The Interpretation of Geologkal Factors for use in Slope Theory By A. >lacG. ROBERTSON*, B.Sc.(Eng)(Rand) SYNOPSIS Joints and other geoloyical discontinuities influence the strength. deformational and permertbility character- istics of a rock mass. They are detectable features and their orientation, spatial distribution and surface properties may be measured in a field survey. The survey data can be processed and analyzed to determine the nature of the joint population of the rock mass and zones of similar.properties or structural regions delineated. A model representing the jointing in a rock mass is constructed and may be used to describe the nature of any failure surface that forms in the mass. Apparent strength parameters describing the strength along such potential failure surfaces, are postulated and a process for estimating the value of these parameters, from the data obtained in the field survey, is developed. I. INTRODUCTION In order to perform an analysis leading to a numerical factor of safety for a slope cut in a rock mass it is necessary to know the properties of the rock mass in numerical form. Of particular interest are those properties relating to shear strengths. deformations and water pressure distributions in the slope. These properties are largely controlled by the geological discontinuities within the mass and, therefore, any such descriptions must take due cognisance of the orientation, ’ spatial arrangement and properties of the discontinuities. Rock mass properties may be determined in the following ways: (a) Large scale /ie/d fesfs which are of such a size that the erects of all geological discontinuities are accounted for. (6) Small sea/e jeie(d rests which cover the effects of smaller geological discontinuities but where the effects of larger scale discontinuities must be assessed by modeling or other means. (c) Direct measurement of the discontinuities and their properties and modeling of the rock mass to determine its overall behaviour. Method (a) would produce an exact answer but the size of test required renders it impracticable, or at best, extremely expensive. Method (b), although less costly, is only suitable where the rock mass of interest has relatively homogeneous properties: in this method reliance must still be placed on the engineer’s ability to model the strength, behaviour characteristics or other properties of the large geological features not accounted for in the field test. Method (c) relies entirely on the engineer’s ability to model the mass behaviour from the properties of the rock material and the geological discontinuities. It presumes that an accurate numerical description of the orientation and spatial arrangement of the discontinuities and the nature and properties of the discon- tinuities and intact rock material can be achieved. In con- structing the model, the engineer must make use of all available information including his own laboratory tests and other suitable results published in the literature. This paper accepts the last method as the one most likely to provide a useful design approach. It outlines a survey technique and data analysis procedures to enable the engineer to determine rock properties both of the discontinuities and of the intact rock. A basis for the model representing both joints and intact regions within the rock mass is presented and a process is developed in order that apparent strength parameters along a potential failure plane in a rock mass, may be estimated. This information is required for the development of a model representing potential planes of failure and for the synthesis of a rock slope stability analysis method given by Jennings”. This approach is based on four of the five geological propositions made by Piteau”, viz: (i) Thar structural di.tcontinuities are detectable features of a rock mass and cat1 be described quantitatively. The exact nature of the rock mass must be known before it can be modelled. Intact rock properties are relatively easily measured and described. The nature and properties of the geological discontinuities, termed joints in this paper, are however more difficult to describe quantitatively. Where these properties are not readily described in this manner it is necessary to resort to qualitative descriptions, but in all cases it has been found possible to assign quantitative values to the effects that these properties have for use in the stability calculations. (ii) That structural regions exist in a rock mass. The model, which may be either physical or mathe- matical, describes but a part of the rock mass. It is only representative of those regions in the rock mass in which the properties of the joints and intact material are similar to those modelled. It is therefore necessary to be able to divide the rock mass into regions of similar properties or slrrrcflrral regions. (iii) That a reliable model representing jointing of a rock mass can be constructed. The spatial arrangement and properties of the joints in a rock mass will influence the nature of the failure planes which form in it. Hence the accuracy with which the model may describe the failure surfaces depends on the accuracy with which it models the jointing. (iv) That failure surfaces will be essentially plane or com- binations of planes. Since geological discontinuities are generally planes of weakness in the rock mass, a potential failure surface will tend to form, preferentially, along them. This implies that a failure surface will consist of planes or combinations of planes formed by the joints in the rock mass. As joints commonly occur in joint Sets which have a common origin and hence similar properties, including joint orientation, a failure surface is often associated with such a set of sub-parallel joints. *Formerly Technical Assistant, de Beer’s Consolidated IMines, Ltd. Presently Design Engineer, Frankipde South Africa (Pty.) Ltd. 55
Transcript
The Interpretation of Geologkal Factorsfor use in Slope Theory
By A. >lacG. ROBERTSON*, B.Sc.(Eng)(Rand)
SYNOPSIS
Joints and other geoloyical discontinuities influence the strength. deformational and permertbility character-istics of a rock mass. They are detectable features and their orientation, spatial distribution and surfaceproperties may be measured in a field survey. The survey data can be processed and analyzed to determine thenature of the joint population of the rock mass and zones of similar.properties or structural regions delineated.
A model representing the jointing in a rock mass is constructed and may be used to describe the nature of anyfailure surface that forms in the mass. Apparent strength parameters describing the strength along such potentialfailure surfaces, are postulated and a process for estimating the value of these parameters, from the data obtainedin the field survey, is developed.
I . I N T R O D U C T I O N
In order to perform an analysis leading to a numerical factorof safety for a slope cut in a rock mass it is necessary toknow the properties of the rock mass in numerical form.Of particular interest are those properties relating to shearstrengths. deformations and water pressure distributions inthe slope. These properties are largely controlled by thegeological discontinuities within the mass and, therefore, any
such descriptions must take due cognisance of the orientation,’ spatial arrangement and properties of the discontinuities.
Rock mass properties may be determined in the followingways:
(a) Large scale /ie/d fesfs which are of such a size that theerects of all geological discontinuities are accountedfor.
(6) Small sea/e jeie(d rests which cover the effects of smallergeological discontinuities but where the effects of largerscale discontinuities must be assessed by modeling orother means.
(c) Direct measurement of the discontinuities and theirproperties and modeling of the rock mass to determineits overall behaviour.
Method (a) would produce an exact answer but the size oftest required renders it impracticable, or at best, extremelyexpensive. Method (b), although less costly, is only suitablewhere the rock mass of interest has relatively homogeneousproperties: in this method reliance must still be placed onthe engineer’s ability to model the strength, behaviourcharacteristics or other properties of the large geologicalfeatures not accounted for in the field test. Method (c) reliesentirely on the engineer’s ability to model the mass behaviourfrom the properties of the rock material and the geologicaldiscontinuities. It presumes that an accurate numericaldescription of the orientation and spatial arrangement of thediscontinuities and the nature and properties of the discon-tinuities and intact rock material can be achieved. In con-structing the model, the engineer must make use of allavailable information including his own laboratory tests andother suitable results published in the literature.
This paper accepts the last method as the one most likelyto provide a useful design approach. It outlines a surveytechnique and data analysis procedures to enable the engineerto determine rock properties both of the discontinuities andof the intact rock. A basis for the model representing bothjoints and intact regions within the rock mass is presentedand a process is developed in order that apparent strengthparameters along a potential failure plane in a rock mass,may be estimated. This information is required for thedevelopment of a model representing potential planes of
failure and for the synthesis of a rock slope stability analysismethod given by Jennings”.
