Cable method of in situ rock testing

Int. J. Rock Mech. Min. Sci. Vol. 4, pp. 273-300. Pergamon Press Ltd. 1967. Printed in Great Britain

CABLE METHOD OF I N S I T U ROCK TESTING O. C. ZIENKIEWICZ and K. G. Sa'A~c School of Engineering, University of Swansea, S. Wales (Received 18 November 1966)

Abstract--One of the problems in applying loading to massive rock in situ is the question of obtaining a suitable 'reaction' point. This is overcome usually by confining such tests to tunnels or slits in rock. An alternative method suggested relies on the use of cables anchored at some depth in a borehole. With this arrangement large forces can be applied to the rock surface directly and in any direction desirable. If two parallel cables are used further advantages accrue. The volume of rock influenced becomes greater and an additional jacking load parallel to the rock surface can be applied between the 'heads'. Thus anisotropy of rock properties can be detected and in the limit, by carrying the test to failure, the shear strength can be found. The analysis based on the Boussinesq and Cerrutti solutions is extended to cover the case of transverse isotropy and pilot field trials are described. 1. INTRODUCTION MANY important types o f large-scale engineering structures are founded u p o n or constructed within rock masses. Examples o f the former are the various types o f concrete d a m ; o f the latter, power and transport tunnels and some hydroelectric power station machine halls. There is also the possibility that in the future some nuclear power stations will be housed in large underground caverns. In all the above examples the rock mass forms an integral part o f the complete structure and the deformation of the rock under load will influence the strain and hence stress distribution in the rest of the structure. Therefore in order to achieve an economic yet safe structural design it is necessary to have some knowledge of the load deformation properties of the associated rock mass. The work of VOGT [l] and the United States Bureau o f Reclamation [2] was largely responsible for outlining methods by which rock foundation deformability effects could be approximately introduced into the analysis o f concrete dams. However at this time it was customary to estimate the rock deformability f r o m the results of laboratory loading tests on small specimens of rock. As a direct result o f these laboratory tests a general 'rule o f t h u m b ' was worked out which said that rocks had a stiffness between one and two times that o f g o o d quality mass concrete. However, subsequent comparison o f the results o f 'in situ' field tests on rock masses with the results of laboratory tests on specimens o f the same rock show that laboratory tests invariably lead to an over-estimate of the stiffness of the rock mass. Such comparisons have been reported from a large number of field sites and indicate that laboratory tests m a y over-estimate the stiffness of the rock mass by a factor of up to 20, factors o f 5-15 being quite c o m m o n . The principal reason for this discrepancy is the presence o f discontinuities in the rock mass. These discontinuities m a y take one or more of several forms, e.g. (a) more or less systematic jointing and bedding, (b) micro-cracks in apparently continuous material; a n d - o f lesser importance in this context--(c) faults and (d) localized 'altered' rock zones. L a b o r a t o r y specimens, for practical reasons, are almost invariably taken f r o m sound rock 273

