves the use of flat hydraulic jacks (Figure 5.1),

consisting oftwo plates of steel weldedaround their edges and a nipple for introducing oil into the intervening space.

The flat jack method determines the stress parallel to and near the exposed rock surface in an excavation. Since stress is measured in only one direction, at least six measurements in distinct directions are required to calculate the stress tensor. This technique consists of either measuring the change of length between pairs of measuring pins when a slot is cut into the rock between the pins or using two hydraulic borehole pressure cells (bpc) in place of pins. A thin hydraulic cell called a flatjack is grouted into the slot and-pressurised until the initial measured distance is obtained. The pressure in the flat jack is then considered equal to the pre-existing stress. This

5.3 Relief Techniques - Undercoring or Overcoring 149

only of the rock mass immediately surrounding excavations and not the virgin stress, althou& this method can provide an estimate of the in situ stress and reveal general trends. This method can be used in highly stressed areas where methods requiring drill core cannot be used. The main causes of error inherent in flat jack method are nonrecoverability of strain under load and the influence of shear stresses in the test area.

Figure 5.1 Typical flatjack (a) and flatjack test configuration @) (afterBickel,1993)

pressure

. flat jack I / II

5.3 Relief Techniques - Undercoring or Overcoring

5.3.1 General aspects

I .--..,"L,YL.-

large area of the flat jack; therefore, local variations in the rock stress are minimised.

con&tion must he reached to obtain valid results. .

The flat jack method does not require any knowledge of the elastic propelties ofthe rock; hence, it is considered a true stress-measuring method. Very often relief of the state of stress seriously disturbs the mechanical properties of the rock, and conse- cpently the rock samples used to determine thesepropettiesmay not be representative of the rock mass. The flat jack design has an averaging effect over a comparatively

This method is also better adapted to measurements in inelastic rock. Because of the difficulty in cutting deep flat jack slots, the flat jack method is

restricted to near-surface measurements. Thus, the determined stress is representative

The stress relief techniques involve initially the placement of the sensing apparatus on the surface or in a borehole, in contact with the rock under stress. The stress is then relieved by undercoring or overcoring. The final step is to measure deformations or strains due to stress relief, and to relate these deformations or strains to stresses.

Fiyre 5.2 illustratesthefollowingthree steps commonly involvedinany overcoring tprhnin,...

1. In Figure 5.2 (a), alargediameter holeisdrilled in thevolume of rock wherestresses have to be determined. The hole is drilled to a distance sufficiently far from the


investigation chamber so that the effect of the chamber itself on the stress measurements can beneglected. A distance at least equal to one chamber diameter isrequired. 2. In Figure 5.2 @), a small pilot hole is drilled at the end of the previous one. An instrumented device mustbeable tomeasure displacements, strains orboth ifrequired. The pilot hole must be long enough not only to neglect the effects of its own ends on the measurements but also to neglect the stress caused by the larger hole. The instrumented device can also be positioned directly onto the flat end of the large diameter hole and no pilot hole is needed. 3. In Figure 5.2 (c), the large diameter hole is resumed, partially or totally relieving stresses and strains within the cylinder of rock that is formed. The resulting changes of displacement or strain within the instrumented device are then recorded.

Rigure 5.2 Steps commonly involved in any overcoring technique.

large diameter hole

insuumented / instrumented \device

(b, I+,. --J' ' '' pilot hale ,, ,

5.3.2 Measurements on surface

Deformations of a rock mass which result from the drilling of a borehole can be used to determine surface stresses. The surface rosette method, which is also called the displacement rosette method or undercoring, determines absolute swess and stress changes on the surface around an opening. This technique uses six pins installed in pairs across three diameters spaced 60" apart (Figure 5.3). After initial measurements are taken across the three diameters using a Whittemore-type deformation gauge, the surface rock stress is relieved by drilling a hole concentric to the six pins. Final measurements are then taken across the three diameters. Based on the elastic infinite plate theory, the change in length of the diametral distances and the elastic modulus of the rock areused to calculate the surface stresses. Generally, rosette pins are placed on 25 cm diameter but can be varied to meet the measurement precision required.

The surface rosette method is a simple and economical technique for determining the absolute stresses and the change of stress on the surface around the opening.


However, the surface rosette method is not suitable for heavily jointed rock or rock that has been severely damaged by bblsting.

Figure 5.3 Surface rosette configuration (after Bickel, 1993)

5.3.3 Measurements in borehole

Exploration of the virgin stress state requires that the measurements are extended beyond the zone of influence of the walls of the excavation. This may be attempted by drilling a borehole into the wall and then making measurements in it. There are two types of measurements usually made in the borehole:-

1 Measurements in the borehole 2 Measurements at the back of the borehole

The first type is not suitable for measuring stresses approaching the strength of the rock because the surface ofthe borehole fails and overcoring becomes impossible due to discing.

The second type has the advantage of avoiding the problems associated with overcoring a borehole using a large diameter coring bit. The major difficulty is that it is necessary to relate the strain relief at the end of the borehole to the stresses in rock.

5.3.3.1 Measurements in the borehole

Borehole deformation gauge

Borehole deformation gauges are designed to measure diametral deformations of a borehole during the process of stress relief by overcoring. The current version of a 3-component borehole deformation gauge (3-C BHG) developed by U.S. Bureau of Mines is shown in Figure 5.4. This measures three diameters 60" apart and has all sensing elements in the same plane. It is designed to measure diametral deformations in an EX-size (38 mm diameter) borehole during stress-relief by overcoring The

152 i 5 Virgin Rock Stresses . ! 5.3 Relief Techniques - Undercoring or Overcoring 153

deformation of a hole, lJin a biaxial stress field and in plane strain can be expressed by the following equation.-

where d = diameter of hole; S. T= applied stresses perpendicular to axis of borehole;

0 =angle measured positive in counterclockwise direction from S to r (radial distance) (Figure 5.5);

E = modulus of elasticity; and v =Poisson's ratio

Figure 5.4 Current version of U S Bureau of Mines 3-component borehole deformation gauge (after Bickel, 1993)

(1) gauge body with tapered mounts 0 2 5 (2) tapered-mounted msducer scale, mm (3) lockng nut (4) s a w plug (5) pistan

Figure 5.5 Cross-section of hole in plate (after Bickel, 1993)

If the deformationsacrossthree diameters, themodulusof elasticity, EandPoisson's ratio, vare known, themagnitudeand direction ofthestressessand Tcan becomputed by the following equations:-

6 ( 0 2 - 71; ) tan 28 1 =

2 ~ J I - (I2 - 113

where S,T= maximum principal stress (S) and minimum principal stress ( 7 ) in the plane perpendicular to a borehole;

Ul,lJz,U; = deformation measurements across diameters 60' apart; and 81 = angle measured positive in counterclockwise direction from S to UI

(Figure 5.6)

Figure 5.6 Cross-section for 60' deformation rosette (after Bickel, 1993)


The method basically consists of (I) drilling an EX-size pilot borehole with a diamond bit and reamer, (2) positioning the BHG in the pilot hole, and (3) drilling a largerconcentric borehole (e.g.150 mm diameter) overthe BHG and recording U I , U ~ , and U3. The core is cut in one continuous drilling operation, and U I , U ~ and U3 are measured at the start, during, and at the end of drilling. Generally, UI , Uz, and U3 are taken every 15 mm of drill operation. This will provide some data if the core breaks prematurely. Drilling is finished when the borehole deformation no longer occurs, indicating that the stress is relieved from the core. The BHG and core are removed from the borehole and theprocedure is repeated. The core orientation and the position of the gauge are marked on the core. In the field measurements, overcoring bits ranging from I00 to 300 mm diameter could be used, but the 150 mm diameter hit is considered the optimum size from both economical and practical standpoints.

The marked core is then tested in a biaxial chamber to determine the modulus of elasticity,Eand anisotropy ofthe rock. The elastic propertiesand U I , U2, and U3 from each overcore are used to determine the tivo dimensional state of stress in the plane normal to the axis of the borehole. The core is optionally tested in a triaxial chamber at the same stress level it expelienced in situ to determine Poisson's ratio, v and a more accurate E. A least square method can he used to calculate the average rock stress components from multiple overcoring tests. If deformation measurements are obtained from at least three non parallel holes, the three dimensional state of stress can be determined. Typical layouts of boreholes and directions of measurement are shown in Figure 5.7.

Figure 5.7 Typical layouts of boreboles and directions of measurement (after Bickel. 1993)


Multielement shain gauge

Figure 5.8 shows an external view of the multi-element (5 elements) strain gauge developed by Central Research Institute of Electric Power Industry, Japan. These five strain gauges are protected by a rubber molding. Four strain gauges are oriented in radial directions of the hole (at 45" intervals) and one strain gaugeis directed axially. With these five strain gauge elements, the hvedimensional principal in situ stress state in the radial plane and a normal stress component in the axial direction can be obtained.

To determine the 3-dimensional in situ stress condition, it is necessary to cany out two sets of overcoring in differently directed boreholes. In practice, it is better for accuracy to cany out the stress relief method in 3 boreholes oriented in different directions.

Figure 5.8 Vtew of the multi-element (5 elements) strain gauge (after Kanagawa et a]., 1986)

panition radial gauges (4 elements) axial pauge (I element)

275.5 1

pipe for canent paste (arrangement of strain gauges) (,,it:

Figure 5.9 shows the outline of the measurement procedure. In an adit, a small 56 mm diameter horehole is drilled in which gauges are inserted and set with a cement paste. Stress in the surrounding rock mass is released by overcoring a large diameter hole of 218 mm diameter. In order to convert the released strains into in situ stresses, strain sensitivity coefficients of the gauges are needed. Rock core including the multi-element strain gauge is cut off after overcoring and conveyed to a laboratory for use in large-scale triaxial tests. Through this test, the strain sensitivity coefficient of the strain gauges can be obtained.

Using the theory of elasticity, the magnitudes and directions ofin situ stresses acting in rockmass can becalculated from thereleased strains in situand the strain sensitivity coefficients of the strain gauges.

156 5 Virgin Rock Stresses 5.3 Relief Techniques - Undercoring or Overcoring 157 !

Figure 5.9 Measurement of released strains by drilling a large diameter hole (overcoring) around the multi-element strain gauge (after Kanagawa et al., 1986).

large borehole for overcoring (218 mm dia)

I 1 lead wire

small borehole(% mm dia.)

(rock mass) multi-element strain gauge large drill core

cement paste

1'1 / (enlargement of pan A) U'

CSIR triaxial strain cell

The CSIR (Council of Scientific and Industrial Research in South Africa) triaxial strain cell is designed to obtain the complete state of stress inrock in a singleborehole. The change in strain associated with the overcoring stress-relieftechnique is detected by the strain gauges mounted in the cell body. The cell (Figure 510) consists of a plastic housing containing three 3-strain gauge rosettes mounted on pistons. The pistons actuated by air pressure allow gluing the strain gauge rosettes to the wall of a pilot hole. The strain changes induced by overcoring read by a standard strain gauge indicator unit and the elastic properries ofthe rock are used to calculate the complete in situ state of stress. Overcoring procedure is similar to that used with the BHG. Dificulties can exit with adhesives in wet holes.

CSIRO hollow inclusion cell

The CSIRO cell, developed by the Commonwealth Scientific and Industrial Re- search Organisation (CSIRO) of Australia is similar in consmction and concept to the CSIR triaxial strain cell (Figure 5.1 I). This cell consists of three three-element rosette strain gauges encapsulated in a thin-walled epoxy pipe. It is constnrcted from epoxy pipe with thegauges precisely orientedat 120' anglesalong thecircumference. This construction ensures that the cell is extremelv rueged and fullv watemroof This . -- cell overcomes some problems encountered by the CSIR cell, including possible malfunction of strain gauges when the borehole is wet, or when a strain gauge makes

malfunction of strain gauges when the borehole is wet, or when a strain gauge makes poor contact due toapieceof drillingdetritus, or ajoint, fracture or some imperfection in the rock forming the wall of the hole.

Figure 5.10 CSIR triaxial strain cell (after Bickel, 1993).

rock discglued to

dummy gauge unit ~ l u ~ e d

'unit

'Front coverunit

This cell allows determination of all components ofthe field stress tensor in a single stress reliefoperation. Therefore, this method isadvantageous for stress determination in a relatively high stress field when overcoring is possible in only one direction.

5.3.3.2 Measurement at the back of borehole

Doorstopper method using rosette strain gauge

This technique consists of epoxing a four-element, 45' rosette strain gauge to the end of the borehole that has been ground flat and smooth with a flat-face diamond- impregnatedbit (Figure 5.12) Overcoringa distanceoftwo or threetimestheborehole diameter is required to relieve the stress at the end of the borehole containing the rosette strain gauge, Because of stress concentrations, a correction factor must be applied to the strain readings obtained during the overcoring. These strain readings and the elastic properties ofthe rock are used to calculatethe stresses in a plane normal to the axis of the borehole. Three non-parallel boreholes are necessary to determine the complete three-dimensional stress field. This method can be used in deep mines, but concern exists as to the adequacy of the epoxy bond of the strain gauge or the cell under conditions of high humidity and elevated temperature. In a downhole, other methods should be considered because of the difficulty of removing water and maintaining a dry borehole. This method using-the rosette strain gaugeand avery fast setting epoxy is a relatively inexpensivemethod for measuring the field stress.

Hemisphericalended borehole technique

This technique developed by Kumamoto University in Japan is a promising method which enables one to determine the stress tensor from a set of strain measurements in


a single borehole. In both coal and metal mines in Japan, many measurements have been conducted using this technique. As illustrated in Figure 5.13, lateral strain and longitudinal strain on the hemispherical bottom of a pilot borehole are measured at thezenithal angle of130". The hemispherical-endedboreholecell consistsof 16 strain gauges encapsulated in a hemispherical-shaped epoxy shell. The procedure in this technique consists of (1) drilling of a pilot borehole (e.g. 76 mm diameter), (2) grindingtheend ofthe pilot borehole with a hemispherical-face diamond-impregnated bit, and (3) drilling a concentric largerborehole(e.g. 150 mm diameter) and recording strains.

Figure 5.11 CSIRO hollow inclusion cell (after Bickel, 1993)

depth-laeating rod

nrbber seals (for downhole type) cement exit holes (far downhale wpe)

shear pins entering plus

main body

: ...... , ...... .*.. 3-rosette strain gauges i A i ; B ! -<c,:~ /__.._i :..__..: ..".

orientation of rosette strain gauges around body of shell

pull wire (for pull-wire type

IZ-canductor cable

Conical-shaped straingauge plug

This technique was first presented by Kobayashi et. a1.(1987). This gauge is a type of mold gauge. Six 5 mm-cross strain-gauges are cemented on the surface of a conical shell at six equally-spaced positions along the generator and perpendicular to it as shown in Figure 5.14. Thus 12 components of strains altogether can be measured. The conical shell is made of epoxy-resin mixed with plasticiser by casting. The diameter of the strain-gauge plug is 54 mm and the angle of the cone is 60'. The procedure

5.4 Hydraulic Fracturing Technique 159

consists of(1) drilling of a pilot borehole (e.g. 56 mm diameter), (2) shaping its end into a conical shape using a taper bit, (3) installing and cementing a conical-shape strain gauge plug at the end of the borehole, and (4) drilling a larger concenhic borehole (e.g. 116 mm diameter) and recording strains. This technique also enables the determination of the complete three-dimensional state of stress using only one borehole. In future, this technique may be the overcoring technique using overcoring with the same diameter as that ofpilot borehole.

Figure 5.12 Strain gauge installation (a) and strain gauge configuration viewed from axis of borehole (b) (after Bickel, 1993).

(a) to bridgecircuit

w ~ l 4s0

gauges

5.4 Hydraulic Fracturing Technique The principles of the technique are illustrated in Figure 5.15. The hydraulic fractur-

ing technique of rock stress determination consists of sealing off a section ofborehole with packers and then increasing the fluid pressure in the section of borehole until the surrounding rock is fractured. The application of sufficient fluid pressure induces tensile circumferential stress over limited sectors of the boundary. When the tensile stress exceeds the rock material tensile strengh, fractures initiate and propagate perpendicular to the hole boundary and parallel to the major principal stress. Simul- taneously, the fluid pressure falls in the test section. After relaxation of the pressure and its subsequent re-application, the peak borehole pressure achieved is less than the


initial boundary fractulingpressure by an amount corresponding to the tensile strength of the rock material. A record of borehole pressure during hydraulic fracturing experiment is illustrated in Figure 5.16. Two important parameters defined on the pressure record are the instantaneous shut-in pressure p , and the crack re-opening pressurep,. The shut-in pressure defines the field principal stress component perpendicular to the plane of the fracture. The crack re-opening pressure is the borehole pressure sufficient to separate the fracture surfaces under the state of stress existing at the hole boundary. These parameters are used to estimate the major principal stresses.