This approach is based on four of the five geologicalpropositions made by Piteau”, viz:
(i) Thar structural di.tcontinuities are detectable features ofa rock mass and cat1 be described quantitatively.The exact nature of the rock mass must be knownbefore it can be modelled. Intact rock properties arerelatively easily measured and described. The natureand properties of the geological discontinuities, termedjoints in this paper, are however more difficult todescribe quantitatively. Where these properties arenot readily described in this manner it is necessary toresort to qualitative descriptions, but in all cases ithas been found possible to assign quantitative valuesto the effects that these properties have for use in thestability calculations.
(ii) That structural regions exist in a rock mass.The model, which may be either physical or mathe-matical, describes but a part of the rock mass. It isonly representative of those regions in the rock massin which the properties of the joints and intactmaterial are similar to those modelled. It is thereforenecessary to be able to divide the rock mass intoregions of similar properties or slrrrcflrral regions.
(iii) That a reliable model representing jointing of a rockmass can be constructed.The spatial arrangement and properties of the jointsin a rock mass will influence the nature of the failureplanes which form in it. Hence the accuracy withwhich the model may describe the failure surfacesdepends on the accuracy with which it models thejointing.
(iv) That failure surfaces will be essentially plane or com-binations of planes.Since geological discontinuities are generally planesof weakness in the rock mass, a potential failuresurface will tend to form, preferentially, along them.This implies that a failure surface will consist ofplanes or combinations of planes formed by the jointsin the rock mass. As joints commonly occur in jointSets which have a common origin and hence similarproperties, including joint orientation, a failure surfaceis often associated with such a set of sub-paralleljoints.
*Formerly Technical Assistant, de Beer’s Consolidated IMines,Ltd. Presently Design Engineer, Frankipde South Africa (Pty.) Ltd.
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2 . SURFACES OF FAILURE
Rock slopes may fail in a number of direrent modes(Jennings’)- these involve concepts of potential failure alonga plane or combination of plane surfaces, formed largelyalong the discontinuities or joints. Two basic forms illustratedin Fig. 1 are recognised.
(iii)
(iv)
(4
the properties of the joints and joint surfaces thatoccur on the failure plane,the properties of the intact rock material on the failureplane, andthe effects that each of these properties have on thestrength of the failure surface.
Fig. 1. Plane and stepped failures
Plane/uilure u b occurs in the plane a b which contains morethan one joint. It applies particularly to geological featuresof great continuity such as beddings or faults and involvesshear failure for the entire surface. Where joints occur alongsuch continuous features, the proportion of joint surface tointact surface may be determined by survey and analysistechniques as described by Miiller” and Pacher”.
Sfepped fuilure c d occurs along a combination of jointswhich scatter abouf the plane c d. They are associated withfailure through homogeneous rock types such as igneous andmassively bedded sedimentary rocks, or with stratified rocktypes where failure is associated with cross joints which mayor may not act in combination with beddings or beddingjoints. Stepped failure surfaces may be observed in manypractical cases of failure: however this may not always havebeen recognised as such.
Neither of these failure surfaces are necessarily planar.An example of a failure surface that would be curved on asection parallel to the slope crest is illustrated by failure alongcontorted bedding planes which plunge out of the slope. Anexample of a failure surface curved m any section throughthe slope is illustrated by failure e d (Fig. 1). The figureindicates the possible selective nature of a stepped failuresurface which selectively chooses cross joints which fail intension at the top of the slope, and joints which fail in shearin the compression zone at the toe of the slope.
Fig. 2 represents a three-dimensional idealized illustrationof the stepped failure surface from which it is apparent thatthe failure surface exhibits both tension and shear failurethrough joints and intact rock. Before the strength along anypoten;ial failure plane can be determined, the proportion ofjoint and intact rock surface as well as the proportions ofthe types of failure (tension and shear) on each must beknown. These proportions may be determined from:
(i) the orientation, spatial distribution and size of jointsin the various joints sets that occur in the rock mass,and
(ii) the properties of the model chosen to represent thegeometry of the failure surface that would occur inthe rock mass.
Knowing the composition of the failure surface the followingadditional information must be available before an assessmentcan be made of the strength along the failure surface:
ESY Shror failure alongjolt-31 surlorr
Fig. 2. Block diagram ojstepped faiture surface
The average joint properties (items (i) and (iii) ) for thejoints of a set define the average or design joint. These averageproperties are obtained quantitatively from an analysis ofjoint data obtained in a field survey of the joints occurringin the rock mass.
The quantitative desi-en joint data from item (i) is intro-duced into a mathematical model (item (ii) ) to calculatequantitatively the composition of any given potential failuresurface. Since the design joint properties (item (iii) ) and theintact rock properties (item (iv)) have been obtained, theeffects that each of these properties has on the strength of thefailure surface, may be measured in laboratory tests (item(v) ). Knowing the effect on strength of these properties andthe composition of the failure surface, an estimate may bemade of the total strength along the failure surfaces.
Jennings’ develops a model to represent the failure surfaceand this aspect is not considered further.
3 . T H E J O I N T S U R V E Y
The third geological proposit ion (Piteau”) stating thaf ureliable model represenring joinling of a rock mass can becomrructed assumes that the nature of the joints and their
-spatial distribution is known or can be determined. Since itis impossible to examine each and every joint in the massthis information must be gained from o sumpIe ojfhe jointswhich are selected in a joint sirrvey during which the variousproperties of interest can be measured.
These samples must be sufficiently large to ensure that theinformation determined from them is sufficiently accuratefor the third geological proposition to be justified. Where thejointing patterns difTer for various regions (i.e. structuralregions) the sampling must be sufhciently extensive to permitthe detection and definition of joint sets and the definition ofjoint properties for each of these structural regions. Thesample data required may be large, as illustrated by the
de Beer’s mine survey where 9 000 joints with the measure-ment of 14 features of interest on each joint were required todetermine the joint population for only one lithologicalstrata type for approximately half of the mine circumference.
The geologist undertaking the. joint survey will haveintimate contact with the rock mass and long hours spentobserving and recording joint data will give him a goodunderstanding of the properties of the rock mass and howthese may vary from place to place. His assessment of struc-tural regions, or regions likely to cause instability, and ofthe effects of other geological features are of utmost impor-tance.
Field surveySampling may be conducted on exposed rock faces of
various forms. If the surface is large the number of jointsexposed may also be large and some form of selection may;have to be applied to reduce the sample size. Techniques,which rely on the geoIogirt’s judgment for recognising thejoint sets of importance can greatly reduce the volume ofdata and al low the effor t to be concentrated on theapparently significant joint sets, but there is always the riskof missing or discounting sets which are nevertheless ofconsiderable importance. This risk is greatly reduced whenusing sampling techniques which sample all the joints that
i intersect a face of limited size or line of limited length. Linejsanrp/itrg, in which all the joints which intersect a given linej are sampled, has the advantage that the mathematics for thedata analysis is simpler and hence more easily extended thanthat for area or surface sampling. For a sampling face orline of given length the latter yields a smaller sample andhence is more economical to implement where extensivesurfaces are available for sampling. A more detailed discus-sion of the method of sampling is presented by Piteaur’ andthe accuracy of area (surface) and line sampling is further
considered in Section 7.The bulk of data in measurements of particular properties,
such as roughness on a joint surface and the complex nature’ of many of these properties. make it necessary to resort toclassitications which are qualitative rather than quantitative.
i Since numerical values are required for use in the stability! calculations, it is necessary to determine quantitatively, bylaboratory testing. the ef%cts that each qualitative classifi-cation has on the strength along the joint. Such testing permitsthe assignment of quantitative values to describe the effectson strength of each class interval of the qualitative classiti-cation of properties used in the field.