274

O . C . ZIENKIEWICZ AND K. G. STAGG

between discontinuities. The presence of the discontinuities, with their considerably lower stiffness, reduces the overall stiffness of the rock mass; the reduction factor is therefore dependent on the physical magnitude and the frequency distribution of these discontinuities. There is no reliable method of predicting in advance the overall stiffness of the rock mass from the results of laboratory tests, so that in situ field tests must always be carried out on site despite the considerably greater expense of such tests. Despite their apparent variety the test procedures derived to date are all variants of three basic procedures. O f these two are essentially static; the classic plate-bearing test and the pressure tunnel test. In both these tests known loads are applied to the rock mass and the resulting deformations are measured. The third method, the sonic pulse method, is ~,, dynamic one in which the velocity of propagation of a vibrational disturbance is measured. Although rock is certainly not homogeneous nor, in general, elastic it is customary to interpret test results on the basis of the theory of elasticity, and to assign to the rock values of appropriate elastic constants such as Young's Modulus (E) and Poisson's Ratio (v). Justification of such a procedure rests on the fact that at moderate loads the stress-strain relations are roughly linear and the creep properties are often secondary. In addition, it conveniently permits the use of established methods of analysis which have to date not been displaced by alternative theories. In the plate-bearing test a part of a flat exposed rock surface is loaded normal to its surface and the resulting rock surface displacements measured. These displacements may be related to the applied load by the well-known Boussinesq solution [3, 4] for stress and strain distribution in a semi-infinite elastic medium due to a point normal surface load. This solution may be readily integrated to give the corresponding solution when the normal load is distributed over a finite area of the surface, as is the case in the present test method. Further details of tests of this type may be found in Refs. [5-19]. A variation of this technique in which a flat jack is used to force apart the sides of a slit in the wall of a tunnel is described in Ref. [14]. The pressure-tunnel method relies on driving a circular tunnel, sealing off a section of the tunnel and subjecting this section to an internal hydrostatic pressure. The resulting diametral deformation of the tunnel is then measured. The relationship between diametral deformation and pressure is considered in Refs. [20--22]. Several small-scale derivatives of this basic approach have recently been introduced whereby special instruments are inserted in relatively small diameter boreholes [14, 33, 34]. These instruments apply some form of radial pressure to the borehole walls and measure the resulting radial deformation of the hole. Depending on the pattern of the radial pressure there may not be an exact analytical solution for the resulting displacement so that the results must be interpreted on the basis of laboratory calibration. Despite some claims to the contrary it is unlikely that this method will supplant larger scale in situ testing, but may well serve as a useful complementary procedure. Of the two static methods outlined above the plate-bearing method is the one commonly used, mainly because it is the easier of the two to design and operate. In order to produce measurable deformations the loads required will usually be of the order of several hundreds of tons. These loads are most conveniently applied by hydraulic jacks and in order to take the jack reaction the tests are generally carried out in galleries driven into the rock, the jack reaction then being absorbed by the opposite wall of the gallery to that under test. Since the results are to be analysed by the Boussinesq method then the size of the loaded

CABLE METHOD OF IN SIT U ROCK TESTING

275

wall of the gallery must be sufficiently large to reasonably justify the semi-infinite solid hypothesis explicit in the analysis. If the results of these in situ tests are to be of the greatest practical value it is essential that the volume of the rock influenced by the load and hence contributing to the observed displacement be sufficiently large as to form a truly representative picture of the rock mass. This implies that the volume of rock influenced be large enough to take into account all the structural discontinuities which differentiate the rock mass from the local rock material. This ideal is often impossible to achieve for economic reasons since the necessary gallery size would be excessive. The relative extent to which the displacements measured in the plate-bearing test are influenced by the deformations of the various zones within the total volume of rock measurably influenced by the load area is discussed in detail in Ref. [5]. The essential details may be observed in Fig. 1 in which the displacement w and stress 8z in the z-direction are plotted

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FIG. 1. Normal displacement and stress [5]. for two separate lines normal to the surface within a semi-infinite solid. These results are based on an elastic analysis for a circular loaded area of radius a, area A carrying a uniformly distributed load P. The solid lines show displacements wp and W3a along the axis of the plate and on a line at radius r ~ 3a from the axis of the plate respectively. Dashed lines show corresponding values of azP and az3a, both in terms of P/A. The curves are based on a value of v = 0.2. It is evident from these results that the displacements under the loaded area are influenced to a much larger degree by surface rock than those at some distance from the point of loading. For example, something like 80 per cent of the displacement of the plate itself is contributed by the material within Z/a less than 4, or a depth twice that of the diameter of the loaded plate. For a point r outside the loaded area (r = 3a) the depth contributing 80 per cent of the displacement is of the order of ten times the diameter. In practice this effect may be of considerable significance since the immediate surface rock may have been modified by the effect of blasting. Since the displacements measured outside the loaded area, whilst necessarily being smaller, are more representative of the rock mass it may well be more valuable to conduct tests with several adjacent relatively small and lightly loaded areas loaded simultaneously rather