Figure 5.13 Hemispherical ended strain gauge (after Sugawara and Obara, 1986).

Figure 5.14 Conical-shaped strain-gauge plug (after Kobayashi et al., 1991)

wire

5.4 Hydraulic Fracturing Technique 161

Figure 5.15 Schematic of hydraulic fracturing technique (after Brady and Brown, 1993).

packer

test sectton

flucd at pressure

packer

Figure 5.16 Pressure history in hydraulic fracturing.

as: breakdown Dreswre

The minimum boundary stress, ammaround a circular hole in rocksubjectto biaxial stress, with field stresses of magnitudesp~ andpz, is given by

a m i n = 3 p2 -pi (5.7)

When a pressurep, is applied to the interior of the borehole, the induced tangential stress croo at the hole wall is

ooo = - p,, (5.8)

The minimum tangential boundary stress is obtained by superimposing this stress on that given by Equation (5.7) i e .

an, ,,I = 3 112 -- 11 I - p,, (5.9)


and the minimum effective boundary stress is

where 11 = pore pressure. The crack re-opening pressurep, corresponds to the state of borehole pressurep,

where the minimum effective boundary stress is zero. Introducing pr i n Equation (5.10)

Because p2 = p,$, Equation (5.11) confirms that the magnitudes of the major and minor plane principal stresses, and p2 can be determined from measurements of shut-in pressurep, and crack re-opening pressurep,. The orientation of the principal axes may be deduced from the position of the boundary fractures. obtained using a device such as an impression packer. The azimuth of the major principal stress axis is defined by the hole diameterjoining the trace of fractures on opposing boundaries of the hole.

Although hydraulic fracturing is a simple and apparently deep stress measurement technique, if is worth recalling the assumptions implicit in the method. First, it is assumed that the rock mass is continuous and elastic. Second, it is necessary for the determinatton of complete stress state to assumc that the borehole is in rhc d~recuon o i one of prtnctpal strc.;ses Third, the induced fracture plal~c is assumed to include the axis of hole.

5.5 Methods using Drill Cores The most reliable method for measuring in situ stresses at great depth is presently

hydraulic fracturing technique. However, since the cost of this method increases greatly with depth and as it is also not applicable if the rock temperature is too high, an inexpensive method which could replace the hydraulic fracturing technique has been required especially in the field of geothermal energy extraction engineering.

For the purpose, recently, the methods using drill cores have been tried. These . . rnerliods ~nslude acoustic L.nltsston r , \E, method, deformation rate anal) sts (DR.&,. (lin'erenual srraln curvsanalysis(DSCr\)nnd anelastlc sua~n recover, (ASK) method Although all these methods are inexpensive and can be carried out in the laboratory, it can not be said that they are sufficiently reliable mainly because the mechanisms behind the methods have not been well understood.

5.5.1 Acoustic emission (AE) method

The acoustic emission (AE) phenomenon which makes stress determination possible i n rock is the so-called "Kaiser effect" (Kaiser, 1953). TheKaiser effect can best be

5.5 Methods using Drill Cores 163

described by considering a simple experiment in which the phenomenon is evident. Consider, for example, the simplified laboratory setup as shown in Fibwre 5.17 in which arock specimen is subjected to two cycleloading. As illustrated in Figure 5.18. in the first load cycle, stress is applied to the specimen at a constant rate up to a value of hax and then returned to zero. In the second cycle, stress is increased in a similar fashion, however, the previous maximum stress (omax) is exceeded. During each cycle, AE activity is monitored and accumulated AE events recorded as a function of applied stress.

Figure 5.17 Testing arrangement for AE method (after Hardy, 1981).

platen 4-4

platen

Figure 5.18 Typical data of the Kaiser effect (after Hardy, 1981).

It can be seen that AE activity is generated at all stress levels during the first load cycle. In the secondcycle, the AEactivity is absent until the stresslevel (hlnX) attained in the first load cycle is exceeded. Thus, the Kaiser effect can bedefined as theabsence of AE activity at stress levels below a previously applied maximum stress.

The testing procedure for determination of in situ stress is quite simple. First, a properly prepared specimen is placed in a testing machine, and AE transducers are

164 5 Virgin Rock Stresses i 5.5 Methods using Drill Cores 165

attached to the test specimen. Specimen is then loaded at a constant rate, and load and total AEeventsarerecordedasafunction oftimeasshowninFigure5.19. Twostraight lines are then fitted to the AE curve, and the previous maximum load (in situ stress) is determined by projecting the point of intersection of the two stratght lines from the AE curve onto the load curve.

Figure 5.19 Typical laboratory AE data for tuff (after Kanagawa et al , 1976)

estimated geo-stress

Time, s

S

0

Thereare some factors influencingon the existence oftheKaisereffect:- time, water

5.5.2 Deformation rate analysis (DRA)

If cyclic loading is applied to the rock, a significant gradient change occurs clearly at an axial stress nearly equal to the peak stress in axial svess-axial strain relation of the third or the fifthloading. Thismeans that a significant gradient change can be seen in the axial stress-axial strain relation during cyclic loading of previously stressed rock. This is a basic phenomenon of DRA method developed by Yamamoto and his associates at Tohoku University (Yamamoto et al., 1990). In order to detect the gradient change more easily and more precisely, the strain

difference function A &j, i ( 0 ) in a general form as defined by Equation (5.12) is introduced in DRA method.

wbere u = the applied axial stress and E, ( u )=the reduced strain for the i-th loading,

Figure 5.20@) shows the curves ofthe strain difference functions obtained from the cyclic loading shown in Figure 5.20(a). The peak values (ui: i 22) of previously applied stresses are indicated by arrows in Figure 5.20@). It can be seen from this figure that the function A si + I , i ( u )is approximated by a straight line with apositive gradient at stresses less than the previous peak stress and that it sharply bends down near the previous peak stress. The negative gradient at applied stresses higher than the previous peak stress indicates that the rock specimen is easier to deform in the first loading than in the second one of two successive loading cycles. The specimen is considered ready to enlarge pre-existing cracks andior to create new cracks only at its first experience of high applied stresses. On the contrary, the positive gradient meansthatthe specimen is more compliant to the applied stress in the second loading than in the first one. This is due probably to the increase in crack density at high stresses of the first loading.

Using the strain difference hnction it is possible to detect more easily a bending point of stress-strain curve to estimate the peak value of previously applied stress.

and tetnperarure. The effects of these variables arc ot'imponance since in situ stress determ~narion bdsed on the Katser effect reou~resthat the associated rock samoles bc 5.5.3 Differential strain curve analysis (DSCA)

~ ~~

cored at the field site, transported to the laboratory, prepared into suitable specimens, and then tested. Obviously a considerable amount oftime may elapse between initial coring and testing. Although the effect of time on the Kaiser effect is dependent on the type of rocks, recent research works indicate that the life time of the Kaiser effect in various geologic materials has been fwnd to range from days to months. Further- more, the temperature and moisture conditions of the test material may change significantly during the drilling, cutting, and grinding processes associated with specimen preparation. According to the recent study on this topic, variations in water and temperature conditions do not appear to markedly affect the reliability of the Kaiser effect as long as specimens are reloaded in a dly condition.

When a core is taken from underground, the drilling operation constitutes a stress reliefmechanism. The physical detachmentofthe specimen from the rockmass allows it to undergo differential relaxation. in most rock types, the mechanism involved is the creation of randomly oriented microcracks. If one assumes that the density of such microscopic cracks is allegedly proportional to the stress change, it is expected that the induced crack spectrum will reflect the stress histoly. This is the basic assumption

"for determination of in situ stress using DSCA technique (Ren and Rogiers. 1983).


Figure 5.20 (a) Schematic illustration for the time history of cyclic loading on an andesite specimen and (b) the obtained strain difference function A ~ j . 8 (after Yamamoto et al., 1990).

(2. 1) 5 p ddiv

(4.3) 5 @ ddiv. \ .. -, 5 @ ddiv.

%oo E: main

0 5 10 15

Axial strerr. MPa

If linear crack closure is assumed under the application of hydrostatic stress, the strain due to the presence of a crack can be obtained by subtracting the average matrix strain from the magnitude of the observed total strain.

The standard procedure of determination of in situ stress can be summarised as follows:-

I . A cubical sample is extracted from the centre ofthe rock core for avoidingthe zone of drilling damage. 2. The specimen is carefully hand-lapped to avoid further crack generation. 3. The specimen is instrumented with 12 strain gauges. 4. The specimen is vacuum-dried and then vented with nitrogen prior to encapsulation i n clear, flexible and impermeable epoxy.

5 6 Exercises 167

5 . After curing, the specimen along with the fused silica standard is subjected to increasing hydrostatic pressure during which output signals from the strain gauges are continuously recorded. 6 . The data generated are analysed in several ways:- (a) Assuming that the direction of the principal stress tensor is known, each plane can be interpreted independently (2-D analysis); (b) If no restrictive assumptions are made, the fully 3-dimensional case leads to 64 combinations.

5.5.4 Anelastic strain recovely (ASR) method

In ASR method (Matsuki, 1991). anelastic normal strains in six independent directions must be measured. For simplicity, the rock is usually assumed to be an isotropic and linearly viscoelastic material with two independent modes of deformation:. shear and volumetric. The basic assumption in ASR method is that the anelastic strain recovery depends on six components of in situ stresses and pore pressure, the change of temperature and the anelastic strain recovery compliances of both deformation modes.

In order to determine the absolute values of in situ stresses. the change of pore pressure and temperature must he compensated first for the anelastic mean normal strain. In addition to this, by measuring the anelastic strain recovery compliance of the volumetric mode and its dependence on the mean stress, which may be done in thelaboratory by using the same core sample, the ahsolutevaluesofthree-dimensional in situ stresses could be determined by an iterating method. If the pore pressure is released stepwise and the change of temperature of the rock can be neglected as in the case of most civil and mining engineering fields, the procedure becomes more simple.

5.6 Exercises 1 What form would the data of the flatjack test assume if the initial stress normal to the plane of the jack were tensile? How could the data be worked to estimate the magnitude of the tensile stress?

2 The U. S. Bureau of Mines overcoring method is used to measure stresses in a borehole drilled perpendicularly to a tunnel wall. The site of the measurement (the plane of the measuring pins) is 12 m deep, in a test gallery 5.5 m in diameter. The measuring borehole hasadiameter of3 1.75 mm. The first pair ofbuttons is horizontal; pair number2 is oriented 60°counterclockwise from button pair number 1; and button pair 3 is 120" counterclockwise from pair I . Deformations were measured as a result of overcoring as follows:-

Pair 1 moved outward 0.003 mm: pair 2 moved outward 0.002 mm; and pair 3 moved outward 0.001 mm. If modulus of elasticity is 13.8 GPa and Poisson's ratio is 0.20 determine the stress components in the plane perpendicular to the borehole, and the


major and the minor normal stresses in the plane and their directions. (Assume the initial stress parallel to the borehole is insignificant.)

3 What is the method for determining the complete state of stress using only one borehole?

4 What is the principle of acoustic emission method for estimating the stress component using rock cores?

5 What is the phenomenon on which deformation rate analysis is based?

6 What is the basic assumption of DSCA technique?

Methods of Analysis for Rock Engineering

T his Chapter consists of2 Sections dealing with 1. Stresses around underground excavations and 2. Analysis of rigid blocks. Numerical methods have not been

covered.

6.1 Stresses around Underground Excavations When an underground excavation is made in a rock mass, the stresses which

previously existed in the rock are disturbed, and new stresses are induced in the rock in the immediate vicinity of the excavation. The rock engineer has to evaluate the degree of stability of the rock structure formed and, if required, consider the installation of support. The basic information required include:-

1. Virgin state of stress. 2 Geometrical characteristics of the excavations. 3 . Stress distribution in the rock mass around the excavatiods. 4 Prevailing geological conditions

Stress fields around sections of underground excavations are conveniently portrayed by:-

1 . Plotting stress trajectories - lines indicating the direction of a particular stress at a particular point. Figure 6.1 shows major and minor principal stress trajectories in the material surrounding a circular hole (excavation) in a uniaxially stressed elastic plate. The relative magnitude of stresses may be judged by the number of lines per unit width. Figure 6.1 shows a crowding of the maximum principal compressive stress trajectories on the side of the hole and a widening at the top and at the bottom. Crowding indicates an increase in compression and wider spacing, a reduction in stress. It can also be seen that the direction of the maximum principal compressive stress is vertical at the top and bottom but deviates considerably from vertical in the vicinity of the hole. 2. Plotting stress loci or contours -lines joining points of equal stress (Figure 6.2).

170 6 Methods of Analysis for Rock Engineering 6 1 Stresses around Underground Excavations

3 . Graphically scaling direction and magnitude of principal stresses at pre-selected points around the excavation (Figure 6.3).

Figure 6.1 Major and minor principal stress trajectories in the material surrounding a circular hole in a uniaxially stressed elastic plate.

6.1.1 Calculation of stress fields

Underground excavations cause readjustment ofthe stress field as stress magnitudes and directions vary to redistribute loads from excavated areas to unexcavated areas. Stress increases (or concentrations) resulting from stress field readjustment are greatest at the wall of the undergound excavation and decrease with distance from the wall.

Once excavation geometry and principal field stress magnitudes and orientations are known, stress fields around an excavation may be calculated. As excavation geometry changes, the changes resulting in stress fields can be calculated.

The methods of calculation are as follows:-

1 . Closed-fom solutions. 2. Numerical methods - (a) finite elements (b) finite difference and (c) boundary elements. 3. Analogue simulations - (a) electrical and (b) photoelastic. 4. Physical modelling.

Figure 6.2 Stress Contours. (a) Major principal stress in multiples of the vertical stress. (b) Minor principal stress in multiples of thevertical stress.

172 6 Methods of Analysis for Rock Engineering

Figure 6.3 Principal stresses plotted to scale at selected points

tensile zone

principal stresses plotted to scale zone containing a tensile pnnnpal stress

6.1.2 Closed-form solutions for simple excavation shapes

6.1.2.1 Circular excavation

Figure 6.4 shows the cross-section of a long circular excavation in a homogeneous, isotropic elastic continuum subject to biaxial stress, defined by ay=p, and ax =Kp. The stress dishibution around the excavation may be obtained from the following equations:-

It is worth noting that the stress dismbutionisindependent of elastic modulus, Eand Poisson's ratio, v.

By putting r =a in Equations (6 I ) to (6 .3) . the stresses on the excavation boundaty are obtained as follows:-

6.1 Stresses around Underground Excavations 173

Figure 6.4 Key diagram for interpretation of stresses around a circular excavation in a biaxial stress field.


For K < 1.0, the maximum and minimum boundary stresses occur in the sidewall (0 = 0)

and crown (0 = ?) of the excavation. Referring to Figure 6.4(b), the stresses are as 2

follows:-

These equations indicate that, for the case when K = 0, i.e. a uniaxial stress field directed parallel to the y axis, the maximum and minimum boundary stresses are :-

These values represent upper and lower limits for stress concentration at the boundary. That is, for any value of K > 0, the sidewall stress is Less than 3p, and the crown stress is greaterthan -p. It is worth noting about the existence of tensile stresses at the boundaq in a compressive stress field.

In the case of a hydrostatic field ( K = 1 ), Equation (6.5) becomes:-

i.e. the boundary stress becomes 2p. independent of the angle 0. This represents the optimum distribution of local stress, since the boundary is uniformly compressed over the complete excavation periphery.

For a hydrostatic stress fie14 Equations (6.1) to (6.3) become:-

In Figure 6.5, the variation of o, and oe is given for vertical ( 8 = 90 ') and

horizontal ( 0 = 0 " )directions in the case of acircular excavation in auniaxial virgin stress field.