The strength along a potential failure surface is primarilycontrolled by its orientation and position in relation to theslope under consideration and by the amount of intact rockthat must be sheared through. Since the size, number, spatialorientation and distribution of joints in the rock massdetermine these factors, these are the joint properties whichare of primary interest. The joint features that determinethese properties should be measured with as great an accuracyas possible. All other features may be coarsely measured bycomparison. Hence, dip and dip direction angles and jointtrace lengths should be individually measured. The otherproperties are grouped into classifications of tive class inter-vals (five finger classification) wherever possible. Such agrouping has been found to be readily followed in the fieldby staff with a minimum of training since the two extremevalues, the middle and the two intermediate values, soonbecome apparent.
Sampiing considerarionsJoint surveys are conducted on rock faces associated with
outcrops, exposed slope faces, trenches, tunnels, shafts, borehole sides and bore hole cores. In all these cases the areasampled is limited by physical limitations, with the area
5
available decreasing in the order mentioned. The jointsmeasured (the sample), which may only be a portion of thejoints exposed (the sampled population), are considered to berepresentative of the joints within the entire rock mass (thetarget population) (Krumbein”‘). Each of these survey facesmay produce biased samples (Terzaghi?‘). Outcrops, as theyoccur, may be controlled by differences in resistance toweathering of the underlying bedrock which includes effectsof jointing. The creation of faces, tunnels and bore holecores disturbs the surface to be sampled, and distinguishingbetween natural and artificially induced fractures may bedifficult. Soft gouge is lost during core recovery and infor-mation relating to the continuity of joints is minimal in borehole surveys. Where economically possible it is desirable touse trenches, slope faces or tunnels which allow careful directvisual inspection of the joints on a fairly substantial face,at depth, in the desired lithological strata where the surface‘effects of weathering are no longer appreciable.
Since the sample may be taken from the sampled popula-tion by any sampling process, inferences can in general bemade on a rigorous basis from sample to sampled population.Any extensions that are made of these inferences to the targetpopulat ion are judgment decis ions on the par t of theengineer. Such an extension does not seem unreasonableprovided sampled and target populations are within the samestructural region. Unfortunately analysis of the sampled datacan only define the points at which the sampled populationchanges, i.e. the structural region boundaries to the sampledpopulation. The extent and limits of these boundaries aboutthe target population remains a judgment decision.
Errors in survey dota
Errors in data collection are of two forms: error of propertymeasurement and error of sample selection. The mostsignificant measurement errors are those associated with theangular measurement of dip and strike. These errors varywith the inclination of the joint, the strike becoming indeter-minate as the joint tends to horizontal (Pincusl’). Re-exposureand re-measurement of joint sets which were of consequenceon the side slopes of Bomvu Ridge mine, Swaziland, indicatedthat the maximum errors that could reasonable be expectedin the measurement of dip and dip direction angles were5” and 10” respectively (Jennings-personal communication).An examination of the de Beer’s results appears to indicatea similar possible error range but no detailed analysis hasbeen made.
Joint trace lengths can be accurately measured providedthe point at which the trace terminates can be accuratelydefined, but such trace lengths are not necessarily a measureof joint size. The errors in classification of other jointproperties, e.g. hardness, should not exceed one class interval.
Errors due to selection are numerous, and some of themost significant are: small joints are often disregarded; verylarge fracture surfaces may be measured more than once;fractures almost parallel to foliation or bedding may beoverlooked; details of sample selection may vary betweenpersonnel; the direction of sampling results in a sample bias(TerzaghiL’).
4 . J O I N T S A M P L E D A T A - M A N I P U L A T I O NA N D A N A L Y S I S
Joints are characterized primarily by their orientation inthree-dimensional space. This is defined by two angles, thedip and dip direction. The dip direction is defined by thehorizontal angle 0 between north and the projection of theline of dip, and varies between 0” and 360” referenced clock-wise positive. Dip is defined by the vertical angle 8 betweenthe horizontal and the line of dip and ranges from 0” to 90”.The only way in which the orientational distribution of
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joints can be described with uniformity of all features is on areferenced sphere. Since it is desirable to present this infor-mation on a planar surface some form of projection is used withthe loss of some of the basic data relationships.
The use of Wulff and Schmidt projections (John’) arecommonly made for the presentation of joint survey data(Friedman’) and has been adapted for computer printout(Rosengren”‘). The use of these stereo plots is a powerfultool in the analysis of jointed rock mass since they preservesome of the properties of the geometric relationships betweenthe joint planes.
The rectangular plot is formed by a cylindrical equal-spacedmeridional projection as depicted in Fig. 3 (Pincusis. 1’).In its final form it is rectangular and is particularly suitablefor computer construction (Broadbent’). In the form usedhere it is a projection only of the upper hemisphere with aconvent ion as i l lustrated in Fig. 4 . The nature of theprojection is shown in Fig. 5, and Fig. 6 provides a c o m -parison of the rectangular and stereographic plots. The formof the printout developed for the de Beer’s project is givenin Fig. 7. Any figure on the table represents the number ofjoints with dip angles which are equal to or less than 5”greater than the value of the row in which it appears andwhich have dip direction angles equal to or less than 10” lessthan the column value in which it appears; the column valuesare those at the top of the plot for the upper half of the plotand those at the base for the lower half of the plot. In Fig. 7for example the single joint circled has a dip direction angleof between 240” and 250” and a dip angle of between 40”and 45”.
Fig. 5. The physical inrerpretalion of the rectangle plot
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Fig. 6. Comparison of the stereo attd rectangle plots (offer Pincus’s)
The use of a plot such as the rectangular plot, that can beproduced by a computer and which displays the data in aform in which it can be rapidly assessed by visual inspection,considerably increases the volume of data that can be pro-cessed.
Fig. 7 displays only the uncorrected data as sampled. Thesedata are not truly representative of the joint population anda correction must be applied to allow for the sample biaswhich results from the relative joint and sample line attitudes.This bias results in a greater number of joints which arenormal to the sample line being sampled, while joints exactlyparallel to the line are missed entirely. The data in any classinterval must therefore be corrected to give the number ofjoints which a line, equal in length to the sample line, would beexpected to encounter had it had an orientation normal tothe plane of the joints considered. Referring to Fig. 8, if thesample line defined by dip and dip direction 61 and Or wasused, and if a number N6#, joints fall into the class intervaldefined by midpoints 6~ and 01 then the corrected valueN’6fej would be
N’Spvj =NV9
.’cos le, - e,1 c o s I(& + 6j) - 901..(I)
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DESIGN JO INT SURYEY FOR DE BEERS XINEUF ALL .lOKNTS
Fig. 8. Three dimensional represenlation of normal IO joint set andsampling direction
Once the correction has been applied to each class intervalof the rectangular plot that plot may be compared with othersample bias corrected plots for the qualitative way in whichthe joints are distributed. It is considered that where thecorrection is greater than 5 the results are no longer reliableand an asterisk is printed against the corrected number. Thedata in Fig. 7 corrected for sample bias is shown in Fig. 9.