276

O. C. ZIENKIEWICZ AND K. G. STAGG

than to test with one large heavily loaded area, especially if deformations between ~he loaded areas are used as the basis for analysis. Further discussion concerning the above techniques can be found in Refs. [23-32, 50, 51. 54, 57]. Both the 'static' methods of testing described fall short of the ideal due to economic limitations which restrict the size of the tests. Considerable attention has been given recently to the possibility of applying the sonic pulse method to rock masses. In this method the elastic modulus is deduced from the speed of propagation of sound waves, and is therefore a development of the seismic method of geophysical prospecting [36-45]. Unfortunately, however, the rock mass is not a homogeneous elastic continuum but a more or less heterogeneous assemblance of granules permeated with fissures and the analysis of wave velocity in relation to 'elastic constants' is not completely understood. The results, therefore, are often difficult to correlate with those of 'static' tests with the result that this method cannot yet be reliably substituted for the static method. Nevertheless the outstanding advantage of this method is that it can explore large volumes of rock at relatively low cost and hence it forms a value adjunct to the static method in obtaining comparative information about differing areas of the test site. It is apparent then that the plate-bearing test is one of the primary tools of the rock mechanics engineer and is likely to remain so for the foreseeable future. One of the principal drawbacks of the method as at present practised is the large expense involved in driving special testing galleries. An alternative test procedure would be to load the plate against a steel cable anchored remotely in a borehole. In this way the test could be carried out at the rock surface as an alternative to excavating test galleries. It is with this aspect of the platebearing test that this paper is largely concerned. In discussions of this method the authors have claimed that its basic advantage is its relative cheapness compared with gallery tests. In reply three main criticisms have been levelled against the method and these are best dealt with at the outset. Firstly it is suggested that galleries will still be necessary to fully explore the geological structure of the rock mass. However with modern borehole camera techniques and borehole closed circuit television apparatus available the necessary geological information will in many cases be available from an examination of the cable carrying boreholes. In addition since it will be practicable to drill more boreholes than drive test galleries a more detailed geological picture may be obtained. Secondly it is argued that the results of surface tests would be affected by the presence of weathered rock at the surface. This point can only be considered in detail with reference to particular rock masses under investigation but it is probable that in most practical cases it will be no more of a problem than the presence of blast-affected rock in a test gallery. Thirdly it is suggested that provision of anchored cables in boreholes is itself an expensive process. Private communications to the authors from a specialist contractor have established that such cables can be installed at a relatively low cost, sufficiently so to make the method financially very attractive. However final costings would depend to a large extent on the location of the test site. Nevertheless the economics of the cable method could be improved further by the development of a recoverable cable anchor, thus allowing the cable to be re-used. A further point which arises in relation to the in situ testing of rock masses is the fact that most rock masses are anisotropic to a greater or lesser extent. In practise attempts are made to determine the magnitude of this by conducting several tests each orientated in

CABLE METHOD OF IN SITU ROCK TESTING

277

different directions. The resulting analysis is only approximate since it is still based on isotropic elastic theory. A more elegant method of conducting this analysis is discussed below. 2. T H E U S E O F A N C H O R E D

CABLES TO APPLY TEST LOADS

The major difficulty in exerting loads on the rock surface is the problem of finding a suitable reaction support. An apparently straightforward procedure, but one which appears not to have been tried hitherto, is to provide this reaction support by a cable anchored in a borehole at a suitably large distance from the surface loading pad. A simple arrangement of this type is shown in Fig. 2.

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FIG. 2. Cable jacking test. A minimum depth of anchorage of the order of 6-8 times the bearing pad diameter is tentatively recommended. The loaded area can easily be made sufficiently large, in relation to the diameter of the borehole cutting through it, to justify neglecting in the analysis the presence of the borehole. goads of up to 1000 tons can be applied in this way using a single cable, thus allowing a large volume of rock to be influenced. Several cables could be used to apply even greater loads, if needed. It seems very probable that tests of this type would be more economical to carry out than conventional tunnel jacking tests, since tunnel driving is an expensive undertaking. The cable approach has other distinct advantages. In particular, the rock can be tested at the exact foundation location and in the directions in which the actual loads of the structure will be exerted. The tests could be repeated at various levels of excavation, using the same cable and borehole, to obtain information about the variation of rock characteristics with depth. With two adjacent cables, loads tangential to the surface can be applied and information obtained about the variation of elastic moduli with direction of load. The first suggestions for the use of cables in the context of rock testing appear to have been made by JAEGER in 1961 [52] and also about the same time by F. F. Ferguson of the Cementation Co. Ltd. In the investigation described in the present report, the original ideas were elaborated and extended to cover the problem of tangential loading; the tests were carried out at a site in Derbyshire, England with the collaboration of the Cementation Co. Ltd.