Figure 6.5 Variation of or and oe (in terms ofp) for vertical and horizontal directions in a circular excavation in auniaxial virgin stress field (after Oben and Duvall, 1967).

Vehcal direction:-

r 1. For the excavation surface, or = 0. Even at -- = 5, or is 10 % lower than the the a

applied stress. r

2. On the excavation surface, oe = - p and it stays negative until - > 1.5. Tensile a stresses in the back of excavations are undesirable.

Horizontal direction:.

3. or changes from zero at the boundary to about 37 % of the vertical applied stress r

at = 1.4, then or. drops to zero at - > 4. a a

r 4. oo at the boundary has a stress concentration of 3 which drops at = 3 to nearly

the applied value.

176 6 Methods of Analysis for Rock Eng~neerin~ 1 6 1 Stresses around Underground Excavations 177

Figure 6.6 gives tangential stresses at the boundary for circular excavations and different horizontal to vertical stress ratios.

Figure 6.6 Variation of ae at the boundary (in terms ofp) for a circular excavation in various stress fields i.e, for different values of K (after Obert and Duvall. 1967).

1. K = 1 :- Stresses all around the excavation are twice the applied stress. 1

2. K= 7:- NO tensile stresses in the back, stress concentration on the side is 2.67

3. K = 0:- Maximum concentration is 3 on the side and, - 1 on top.

The effect of the horizontal stress, a,*= K p is to reduce the stress concentration at the sides by an amount equal to K i.e. Stress concentration factor = ( 3 - K ) .

The tangential stress. a() at the roof of the excavation is equal to:-

so that the tension at the roof vanishes for K > 1 3

For an underground excavation, the tangential stresses at the excavation boundary are the most important

6.1.2.2 Elliptical excavation

In 3 b~axial snrss field, the rdngential boundary strcsscs at rhc cnd of thcaxes of an ell~plical excavation ( H height, H'L width )are glven from elastic theory by the following equations (Figure 6.7):-

Figure 6.7 Key diagram for interpretation of stresses around an elliptical excavation in a biaxial stress field.

Note that the axes of the ellipse are oriented with the virgin principal stress axes There is no tension at the crown when.-

It is desirable to have a,$ = 08, therefore:-


After simplification,

W . The tangential stresses are identical if the ratio of --- 1s that of K H

6.1.3 Complex profiles

In civil and mining engineering, underground excavations are often not circular and more complex analysis using numerical methods with the aid of computers needs to be carried out. Stress concentrationsfor arectangular excavation with rounded comers

1 are given in Figure 6.8. The comer radii are - of the smaller of the dimensions W 6

(width) or H (height) The peak of the-stress concentration for the tangential stress occurs in the comers.

Table 6.1 lists critical values of stress concentration factor. Hieh horizontal stresses - and narrow spans reduce the development of zones of tensile stresses in the roof.

6.1.4 Size of an excavation

From the point of view of stress concentrations, excavation stability is independent of size. However, as there are considerable variations of rock strength with increased size or excavation span, stress concentrations and excavation stability for a pdcu la r geometric shape do not go hand in hand.

6.1.5 Preferred shapes for two different stress fields

InFigures 6.9 and 6.10, different shaped excavafions arelisted in order ofpreference for use in two different stress fields.

6.1.6 Multiple excavations

Stresses around multiple excavations have been determined using models and numerical methods. Under elastic conditions, two excavations will interact with each other if separated by a pillar of rock with width less than two times the sum of their dimensions in the direction parallel to the separation. As the excavations approach each other, the average stress in the pillar between them increases and approaches the maximum tangential stress.

The stress distribution around an infinite row of equal-sized circular openings equally spaced in an infinitely wide plate subjected to a uniform stress, either normal to, or parallel to, the line of openings has been studied theoretically and the results are summarised in Figure 6.11. From Figure 6.11(a), it can he seen that the maximum stress concentration occurs at the opening boundary along the horizontal axis with a magnitude of 3.26 which is larger than the maximum of 3.0 for a single opening.


However, the tensile tangential stress concentration is 0.7 at the boundary along the vertical axis and is less than the value at the corresponding location for a single opening.

Figure 6.8 Stress concentrations for a single rectangular excavation with round 1

comers offilletratioof; ofminimum ofwidth, Worheight, H (after ObertandDuvall,

1967).

7 r

(a) $ 3 0 2 5 (b) ~ = 0 5 W (c) ~j W = I 0

7 r


Table 6 1 Critical values of stress concentration factor, C on a rectangular boundary 1

in different stress fields. Ratio of fillet radius to short dimension isg-

-- -~ ~-.

F?=o K = - 1 K = l 3

W -- TOP Comer TOP Comer corner" 7 .

p~~ -~ ~ ~

0.12 - 1.0 + 2.3 + 0.5 + 4.2 + 9.3

0.16 - 1.0 + 2.5 + 0.3 + 3.7 + 7.6

0.25 - 1.0 + 2.5 0 + 3.5 + 6.2

0.33 - 1.0 + 2.6 -0.1 + 3.3 + 5.2

0.50 - 1.0 + 2.7 - 0.2 +3.1 +4.7

l (square) - 1 . O +3.1 -0.3 +3.1 + 3.8

2 - 0.8 + 4.0 - 0.1 +4.1 + 4.7

3 - 0.8 + 4.6 - 0.4 + 4.7 + 5.2

4 - 0.9 + 5.4 - 0.4 + 5.6 + 6.2

6 - 0.9 + 6.8 - 0.4 + 7.0 + 7.6

8 - 1 .O + 8.6 - 0.5 + 8.7 + 9.3

Horizontal stress *K=- Vertical stress

** For top and side, factors positive and smaller

From Figure 6.11(b), it can be seen that the stress concentrations jn the pillar are very small while the maximum tangential stress concentrations at the boundaries are 2.16 at the vertical axis and - 0.39 atthe horizontal axis versus - 1.0 and 3.0 in asingle opening at the vertical and horizontal axes, respectively.

From Figure 6.11(c), it can be seen that tangential tensile stress at the vertical axis 1 reduces and becomes compressive when K is slightly greater than -, while the 3

maximum compressive tangential stress at the horizontal axis reduces gadually and continuously as the confining pressure increases.

wo From Figure 6.1 l(d), it can be seen that for a constant (Wo = opening width; Wp W"

= pillar width) the maximum stress concentration increases with the number of openings but becomes stabilised when that number is larger than 5. This is true for


w o W" openings of different --, exceptthatforlarger -- ,the maximum stress concentration WP WP

increases more sharply as the number of openings increases

Figure 6.9 Different shaped excavations listed in order of preference for use in a unidirectional stress field (after Obert et al., 1960).

f t t t t f t t f (a) (b) (c)

t t i t t t t i 4 4 + + i t +

OD0 t t E t t f t f t t E E t f t

(d) (e) 0

Figure 6.10 Different shaped excavations listed in order or preference for use in a hydrostatic stress field (after Obert et al., 1960).

t t t (c)

t t t (4


wo Figure 6.11 Infinite number of circular openings, = i (Wo = opening width; Wp 1. I ,

=pillar width). (a) Stress concentrations for a row of circular openings; applied stress normal to line of centres. (b) Stress concentrations for a row of circular openings; applied stress parallel toline of centres. (c) Boundary stress concentration for infinite row of circular openings. (d) Increase in maximum stress concentration with number of circular openings; applied stress normal to line of centres. (after Ohert and Duvall, 1967)


Stress distributions around a row of five ovaloidal openings (excavations) in a plate subjected to stress normal to the line of centres of the openings has been studied by the photoelastic method. Figure 6.12 shows the stress concentrations at points where the horizontal axis intersects the boundaries of the openings, for various opening widthlpillar width ratios. As might be expected, the maximum stress occurs at the boundary of the middle opening and stress concentrations increase as the opening widwpillar width ratio increases.

Figure 6.12 Stress concentrations as a function of opening width to pillar width ratio; five ovaloidal openings with height towidth ratio of0.5 and applied stress field normal to line of centres (after Obert and Duvall, 1967).

position E

position B

.- position A E .. 8

Ratioofopening width to pillar width


Figure 6.13 shows the increase in critical compressive stress concentrations which W" W . takes place with increased -- ratios for circles and ovaloids of different - ratios, WP H

Figure 6.13 Critical compressive stress concentration for multiple openings; uniaxial stress field i.e. K = 0. C., = maximum stress concentration in pillars; C = maximum stress concentration around a single opening for a unidirectional stress field (after Obert and Duvall, 1967).

Equation of curves w o C ~ = C + O . O ~ [ ( - C ~ ) ~ - ] WP

8 ! 6 z' .- E - 5 5

8 c3

3 2 4 Y .- m m

e! g 3 0

3 :s 2 G Experimental data

5 ovaloids WdH, = 2.0 5 circles WdHo = 1.0

1

0 I 2 3 4 5

Ratio of opening width to pillarwidth

The following equation has been derived from experimental data:

where (.',,, = maximum stress concentration in pillars; ('=maximum stress concentrationaroundasingleopeningfor aunidirectional

tield: W , = width of opening; and

stress

6.2 Analysis ofRigid Blocks 185

Wp = width of the pillar

wo Figure 6.13 also shows the average pillar stress concentration as a function of ~-

WP - n e average pillar stress is obtained by assuming that any one rib pillar uniformly supports the weight of the rock overlying the pillar and one-half the opening on each side of the pillar Thus,

where op = average pillar stress and a" = average vertical stress.

6.2 Analysis of Rigid Blocks The discontinuities present in the rock mass create discrete blocks of rock. The

blocks lying close to a free rock face have the potential to fall, to slide or to topple from the face. A detailed treatment ofthis topic is beyond the scope of this book. Hoek and Bray (1 981), Goodman and Shi (1985) and Priest (1993) dealt this topic in detail. A two-dimensional single plane sliding is analysed here.

6.2.1 Two-dimensional single plane sliding

The geometry of nvo cases of a single planc sl~ding discussed by Hoek and Brown (1981 ) I S aiveiin Fieure 6 14 Csuallv. a tension crack delimirj the t o ~ o f the slideat ~ , - - <.

a uoint bevond the crest of the slope (Fimre 6.14(a)); sometimes the tension crack intercepts ;he slope of the face itself (Figure 6.14@)j.1fthe tension crack is filled with water, it can be assumed thatwater seeps along the sliding surface, losing head linearly between the tension crack and the toe of the slope.

The condition forlimiting equilibrium is reached when the shear force directed down the sliding surface equals the shear strength along the sliding surface; that is, failure occurs when

where = angle of dip of the sliding plane; c = cohesion of the sliding plane; q~ = angle of friction of the sliding plane: W = weight ofthe sliding wedge; A = length (area per unit width) ofthe sliding plane; l J = force due to water pressure on the sliding plane: and F'= force due to water pressure along the tension crack.


Figure 6.14 Geometry for analysis of plane failure (after Hoek and Bray, 1981).

The factor of safety may he defined as a ratio of the terms on right side to the terms on left side of the above equation.

From Figure 6 14,

6.3 Exercises 187

If the tension crack intercepts the crest of the slope (Figure 6.14(a)):-

If the tension crack intercepts the face Figure 6.14(h)):-

From a sensitivity analysis of parameters, Hoek and Brown (1981) came to the following conclusions:-

1. A reduction in c affects steep slopes more than flat slopes. 2. A reduction in reduces the factor of safety of high slopes more than low slopes. 3. Filling a tension crack with water reduces the stability of all slopes. Drainage is frequently practiced in stabilising rock slopes that exhibit tension cracks and other signs of incipient movement.

6.3 Exercises 1 Calculate the magnitudeoftangential stress component, set up by a uniaxial loading condition at a point on the circumference of a circular opening, if the stress concentration factor at this point is is found to be 2.7. Assume that the opening is at a depth of 300 m and that the increase of vertical virgin stress component is 25 kPdm depth.

2 Sketch the distribution of radial and tangential stresses in the rock along the radial line drawn from a circular shaft subjected to a hydrostatic pressure.

3 Calculate the expected boundary stresses atthe horizontal axis level and at the crown of a circular horizontal tunnel, at a depth where the vertical stressin undisturbed rock is 5 MPa, and the horizontal field stress, normal to the tunnel axis, is:-

(a) 1 MPa (b) 2 MPa (c) 5 MPa (d) 7 MPa (e) 15 MPa

I 4 A horizontal circular roadway of 2.5 m diameter is to be driven 300 m below the surface. Assuming that the ratio of the horizontal to vertical stress due to the

- , I 7 L = , y,,,zL,

b

v overburden is - - . where v is the Poisson's ratio, determine the tangential stress at I - v


the surface of the roadway on the vertical and horizontal diameters Sketch also the form of the radial and tangential stresses with increasing distance on those diametral axes. The average density ofthe overburdenmay betakenas2300 kg/m3and~oisson's ratio as 0.20

5 Would a 3 m diameter roadway driven in massive, elastic, and homogeneous rock be self-supporting, andif so, what would bethe factor of safety? Assume the following properties:-

Density of rock --- 2400 kg/m3 Compressive strength --- 103.4 MPa Tensile strength --- 5.5 MPa Poisson's ratio --- 0.25 Depth of roadway from surface --- 600 m

6 A long opening of circular cross-section is located 1000 m below ground surface. In the plane perpendicular to the tunnel axis, the field principal swsses are vertical and horizontal. The vertical stress, av is eq& to the depth stress, and the horizontal stress, ah is equal to 0.28 a". The unit weight of the rock mass is 27 kN/m3, the compressive strength is defined by a Coulomb's criterion with c = 20 MPa, q = 2S0, and the tensile strength = 0. (a) Predict the response of the excavation peripheral rock to the given conditions. @) Propose an alternative design for the excavation.

7 What would be the probable magnitude of the water pressure in a vertical fissure in an otherwise impermeable rock mass, at a depth of 300 m below the surface? If the rock had an average density of 3000 kg/m3, a horizontal/vertical stress ratio of 0.5, an unconfined compressive strength of 22 MPa, and a q value of 45', investigate the effect of such a water-filled fissure, passing parallel to and close to the sidewall of a circular tunnel 300 m below the surface. Is failure likely?

8 The field stresses in a particular rock mass are as follows:

Vertical stress --- 25 kPdm depth Horizontal stress in north-south direction -- 4 times vertical stress

1 Horizontal stress in east-west direction -- - hmes vertical stress 3

(a) Calculate the principal stresses in thevertical, north-south and east-west directions, as functions of depth below the surface. (b) Calculate the critical boundary stresses likely to be induced on the surfaces of

6.3 Exercises 189

(c) At each of several levels of a mine, longtunnels of circular cross-section are to he driven in north-south and east-west directions. The rockmass has alinear Coulomb's envelope with c = 4 MPa and m = 40'. Over what range of depths will the boundary stresses cause failure for tunnels in each direction?

9 What is the preferred shape of mine opening in a hydrostatic stress field?

10 If the horizontal, virgin principal stresses differ in magnitude, what is the ideal shape of a shaft and what orientation of this shape would be most suitable from the rock mechanics point of view?

11 Calculate the maximum compressive stress around an elliptical underground 1 2 opening twice as high as it is wide, for values of K equal to 0, - -. , 1.2, and 3 The 3' 3

tunnel is 300 m deep. Indicate in each case where the maximum stress occurs.

12 A vertical, elliptical shaft, with an axis ratio of 2 1 , is to be sunk in strong elastic rock where the following information on principal stresses is available:-

7=~-T--~:..~c.-A .- -- Principal stress . -_ .- M- ----- -- l&&en nnnnnn

Maximum 34.5 MPa North-South

Minimum 13.8 MPa East-West

Vertical , . ~

(a) How would you orient the shaft? @) What would be the magnihlde of the maximum and minimum tangential stresses developed?

13 A single rectangular opening 3.0 m in height is driven in rock having compressive strength of 124 MPa and tensile strength of 10.3 MPa The density of rock is 2300 kgim .The opening is located at a depth of 600 m in a stress field of no lateral pressure

1 and has a fillet ratio of - - 6.

(a) Determine if the opening will fail when its width is 6.0 m. (b) Is there any benefit to reducing the width to (i) 3.0 m? (ii) 1.5 m? (c) What is the maximum safe width of opening?

14 An attempt was made to drive headings 8.5 m in width in a 1.1 m seam of coal, but failure occurred. Investigating from a critical stress standpoint. what is the maximum safe width of heading (in even multiples of side ratio) that can he driven?

circular tunnels driven in the north-south and east-west directions, as functions of I The fillet ratio is i. 'The following conditions prevail:. depth (assume elastic behaviour).