(hrnp,= ,ine is horironu, -It* dip d i r e c t i o n of 90°1
Fig. 9
The effect of the correction of the data shown, is to increasethe apparent density of jointing with dip directions in theregion of 90” or 270”, i.e. normal to the direction of sampling.Instead of showing a distinct joint set centred about dipdirection 350” to 10” as in Fig. 7, the plot in Fig. 9 indicatesa more general dispersion of near vertical joints with a lesserconcentration of joints about dip direction 0”.
To facilitate comparisons between surveys of differentlengths (L) a correction to a standard survey length (15s) isrequired.
In order to compare the density of jointing in the variousjoint sets quantitatively, a correction must be applied to allowfor projection errors: _
N” sjejN”‘Sj0, = ~ . . . . . . . . . . . . .._
cos sj
The corrected rectangular plots may be rapidly construc-ted on the computer and have proven to be extremely usefulfor the visual definition of joint sets, while the comparisonof different plots indicates whether the sampled data on eachwas from the same population, i.e. the same structural region.
5 . D E T E R M I N A T I O N O F S T R U C T U R A LR E G I O N S
Many workers in the field of the mechanics of jointed rockrecognise that a rock mass may be divided into zones of‘similar jointing’ or structural regions. Since jointing to a
with the earlier parts (i.e. by folding away the suspectedtransition zone). Adjacent sections of tunnel may be joinedtogether but where the bearing of the sample line changesby between 60” and 120” it is difficult to compare the res-pective sequential plots since the area of one, which showsadequate sampling, is the poorly sampled area of the other.Each portion of survey line, or data set, between apparentchanges in jointing pattern and changes in survey line bearing,where uncertainty exists, is termed a section. A set of correctedrectangular plots may be made for each section and theseare compared for differences in jointing patterns. Figs. 11 and12 are two such samples taken from survey lines at rightangles to each other. The similarity between these plotsindicates that they are from the same structural region.Comparison of these figures with Fig. 9 indicates that thedata represented by Fig. 9 was sampled in a different struc-tural region.
large extent controls the mechanical behaviour of rock, thesestructural regions are zones of ‘similar strength’ i.e. they are,in a statistical sense, homogeneous with regard to strength.This requires that jointing in a structural region must benot only similar throughout with regard to dip, dip directionand intensity of jointing, but also similar in such factors asroughness, waviness, thickness, type of gouge and the otherfactors which determine the strength along joints.
Initial demarcation of boundaries to structural regions isgiven by the geologist on the basis of his experience andknowledge of the jointing patterns gained from visual inspec-tion of the various exposed rock surfaces. They usuallycoincide with major geological features.
The use of graphical methods of joint representation suchas the stereo plot and rectangular plot is extremely laborious,and the accuracy of determination of structural regionsboundaries depends on how quickly the person plotting candetect a change of pattern on the plot. To overcome thislimitation a sequential plot of the form illustrated in Fig. IOis used. The central column of the computer printout givesthe joint identification, referring to the joint survey sheetnumber and the line on that sheet which contains the jointdata being plotted. The left hand portion of the printoutgives a plot of the angle of dip direction, while the right handportion shows the dip of each joint. The class intervals are5” wide and joints with dip direction between 0” and 180”plot as X’s and fall in the left hand half of the dip plot whilethose with dip direction of 180” to 360” plot as O’s and fallin the right hand half of the dip plot. Faults or majorgeological features are indicated by special symbols. The plotsrun sequentially, without break, for the full length of allsurvey lines. The direction of the horizontal normal to thesurvey line bearing is drawn on the dip direction plot toindicate the angular region where sampling is inadequate.This region of poor sampling in the vicinity of the linerepresenting the normal to the tunnel bearing is apparentin Fig. 10.
S E C T I O N 4Lsrruc,ura, Rcpron El
DIP DIRECTION A C R O S S A N D D I P A N G L E DOWN0.0 20.0 uo.0 60.0 80.0 100.0 120.0 140.0 160.0 160.0
In general corrections seem excessive, indicating that thesampling is not quite as biased as the mathematics predict.Intensities of jointing may be compared on the plots correctedto standard lengths of survey line. If there is any doubtabout the variation of the joint properties of a joint set ina structural region, then the various properties of the jointsfor different parts of the region may be separated out, asdescribed later, and compared for any differences,
For de Beer’s mine the geologist indicated ten structuralregions in the volume of interest. After survey data analysis,thirteen regions were defined with relatively few changes tothe boundary positions originally indicated by the geologist.
Fig. i0. Initial portion of a seylrential plot
The sequential plots, which are layed out on the floor orany other large flat surface, are examined visually and thepoints at which there is a consistent change in trend of the dipdirection and the dip angles is noted. Fig. 10 is only theinitial portion of a plot and of insufficient length to indicateany change in the trend. Often such changes take placegradually, through a transition zone, and in such cases itmay be necessary to compare the latter parts of the plot
60
The follo\c ing general conclusions were reached:
(i) Structural regions or volumes of rock mass in whichthe jointins population may be described as havingsimilar joint properties, orientations and intensity,can be delineated.
(ii) Transition from one zone to another may range fromgradual to abrupt.
(iii) In most. but not all cases, boundaries appear tocoincide with major geological features.
(iv) Large geological features may have associated withthem large zones in which the fracture pattern differsfrom the fracture patterns of the adjacent regions.In some instances, however, the effect of the geologicalfeature is evident for only a few inches on either sideof that feature.
(v) The greater the shear movement along a fault, thegreater the effect on the fracture pattern adjacent tothat fault.
(vi) Fracture pattern disturbances are greater for faultsthan for dykes.
(vii) At de Beer’s mine most of the structural regionboundaries radiate from the pipe.
structural region. The data for each subset are fed into thecomputer. Uncorrected as well as directional bias, standardlength and projection error corrected rectangular plots arethen made. The values reflected on all the plots of a similartype (i.e. having similar applied corrections) for each struc-tural region are then summed and accumulated plots made.The most useful of these plots is the accumulated plot for thestructural region in which the subsets of data have beencorrected for directional bias and to standard lengths. Pro-
-vided at least two approximately perpendicular sections ofsufficient length were sampled in a structural region thiscumulative rectangular plot should be representative of thetotal joint population in the structural region. Such acumulative plot is shown in Fig. 13.
Joint sets are defined from the cumulat ive plots byinspection. To aid this inspection the plots may be contouredaccording to density of joints. For convenience of describingthe range of dip and dip direction angles of the joints in aset, the boundaries of the sets are shown as straight lines(see Fig. 13). The other plots indicate the effects of the variouscorrections and may influence the definition of joint sets.When defining joint sets the engineer should bear in mindthe direction of the slope of interest, and should exerciseconservatism when defining joint sets in the area representedby dip and dip direction angles which may form planes off a i l u r e .
Joint sets about de Beer’s mine were characterized bylarge dip and strike variations. Where these variations weregreater than 60” on strike and 30” on dip the set was treated,for design joint determination purposes, as two or more
6 . D E T E R M I N A T I O N O F JO[NT S E T S A N DT H E I R P R O P E R T I E S
The data are divided into data sets for each structural region.Since sampling bias correctlons are different for differentsample line bearings, the data sets are sorted into subsetsfor the various straight lengths of sample line within each
61
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‘overlapping’ sets, each with these maximum variations ondip and strike angle. The midpoints of each of these over-lapping sets were arbitrarily defined with due considerationfor the slope faces in the vicinity and for any potentialfailure surfaces.