278

O. C. ZIENKIEWICZ AND K. G. STAGG

2.1 Double.cable loading and directional variation of rock properties In Fig. 3(a) the diagrammatic arrangement for double-cable loading and the system of application is shown; Figure 3(b) shows the essential characteristics of the loading which is being applied to the surface of the rock. Cable head

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FIG. 3. The 'double cable' test. If no directional variation in the properties of the rock occurs ('isotropic' rock), it is evident that the elastic modulus can be obtained by the theoretical reasoning given in the previous section dealing with pad tests in general. Even then, however, an advantage of double-cable loading is evident; the displacements of the ground between the loading pads will be increased and, by taking test measurements there, the elastic properties of a much larger mass of rock will be determined than in tests using a simple pad. In such isotropic rock, no additional information would be obtained by applying tangential loads and these clearly would not be necessary. However, in the majority of real rock materials a considerable variation of properties occurs with the orientation of the applied force. Such rocks are termed generally 'anisotropic', and here the results of both normal and tangential loading tests will be required. In a completely anisotropic situation it has been shown in many texts that 21 independent elastic constants may exist [46]. In stratified rock the problem can often be simplified as the

C A B L E M E T H O D OF IN SITU R O C K T E S T I N G

279

properties in the plane of the strata may be considered to remain constant in any direction ; thus a specification of properties within this plane and in a direction normal to it suffices. Such materials, for which the term 'transversely-isotropic' is used, are characterized by five elastic constants: namely, two values of the elastic moduli E associated with specified directions, an independent shear modulus and two Poisson's ratios (see below). In the conventional rock tests, values of Poisson's ratio are assumed and a further simplification regarding the relation of the shear modulus is suggested in Refs. [48] and [49]. If both sets of assumptions are made, the rock properties can then be adequately described in terms of only two constants, which in turn could be obtained from two tests if the appropriate theoretical relationships were known. A basic solution to this general problem was described by MITCHELL[47]. This theory is elaborated in the next section of this paper to give expressions for displacement under distributed normal and tangential loads. With these assumptions it can be shown that if the elastic modulus in the direction perpendicular to the surface is E (the surface is taken to be approximately parallel to the plane of isotropy), and if n × E is the elastic modulus in the plane parallel to the surface, then the formulae for the average displacements of a single square pad with a side a are: r~ z 2"97A1 -Pa

(1)

where ~ is the average displacement of the pad in a direction perpendicular to the surface and

fi -- 2.97B~ Q a

(2)

where fi is the average displacement of the pad in a direction tangential to the surface. In these formulae it is assumed that all Poisson's ratios are zero and that the shear modulus G ~ [n/(n q- 1)]E. A1 and B1 are functions of E and n, which are plotted in Figs. 19-22 and from these the appropriate constants can easily be obtained. The background mathematical theory is set out as an Appendix. 2.2 Details of fieM tests The field tests were carried out on limestone rock at the Earl's Cement quarry at Hope in Derbyshire. The rock was well jointed with two clearly defined major sets of joints, each having an average spacing of about 3 It, the two sets being inclined to one another at approximately 75 °. Some of the joints were up to 0.5 in. open and were filled with clay. The joint planes were slightly inclined to the vertical. Less well defined bedding planes, substantially horizontal, occurred at intervals of approximately 3 ft with major bedding planes at varying intervals of the order of 30-60 ft. The rock had a coarsely crystalline texture and contained scattered fossils, mainly crinoids. The site was chosen on level ground above and behind the top quarry face. This position was preferred to alternative sites on the quarry benches where the rock floor had been considerably disturbed by blasting, it being the practice of the quarry to drill the blasting holes to below the level of the resulting benches. The site chosen suffered the alternative disadvantage, however, of being influenced by any weathering of the surface rocks. The soil overburden of approximately 2 ft had been removed and the top 3 ft of rock cleared

O.C.