Depth --- 300 m Density of overburden --- 2300 kgim3

I 01, = - 0" 3 Strength of roof strata --- Compressive --- 124 MPa

Tensile --- 8 3 MPa Strength of coal -- Compressive --- 41 4 MPa

Tensile --- 2.8 MPa

15 It is planned to drive a row of 9 m diameter circular tunnels spaced on 13 5 m centres at a depth of 240 m in a massive rock formation having the followtng properties.-

Density --- 2600 kg/m3 Compressive strength -- 103 MPa Modulus of rupture --- 5.5 MPa Poisson's ratio --- 0.25

Determine whether the tunnels can be driven safely

16 A horizontal circular roadway of 4.5 m radius is to be excavated in granite at a deuth of 150 m. Thevertical stress is eaual tothedeuth stress. and the horizontal stress is kqual to 1.5 times the vertical stress'. A discontinuity strikes parallel to the axis of the roadway and at its closest point it is 3.0 m from the roadway wall. It dips 60' as shown in Figure 6.15. (a) Assuming linear elastic behaviour, determine expressions for the normal and shear stresses on the discontinuity (b) Show that the shear stress has its maximum value when a = 60'. (c) Determine whether the discontinuity will be disturbed by shear failure, taking the discontinuity to have c = 0.02 MPa, and = 24*.

17Forthecase shown in Figure 6.16, calculate the factor of safety ofthe block ABCD against sliding.

6.3 Exercises I

Figure 6.15 Roadway adjacent to a discontinuity, Exercise 16.

/ discontinuity

Figure 6.16 Stability analysis of block ABCD, Exercise 17

top ofslope

c and o of discontinuity = 10 kPaand 3Oorespeeivcly

Unit weights of mck and water -27 and 9.8 W I ~ ' respectively

Rock Reinforcement and Support

T h ~ s (:liaprer :overs ~~ieclian~sms ot in~lure ill rock c.nginer.rll~: slnl~r-les rock ! loads, supponln,: and rr.~nti>r:inp menll)ers and des12n oi,ul~pon and rc~niorce-

I ment systems.

7.1 Mechanisms of Failure in Rock Engineering Structures

7.1.1 Underground openings

The modes of failure likely to take place in the vicinity of underground excavations may be classified, depending upon the stnrcture of the rock mass as follows:-

1. Failure modes involving only intact rock. 2. Failure modes involving discontinuities and intact rock. 3. Failure modes involving only discontinuities.

The failure modes under this classification are given in Figure 7.1


Rockbursting - This type of instability is a result of the combined action of initial shearing and subsequent splitting resulting in sudden detachment of rock slabs with ahighvelocity. It isusually observed in brittle hard rockssuch asunweatheredigneous rocks and siliceous sedinentary rocks. As the rock becomes less brittle, the rockbursts become less severe.

Squeezing - This type of instability is a result of complete shearing of rock surrounding the excavation. 'It can be observed in ductile materials such as rock salt, thickly bedded mudstone, halite, chalk etc.


Bending - This type of instability is usually obselved in sedimentary rocks due to gravitational forces when layers are generally parallel to roof and in situ stresses parallel tolayeringarerelatively low. It isassociated with the tensilestrenyh oflayers at the early stages of failure.

1 94 7 Rock Reinforcement and Support I i 7.1 Mechanisms of Failure in Rock Engineering Structures 195

Figure 7.1 Classification of modes of instability in underground excavations

Failures involving only intact rock

Rockbursting Squeeing failure

Failures involving only discontinuities

6

Falls Sliding Toppling

Buckling - On the contrary to bending failure, this type of instability is observed when high in situ stresses parallel to layering are present and the thickness of layers in comparison with the span is relatively small. It is i~sually observed in metamorphic rocks and thinly layered sedimentary rocks.

Flexnral toppling - This is a localised form of instability and it can be observed partici~larly in roofs and sidewalls of openings in sedimentan/ and metamorphic rocks. Layers of rock bend and fail like interacting cantilevers fail in flexure.

Shearing and sliding -This type of failure involves sliding along discontinuities and shearing of intact rock. It is most likely to be seen when in situ stresses are higher than the compressive stren@h of rock and buckling failure is not possible.

7.1.1.3 Failure modes involving only discontinuities (blocky medium only)

These types of failure can occur at any depth as long as the rock mass has two or more discontinuity sets.

Block falls - This type of failure is observed in the roofs of openings due to gravitational forces.

Sliding- This type of failureis observed when one ofthe discontinuity setsdaylights near the toe of sidewalls and the disturbing forces are greaterthan its shear resistance.

Toppling -The inclination of the critical discontinuity set, on which toppling will occur, should be such that no sliding failure is possible.

Sliding and toppling -This type of failure is observed when the conditions for both types of failures are satisfied.

7.1.2 Slopes

As in the case of underground openings, a similar type of classification can be made in the case of slopes (Figure 7.2).


Shear failure - This type of failure is observed in cases such that the slope angle and height are sufficient to cause shearing of the intact medium in continuous or tabular or blocky medium. In tabular or blocky medium, the internal structure and slope geometry should be such that no other forms of instabilities are possible. Depending upon the slope angle, tensile cracks at the top of slopes may appear and the failure of slopes. therefore, can be due to a combination of shearing and tensile stresses.

Bending failure - This type of failure is likely to be seen in the case of slopes with a toe eroded. The mode of failure is similar to that of cantilevers. The failure is otien observed in cliffs near sidzs or river embankments. For this type offailure, the ratio of the erosion depth to the slope height should be sufficient to cause bending failure rather than shear failure.


Combined shearing and sliding failure -This type of failure can occur when one of~hzdiscontinu!n* sets has ail in~lindtlon cjual lothedope angl.'lrrld nootl~er l'urnls or iail~~~carcoussihle 'Tills riglure m ~ n i i e ~ t s itselfas slid~nc alonqa ~nrical ulanrnnd - - shearing of intact rock near the toe of the slope.

Buckling - This type of failure occurs when the slope angle is equal to that of the discontinuity set and the ratio of discontinuity spacing to the slope height is relatively small.

- T

- -. -

196 7 Rock Re~nforcement and Support I I

7 2 Rock Loads 197

Figure 7.2 Classification of modes of instability in slopes.

Failures involvine only intacl rock ~ ~ - .

. . Bending failure

Faiiurer involving discontinuities and intact rock .

Flexural toppling failure

Failures involving only discontinuities

a B 4 Plane sliding Wedge sliding

Toppiing

Flexural toppling -This type of failure occurs in the case of slopes excavated in sedimentary or metamorphic rocks

7.1.2.3 Failure modes involving only discontinuities

Sliding fiilure -There are two types of sliding failure These are -

i . Plane sliding - This involves only one set, the strike of which is parallel or nearly parallel to the slope axis. and occurs along a critical plane daylighting near the toe of tlie slope. 2. Wedge sliding - This involves two throughgoing discontinuity sets and occurs wllen the intersections of two sets daylights near the toe of the slope

I Toppliugfailure -This type of failure occurs when one of the discontinuity sets, the strike ofwhich is parallel or nearly parallel to the axis of slope, has aninclination such that no sliding is possible.

Combined toonling and slidinefailure - ThistvDe offailure is observed when both .. - - conditions for toppling and sliding are satisfied

7.2 Rock Loads Rock load may result from the dead weight of some potentially unstable layers or

blocks of rock as well as from the so-called ground pressures. The following parameters are to be evaluated:-

1. The structure of the rock mass. 2. The initial state of stress. 3 The mechanical properties of rock element. 4. The mechanical properties and spatial distribution ofdiscontinuities. 5. The mechanical response of support members.

7.2.1 Empirical approaches- Rock classifications

Regarding the empirical approach to determine rock loads. the rock classifications can be counted for The classifications proposed are many and differ from country to country. Nevertheless, the classification proposed by Terzaghi (1946) is the most fundamental one among all proposed rock classifications and it is the basis of other classifications thereafter (Table 7.1). His classification is based on his own experi- ences in tunnels driven through Alps in Europe and America and trap-door experiments in laboratory. The modem versions of this classification are those proposed by Deere et al. (1969). Bieniawski (1973,1974b) and Barton et al. (1974). A comparison of rock loads in terms ofa dimensionless parameter suggested in their classifications is given in Table 7.2. The rock loads to be carried by support members results from the structure of rockmassup to class 6 in Terzaghi's classification and the true yielding of surrounding medium in the rest of rock classes.

7.2.2 Semiempirical approaches

The old and still widely used approach is to assume a loosening zone created about the opening following the redistribution of stresses due to excavation. According to this approach, it is assumed that a ground arch about the opening occurs and the boundaries ofthe loosening zone are the lower side of this ground arch and the upper surface ofthe opening (Figure 7.3). The fundamental reasoning for this assumption originates from the fact that the rock mass is incapable of resisting to tensile stresses and has a sufficient resistance against the compressive stresses. To determine the boundaries of the loosening zones, one can find numerous semi-theoretical and empirical proposalsin the literature (see Szechy (1973)for thelist ofthecompilation).

198 7 Rock Reinforcement and Suppon I I 7.2 Rock Loads 199

Among all semi-theoretical proposals, it may be worth mentioning the proposals of Terzaghi (1943) and Protodyakonov (see Szechy, 1973)

Table 7.1 Rock load classification of Terzaghi (1946). - -- -p.--p-......p--. .. -. -~ --.--p----..-..-p-p...-... ~ ~-

Class number Rock condition -- -. M g h ~ f L o ~ s e ~ ~ ~ ~ ~ ~ 1 Hard and intact 0

2 Hard stratified or schistose 0 - 0.5 B

3 Massive moderately 0.25 - 0.5 B jointed

4 Moderately blocky and (0.25 - 1. I) (B + Ht) . seamy

5 Very blocky and seamy (0.35 - 1.1) (B + Ht)

6 Completely crushed but 1.1 (B + Ht) chemically intact

7 Squeezing rock - moder- (I. 1 - 2.1) (B + Ht) ate depth H c 100 m

8 Squeezing rock - great (2.1 - 4.5) (B + Ht) depth H > 100 m

9 Swelling rock U p A?--P B = Tunnel width Ht = Tunnel height H = Overburden height

The original theory was proposed by Janssen (1895) for silos and was applied by Terzaghi (1943) to mnnelling. For a cohesive ground, the rock pressure for a unit width, P,9 to be carried by the supports is given by

yB-2(; p - -~ ( 1 -e-".3+por-""= 2 KO tan 9

(7.1)

where y = unit weight of rock; B = opening width;

(',, = cohesion: K,,= lateral pressure coefficient; v = angle of friction of rock; -

2 K,, tan Q, A,> = ~

H ' :=depth from surface or assumed level; and

f'(, = applied surface pressure.

Table 7.2 Comparison of classifications.

I D e e r e i Bieniawski l~ar ton

r-

, ~ -

I8 i DB 0.6-1.3

0.6 - 0.8 7 0.77- 1.66

-

9 3.57 - 7.7

TBM - Tunnel boring machine; DB -Drill & blast.

Figure 7 3 Loosening zone concept.

200 7 Rock Reinforcement and Support

Replacing P , by y h and taking the limiting value results in the following limiting value for the height of loose ground:- ,-

Protodyakonov (see Szechy, 1973) assumes that the yielding takes place in the horizontal direction, then the resulting formula for the maximum normalised height h -of load takes the following form-- B

The parameter f in the above equation is given by

f = tan 9 for frictional media

=tan +& = for cohesive and frictional media Bc 2 COS

(7 4)

where C T ~ = uniaxial compressive strength and C U -- - 0 2 cos 9

The total load to be supported is also given by the following equation for a unit width:-

Since rock mass is usually not capable of resisting to tensile stresses, the rock mass within the region subjected to tensile stresses about the openings will tend to fall into the opening under gravitational forces. The most optimum opening shape in an arbitrary biaxial compressive stress state is an elliptical opening whose major axis coincides with the major in situ stress and the ratio of its axes is the same as the ratio ofinsitustresses Forthisparticularcondition, theopening perimeterwill besubjected to a uniform compressive tangential stress, GO:-

where BI =maximum initial in situ stress:

7.2 Rock Loads 201

03 =minimum initial in situ stress.

For a medium without any tensile strength, the opening must have the following dimensions (W= major axis; H = minor axis) in order to be stable (assuming that the major axis ofthe elliptical opening coincides withthe direction ofthe maximum initial in situ stress and tangential stress, oe is zero at the crown):-

W I - K H - 2 K

If the opening is enveloped by this ellipse, regions, which are free of the true ground stress field, will appear about the opening. The regions, which are bounded by the opening shape and this ellipse, are herein termed as the loosening zone. Among the regions, the region above the roofwill be potentially unstable under the gravitational forces.

7.2.4 Structural defect approach

This approach basically involves the determination of the dimensions of potentially unstable blocksorlayers ofrockin relationto the spatial orientation ofdiscontinuities and openinggeometry. The total potential volume of rock prone to fall into the opening will differ depending upon the number of discontinuity sets and the tensile strength nf rock andlor shear strennth of discontinuities. ..... ~ u

When the rock mass is th~nlv laycred, the layers may not be strong enough to resist to rrns~le bt~esier due to bending under the gravitalional forces In such cabs. trnjlle ~~~ ~ ~ ~- -~ ~ ~

stresses in layers should be reduced below their tensile strength, if the stability is required.

In the construction of rock engineering structures in layered rock mass, the layers may be also subjected to loading conditions such as encountered in cantilever beams in structural engineering. To prevent the failure of layers subjected to such loading conditions, the support members should act in a way so that the magnitude of the disturbing moments acting on the columns is reduced.

When the number of discontinuity sets is two or more, the determination of potentially unstable blocks, which may fall, slide or topple into the excavation space, can be carried out by usingthe block theory developed by Wittke (1964). John(l970), Londe et al. (1970) initially and elaborated by Goodman and Shi (1985). This theory can effectively evaluate the possibility of sliding or falls of blocks and resulting rock loads due to these types of failures can also be effectively estimated by this theory. However, it will be impossible to determine the rock load$ which may arise due to the toppling type of failures.

202 7 Rock Reinforcement and Suppon

When the discontinuity pattern of rock mass is cross-continuous, the volume of potentially unstable region consisting of blocks of rocks prone to fall into opening will be defined by two critical intersecting discontinuity sets which are either tangent to the curved geometry or emanating from the comers of the comered geometry of the opening at shoulders (Figure 7.4). For an arched roof, the rock load can be easily shown to be for a unit thickness as:-

tan a1 tan u2

whereI. = width of opening, Ko = radius of arching,

0 = angle of arching; and a, = inclination of sets in the plane of the cross-section

Figure 7.4 Two critical intersecting discontinuity sets in the roof.

If the sliding type of failure is likely to occur in sidewalls (Figure 7.5). the weight of the sliding-prone body can be also calculated from theabove expression. However, the rock load will be less than that in the case of the roof due to shear resistance of the set, on which the sliding is possible.

Let us consider the limiting equilibrium state ofthe body in the sidewall as shown in F ig re 7.5. The equilibrium equations are:-

7 2 Rock Loads 203

Figure 7.5 Two critical intersecting discontinuity sets in the sidewall

where s and i z denote the directions parallel and normal to the sliding plane and

,'; Nand S stand for total, normal and shear forces, respectively

Assuming a frictional resistance on the plane a,, the limiting state is then given by

S N = tan 9, (7 9)

The required resistance from the support member then takes the following form:-

f, P - - ,+, ?ni 1-9~) cos (p

where W= weight of sliding body; ai = inclination of the set, on which sliding is to occur: and ~i = friction angle of the set.