A design joint is defined as a joint which has propertieswhich are the average properties for any particular joint set.It will have dip and dip direction angles equal to those of themidpoint of the joint set, and all joints in the set which liewithin the dip and strike variation mentioned above areconsidered in determining the design joint properties.
It is convenient to plot the defined joint sets on rosettesof the form illustrated in Fig. 14 using the-data from therectangular plots. The dip direction is represented by thebearing of any radius, clockwise positive, with the uppervertical as bearing 0”. Dip decreases along the radius from90” at the centre of the rosette to 0” at the outer edge. Shownin Fig. 14 are the joint sets defined for four of the thirteenstructural regions about de Beer’s mine. In some instancesthe angular range of the joint sets as defined are large but inmany of these the majority of the joints were contained ina region of dense jointing of much smaller angular variationnear the centre of the defined set.
In determining the properties, other than dip and dipdirection, of the joints in the various sets. e.g. roughness ofthe joint surfaces. the data for each structural region isprocessed as a whole. The data ‘are sorted into rectan+arplots according to the various properties of the joint, r.e. ajoint having roughness 2 (see Piteau”) will be allocated to aplot in which all joints have roughness 2; similarly for allthe properties required. From these plots the average proper-ties of the joint sets are determined. The properties determinedfor the joint sets about de Beer’s mine are shown in Table 1.
Examples of the distribution of joint trace lengths obtainedare shown in Fig. 15. The method of joint irace length recor-ding adopted, using a category system for recof_ding theobserved lengths, results in some dificulty in the placing ofjoints in class intervals and only four classes were defined.Since traces may be continuous into walls, roof or floor ofthe tunnel some of the joints in the larger class interval maynot have been placed in the correct category. A method,whereby more exact trace lengths or more class intervalsmay be obtained, should yield a more accurate distribution.
ium, 0in lnsti’
gnd he
s o m e : ;
Id expt
ady de
ining a
eeper F
rital thl.
anning:early d
imined.;orum w’
Y and Iies cou
d problt
.tion of ’>osiumeers in\5peCt O f
REGION 0 JOINT SET b
n- 320- observed
a---a- f i t t e d d = 03L3
S T R U C T U R A L R E G I O N I
Trace Length ttt.1
REGION B JOINT SET a
S T R U C T U R A L R E G I O N K
‘RUCTURAL R E G I O N LSl
4.0 1 2 3 L 3 6
S T R U C T U R A L R E G I O N MTrace t ength trt.j
Fig. 14. Roserres represetrfitrg voriafion of joints sets Fif. 15. Disrriburion of joint trace lengfhs in a joktt set
62
.
Various functions were tried and it was found that anexponential function of the form
appeared to be the most suitable. In the distributions illus-trated the tit of the upper curve is considered to be good andthe lower poor. Of the joint sets for which trace lengthdistributions were plotted, all except for two could be fittedwith exponential function curves having good fits.
Determination of joint sizes, from length of dip and striketraces visible in the tunnel, is possible only if the generalshape of the joints is known. Bivariate plots of dip tracelength against strike trace length indicated that, in the homo-geneous igneous rock of de Beer’s mine, these were approxi-mately equal (Robertson and Stamerrg), and joints wereconsidered to be round. Calculation of the joint sizes of cir-cular joints from trace lengths underestimates the joint sizesin the relationship (Robertson and Stamer”P):
where A’ is the joint area calculated from the visible tracelengths. From the fitted exponential curve and the aboveconsiderations the average joint size, Lf, of a set and thesize of the largest joints, Lfl, in that set that are likely tooccur in a slope of given dimensions may be calculated.The size of the largest joints that were likely to occur in theiide slope of de Beer’s mine was arbitrarily defined as the
Structuralreg.
A
B
C
D
E
! FI
loint %set Gouge
a
a
a
a
23
23
I
/ G,
H
I
K
L
M
size of the joint which is expected to occur only once on a300 ft length of sample line taken normal to the plane of thejoints in the set considered.
The average distance between the adjacent joints of a set(d”,) is determined from equation:
I cos (6d) cos (ed)dm =
Iv. . . . . . . . . . . . . . . .._ (6)
where N is the number of joints from the joint set sampledon a survey line length f. 6d and ed are the dip and dip
_ direction angular differences between the sample line and thenormal to the joint set.
i
I
Since some of the joint sets may have developed simul-taneously as a result of a general regional stress, they mayhave similar characteristics although they occur in differentstructural regions. To test this hypothesis comparisons weremade between trace length distributions for joint sets indifferent structural regions but with similar orientations.The Kilmogorov-Smirnov test for two samples was usedunder the null hypothesis that the two observed trace lengthsamples came from the same population. These tests indicatedthat although some of the samples could have come from thesame population, the majority were from different popula-tions. The results of these tests supported conclusions reachedwhen comparisons of (L, Lf and Ljl were made. It was con-cluded that the majority of the joint sets must have developedunder different conditions of stress.
The various joint set characteristics taken from the rec-tancular plots for the various s t ructural regions aboutde Beer’s mine are shown in Table 11. Tables I and 11,together with the average dip and dip direction angles forthe various overlapping sets, define the design joints aboutde Beer’s mine.
7 . T H E A C C U R A C Y O F J O I N T P O P U L A T I O NE S T I M A T I O N F R O M A S U R V E Y S A M P L E
The most critical factor determining the strength along a givenpotential plane of failure is the continuity of joint surfaceon it. As stated earlier this continuity is determmed largelyby the size and spatial arrangement of the join&s of the setwhich occur on the plane and in particular by the densityof joints in the set. The accuracy of the determination ofthese continuity factors depend on the accuracy with whichthe joint characteristics, including density, can be determined.It is therefore necessary to be able to make an assessmentof the rel iabil i ty that can be placed on the joint setproperties determined.
The extension of inferences, regarding the sampledpopulation, to the target population and the determinationof the accuracy of any estimates made, require either theknowledge or an assumption of the spatial distribution ofjoints in the rock mass. A model in which joints occur in aspatial arrangement which is anisotropic and homogeneousin a srarisficaf sc~zse is accepted here. This is distinctly differ-ent from assuming a spatial arrangement in which the jointsoccur in a regular pnt/ern which is anisotropic and homo-geneous. The difference between these spatial arrangementsis illustrated in Fig. 16.
On this model, describing the spatial distribution of joints,is based the more extensive model describing the entirejointpopulation. In order that the third geological proposition,which states that a reliable model representing jointing of arock mass can be constructed, can be satisfied this modelmust be a realistic representation of the spatial arrangementof joints in the rock mass considered. The statistical dis-tribution of the first mentioned spatial arrangement is con-sidered to be representative of jointing in homogeneous rockand of jointing other than joints in the plane of the beddings
Lj is the average Joint length m a setN’ is the corrected sample size,511 is the length of the joint which is expected ro occur once in every 300 ft length taken normal to a joint setdm IS the average distance between adjacent Joints in a set
64
in stratified rock types. These joints form failure planes ofthe step type and this model is accordingly suitable fordetermining strength parameters and behaviour character-istics for failure planes of this type. For plane failure surfacesassociated with failure along single planes of weakness suchas bedding planes, which contain bedding joints, a modelwhich permits more than one joint on any single plane mustbe used. A survey and analysis technique to suit such a modelhas been described by Miiller” and Pacher’?.