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Z I E N K I E W I C Z AN[) K. d . STAGG

away from the test site. This top layer contained the major part of the weatilered surlace rock. Four boreholes of 4½ in. diameter, positioned at the cornet's of a square of side lenglll 6 It, were drilled by percussion drill to a depth of approximately 40 ft. The hole positioning was such that one pair of parallel square sides were parallel to one of the joint system~;. A multi-strand steel cable was anchored in each borehole. The site layout is shown diagran',matically in Figs. 4 and 5. The maximum permissible load varied slightly for each cable bul ._

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was always in the range 250-290 tons. In each case there was at least 15 ft of free cable below the hole collar and above the anchorage grout. A square, reinforced-concrete loading pad was cast in place at the surface above each hole and centred on the hole. The cable passed freely up a hollow tube through the centre of the pad. The pads had a side length of 3 ft and a depth of 3 It, all the pads being orientated in the same direction so that the sides of adjacent pads were parallel. Square pads were chosen in preference to circular ones to facilitate horizontal loading arrangements.

CABLE METHOD OF IN SITU ROCK TESTING

281

The pads were each loaded vertically by three 100-ton hydraulic jacks, placed symmetrically about the cable on the top of the loading pad and acting against the under surface of the circular cable head. The cable heads comprised a 2 ft 6 in. diameter steel cylinder 1 ft deep resting on a ~-in. thick steel disk of the same diameter, the whole being filled with concrete. The free end of the cable passed up through the steel disk and was anchored by the concrete. A load cell was placed between the base of each jack and the top surface of the loading pad. The load cells were sealed and calibrated Freyssinet-type jack cells, fitted with a hydraulic pressure gauge and having a bearing surface of 1 ft diameter. Provision was made for horizontal loading between adjacent pairs of loading pads. A horizontal 100-ton capacity hydraulic jack was placed between the first pair of pads to thrust them apart, together with a Freyssinet load cell and a distance piece constructed of a concrete-filled steel tube; the jack was positioned as low as possible but clear of the ground surface. The second pair of loading pads were pulled together by a pair of l~-in, diameter Macalloy prestressing bars together with two Macalloy jack sets, the total available load being 70 tons. Hydraulic pressure for the vertical jacks was provided by means of six independent rotary hydraulic pumps, each coupled through a variable speed drive to a common shaft driven from the power take-off of a Land Rover. A separate pressure control valve was in circuit with each pump together with a pressure gauge. This pressure gauge could be used to check the reading of the load cells, allowance being made for the presence of jack friction. Pressure for the horizontal jack was provided in each case by hand-operated pumps. A measuring datum was constructed, consisting of a 20 ft long 5 x 3 in. R.S.J. and a a 20 ft long 6-in. steel channel section. These were positioned parallel and 2 ft 6 in. apart between the two pairs of loading pads, as shown in Fig. 4, at a height of approximately 6 in. above the ground surface. The girders were bolted at each end to steel plates, welded on to steel tubes and grouted into vertical holes in the rock. (Fig. 5). The girder anchorages were therefore situated approximately 7 ft from the centre of the boreholes. The two girders were braced together by six members made from 2-in. angle section. The vertical movement of each loading pad was recorded by means of two 0.0001-in. dial gauges positioned at points X, Fig. 6, the dial gauges being held in position by rigid arms bolted on to the measuring datum. The dial gauges rested against brass plates fixed to the pad surface by adhesive. Horizontal movement of the pads was measured using a 0.0001-in. dial gauge positioned at points IT. Ground stations wer established in the rock at point shown in Fig. 6. Those marked 'type A' consisted of short lengths of square-section bar grouted into small holes in the ground (Fig. 5), the holes being centred 8 in. apart, along the lines shown. Relative horizontal movements between these pegs were measured using an 8 in. gauge length DEMECdemountable strain gauge. Vertical movements of the pegs were measured using the 8-in. DEMECgauge placed between the peg and a vertical steel plate, held vertically above the peg by a steel channel section measuring arm cantilevered out from the measuring datum. A secondary system for measuring surface movements was established at points B, Fig. 6. These measuring stations are illustrated in Fig. 7. The measuring datum is established by a steel rod passing freely down a small borehole and anchored at a depth of 6 ft. At the surface the rod passes freely through a metal collar grouted to the ground surface. Vertical movement was measured between the collar and the measuring head on the free end of the datum rod, using a 2-in. gauge length DEMEC gauge. It is assumed that the datum rod is