7.2.4 Rock-support interaction approach

Another approach to determine the rock loadsis based upon the interaction between the rock mass and support members. The.basic idea of this approach was first

204 7 Rock Reinforcement and Suppon i 7.2 Rock Loads 205

suggested by Fenner (1938), and Rabcewicz (1964. 1965) was first to introduce it to the real tunnelling practice with aclose collaboration of Pacher (1964) The approach is the basis of the modern tunnelling philosophy known as the New Austrian Tunnelling Method - NATMinmany countries, except in England and France where it is called Characteristic Cuwe Method and Convergence Confinement Method, respectively. The principles of this approach are well illustrated by a ground response curve - support reaction curve. For this purpose, let us assume that a circular underground opening is situated in a hydrostatic state of stress and the behaviour of rock is to be elastic/britlle-plastic. In addition, the gravitational force is assumed to come into existence following the plastification of rock. For the roof of the opening. thenormalisedintemal pressurecan be shown tobeby usingtheMohr-Coulomb yield criterion as :-

Ps where f i = -. 00

p . ~ = internal pressure provided by support member; 00 =in situ stress;

I +sin*. (1 = 1 - sin q '

oe = uniaxial compressive strength of rock; a = radius of opening;

= radius of elastic-plastic boundary;

I +s inq*, 'I = *'

I - sin q y = unit weight of rock; q = friction angle of intact rock; and

q~* = friction angle of plastified rock.

The above equation consists of two parts, the first one resulting from the trueground pressure and the second one resulting from the weight of the rock in the plastified zone. These functions are plotted in Figure 7.6. As it is noted from the above expressions, the function is a monotonically decreasing function while the function I.& is a monotonically increasing function. The principle, which is put forward by Rabcewicz (1964). states that the support should be installed before ;he minimum of the function I., is achieved. The load to be carried out by the support

members will be the one at which the ground response curve intersects the support reaction curve. As noted from Figure 7.6, the load on supports results from the restriction of the inward movement of the surrounding rock, and there is an optimum suppon load to be countered by the support system which corresponds to the intersection point of the reaction curve numbered (2) and ground response curve.

Figure 7.6 Ground response -support reaction curve.

7.2.5 Unified approach

The approaches described above to determine the rock loads are justified as the concepts, they are based upon, are realistic. Nevertheless, a unified approach should be established to include the above approaches.

Disregarding the empirical approaches, the semi-theoretical approach, the structural defect approach and the rock-support interaction approach can be unified to suggest a more generalised approach. To illustrate thisunification, it is proposed that the three zones (i.e the plastic zone, the loosening zone and the zone due to structural defects)

___- - -

; / / . / . - I

0

o

0.4 0.8 1.2 S W n at opening wall

206 7 Rock Remforcement and Support 7 3 Supporting and Re~nforcement Members 207

can occur about the opening. The conditions for the appearance of such zones are (Figure 7.7):-

1. The zone due to structural defects can occur when rock has one discontinuity set or more, and at least one of which daylights on the surface of the opening. 2. The loosening zone can occur when the initial in situ stress ratio and the geometry of the effective opening shape defined by the geometrical orientation of structural defects (discontinuity sets) are such that tensile stress regions about the opening are tooccur. Notethat the term oflooseningzoneis herein associated with thezone caused by tensile stresses, which the rock mass can not sustain and become free from theme ground stress field about the opening. It is distinguished from the zone created by the plastification of rock due to compressive stress field. 3 . The plastic zone can occur ifthe redistributed stress state is such that it is sufficient to cause the yielding of rock in the region outside the possible above two zones.

Figure 7.7 An illustration of the plastic zone, the loosening zone and the zone due to structural defects.

All these three zones may not be observed in every excavation and their occurrence will depend upon the conditions such as the geological structure of rock mass, the geometry of the opening, the initial in situ stress field and the mechanical properties of rock. Therefore, there may be a number of varieties in real rock engineering practices.

7.3 Supporting and Reinforcement Members There areanumber of support members availablein geotechnical engineering field.

These are:-

1. Steel liners. 2. Steel ribs. 3. Concrete liners. 4. Shotcrete. 5. Rockbolts (including rock anchors).

The above support members may be superior or inferior to each other depending upon thegivengeological, hydrological, and environmental conditions, and economical and constructional advantages and disadvantages.

7.3.1 Steel liners

Steel liners are generafly used as sealants against water inflow-outflow and/or to reduce the frictional resistance to the flow of fluids through the opening in many rock engineering projects It is hardly used to resist rock pressures as it is highly expensive regarding the storage, transportation and installation. When the steel liners are used as a support member, the loading conditions should be uniform Otherwise, they may be very weak under non-uniform stress field and may buckle under high compressive stress fields as they are usually thin in relation to the dimensions of openings

Steel liners are structurally modelled as thin tubes (shells) in some simple theoretical analyses. In numerical analyses, they are represented by shell or plate elements.

7.3.2 Steel ribs

Steel ribs are one of the most conventional support members used in geotechnical engineering structures for a long time especially for resisting rock loads. They are generally employed togetherwith wood lagging. However, the recent tendency is to use the ribs together with shotcrete and/or rockbolts in difficult rock conditions.

The ribs are structurally modelled as thin ribs subjected to axial and bending loads. In numerical analyses they are modelled as beam elements.

7.3.3 Concrete liners

Concrete linen can be in-place cast or segmental with or without reinforcement in rock engineering structures, the liners are usually employed to resist fluid pressures.


They are regarded as a supplementaly support member against rock loads as their installation is usually delayed. They reduce the fktional resistance against water or air flow by smoothing the surface or they keep the excavations dry They are usually regarded as a good sealant when they have no cracks due to thermal stresses during hydration of concrete.

When theloadsact on the linersuniformly, they are usually designed as thick-walled cylinders. If the expected loads are non-uniform, then they have to be reinforced to resist bending stresses. In such cases, segmented liners are usually used in order to reduce the bending stresses in the liners.

7.3.4 Shotcrete

Shotcretefirst appearedas gunit in the world oftunnelling and it has become popular as the NATM has become one of the most popular tunnelling methods. The shotcrete has four important effects besides its internal pressure effect:-

1. Preventing rock at the excavation surface from becoming exposed to air and moisture changes directly. This is quite an important effect in some rocks such as mudstone, shale, serpentine and anhydrite which weaken in strength as they loose ~. their moi&re. 2. Preventing rock near the excavation sul-face from relaxation. This is particularly imoortant in the case of some sedimentary rocks, having hard-sol3 intercalated layers w i c h weakenin strength duetothe interlayer sliding causingfractures in rockparallel to the excavation surface. 3 . Initiating an arch action within the rock mass through restraining interblock sliding and allowing interblock rotation. 4. Initiating a wedging action by filling up the open discontinuities which prevents rock mass from loosening.

Therefore the supporting effect of shotcrete, although it is small structurally when they are thin, is an indirect one and manifest itself as the rock mass mobilising the utmost use of its available resistance. The shotcrete isusually modelled as thin tubes in theoretical analyses and as shells in numerical analyses.

Rockbolts are nowadays one of the most popular support members in the rock engineering works. They are generally grouped into two (Figure 7.8):-

1. Mechanically-anchored rockbolts 2. Grout-anchored rockbolts.

Mechanically-anchored rockbolts utilise the frictional resistance between the rockbolt and rock mass through the radial expansion ofthe bolt shank. Theearlierversions of the boltsutilise expanding wedges at their anchorages The modemversions ofthis

7.3 Supporting and Reinforcement Members 209

kind bolts such as Split Sets or Swellex bolts are expanded for their entire lengths. As compared with the earlier versions, the modem ones can be also installed in much weaker rocks and they provide immediately a support pressure They are regarded as a temporary support measure as they will corrode in long-term and loose grip due to the creep deformation of rock around the holes as aresult of applied pressures by the expansion ofthe wedges.

Figure 7.8 Rockbolt types; (a) Mechanically anchored rockbolt - Expansion shell anchor; (b) Friction anchored rockbolt - Split Set; (c) Friction anchored rockbolt - Swellex; (d) Grouted rockbolt - Rebar; (e) Grouted rockbolt - Dywidag steel; (0 Grouted cable bolt - Fleximpe.

Grouted rockbolts utilise a cement or resin type grouting agent. Cement type grouting agent is commonly used as the resin could be expensive as well as difficult to handle with. The deformed steel bar is used as the main load bearing element. Glass-fibre bars are also used particularly in civil engineering works. They are regarded as permanent reinforcement members since they are highly resistant against corrosion and they do not apply internal pressures, which cause high tensile stresses in rock, so that the creep deformation of the bolts is negligible. Grouted cables utilise

--- V

7 Rock Reinforcement and Support 7 4 Des~gn of Support and Re~nforcement Systems 210 21 1

a steel cable instead of steel bars as in the case of grouted rockbolts. They may be inexpensive when the old cables used for skips in mining shafts are utilised.

7.4 Design of Support and Reinforcement Systems

7.4.1 Reinforcement of continuum by rockbolts:. Pattern rockbolting

In continuum, it is often reported that rockbolts improve the apparent mechanical properties (i.e deformationmodulus, strength, etc.) of the medium, in which they are installed, and offer a confinement effect. In the following Sub-sections, it is shown how to evaluate these contributions to the apparent properties of the medium and the confinement effect of rockbolts.

7.4.1.1 Conhibution to the deformation moduli of the medium

The main element in the rockbolt system is the bar which has axial and shear stiffnesses. Although these stiffnesses may be disregarded in some certain directions, the stiffness against axial loading parallel to the bolt axis and the stiffness against shearing perpendicular to the bolt axis have to be properly taken into account, In any given problem these stiffnesses will be mobilised up to a certain extent. To evaluate the contribution of these stiffnesses to deformational properties of a medium, it will be necessaly to carry out an averaging procedure in which the geometrical dimensions of a representative element must be specified as in the theory ofmixtures or composite materials. Thus the contribution of rockbolts to the medium will become a relative quantity. Specifically, the axial and shear contributions of the bolts to an isotropic medium, in a local cartesian coordinate system (oxyz) may be written as:-

whereEr= elastic modulus of medium; /<h = elastic modulus of bar; (;,- = shear modulus of medium; (;A = shear modulus of bar; n = A~IAI;

Ah = cross section of bar; and Al = cross section of representative volume

Asterisk refers to properties in bolted state.

It should be noted that the equivalent characteristics of thereinforced rock mass will be anisotropic even though the rock mass itself is isotropic.

7.4.1.2 Contribution to the strength of the medium

Although the contribution of deformational properties of the bar to a continuum can be deterministically evaluated. the contribution of bolts to the strength of the continuum could not be done so. This is because of the difference in the yielding strain of the geomaterials as compared with that of the bar as shown in Figure 7.9. Although the yielding strain of steel is less than that of geomaterials, the axial and shear contribution of the bar can not be fully mobilised at the time of of geomaterials since the response of bolts are closely associated with the response of the minimum principal strain, ~1 in real engineering situations such as in underground excavations and slopes. Therefore, the strength of the bar becomes mobilised after a certain amount of straining of the geomaterials has taken.place, provided that the medium behaves as continuum. As aresult, this fact makes the apparent behaviour of bolted geomaterial bodies more ductile as compared with that of unbolted geomaterials.

Figure 7.9 Comparison ofthe uniaxial behaviour of rocks with that of steel

Z 0.5- -steel.

---- tuff [Oya) gneiss

, concrete ---granite

I 2


To visualise the contribution of rockbolts to the strength of rock, let us consider a bolted rock pillar subjected to a triaxial stress state as shown in Figure 7.10. The

normal stress, 0; and shear stress, s: on a given plane inclined at an angle a to the minor principal stress direction can be shown to be (stresses with asterisk refer to the stresses in bolted state)

a a i - o i om=--+- 2 2

cos 2a

* * * 0 1 - 0 3 . %=2 s ~ n 2a

where 0; = 03 + A 0 3 and

o?=ol+Aoi.

Figure 7.10 Notations for a bolted rock pillar

7 4 Design of Support and Reinforcement Systems 213

The bolts crossing the plane a, the axial component oa and shear component TT are related to the stress oh in bolt in the direction of plane a by

00 = ah cos a r, = oh sin a

The contribution of the axial response of the bolts to the lateral confining stress 0 3

may be written in the following form -

where n = bolting density parameter and m = number of bolts.

The action of bolts will tend to reduce the magnitude of the shear stress on the plane a. This reduction may be evaluated by considering the projection of the stress components in the bolt on the plane a in the following form:-

where A ra stands for the shear stress decrement due to bolt resistance.

The geomaterials are generally cohesive and frictional and obey the Mohr-Coulomb yield criterion. The failure may take place along a single plane or two conjugate planes. The Mohr-Coulomb criterion is given by

where c = cohesion and q = friction angle.

By rearrangement

where cc = uniaxial compressive strength of rock; A DC = increment in uniaxial compressive strength due to the strength of the

bolt: and I + sin

(1 = tnaxial coefficient = I - sln q


Thus, the contributions of rockbolts to the strength of rock pillar are

2 Bb C O S 2 AOC;.=nm - - 1 - sin cp

The strength ob ofthe bar at yielding can he calculated from the Von Mises criterion given as

where o r = tensile yield strength of bar and 9 =angle between the bolt axis and the normal of plane a for translational

type movements and angle between the bolt axis and the plane a for separational type movements.

7.4.2 Reinforcement of discontinuum by rockbolts

In a medium with discontinuities, the discontinuity planes may exhibit various displacement patterns, the fundamental behaviours are

I. Separation (opening-up) 2. Closing. 3. Translation.

These displacement patterns are observed either individually or combined. Some of these are illustrated in Figure 7.11. Rockbolts are mainly required to prevent the opening andlor translational movements of discontinuity planes. Depending upon the direction and the nature ofdisplacements of discontinuities andthe installation panern ofrockbolts, thereinforcement effects will result from the axial and/or shearresponses of the bolts to these displacement fields.

Discontinuities are usually planar in large scale but wavy in small scale. While the discontinuities can not transfer tensile loads, they are usually capable of transferring very high compressive normal loads. As for the shear loads, their resistance will he largely influenced by their surface configurations, the properties of wall rock and its frictional properties. The required reinforcement effects of the bolts will be

1 To provide a tensile response to transfer the load from one side to another in the case of the separation type movements and/or 2 To contribute to the shear resistance of the discontinuity by providing an additional shear strength through its own strengh and resisting shear and frictional loads

7.4 Design of Support and Reinforcement Systems 215

Figure 7.11 Types of movements in discontinuum.

separation closing-up

7.4.2.1 Increment of the tensile resistance of a discontinuity plane by a rockbolt

The cases, which necessitate a tensile resistance, arise when the discontinuity walls tend to separate from each other The tensile resistance Trr offered by a rockbolt can be given in the following form (Figure 7.12):-

where Ah = cross section of the bar.

The resistance a h of the bolt can he obtained from Equation (7.21). The angle 9 in Equation (7.21) must be taken as the angle between the direction of separational movement and the bolt axis. Themaximum resistancewill beobtained when theangle becomes 90" since the tensile strength ofthe bar is greater than its shear strength.


Figure 7.12 Typical examples where tensile resistance of bolts are required

7.4.2.2 Increment of the shear resistance of a discontinuity plane by a rockbolt

Rockbolts contribute to the shear resistance of discontinuities depending upon the magnitude and the character of stress components in the bolts and their orientation with respect to the sense of movement along the discontinuity plane. These contributions are

1. Reinforcement due to axial response of bolts - (a) Frictional component and @) Shear component. 2. Reinforcement due to shear response of bolts (Dowel effect).

To investigate all these effects one by one in detail, consider two reinforced blocks of rock put together so as to create a throughgoing discontinuity plane which has a constant friction angle q? as shown in Figure 7.13. Let the angle between the bolt axis and the normal of the discontinuity plane be 0. The axial and shear stresses ou and z, in bolt resulting from the translational movement along the discontinuity can be given in terms of the stress a h in the bolt. acting in the direction of movement as

o a = a h s i n e r, = a,, cos e

In the following Sub-sections, the reinforcement effects due to the axial and shear responses of a bolt are separately discussed first so that the similarity and the difference between the mechanically anchored bolts. partially grouted bolts and fully grouted bolts can be clearly shown Then, the total reinforcement effects of bolts are presented


Figure 7.13 Effect of installation and the character of axial stress in botts on their reinforcement effect.

I ob '..j ab compression (-) tension (+) 1-1

(a) Reinforcement due to axial response of bolts

The axial stress will create normal and shear force components on the discontinuity plane. Assuming that these forces are uniformly distributed over the shearing areaAs. one gets the following expressions for normal and shear stresses resulting from bolts as

i a,,h = n a. cos 8 , k h = 11 a" sin 0

218 7 Rock Reinforcement and Support I 7.4 Design of Support and Reinforcement Systems 219 i

1

Introducing the friction law of Amonton (Bowden and Tabor, 1964) which is given 1 with that of the axial response. Therefore, the shear response conrribution should

as i always be taken into account as:- ! I

7 I K,, = r, cos 0 -- - (7.28)

-tan q, (7.26) I

the total shear resistance offered by the axial stress component of a rockbolt can be written in the following form:-

where (+) and (-) stand for adjectives tensile and compressive, respectively

To see how these effects can contribute to the shear resistance of discontinuity, it is necessaly to know the character of stresses in the bolt and its installation angle. A sample calculation is carried out by varying the installation angle of the bolt for various friction angles of discontinuity plane for two cases as shown in Figure 7.13.