Spatially Honrogenous andAnisotropic Jointing in aStatistical Sense.
If joint sets are considered as data sets within a jointpopulation, then most forms of sampling will show a ten-dency to sample more of some sets than others. Hence, it isnecessary to determine the accuracy of estimation of eachset individually. A more detailed consideration of the assump-tions and limitations inherent in this model is given byRobertson”.
Projected joint
Fig. 17. Illustration of end projection
Fig. 17 shows an arbitrary defined rock mass with end face,normal to the sampling line, of area A and having a length Lin which there are n randomly distributed joints of averagesize am of a particular set. All joints intersecting a surveyline passing through this volume would be. sampled; say thisnumber of joints is ln, the subscript 1 indicating a line survey.The average area of projection of a joint on the end facewill be
(I’m = Um CoS ed . cos 8d . . . . . . . . . . . . . . . . . . (7)
w h e r e 4d and 8d are the horizontal and vertical angulardifferences between the normal to the joints set and thesample line.
6.5
The probability of any joint being intersected by the sampleline, which is normal to the end face, is on the average
,
lP = a5. . . . . . . . . . .._.__.__._....
provided a’,,, < < A,
Since the joints are randomly distributed, the probabilityof intersecting ln joints follows the binomial law, viz.
: $;-t..
P m = ; 1P0
1” n - ,”
. 14 . . . . . . . . . . . .
mAn unbiased est imate of n is n^ = -
lP
ln . Azz . . . . . . . .
am . cos 8d _ cos f&. .(lO)
And the density of jointing is given byA
m
L _ am cos 8d . cos t&J
-. . . . . . (11)
The variance of the estimate 2 is
lVnr(i) = (-L)I.R($) . . . . . . . . . . . . . .
and the coefficient of variance (C. V.) is given by
@ar (fi) x ,ooyrC.v. (%) = v 0D
-=
J; x 1 0 0 % . . . . .
.w
. . . . . . . (13)
Hence, the confidence limits of the estimate 6 may be statedas follows:
The true density (D) lies in the range
6 f k,. rC.V.% . . . ..__..._............. (14)
with a confidence of 1 - I where z is given approximately by
- k,2
==z Ie-“‘* . dt f o r ln > 10
--co
1 - t’lz ,where - e is the standard normal deviate.
2x
Equation 14 indicates that about 100 observations of jointsfrom a set are required to estimate the population density ofthat joint set with 95 per cent confidence to within 31 20per cent of the true value. The-variation of 1C.V. per centwith observed number of joints is shown in Fig. 18.
In a similar manner equations for estimating joint popu-lations from other forms of joint surveys may be derived.The ratio of the coefficients of variance is a measure of therelative accuracies of the various forms of surveys. For lineand area sampling
(1 + tan’8d. sec’od)~
. . . . . . . . (15)where h is the height of the sampling face.
t 9 mpos
BeckS’o’
I-
\20
Itim
6
Iate 95 % Confidence
her of Jcnt:L10 II
1
‘se.I-i0
6tI_
3-
0
\
Appro)6 =
.imits f o r
If (I GivI
120
pelOPme
, Ind the t
; ,ts ItWNi
hat the P’
q such P’. describer
,&The SY,
I where Wd practice:
3&j disc1+!rllS invj
of the Prc.,,,, providl;nvolved: of minin
z!Len Set
1LO 11
Ob Its a
1
rved N u r
e4 e 1 20(I” 1
Fig. IS
For the particular case where joints are normal to the lineand face of sampling (and the dip is parallel to the hdimension of the area sampling face) this equation reduces to
This relationship has been plotted in Fig. 19 for h = 7 ftand indicates, for this particular orientation of joints, thatarea sampling is somewhat better than line sampling forsmall joints but that the advantage is small for larger joints.
A given fine sample is inadequate if either 6d or &$ tendto zero, while an area sample only becomes inadequate ifboth 6d and sd tend to zero.
To ensure that all joint sets are ‘discovered’, either areasampling on two faces which are approximately perpendi-cular to each other, or line sampling on three approximatelymutually perpendicular lines, is required. Since in manyinstances joint sets of specific orientation are not of par-ticular interest or jointing patterns are readily apparent, thismay be reduced to one suitably chosen face or two suitablychosen lines of survey.
The necessary mathematics to determine confidence limitsfor the estimates made of other joint properties has yet tobe developed. Until this is achieved the confidence limits forthe joint density must serve as a guide as to the reliabilityof these other properties.
Having determined the joint properties of interest and theconfidence limits for the density estimate, the engineer mustdecide whether or not he can place sufficient, reliance on theresults. If not, further sampling will have to be conductedwhich may involve further rock surface exposure e.g. tun-neling or boreholes. In some instances the additional accuracy
66
may be achieved by conducting area rather than line samplingon the exisitng exposed surfaces. This can be determinedby comparing the relative accuracies using equation (15).Where addit ional surface exposure and/or sampling isrequired the optimum direction for this sampling is readilyapparent from the various equations.
8 . A P P A R E N T S T R E N G T H P A R A M E T E R SF O R R O C K M A S S E S
For any point on a potential failure plane, the strength ofthe rock material or the joint surface at that point is dependenton the state of stress that exists there. The joints, which arethemselves planes of weakness, create stress variationswithin the mass and hence the pattern of stress-strengthrelationships at various points in the mass must be complex.If failure is defined as that condition which exists when thedisturbing force (or stress) at a point just exceeds thestrength of the material at that point, then this complexpattern must produce local points of failure long before therock mass as a whole can be considered to have failed.Rock mass failure is a progressive phenomenon and mustdisplay a history of stress variation and failure within themass which ultimately leads to the formation of a surface orsurfaces of rupture through the mass. In slope stability it isthis ultimate formation of rupture surfaces through the rockmass which is viewed as a slope failure.
Many workers have argued that, since failure is a progres-sive phenomenon, the analysis of jointed rock slopes by rigidbody mechanics and a failure criterion of the Mohr-Coulombtype is not valid. This argument can be extended IO the failureof ‘intact’ rock materials which must also display progressivefailure as a result of the pre-existing micro-features etc.:
R1.6
l.L
1.2
l-0 -150 200 250 300 350 ioo cso 500 550
PROJECTED AREA OF JOINT G(squarc feet)
Fig. 19
/et numerous workers have shown that intact rock displays,trength characteristics which are sufficiently accuratelyiescribed by failure criteria of the Mohr-Coulomb type.fhus it is argued here that the strength of a rock mass mayJso be described by criteria of this type, i.e. that the jointsn a rock mass have a similar effect on the strength of the rocknass as do the micro-features on the strength of intact rock
specimens. Large geological features which are sufficientlylarge to singly effect the strength of the rock mass must beconsidered separately. However, since the strength along such
! features and of the intact rock may both be described by/ Mohr-Coulomb type criteria it would appear reasonable to1 expect that the strength along a failure plane, passing through) both intact material and joint surface, may also be describedby these criteria. The apparent strength parameters of the
1 rock mass will differ from both that of the intact rock and! the discontinuities within it.