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CABLE METHOD OF IN SITU ROCK TESTING

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anchored in a region in which the effects of both the surface loading and cable anchorage reaction are negligible. Relative horizontal movements were measured between adjacent collars, using the 8 in. DE~EC gauge. The entire measuring rig was set up for tests on holes 3 and 4 and then turned through 180 ° for tests on holes 1 and 2. The whole of the test area was covered by a shelter constructed fi'om tarpaulins fixed over scaffolding. The principal purpose of this shelter, whilst providing protection from snow, rain and wind, was to shield the measuring datum from the sun's rays. If this precaution is not taken, local heating due to absorbed radiation can easily produce severe warping of the measuring datum. The day-time air temperature at the site varied little throughout the test period, being in the range -- 4°C to -- 2°C. The test programme was to some extent determined by the primary anchorage test programme, but basically it consisted of applying successive loading-unloading cycles with a gradually increasing peak load. In general the pair of cables under test were loaded simultaneously and, when both the cables were under maximum vertical load, cycles of horizontal load were applied. The load cycling process was inevitably a slow one due to the need to halt the loading frequently, to record measurements relating to anchorage behaviour, these measurements being taken at load increments of approximately 10 tons on each cable. A complete set of measurements of ground and pad movements was taken at vertical load intervals of approximately 30 tons. Since the tests were carried out at midwinter the daylight hours were short; testing was continued after dark with paraffin floodlighting but the speed of observation was seriously reduced under these conditions. This meant that the tests sometimes had to be left overnight at an intermediate load position, and when this situation occurred readings of load and deformation were taken at the end of one day and repeated at the beginning of the following day; in these circumstances the load generally remained constant, but some creep of the rock was always found. Conditions during the test were far from ideal for using a sensitive instrument such as a DEMEC gauge, and each measurement was taken three times with separate applications of the DEMEC gauge, the average value being recorded. The reproducibility was poorer than would normally be expected with these instruments, but the total spread at each measurement was generally not more than 0.0003 in. and this was considered adequate. Maximum total vertical movement of the loading pads was of the order of 0.1 in. including consolidation and creep. This movement was checked independently for one loading pad by means of a surveyor's level, mounted about 30 ft away from the test area and focused on a scale attached to the pad. Initially measurements were taken at all measuring points during each observation period. It was found, however, that vertical movements at the outer three or four ground stations were difficult to distinguish from the experimental spread of measurements. Systematic movement was in all cases undetectable at the outermost stations at all times and also at the penultimate one in every case but one. Readings at the outer three or four stations were therefore only taken at maximum and minimum load stages and occasionally at an intermediate stage. These readings were taken as a check on the stability of the measuring datum and, as such indicated satisfactory behaviour of the datum. This is not complete proof of the stability of the datum, but is good indirect evidence of it, and the reproducibility of the measurements was generally better at these outer stations. Vertical movement generally appeared to cease abruptly at a certain distance from the load pad, appreciable vertical movement being found at one peg and little at the next one away. In every case this could

284

O. C. Z I E N K I E W I C Z

A N D K. G . S T A G G

be correlated with a fissure running between the two stations. The type B stations indicated substantially the same movements as the type A ones associated with the same loading pad, Horizontal movement measurements were less satisfactory. The reproducibility of the measurements was better than for the vertical movements, the rigidity of the gauge points being greater. At the ground stations there was not a smooth pattern of movement away from the loading pad. The movement occurred in a random manner between scattered pairs of ground stations and could be correlated with well-defined joints between those stations. It was not practicable to deduce horizontal modulus values from any of the horizontal measurements except those of the loading pads themselves. 2.3 Results of field tests 2.3. I Moduli derived at loading pads--(a) Vertical moduli. Cables 3 and 4 were the first pair to be tested, followed by tests on cable 1 and 2. The load cycles were applied simultaneously and, as far as possible, in step to each cable. The vertical movements of the load pads plotted against vertical loads are shown in Figs. 8--11. In each case the movement shown f