When the axial stress in bolts is tensile for the range - 90' < 0 < oO, the frictional contribution is positive while the shear contribution is negative. On the other hand, when the stress is compressive, a reverse sihlation appears. For the range 0" S 0 < + 90". both effects will be contributing positively to the shear resistance of the discontinuity. Thus the installation angle of rockbolts must be in between 0' and + 90" in order to make their effective use. Figure 7.13 implies that the bolt will be most effective when they are installed at an angleof 90' - 9 as long as their axial stress contribution is considered.

When the bolts are installed at an angle between - 90' 0 < 0' there will be a tendency of developing compressive stresses in the bolts due to the direction of loading. If the bolts are of mechanical type and prestressed, after acertain amount of displacement bolts will become loosein the hole. However, if the bolts are of grouted type, the axial stress in bolts will be compressive. On theother, when they are installed at an angle in between 0" 5 0 5 + 90° the tendency will be oftensile type. Mechanical and grouted rockbolts will all experience tensile stresses within this range.

(b) Reinforcement due to shear response of rockbolts

The effect of shear response of the rockbolts may be considered in a similar manner to its axial contribution. However, such a consideration will make the resistance offered by rockbolts to become independent of the character of the axial load caused in rock. As experimentally confirmed that the reinforcement effect is mostly affected by the character of the net axial stress, that is to say, ifthe stress is of tensile character. the largest resistance is offered, otherwise, the least resistance is offered. The impli- cation is that the frictional component due to the shear stressin bolt should beomitted as it can not be distributed over the large area of the discontinuity plane as compared

The shear response of mechanically-anchored rockbolts comes into effect after a certain amount of displacement takes place since a space exists between the bar and borehole. On the other hand, the situation is different in the case of grouted rockbolts and theresponse ofthebolt comes into effect as soon as the relative shear displacement along the plane takes place.

(c) Total shear reinforcement offered by rockbolts

The total shear reinforcement offered by rockbolts to a discontinuity plane will depend upon the character ofthe axial stress in bolts, the installation angle, their type as well as their dimensions. For mechanically-anchored rockbolts the contribution of the shear resistance will be due to the axial load in bolt and its mathematical form will be the same as Equation (7.27). On the other hand, the grouted rockbolts will offer a shear resistance not only from their axial response but also their shear response and it will take the following form for a rockbolt:-

where n = bolting density and ob = strength of the bolt given by Equation (7.21).

7.4.3 Suspension effect of rockbolts

One of the reinforcement effects of rockbolts is the so-called suspension effect. As the term implies, the rockbolts function to suspend an unstable volume to the stable region further back. In other words, the bolt has a role of transferring the load to the stable region (which may be also regarded as the bolts make the rock mass carry the load resulting from itself). However, this transfer is only possible if the bolts are well anchored in the stable part. In rock engineering, there are a number of cases, in which the suspension effect of rockbolts is required (Figure 7.14). The design of rockbolts for their suspension effect involves three main steps:-

I . The determination of the extent of the zone to be suspended and the mabnitude of the load. 2. The stable zones, to which the bolts are anchored. 3. Dimensioning the bolts, installation pattern etc.


Figure 7.14 Some typical cases by which the suspension effect of bolts is required.

Layer falls Block falls

In the first step, one of the approaches, which are explained in Section 7.2, can be used to determine the extent of the zone to be suspended and the magnitude of the load resulting from this zone.

In the second step, since the extent of the zone to be suspended becomes h o r n , then the zones, to which the bolts are to be anchored, are easily determined from the geometry of the opening and the zone to be suspended.

The third step involves the geometrical dimensioning of the bolts such as diameter, length and their installation pattern and the number of bolts. The usual procedure is first to choose a steel bar of a given diameter with given yield and ultimate tensile strengths, and then, to calculate the number of bolts from the following formula, by assuming that every rockbolt is subjected to the same intensity of load.-

where N = number of bolts; B = span of opening; h = maximum height of the zone to be suspended: q = total load per unit width for a given span R;

SF= safety factor; a? = yield strength of bolt; and

C,,IZ = ultimate strength of bolt.


where I,!= total length of bolt; 1," = length of anchorage in stable zone; and LUs = length in zone to be suspended.

One of the problems involved in the design of rockbolts for their suspension effect is probably the determination of the minimum anchorage lengtb within the stable region. As the anchorage capacity of rockbolts will depend upon the geometry of the bolt and borehole;elastic properties of rockholt and grouting materials and shear strength of interfaces between bolt-grout and grout-rock. The simple procedure to determine the necessary anchorage length is to cany out some in situ pull-out tests. Another procedure is to employ one of empirical or theoretical formulas available in literature. The following expression can be used to estimate the anchorage length of rockbolts for the practical purposes:-

Case I :- The bolt-grout interface is critical

r - Case 2:- The grout-rock interface is critical

1 1 arKa I,, = - tanh- (

a r,, $ ~b a )

where Kg = --

ro = rigid boundary radius (half of spacing); rl, = radius of hole; G, = shear modulus of rock; (;p = shear modulus of grout;

.rig= shear strength of bolt-grout interface, and

The bolts must have their anchorages within the stable region. In other words, their length should exceed beyond the potentially unstablezone. Therefore, the total length of rockbolts can be expressed as


7.4.4 Beam building effect of rockbolts

Sedimentary rocks in nature are usually layered varying in thickness from place to place due to the process they underwent during their formation. The layered rocks have a continuous discontinuity planes called bedding planes. These bedding planes may bevery smooth orvery roughandmay contain soft infilling materials. Their shear properties will depend upon this roughness and infilling materials, if they have. On theotberhand, they almosthavenotensile strength. When the openingsare excavated in such rock formations, the stability of openings will be largely governed by the behaviour of these planes. If the layering is horizontal or nearly horizontal, and the horizontal stresses are low in comparison with the vertical stresses, the layers above the roof will tend to sag into the opening. These layers will act as either beams or plates under the gravitational forces. If the resulting tensile stresses are high enough to cause cracking of the layers, the layers above the roof will eventually cave into the opening. This is a fairly common form of failure experienced in many mining adits driven in sedimentary rocks. To prevent this type of failure, it is essential to reduce the tensile stresses within the layers below the tensile strength of rock. To do so, the layers should he stitched toeach other so as to obtain a thicker beam resultingin lower tensile stresses in the layers. The stitching can be obtained by shear keys in ordinary beams. However, the only means of stitching in rock engineering applications will be through rockbolts in layered rocks. First fundamental studies on rockbolts were initiated by Panek (1956a, b, c, 1962a. b). These involved both theoretical and model studies. Since then, a great number of researches has been undertaken by various researchers (Fairhurst and Singh, 1974; Snyder, 1983; Roko and Daemen, 1983).

Let us consider three beams of the same thickness f and rigidity, put together in a way as shown in Figure 7.15, and also another beam with a thickness of 31. Assume that intelfacing sides of beams are smooth and they are subjected to their own weight andare simply-supported. These beamswill actindependently fromeach other. When we compare the maximum tensile stresses, the stress in the monolitfic beam of the thickness 3t will be only one third of the beam of the thickness of 11. Thus, if a multilavered beam is made to act as an eauivalent monolithic beam. the tensile


Figure 7.15 Three beams of same thickness I put together and one beam of thickness 31.

Shear stress distribution

stresses, which are clitical forthe layers, can be drastically reduced. As also pointed 1 The real situation in the roof of underground openings in layered rock mass is quite out in the beginning of this Sub-section, the reduction of the tensile stresses in the similarto that of built-in beams or plates. Therefore, the shear stress for atypical layer layers can be attained through a means of an internal shear reinforcement so as to

I can be given in the following form:. obtain a thicker beam or plate. The largest shear stress distribution within the beams take place at the neutral plane of the beam and is independent of the thickness of the I

I beam. This implies that if, for example, two beams of the same thickness are stitched I, 1 toeach other and act as amonolithic beam, themaximum shear stress will be observed at the plane at which the beams face each other and the stitching material must be ! resisting to these stresses. If the stitched beams are required to act as an equivalent i h t'

i wherelis - - forbeams with a rectangular cross section. When beams are subjected monolithic beam, then the shear resistance Trto be orovided bv the stitchine material I2 - must be equal to or greater than the integrated shear force distribution, which isgiven ' only to the gravitational force, q can be taken as rf = y h t where y is the unit weight as

I j of beam material.


Assume that the stitching material is rockbolts The total shear reinforcement provided by n bolts per unit width b can be written, as shown in Sub-section 7.4.2.2 (Equation (7.29):-

where 56 =axial stress in bolt i;

~b = cross section of bolt i;

9' =inclination of bolt i with respect to the normal of layers; and q = interlayer friction angle.

Assuming that bolts are installed in a pattem so that each.bolt is subjected to the same axial load, having the same inclination angle, then the total number of bolts required may be obtained from the following expression:-

As noted from above equations, information on the stresses, to which the bolts are subjected, is required for dimensioning the bolts. To simplify the problem further, it is assumed that bolts are subjected to a prescribed load or their yield strength which can he obtained from Equation (7.21).

The effective way of installing the bolts is to obtain the minimum amount of sliding along beddings at an angle which is most suitable for such a purpose, and to increase the density of bolting where a large amount of relative sliding displacements is expected. The amount of relative sliding displacements between two beams of different thicknesses under their own weight with the frictionless interfacing sides can be calculated from the following relationship, which is derived from the sum of the deflection of the upper side of the lower beam and the deflection of the lower side of the upper beam (Figure 716(a)):-

A.Y(x)= dtr1 1, + dl? 12 & 2 d r 2

&I' dl,Z where - and are derivatives of deflection curves u' and I? of beams of & r l ;

thicknesses 11 and 12 with respect to x.


Figure 716(b) shows-the disfribution of the relative amount of sliding along interfacing sides of the two beams in the case of no reinforcement qualitatively. The maximum amount of the relative displacement occurs at the end of spans for simply- supported beams and at distance of 0.21 L from the ends in the case of beams with built-in ends. Once the tensile strength and thickness of the layers and the load q are known, the required thickness of the equivalent beam can be calculated for the allowable tensile stress in the layers. This thickness can be assumed to be equal to the minimum required rockbolt length. Then, the remaining problem is how to determine the number and pattem of the bolting. This can be achieved by using the principles described above.

Figure 7.16 Definition of relative sliding and distribution of relative sliding along interfacing sides of layers.

Simply supported

7.4.5 Arch formation effect of rockbolts

Arching phenomenon is a well recognised phenomenon in many soil and rock engineering works. To illustrate this phenomenon, let us consider two blocks with a gap S as showninFigure7.17inagravitational field. Iftheseblocks arenot restrained, they will tend to fall freely. On the other hand, if they are restrained as shown in the figure and allowed to freely rotate, the so-called arching action will come into existence. The arch will be stable unless the blocks are cracked by the induced stress state. Therefore, the arching is only possible with necessary restraining, the allowance of rotation and the sufficient strenb* of the block material. The most suitable supporting material is the one that satisfies the necessary conditions ofthe arch action. As rockbolts superbly satisfy these conditions, they will be more superior to other support members. It should be noted that the main load bearing element is rock itself not the support member which helps the rock to mobilise its intrinsic resistance.


)

Figure 7.17 A simple model to illustrate arching phenomenon

To design the bolts, two possible failure foms of the resulting arches must be considered (Figure 7.18):-

1. Compressive failure at crest or abutments. 2. Shear failure at abutments or on a discontinuity plane within the arch

In thefollowingpresentations, the arch is assumed tobeequivalent to athree-hinged arch from the structural engineering point of view (Figure 7.19). For agiven symmet- rical loading function F2 ( x ) - FI ( x ) = , f ( x ) with respect to the centre lineof the opening per unit width, one can write the following equilibrium equations for the adopted coordinate system:.

7.4 Design of Suppon and Reinforcement Systems

Figure 7.18 Failure modes of rock arches.

Compression failure at crown and abutmen@

Vertical shearing at abumene

Horizonti shearing at abutments

Sliding along a discontinuity within the rock arch

where WZ = [ y . f ( x ) &;

y = unit weight of loose medium: F.., = reactions in .?-direction at respective points: F.,, = reactions in y-direction at respective points;

H = horizontal thrust; and I. = span.

Taking the moment with respect to point (' (left or right), one can write the following:-


Figure 7.19 Structural model for rock arches.

L

f f ( x ) x A where jlc = L = average distance of the load vector from the origin; and

p f ( x ) d o

yc = rise of arch

Accordingly, the horizontal thrust H is obtained from Equations (7.39) and (7.40) as

Taking the moment with respect to point l> (to the left) results in

~ M ~ , = I ~ U ~ y - l ~ ~ x + ~ ~ ( x - j l r ) ) = ~ (7.42)

where ~ r = ( ~ f ( ~ ) r l ~

7.4 Design of Suppo~t and Reinforcement Systems 229

- xD = [ f ( x ) x k . -. -- , and

[ f ( x ) m

x = distance of point D from the origin

Re-arranging the above equation yields the position y of the thrust line as

Bending in the arch results from the deviation between the centre line of the arch and the influence line of thrust T. Thus, the distribution of the axial stress in the arch per unit width can be calculated from the following expression in terms of an average -.

I stress, defined as o - ----

O - I .

where t = thickness of arch, e = distance between centre line and influence line; y = distance of a point from the centre line; and T = thrust.

Denote the maximum compressive stress by a,, which occurs at the upper side of f e

the arch (!J = T ) and by 11,. Then, the relation between ocr and oo becomes

Ocr 00 = (7.45)

1 + i l p

At the centre C of the span, the following relation must hold:-

acr t As T = oo I = and H i s given explicitly in Equation (7.41), the following 1 +n,

relation can be written:-


f lntroducing a nonnalising parameter E, = - between the thickness 1 of arch and the 10

niaximum height I,, the riseye of the arch is expressed from the geometry of the arch as

lnsetting Equation (7.48) into Equation (7.47) and re-arranging the resulting expression yields the maximum height of the arch as

Derivating this expression with respect to E, yields

Substituting the above result in Equation (7.48) gives

Then, thickness t of the arch is obtained by insetting the above expression in Equation (7.41) and re-arrangement of the resulting expression as

The length of rockbolts may then be assumed to be equal to themaximum height of the arch. The thicknessofthearch can bespecifically determined by introducingsome criteria for the stability of arch against various forms of failure. These criteria are as follows:-

I . Compressive failure at crest:. This condition is introduced by equating the compressive strength of the rock to the maximum compressive stress at the crest of the arch:-

or ' Ocr (7.53)

7.4 Design of Support and Reinforcement Systems 23 1

In this particular case, the action of rockbolts is just to provide a restraining effect and to allow block rotations. The strenyh of bolts can not contribute to the resistance of the arch. Therefore, the determination of the necessary bolt number is indetermi- nate. Nevertheless, the following formula is suggestedto calculatethenumber ofbolts (it is assumed that the bolts should suspend the rock mass bound by the opening geometry and the lower side of the arch) (Figure 7.20):-

where FI ( x ) = equation of roof line; F2 ( X ) =equation of the lower side of the arch;

Th = pull-out capacity of a rockbolt, and y =unit weight of rock.

Figure 7.20 Illustration of the load to be suspended by bolts to rock arch.

I a load,ta be suspended to 'W arching zone by bolts

2. Shear failure a t abutments or along a discontinuity plane:- The stability of arch should be checked against vertical and horizontal shearing at abutments. For each case, the necessary rockbolt number is calculated from the following expressions by assuming that rock obeys to the Mohr-Coulomb yield criterion and bolts are subjected to the same magnitude of bolt stress:-

232

(i) Shearing a t abutments:

(a) Vertical shearing:.