Slopes fail under the action of gravity and other forceswhich produce stress patterns in the slope. The presence ofthese stress variations, and not the local stress variationsdue to discontinuities, suggests that an analysis techniquewhich takes the stress-strain characteristics of the mass intoaccount should be ultimately employed for slope analysis.Since the behaviour characteristics of the rock mass is notyet sufficiently understood to justify such an analysis, useis made of rigid body mechanics and apparent strengthparameters for slope stability determination.
Tests on specimens of the rock mass sufficiently large toinclude representative joint populations for strength para-meter determinations are prohibitively costly. Further, it isnot possible, as yet, to predict the strength of a rock massfrom the known strengths of the intact rock material andstrength along and spatial distribution of the joints. However,
as stated earlier, a strength assessment can be made for anygiven plane through the mass provided the nature of thesurface is known.
Fig. 20. Plane failure along a surface which includes pre-existingjoints
By way of illustration, Fig. 20 shows a slope through whicha potential plane of failure has been defined. In the total
surface AE portion 5 hz will pass through intact rock and !?I azI x-1
will pass throughxtte joint surface. If both the strength o nthe joints and s t rength through intact rock could bedescribed by equations of the Mohr-Coulomb type,
SJ = cj f on tan 6~ . . . . . . . . . . . . . . . . . . . *..(17)
67
The SYIECal Backc37 Pit Mir
? to SlopeJ. van ReJournal (litute of 1
xganised
itute of f~
Id at Joh300 defeg
m from n
slopmentd the trenS it wasthe print;!
ch Pmjecribed and
e w-npos,e world arice of pla"SCUSS anonvobed.
sroceedincIdes a V~C
in this
‘9.
535 00)
sm = cm + aII tan +m . . . . . . . . . . . . . . . . . . (18)where sf and sm are the strength of the joint and intact rockmaterial respectively,
cl and cm are the cohesive strength parameters,41 and +m are the friction angles, andon is the normal stress on the surface of consideration,
then, provided rigid body mechanics apply and sj and s,,,are mobilized simultaneously, the strength-of the failuresurface would be given by
s = (I - k) (crfz + on t a n +*) + k (cj + a11 t a n 4j) _ .(19)
where k is defined as the continuity of joint plane on thefailure surface, and is given by
It is recognised that the assumption that sj and srn developsimultaneously may not be justified. If, as in soil mechanics,an idealized strength-strain curve such as curve abc in Fig. 21could be assumed, then at failure the various points alongthe failure surface would all lie (at peak strength) alongthe portion bc of the curve. This does not hold for eitherintact rock material or clean joint surfaces, since they haveidealized stress-strain curves which show a reduction instrength after peak strength has been reached and may berepresented by idealized curve adef. The strain at which peakjoint strength and intact rock strengths are reached differ,as does the slope of the portion ad of the respective stress-strain curves. As yet insuff icient is known about therelative contributions to strength from the intact rock andfrom joint portions of the failure surface to produce thecombined stress-strain curve. The approach adopted here isto accept mobilization of peak strengths from all sourcesand to design slopes to a suitable factor of safety.
S
Fig. 21. Idealized stress strain relariomhips
The process for determining the apparent strength para-meters for a potential failure surface from the parametersdescribing joint and intact rock strengths requires con-siderable further research and development.
9 . T H E E S T I M A T I O N O F cm, I A N D &,F O R I N T A C T R O C K M A T E R I A L
ti’here the rock is reasonably homogeneous cm and +m maybe determined by the testing of specimens in the laboratory.Since ctn and 4, are of secondary importance to the strengthalong a failure surface compared with the continuity of joint
surfaces along it a relatively coarse measurement of themwill suffice. cm may be estimated from the rock hardnessclassification (Jennings and Robertson’) and bm estimatedfrom test results quoted in the literature. This is probablythe most satisfactory procedure where the natural variabilityof the rock is large, since a large number of estimates canbe made and the average properties determined. Apart fromthis it also represents a considerable saving in laboratorycosts since any further testing is confined to particularareas of interest.
The classification of consistency for soils and hardness forrock types as used in the joint survey is described by Piteau”.The results of unconfined compressive tests conducted onspecimens which have been classified according to this systemare shown in Fig. 22. An envelope of minimum strengthagainst consistency classification may be drawn. Line ECDrepresents the minimum strength envelope for the results oftests conducted in South Africa (Robertson and Jennings”)but ABCD represents the minimum strength envelope usingall data, published in the literature. Knowing the averagehardness classification for the material (varying from verysoft soil to very hard rock) a conservative estimate of theunconfined crushing strength (qu), may be made by usingthe envelope ABCD.
5.1Very sot! sod
SbfFfoil0
53Firm soil
SlS t i f f roif
SSVery stifi soil
R3Hard rock
RLVery hard rock
RSVery very
hard rock
.
i
iM
,
hI ’-L
I100
4\
1
REFERENCES
I .75G-G5
10UNCONFINED COMPRESSIVE STRENGTH lplll qu=2Cu
Fig. 22. Relatioruhip between consistency and unconfined compressivesrrengrh
The cohesive strength (cm) and tensile strength (1) of theintact rock may be estimated from qu using the followingrelationships (Robertson and Jennings”):
(iv) The hardness of the rock forming the sides of thejoint,
(v) The shear movement that has occurred.
(vi) The presence, nature and thickness of gouge materials.
(vii) The effect of water.
The reridml angle of friction characteristics of various rocktypes (factors (i) and (v) in previous paragraph)
If a joint surface is sheared under a given normal load,the shear stress differs with strain. Now if the surfaces are
-initially smooth and polished the shear stress increases withstrain, (Hoskins ef al’) but if they are initially rough the shearstress decreases (Krsmanovic’). At large strains, the para-meter 4 tends to a constant value characteristic of the rockmaterial and grain size. This value of the residual angle offriction is considered to represent the lowest frictional anglefor a clean joint, before modification by other factors, thatis likely to be encountered in the field (Jaeger and Rosengren’).The friction angles obtained from tests on smooth but un-polished surfaces, such as obtained from saw cut specimens,are accepted as being representative of the residual angle offriction.
The efict of waviness (factor (ii) )
Since the undulations of the joint surface are of such amagnitude that they are unlikely to be sheared off duringfailure along the joint, they modify the direction of movementas illustrated in Fig. 23. Waviness is measured in the fieldsurvey in terms of the amplitude and base length of thewaves. The determination of the wave angle X, as definedin Fig. 23, depends on the shape of the wave. The waves onjoints do not appear to follow any definite mathematical shape(Robertson and Jennings”) and it was accepted that 1 maybe determined from the following euqation:
01 = tan-‘2 -
L. . . . . . . . . . . . . . ..__._...... (23)
where 0 is the amplitude of the wave and L is the base length*
i
i These are approximate and discretion and caution shouldi be exercised in their application.
j Estimates of the values for the internal angle of friction,! tan 6 or coefficient of friction .u is made from the numerous1 results available in the literature. Table III lists the results of, strength parameter tests obtained in this way for the rocksr about de Beer’s mine.
E1
I O . T H E E S T I M A T I O N O F cI A N D 4~F O R J O I N T S U R F A C E S
1 lt has besn shown by Hoskins ef al’, Jaeger and Rosengrens,s Pattonls and others that for the low stress range associatedi with slope stability, the strength along a joint may be
described by the Mohr envelope in a straight line form i.e.equation (17). Withers?: and Jaeger and Rosengrenj foundthat Amonton’s friction laws appear to hold i.e. the frictionalforce is independent of the area of contact, and dependsdirectly on the normal load, and that the two are linearlyrelated.