@) Horizontal shearing:-

7 Rock Reinforcement and Support

where c = cohesion of rock; q = fr idon angle; and

1 k= coefficient for the assumed stress distribution = ---

( l + n e )

(ii) Shearing a t one of discontinnity planes:-

The most critical plane in the arch with well developed discontinuity sets is the one, which emanates from the abutments and inclined at a low angle to the thrust line (Figure 7.21). The required rockbolt resistance can be calculated from the following expression:-

JZp, H 1

- - ( ~ o s ( a - a ) - s i n ( a - ~ ) t a n q ) < ~ ( 7 b ( l + - t anqs in28 i ) ) cos !y 2

i=I (7.57)

where y = inclination of the thrust line from horizontal; q = friction angle of discontinuity plane; u = inclination of critical discontinuity plane; H = horizontal thlust;

!I,,/= number of bolts crossing the critical plane;

7h = force acting in the bolt; and 8 . - ' , - ~nclination of bolt with respect to the normal of the critical plane.

To dimension the bolt length, the height of the arch should be determined. For this purpose, the procedure given here can be followed. For the overall stability of the arch, the following condition must be satisfied:.

7 4 Des~gn of Support and Reinforcement Systems 233

I i I

Figure 7.21 Rockarch having well defined discontinuity sets and force system acting I on a discontinuity plane.

(7.58) where yo = inclination of the thrust line from horizontal at abutments and

11 =total number of bolts per unit width.

If bolts are assumed to be strained to the same stress Level Th= Ti and their inclination with respect to the critical discontinuity set remains same (8; = 8), the rise y, of the arch can be found by inserting H given in Equation (7.41):-

~ ~ ~ ~ a ( q ~ ~ ~ - I n ( c o s ~ ~ ~ ~ ) t a n q ) - s i n a ( l n ( c o s y n ) + ! ~ ~ o t a n r p ) - ~~ .... c - - I

II Th ( 1 + - tan rp sin '3 ) 2 (7.59)

Derivation of the expression for thrust liney with respectto x at x = 0 yields qru as

- I dq' !I10 = tan ( ) = tr (7.60)

dx

734 7 Rmk Reinforcement and Suppon

The above expression will be afunction ofy, and 1. only. For example, if the load x X W, is a uniformly distributed load (i.e. WY = y x I, ), then y = 4 . y ~ ( t ) ( 1 - - ) The L

4 .VC 1, tan \YO derivation of the above equation at x = 0, tan tyci = - ory, = L 4

q10 is tirst obtained by applying an iteration scheme, such as the Newton-Raphson technique, to the function defined as

g =ye (Equation (7.59)) - yc (Equation (7.60)) = 0 (7.61)

for a given set of n, Ti, 0 ,a, Ax) and L. Once ivo becomes known, then y,, and subsequently 1, and t can be easily calculated.

7.4.6 Stabilisation against sliding

Thereareanumber of casesin which support measures arenecessary against sliding. Typical examples are illustrated in Figure 7.22. The sliding type of failures involves at least one or more planes on which the sliding occurs. In the examples depicted in Figure 7.22, it is assumed that a sliding takes place along plane a, only and a

separation occurs at plane az. The specific forms of the resistances C%' , Cg2 offered by rockbolts on planes a , and az can be given in the following forms on the basis of discussions in Sub-section 7.4.2 provided that the angles Ba, , Ba2 between bolt axis and the normals of critical planes remain to be the same at the respective planes.

On plane ailfrom Equation (7.29)):-

C 7g' = IIU, AR' at1 ( 1 +I sin 20a1 tan v ) (7.62)

where ,fix, and A X ' . 0%' are the total number of bolts and their cross-section and strength, respectively.

On plane ar(from Equation (7.22)):-

C 7X' = I I , ~ AX' D):' (7.63)

where 1 1 , ~ ~ and AX^, err' are the total number of bolts and their cross-section and strength, respectively.

Then, the factor of safety against sliding takes the following form after some algebraic manipulations:-


Figure 7.22 Typical examples in which reinforcement against sliding is necessary and notation.

L:ndergrntind o;lcnings

Slope Foundation

A c r c r c

where Wis the weight of the sliding body, which can be easily calculated from its geometry together with the unit weight of rock.

7.4.7 Support of a circular tunnel by shotcrete, rockbolts and steel ribs

To start the derivations, the following are set -

I. The problem is axisymmetric and the installation of rockbolts does not violate this assumption. Then, the governing equations for bolted and unbolted sections are (Pisure 7.23):-

* * c/o? + ;la- - ;lo d 1.

bolted section r (76.5)


Figure 7.23 Notation for a reinforced circular tunnel

1

d a r + a ~ - a Q - ~ unbolted section d r r

wherer = distance from opening cenee,

a; = radial stress in bolted section;

ai) = tangential stress in bolted section, or = radial stress in unbolted section; and

= tangential stress in unbolted section.

2. In bolted section, the radial stress is shared by rock and bolts which is expressed- by

where or. and orb denote the radial stresses shared by rock and bolts, respectively. orh is related tothe axial stressah in each bolt at agiven point from the opening centre by the following expression with the consideration of effective bolted area Ar ( r ) at a given point:.

7 4 Design of Support and Reinforcement Systems 237

where Ah = cross section of bolt; a = radius of opening;

CO = el er; Cr = transverse spacing; and ei = longitudinal spacing.

3. Equivalent radial and tangential stresses and displacement u* in bolted section and those in the unbolted section satisfy the following conditions at the boundary of the bolted-unbolted sections (r = h , h = a + lo, I, = bolt length):?

! 4. The behavioqrs of rock and bolts are assumed to he elastic-brittle plastic and i

elastic-perfectly plastic, respectively (Figure 7.24).Rock oheysto theMohr-Coulomb 1 i I yield criterion:-

i al=qoj+oc intact rock

! (7.70)

i 0 1 = q" a3 +a: plastified rock i

(7.71) l + s i n q . l + s i n V *

j q=-- I ,q = 1 -sin q 1 - sin 0," j 6 i where oi =maximum principal stress; i 03 =minimum principal stress;

i ac = uniaxial compressive strength of intact rock;

o: =uniaxial compressive strength of plastitied rock;

1 O, =internal friction angle of intact rock; and 1 i o," = internal friction angle of plastitied rock. I 1 5. Bolts are assumed to be installed after a certain amount of displacement a:, has ! taken place in order to account the face advance effect. Accordingly, the resulting

i internal pressure effect of rockbolts is related to the differential displacement between

t installation and equilibrium states. The governing differential equation for a ypical

i rockbolt with agiven differential displacement A 1117at the grout-rock interface can be 8 written in the following form:- t i

238 7 Rock Reinforcement and Support 7 4 Design of Support and Reinforcement Systems 239

whereq=r-u , ri, = radius of borehole; rh = radius of rockbolt; and Gg = shear modulus of grout

Figure 7.24 Behaviour of rock mass and support members

= I rock

The procedure how to evaluate the differential displacement A ui, is shown later.

6. The constitutive law between stresses and strains of rock for elastic behaviour is of the following form:.

Dz=- & v, ( l + v F ) ( l - 2 v r ) '

E,= elastic modulus of rock; and vr= Poisson's ratio of rock.

7. Relations between strains and radial displacement u are

d l * Ey = --

dr

where 4, E: = radial strains and

EO .E$ =tangential strains.

8. Shotcrete is assumed to be installed after a certain amount of displacement II,, has taken place in order to account the face advance effect. Accordingly, the resulting internal pressure effect of shotcrete is related to the differential displacement A a;, between installation and equilibrium states. 9. The assumptions for steel ribs are the same as those for shotcrete.


7.4.7.1 Pure elastic behaviour of rock

The stresses and displacements in the bolted and unbolted sections can be obtained by solving the governing equations (7.65)-(7.66) together with the constitutive law (7.73) and relations (7.74)-(7.75) and the boundary conditions.

(a) Bolted section

o : = P , a t r = u

o F = P h a t r = h

(b) Unbolted section

o T = P b a t r = b s r = o o a t r = m

The resulting expressions for displacement and stress fields me as follows.-

(a) Bolted section


(a) Bolted section

o 2 u;=oo-(ao-l ' i )- 2

U 2

0; = 00 + ( 0 0 - Pi ) r-


b2 5, = o,, - ( 00 - I'h ) -2 (7.80)

r

7.4 Design of Support and Reinforcement Systems 24 1

7.4.7.2 Elasto-plastic behaviour of rock

(a) Bolted section

I . Plastified rock section:- Inserting the yield criterion (7.71) in the governing

equation (7.65) with 0 3 = O F and ol = og and solving the resulting differential equation together with the boundary condition yields the stress fields for the plastic region as

The radius of elastic-plastic boundary is obtainedfrom the yieldcriterion (7.70) and the identity or + oo = 2 oo together with the continuity condition.of radial stresses at the boundary as

The derivation for the displacement of the plastic zone will be briefly described as follows:-

The compatibility condition for total strains is

Assumethat the following relation holds between the total radial and total tangential strains:-

242 7 Rock Reinforcement and Suppon 7.4 Design of Support and Reinforcement Systems 243

ef =-f. (7.86) €5

where f = a physical coeficient determined from experiments

The solution of the above differential equation is

The integation constantA is determined from the continuity of the tangential strain at elastic-plastic boundary r = Hp as

Using the above relations, the displacement of ground is obtained as

2. Elastic rock section:- As the derivation of expressions for stresses and displacement is similar to those described previously, thefinal forms of expressions are only given here.

(a) Bolted section

II ~ f = o ~ - ( o ~ - o ~ ~ ) ( ~ ~ ) ~ r (7.90)

"P )Z o ~ = o o + ( o o - ~ ~ ) ( - - r (7.91)

l + v r u = ( 00 - ovp ) R: (7.92)

r


h 7 or = a,> - ( c,, - l'h ) ( )- (7.93)

2 oo = a,, + ( c,> - 1% ) ( ) (7.94)

where 1'1 = internal pressure applied at opening wall and Rp 2 P ~ = o ~ > - ( o o - G ~ ~ ) ( ~ )

7.4.7.3 Modelling of rockbolts

Rockbolts are assumed to behaveelastic-perfectly plastic as shown in Figure 7.24(c) and the gweming equation of rockbolts given by expression (7.72) is related to differential displacement between the installation state and equilibrium state. The procedure for the evaluation of the differential displacement for each respective behaviour is given here.

Procedure for evaluating the differential displacement The form of A uh given in Equation (7.72) depends closely to the states of rock at

the timeofinstallation ofrockbolts and atthe final equilibrium state(the stateat which the inward displacement of rock ceases to occur). There are a number of possible combinations. The general procedure to determine the form of A uh is to take the difference of displacement hnctions at the time of installation of bolts and the equilibrium state as

where rq and in stand for equilibrium and installation states, respectively. In the following presentation, the procedure will be explicitly shown forthe pureelastic case only. As the procedure is the same for all the cases, only the final form of expressions will be given. Assuming the bolt-system remains elastic. the most possible combinations and associated functional forms of 4 uh are as follows:-

I --:- Rock is in elastic state at the installation and equilibrium states.

Denoting the internal pressure Pi by P,,, at the time of installation of rockbolts, the displacement at the tunnel wall is obtained from Equat~on (7 76) as

2 I + v r N;,, = - ( a,, - Psr ) -~ (7.97)

hr r

Similarly, denotingthe internal pressure Pi by PJy, the displacement at the equilibrium state can be written from Equation (7.76) as


a l + v r n 2 lieq = -- ( - peq ) ;~ E,

(7.98)

Taking the difference, one gets A uh as

I +vr A uh = ---- '2

(I'i,? - Peq ) 7 (7.99) E,

I1 - W:- Rock is elastic at the installation state and becomes plastified at the equilibrium state.

1 +vr A UII = --

R a 2 r [ ( ~ o - ~ r ~ ) ( - . : P ' - ( ~ o - ~ i n ) ( ; ) 1 (7.100) G

II -4vLi:- Rock is plastified at the installation state and remains plastified at the equilibrium state.

Procedure for evaluating the stress field of rockbolts The solution of Equation (7.72) with the use of the above relations is cumbersome

since the non-homogeneous part of the resulting differential expression involves improper integrals which could not be directly integrated with elementary methods and the use ofnumerical techniques will be necessluy. The approximation of the above functional forms by exponential type functions will be introduced since they cause little error as :-

The constants of the above functions for each respective region are as follows:-

( Z = A I,"

l n ( i ) , = - - ~

h - n I +v,.

A 11" = . ];,I. ( f J j n - n


@) Elasto-elastic behaviour of rock

Plastic region

= A I,"

A tia In ( --- )

A II {>U = Rp-a

Case 2

Case 3

Elastic region

Case 2

I +v r R z a 2 A"&=---- h [ ( ~ r ~ ~ - o ~ ~ ) ( - ~ ) -(~(,-f:,,)(;) ] l;7 h

Case 3

. 246 7 Rock Reinforcement and Suppon 7.5 Exercises 247

The general solutions of the non-homogeneous differential equations are obtained as follows:-

(a) Axial displacement in bolt

~ b = A l e - * ~ + A 2 e ~ ~ + ~ , e - ~ ~ (7.103)

(a) Axial stress in bolt

The integration constants for each respective case are obtained from the boundary conditions and continuity conditions of axial and shear stresses.

7.4.7.4 Modelling of shotcrete

Shotcrete is generally modelled as a thin-walled tube or thick-walled tube in literature. As shown in the previous Sub-section, the relation between the radial displacement of the tube with an outside pressure P,, and a zero internal pressure P, = 0 at the adjacent side to the tunnel wall becomes:-

The incremental form of the above expression is

Inversely, we have

A Pi.7 = K,T A u

If the thickness of shotcrete is relatively small compared with the excavation radius, then the above expression can be further simplified to the following form:-

E I where Kr = - --- 2 2 I - vr a,

The above expression is equivalent to the expression for thin-walled tubes.

7.4.7.5 Modelling of steel ribs

Steel ribs can be modelled as a one-dimensional circular rib. The radial deformation of the rib can be shown to be

The incremental form of the above expression is

Inversely, we have

Elb Arb where Krh = -- 2 el

7.5 Exercises 1 Spacings of rockbolts with a diameter of 25 mm are 1 m x 1 m. Assuming that the elastic moduli of rock and rockbolts are 1 GPa and 210 GPa respectively, determine the increment of elastic moduli of rock due to rockbolting.

2 A rock pillar with a height of 3 m and a width of 2 m is reinforced by 3 grouted rockbolts. The cohesion and the friction angle ofrock are 1 MPaand35" respectively. Assuming that the axial stress of the rockbolt with a diameter of 25 mm is equal to the yield strengthof steel, which is450 ma, determine thestrengthincrement of rock pillar due to bolting.

3 The rock mass is classified as Class 6 in Terzaghi's classification. The unit weight of rock is 20 kN/m3, and the span of opening is 10 m. Assuming that the axial stress of the rockbolt with a diameter of 25 mm is equal to the yield strength of steel. which is 450 MPa, determine the number of rockbolts to suspend the loosening zone to the stable zone.


4 There aretwo structural weakness planes emanatingfrom the comers of theopening with a flat roof. The inclinations of the discontinuities are 60' and 120' respectively. Theunit weight of rock is 20 kN/m3, and the span of opening is 10 m. Assuming that the axial stress of the rockboltwith a diameter of 25 mm is equal tothe yield strength of steel, which is 450 MPa, determine the number of rockbolts to suspend the loosening zone to the stable zone.

5 There are two structural weakness planes emanating from the comers of the sidewall of the opening. The inclinations of the discontinuities are 60' and 120' respectively. Theunit weight of rockis 20 kN/m3, and the height of the opening is 15 m. Assuming that the axial stress of arockbolt with a diameter of25 mm is equal to the yield strength of steel, which is 450 MPa, determine the number of rockbolts to resist against sliding of the rock in the sidewall.

6 The span of the underground opening is 10 m and the roof is flat. The height of 3

loosening zone is 7 m and the unit weight of rock is 26 W m .

(a) Assuming that the uniaxial compressive strength of rock is 10 MPa, and the axial stress of the rockbolt with a diameter is 25 mm is equal to the yield strength of steel, which is 450 MPa, determine the number and length of rockbolts required againstthe compressive failure at the roof.

(b) Assuming that there is a vertical joint set with a friction angle of 35' and zero cohesion, and the axial stress of the rockbolt with a diameter of 25 mm is equal to the yield strength of steel, which is 450 MPa, determine the number, length and the optimum installation pattem of rockbolts required against the shear failure at the abutments.

7 Two rock beams of 0.5 m thick are put upon each other and they are subjected to built-end boundary conditions. The unit weight, elastic modulus and tensile strength of rock are 26 k ~ / m ' , 3.15 GPa and 0.05 MPa respectively Assuming that the span is 3 m, determine the number and the optimum installation pattem of rockbolts required against the bending failure of the beams.

Monitoring of Structures

I n a general geomechanics context, monitoring of structures may be canied out for the following four main reasons.-

1 To record the natural values of, and variations in, geotechnical parameters such as water table level, ground levels and seismic events before the initiation of an engineering project. 2. To ensure safety during consmction and operation by giving warning of the development of excessive ground deformations, ground water pressures and loads in support elements, for example. 3. To check the validity of the assumptions, conceptual models and values of rock mass properties used in design calculations. 4. To control the implementation of ground treatment and remedial works such as tunnelling through water-bearing ground, grouting, drainage or the provision of support by tensioned cable.

In mining rock mechanics, most monitoring is canied out for the second and third reasons. Monitoring the safety of the mine structure is a clear responsibility of the mining engineer.

Monitoring systems used in conjunction with modem large-scale underground mining operations can be very sophisticated and expensive. However, it should be remembered that valuable conclusions about rock mass response can often be reached from visual observations and from observations made using very simple monitoring devices Items that may be monitored in an underground mining operation include.-

(a) Fracture or slip of the rock on the excavation boundary (observed visually) (b) Movement along or across a single joint or fracture (either monitored by a simple mechanical "tell-tale" or measured more accurately). (c) Relative displacement or convergence of two points on the boundary of an excavation (d) Displacements occurring within the rock mass away from the excavation periphery

250 8 Monitonng of Structures

(e) Surface displacenlents or subsidence. (f) Changes in the inclination of a borehole along its length. (g) Ground water levels, pressures and flows. (h) Changes in the normal stress at a point in the rock mass. (i) Changes in loads in support elements such as steel sets, props. rockbolts. cables and concrete. (j) Normal stresses and water pressures generated in fill. (k) Senlements in fill. (1) Seismic and microseismic emissions. (m) Wave propagation velocities.

8.1 General features of monitoring systems The instrumentation system used to monitor a given variable will generally have

three different components. A sensor or detector responds to changes in the variable being monitored. A transmitting system which may use rods, electrical cables, hydraulic lines or radio telemetry devices, transmits the sensor output to the read-out location. A read-out andlor recording unit such as a dial gauge, pressure gauge, digital display or magnetic tape recorder, converts the data into a usable form and presents them to the engineer.

In order that the monitoring system should fulfill its intended function economically and reliably, it should satisfy a number of requirements:-

(a) Easy installation, if necessaly under adverse conditions. (b) Adequate sensitivity, accuracy and reproducibility of measurements. (c) Robustness and suitable protection to ensirre durability for the required period of operation. (d) Ease of reading and immediate availability of the data to the engineer (e) Negligible mutual interference with mining operations.

8.2 Monitoring systems

8.2.1 Convergence measurement

Convergence, or the relative displacement of two points on the boundary of an excavation, is probably the most frequently made underground measurement. The measurement is variously made with a telescopic rod, invar bar or tape under constant tension, placed between two measuring points firmly fixed to the rocksurface(Figure 8.1 ). A dial gauge, micrometer, or an electrical device such as LVDT, is used to obtain the measurement of relative displacement.

Figure 8.2 shows a high precision convergence measuring system developed by Kovari et a1.(1974). The displacement gauge has a readability of 0.01 mm and range of 100 mm. The overall accuracy of the convergence measurements is 0.02 mm.

8.2 Monitoring systems 251

Figure 8.1 (a) Convergence between roof and floor measured with arod convergence gauge, (b) Typrcal five-pornt convergence array (after Brady and Brown, 1993)

Figure 8.2 The distometer ISETH, a high-precision mechanical convergence measuring system (after Brady and Brown, 1993).

! tensioning displacement device gauye

I I joint

\ Invar ","re j0l"t

/ m n g bolt dynamometer

i

i 8.2.2 Multi-point borehole extensometer

Among the most useful measurements of rock mass performance are those made using multi-point borehole extensometers (MPBXs) (Figure 8.3). A multi-point extensometer can give the relative displacement between several points at different depths in the borehole. In this way,the distribution ofdisplacements in relatively large volumes ofrockcan be recorded. These dataaregenerally moreuseful than the results of convergence measurements which only give relative surface displacements and may be influenced by surface conditions. Sometimes convergence measurements are made between the heads of MPBX installations as illustrated in Figure 8.3.

Suggested methods for monitoring rock movements using borehole extensometers are given by the International Society for Rock Mechanics Commission on Stand- ardisation of Laboratory and Field Tests (ISRM. 1978~). For large undersround

252 8 Monitoring of Structures

excavauon;, the minimum measuring range shuuld be 56 mm (700 mm with rciet). the precision should bcin therange0 25.2 5 nlm sndthclnclNmunt senstttvlt\ should be typically 0.25-1.00 mm.

The multi-point extensometer measures the relative displacement of the wires, which are fixed in the ground along the axis of a borehole as shown in F ig re 8.4. During installation each wire is tensioned by spring cantilevers in the measuring head. As the ground adjacent to the borehole deforms, the distance betvieen the sensor head and each fixed point changes.

Figure 8.3 Multi-point borehole extensometer (MPBX) installations with convergence measurement between MPBX heads (after Brady and Brown, 1993).

Figure 8.4 Multi-point extensometer (after Hanna, 1985).

extensometer head


F ip re 8.5 illustrates the multi-rod extensometer This extensometer may also be placed in down or up boreholes and can be mounted on theground surface to measure displacements across cracks or joints.

Figure 8.5 Multi-rod extensometer (aFter Hanna, 1985).

dial depth gauge

reference head Concrete or resm

anchor

unit k i 8.2.3 Hydraulic pressure cell

The hydraulic pressure cell consists of a flatjack connected to a hydraulic or pneumatic diaphragm transducer which in turn is connected by flexible hbing to a read-out unit. Normal stress transferred from the surrounding rock or concrete is measuredby balancingthefluid pressurein thecell by apressureappliedtothe reserve side of the diaphragm. Hydraulic pressure cells are used to measure changes in total normal stress in materials such as fill, or at interfaces between materials, e . g at a rock-shotcrete interface. If effective stresses are required, a piezometer should be installed alongside the pressure cell.

The most widely used hydraulic pressure cell is probably the Glotzl cell which is described by Franklin (1977). The installation ofthe pressure cell to measure circumferential stress in a shotcrete lining and normal stress at the shotcrete-rock interface is illustrated in Figure 8.6.

The fluid used to fill the cell depends on the material in which the cell is installed. The compressibility of the cell should be similar to that of the sunounding material if the cell is to give a correct measure of the undisturbed normal stress in material. A

254 8 Monttoring of Structures 8 2 Momtoring systems 255

cell that is too stiff for its surroundings will register an excessive pressure, and one that is insufficiently stiff will register a pressure that is too low.

The fluid pressure is measured by applying an air or oil pressure to one of the twin tubesthat connect the hydraulictransducertothe read-out. When this applied pressure is sufficient to balance the pressure in the cell, a return flow of air or oil will he registered at the read-out unit. The normal pressure is then given as

where P = normal pressure; Pr = indicated pressure; p - ' " , - lnthal cell pressure; Ph= static head correction forthe differencein elevation between the cell and

read-out; Pf =correction for friction losses in the delivery line, and E = multiplication factor (less than 1 .O) to compensate for cell edge effect

8.2.4 Stress change measurement

A wide range of instruments has been developed for monitoring the stress changes induced in rock by miningactivity These have included photoelastic plugs and discs, instruments based on the hydraulic pressure cell, the vibrating wire stressmeter, and rigid inclusion instruments using electric resistance strain gauges. These are usually borehole instruments and suffer from the disadvantage that they monitor the stress change in one direction only.

Thevibrating wire stressmeter is used widely for stress monitoring, especially in the U.S.A.. The main components of the stressmeter are shown in Figure 8.7. The instrument consists of a hollow, hardened steel body which, in use, is pre-loaded diametrically between the wall of a 38 mm diameter borehole by means of a sliding wedge and platen assembly. Gauges have been installed in boreholes up to 100 m deep. Stress changes in the rock in the pre-load direction cause small changes in the diameter of the gauge cylinder. These changes are measured in terms of the change in the frequency of vibration of a high tensile steel wire stretched across the cylinder in the pre-load direction.

If only pressure differences are to be monitored, and P; is set after installation and compensation, P can be calculated as:

Figure 8.7 Exploded view of a vibrating wire stressmeter (after Brady and Brown, 1993).

Figure 8.6 Glotzl pressure cell installation in a shotcrete lining (after Brady and Brown, 1993).

-/ read-out

I I R' L r o l l pin

cable clam; \ compound

256 8 Monitoring of Structures

8.2.5 Microseismic aciivitylAcoustic emission monitoring

Rocknoises are often heard by miners working underground and taken as a warning of imminent danger from rock failure. Laboratory and field studies have shown that these audible noises are preceded by subaudible energy emissions from the failing rock. Themonitoringof suchmicroseismicand acousticemissionsinduced by mining activity forms an essential part of the monitoring programs in a number of mines, particularly those susceptible to rockburst activity.

Audible or acoustic wave frequencies are in the range 20 Hz to 20 kHz The frequencies of waves radiated by events associated with mining activity range from less than 1 Hz to more than 10 kHz That part of this frequency range in which most of the energy is concentrated depends on the size of the event. An individual acoustic emission may contain aspectrum of different frequencies and the form ofthe spectrum is produced by two separate factors, (i) the spectrum of the emission at the source and (ii) modifications during propagation through the ground to the transducer location. This attenuation plays a major role in modifying the source spectrum. As a general rule attenuation increases with frequency. Consequently only the low frequency values will be observed at large distances from a sourceof emission. Also, ifthe source spectrum contains no significant low frequency components, there will be a critical distance beyond which the emission cannot be detected. An idealised frequency-range relationship is shown in Figure 8.8.

Figure 8.8 Range versus frequency for acoustic emission signals (after Hardy, 1981)

Frequency, Hz

Recording of acoustic emissions relies on the useof suitable sensors which will pick up any emissions that occur In addition to the sensor, an amplifying, filtering and recording system will be required. Figure 8.9 illustrates a typical system. The most common sensors are geophones, hydrophones, accelerometers and acoustic emission transducers. The last three are more sensitive than geophones.

In order to delimit the source of an acoustic emission, a number of transducers is located throughout the area. Acoustic emission is detected by each transducer at a


different time depending on the distance ofthe transducer from the acoustic emission source. Thus the position of the source of acoustic emission may be determined knowing the velocity of propagation and the geometry of the fiansducer layout.

Figure 8.9 System for acoustic emisston recording (after Hardy, 1981)

transducer

bandpass filter

i mms chan condition recorder

The design of an acoustic emission monitoring must pay particular attention to:-

(i) The most suitable transducer. (ii) An efficient installation procedure and transducer layout (iii) Connection of the transducers to the recording system. (iv) Design of a sensitive yet suitable system.

Two forms of monitoring may he used:- general and location. With general monitoring, the aim is to check if acoustic emission is being generated in the area under consideration. General monitoring may be complicated by other effects including traffic, blasting, electrical transient and low-level seismic activity. In contrast to general monitoring, source location can only be achieved with a suitable array of transducers. For location in a horizontal plane, for example, four transducers are needed and five are required for a threedimensional fix.

Acoustic emissions may be monitored by measuring the disolacements. velocities ~ ~-

or accelerations generated. Where there are high frequency components in the signal (2000 Hz), accelerometers aregenerally used, whilst for low frequency signals (1 Hz) displacement gauges are used. Velocity gauges are used to detect signals between these extremes.

i

- 258 8 Momtonng of Structures 8 3 Examples ofmonttonng structures 259

8.3 Examples of monitoring structures An example showing the changes in the relative horizontal displacements in the wall rocks with the progress of excavation is given in Figure 8.12. Extensometers can

8.3.1 Monitoring of rock mass behaviour around the caverns during usually be installed only after the excavation of the main cavern has reached the

excavation desired installation depth. Measured values, therefore, refer only to changes in rock deformation due to excavation of the main cavem below the point at which these

Figure 8.10 shows an example of the layout o fa monitoring system. Extensometers instruments are installed. are installed at small intervals near the excavation walls, so that the sizes of relaxed zones can be detected. An example of an extensometer is illustrated in Figure 8.11

Figure 8.12 Measured relative horizontal displacements obtained from extensometers (after Hibino and Motojima, 1993).

Figure 8.10 Layout for monitoring system (after Hibino and Motojima, 1993)

arched concrete lbning

extensometen

- reenforcement gauge

A concrete suam gauge

iCi thermameler

Figure 8.11 Extensometer used in measurements (after Hibino and Motojima, 1993)

cement mortar

supply lube

~ n v a r rod return lube

What is characteristic in Figure 8.12 is that the horizontal displacements of the cavern walls around the centre (R-17 to R-19) were much larger than those higher up (R-10 to R12) There may be two reasons for this. Firstly, since rocks in the higher pan of the cavern are located close to the arch their deformations are suppressed by the 3-dimensional strengthening effect in the comers, while rocks in the central part experience a smaller surrounding restraint. Secondly, the whole cavern has a vertically long and horizontally narrow shape, which is mechanically unstable. If the cavern excavation was completed up to the main part (section 2). the ratio of the height to the width of the whole cavem would be close to one, thus making it mechanically stable. The amount of deformation would also be reduced. To excavate such a vertically narrow cavern it is therefore necessary to pay special attention to the stability ofthe cavern when the lower half is excavated.

2

8.3.2 Monitoring of excavationinduced microseismic activity

- 2

An array of 16 triaxial accelerometers was installed to monitor the microseismic events associated with the excavation as shown in Figure 813. The accelerome- tenwith afreauencv resnonse from 50 Hz to 10 id-Iz. were &outed in dace at the end

1YIa Feb.

~ ~ . , - of diamond-drilled boreholes b he sequence of the chnshuction schedule for the test

Mar

I

Apr

z May

3

lun.

4

Jul.

5

Aug.

excavation sfage

Sep. Oct.

- ~ - ~ ~ ~ . . . ~ ~~ ~ ~ ~ ~~~.

260 8 Monitoring of Structures 8.4 Exercises 261

tunnel provided about 12 hours of quiet time for monitoring after the initial perimeter drilling and about 12 hours of quiet time for monitoring after mechanical breaking of therock stub. This provided a total ofabout24 hours of monitoringperround ofhlnnel advance.

Figure 8.13 Location of the Mine-by test tunnel and the microseismic triaxial accelerometers (after Martin and Young, 1993).

4.6 m diameter -

Some 25,000 events were source located. Inspection ofall 46 rounds showed similar trends. Figure 8.14 shows the 47 microseismic events recorded over a 10-hour period immediately after the perimeter drilling was completed. At this point the events do not show strong clusterin& although there is a slight grouping of events where the first breakout eventually observed. After the rock stub was removed, 52 new rni- croseismic events were recorded during a 16-hours monitoring period (Figure 8.15). These events show strong clustering in the roof, particularly where the breakout eventually occurred. It would appear that the concentration of events is defining the region where the breakout geometry will appear Another feature ofthe excavation rounds investigated is the induced seismicity occurring ahead of the tunnel face (Figure 8.15) Presumably this damage is occurring because of the stress concentrations caused by the flat face.

8.4 Exercises I Which monitoring should be canied out in mining rock mechanics? Describe reasons for the monitoring.

2 Which technique should be usedfor monitoring displacements occurring within the rock mass away from the excavation periphery? Describe, in detail, the technique

3 What is the technique for monitoring stress and stress change in material?

4 What are the frequencies of waves radiated by events associated with mining activity?

5 Describe the two forms which are used for microseismic activity/acoustic emission monitoring.

6 What are the points, youmust pay attentionto, in the design of an acoustic emission monitoring?

Figure 8.14 Location of microseismic events at the end of the perimeter drilling of Round 13. Note the slight clustering of events in the roof where the first breakout eventually appeared (after Martin and Young, 1993).

-. i ..c ...... : . .. Section

vutukuri katsuyama introduction to rock mechanics

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