Tests by Jaeger and Rosengren’ and by Hoskins er al’indicate that cohesion values are appreciably large, even for‘clean’ joints and may have values between 50 lb/in’ and200 lb/in’. Patton I3 explains this apparent cohesion for cleanjoints by showing that the Mohr envelope is in fact curvedat very low normal stress. A technique for the assessment ofcohesion from field classifications ofjoint surfaces has not yetbeen developed and, since any cohesion is rapidly destroyedas small movements take place (Krsmanovic9) the cohesionfor joints which are clean are assumed to be zero.
I
; Summarizing the findings of the many workers who have:
I
considered the strength along joints. Robertson and Jennings’*conclude that the main factors affecting the strength along
/ joints may be defined as:
i (i) The composite frictional angle of the minerals forming
i _.the walls of the joint.
(II) The effect of large, or first order, asperities termedc waviness.I
(iii) The effect of smaller asperities or the rorr&ness effect.
TABLE III
STRENGTH PARAMETERS FOR THE ROCKS ABOUT DE BEERS MINE
t::
(0.73) 1.0
(0.73) 1.0
1.0
Rock 1 De$:~low 1 Description of rock typeI bT%
Hardnessclassifica-
tion
RI-R2 350 56R3 1500 240
Type doleritecontact
1 Surface to Medium grained dolerite (weathered) with fine grainedcontact -0’ intrusions . _ . . . . . . . . . . . . . . . . .
hined. 1 :urn wf :and prs coulcprobleron of tl6ium p31s invclect of 1
Fig. 23. The efect of waviness on the vector of I mvemenli
The eflecr o/ rofrghness (factor (iii) )
size) resulted in increased initial friction antests on both small and large samples clearincreases in dj were dependent on the sizeforming the roughness of the sliding surfacerelative size of the asperities compared withSince theory predicts (PattonIs) that this incre
natural joints from dein Table IV. From t hroughness category aalong the joint by apthe range 50 lb/in* to
The effect of hardness (factor (iv) )
The difference in frictional characteristicmodels tested under high and low normal sapparent. For low normal stresses the fewasperities on the surface produced an apparenangle. These asperities shear under higher nthe more numerous lower angle asperities givlower friction angle, Hence, the effect of rotake due cognisance of the strength of thehardness) and the stresses that wit1 bejoints.
The increase of 4” in the angle of friction focategory was found from models made of pRl) which is softer than the rocks about(hardness Rl to R5). Since the range of nthe models were subjected to (50 Ib/in* tapproximately that which could be expect
in the side slopes of de Beer’s mine, these results should beconservative.
The eficfs o/gouge (factor (v) )
Where gouge is sufficiently thick to prevent the walls ofjoints from touching, it controls shear strength entirely, andthe joint strength parameters are those of the gouge. Verythin gouge deposits are likely to modify only the residualangle of friction, and the apparent increase in friction due toroughness must still occur. Between these two exiremes acomplete range of intermediate effects must occur.
Since the gouge hardness is measured in the joint survey,an estimate of the gouge cohesive strength may be madefrom Fig. 22 and equation (21). Alternatively. both t h ecohesion and the angle of internal friction of the gouge maybe determined by laboratory testing samples.
In general, only a portion of the joints in a joint set containgouge and of these some may have thick deposits while othershave only traces. The assessment of the average strength isessentially a judgment process and the following points arenoted to aid this process:
(a) The greater the average size of the joints in a joint set,the greater the proportion of joints containing gouge.
(6) The greater the joint size the greater the chance of itcontaining gouge, and the greater the thickness of thegouge that may occur.
(c) The larger joints form a greater percentage area of anyfailure plane than the percentage their numbers formof the total joint population.
(d) Any failure plane must have a tendency to selectjoints of lower shear strength.
As a result of the above considerations the followingmethod of estimating the joint shear strength is suggested:
(i) Determine the cohesion (CJJ and friction angle (41~)for a clean joint and the cohesion (cjJ and frictionangle ( +jp) for a joint with gouge of sufficient thick-ness that it controls shear behaviour.
(ii) Based on the normal stresses on the plane of failurecalculate the approximate average strength that eitherof these joints would have on any potential failureplane. Let these values be sjc and sfg respectively.
(iii) If sjc > sfe then determine cf and #j as follows:(a) If the percentage of joints containing gouge is
greater than 30 per cent then
Cl = CI#
>
. . . . . . . . .._.._..+I = h#
(24)
TABLE IV
FRlCTlON ANGLES FOR PLASTER MODELLED JOINTS OF VARIOUS ROUCHNESSES
Roughness escription of 4 at low normal 4 at high normalcategory -0int surface stresses stresses
(iv) If sic < ~1~ then calculate cj and 4j as follows:
(u) If percentage of clean joints is greater than 70per cent:
Cj = cjc . . . . . . . . . . . . . . . .h = +jc
(b) If percentage of clean joints is x per cent where* x per cent < 70 per cent then
70-x%CI = cjc + (Cjd - Cjc) 70
I
. . . . (27)
4j = djc + (bjg - 4jc) 7*
These rules are rough and are based essentially on ajudgment assessment. The effects on strength and distributionof gouge filled joints along failure surfaces requires furtherattention.
J11. C O N C L U S I O N
From the data collected in a field survey of the rock andjoint types and characteristics the population of the mechani-cal discontinuities in the rock mass may be determined.The volumes of rock masses of s imilar physical andmechanical properties can be delineated. These are thestructural regions. The strength and behaviour characteris-tics of the various elements that form the rock mass can beestimated and, if particular planes of failure are investigated,the apparent strength parameters relating to these planesmay be estimated.
Thus the interpretation of geological factors can yield,with a minimal amount of laboratory testing. sufficientinformation to enable the engineer to proceed with the deter-mination of the stability of a slope. This information may atthe present level of knowledge be relatively inaccurate, butit permits a first assessment of the stability of a slope.
The regions and planes of possible instability are definedand these lesser regions may be more intensively investigated,with laboratory and field tests being conducted whereadditional or more precise information is considered neces-sary.
The most important single factor affecting the strengthalong a potential failure plane is the continuity of the jointson that plane. Improved techniques for determining jointproperties which influence the continuity value calculatedand more precise mathematical models to represent potentialfailure planes, would greatly increase the value of thisanalysis technique.
A further aspect that must be resolved is the exact natureof the apparent strength parameters for the failure surface.particularly the contributions to the apparent strength fromthe inT%zt and joint surfaces for various strains.
The stress-strain behaviour of a slope and the progressivemobilization of strength and subsequent failure will ulti-mately have to be accounted for.
As the knowledge of the effects of the various influentialfactors increases the analytical approach of slope stabilitydetermination will become more precise. Confidence in design
71
A C K N O W L E D G E M E N T S
The author wishes lo thank de Beer’s Consolidated Minesfor permission lo reproduce certain information and resultsin this paper and to express his appreciation to ProfessorJ. E. Jennings of the University of the Witwatersrand underwhose direction most of the work described was conducted,for his enthusiasm, encouragement and assistance. Theguidance of R. Stamer of Operation Research Bureau (Pty.)Ltd. on the statistical concepts, and of D. R. Piteau, on theengineering geological aspects, is acknowledged with appre-ciation and thanks.
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