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~00 -HOrizohtQI

200 o

TY

tOO

0

~~5

75

50

V~r tical dispiacement,

103

inXlO

FIG. 11. Vertical movement of pad 2. is the average of the two dial gauge readings and the loads are the sum of the three jack loads. Total vertical movements measured by two gauges on the same pad never differed by more than 7 per cent, and the values of the secant slope of the second curve to full load from the same measurements never differed by more than 4 per cent. These differences were partly attributed to slight tilting of the loading pad due to the presence of the adjacent loaded pad, and partly due to the asymmetric distribution of the jacks relative to the measuring gauges. In all cases points on loading curves are indicated by circles and those on unloading curves by crosses. The average vertical displacement at the rock surface will be less than that measured at the top of the loading pad by an amount corresponding to the vertical compressive strain induced in the loading pad by the load. A correction must therefore be applied to the measured displacements. Assuming that the concrete o f the loading pad had a modulus of 5 × 106 psi then the mean vertical compression produced in the loading pads was 0-0000125

286

O. C. Z I E N K 1 E W I C Z A N D K. G , S T A G G

in. per ton of load. As this represents only about 3 per cent of the total deformation the accuracy of the assessment of this correction was deemed adequate. The modulus values derived below have been calculated on this assumption. From the vertical movements of the loading pads the following moduli of the rock mass in the vertical direction were found: Pad3 E=:3.08 Pad4 E--3.02 Pad 1 E - ~ 3 . 2 1 P a d 2 E=~3.10

× × × ×

105 psi 10~psi l0 s psi 105 psi.

(b) Horizontal moduB. Figures 12 and 13 show the horizontal displacements of loading pads 3 and 4, and 1 and 2 respectively due to horizontal loads. The horizontal loads were applied at vertical load values indicated on the vertical load graphs discussed above. The

t-

o

b

I0

i5

?0

~

Horizonta! displac~;~e,~t,

b

I0 in X I 0

15

20

25

~

FIG. 12. Horizontal displacements of pads 3 and 4.

maximum horizontal load was limited to approximately 60 tons to avoid shearing the loading pads off the rock surface as such movement might have had an adverse effect on the anchorage tests. A similar solution to the Boussinesq one but relating to a tangential surface load is due to CERUx'n, see Refs. [4, 53]. In a similar manner this solution may be developed for the case of a distributed tangential loads and gives a solution of the same forms as that for normal loads. It is most convenient to analyse the results of the horizontal tests by assuming that the load is applied independently to each of the loading pads. A correction to the displacements will then have to be made for the component of the displacement due to the load on the adjacent loading pad. Examination of the Boussinesq and Cerutti equations shows that if r is large compared with the radius (or in this case half the side length) of the loaded area then the load can be assumed to be concentrated at the centre of the loaded area with negligible loss of accuracy.

287

CABLE METHOD OF I N S1TU ROCK TESTING

~o~

Po~,

/~

Po~ 2

/17/' / /

/

/, 1 ___J

r;,

:,

lO

15

20

IO

.r]

Hofizonlol displacemen!,

15

20

25

m X Io 5

Fla. 13. Horizontal displacement of pads 1 and 2.

A correction was therefore derived on the assumption that all the load was concentrated at the position of one borehole, and the resulting displacement at the adjacent borehole was taken as a mean correction to be applied to the displacement of the pads. Again using the secant slope to the second curve to full load, we find from the horizontal pad displacements P a d 3 E - 2.41 × 105 psi P a d 4 E = 2 . 3 4 × 105 psi Padl Pad2

E=2.14 E=2.04

× l0 apsi × 105 psi.

Since the horizontal tests give a lower modulus than the vertical ones, the rock is nonisotropic and therefore neither in the vertical nor in the horizontal directions is it permissible to use the isotropic expressions of Boussinesq or Cerutti and the expressions (1) and (2) now govern displacements. It is possible to correct for this by noting that effectively: Ehorizontal

A1

EverUcal

B1

(3)

Using mean values of the horizontal and vertical moduli for each pair of loading pads we find by inspection of Figs. 19-22 that: for Pads 3 and 4 : n -- 0.55 and for Pads 1 and 2 : n = 0.45. The mean vertical modulus can now be corrected for the effect of anisotropy, and we have: Eeorrected = E v = Emean vertical X A1

and Eh -- hey.

R.M.--T

288

O. C . Z I E N K I E W l C Z

A N [ ) K, G . S T A G G

Applying this correction we have: for Pads 3 and 4 : E,, = 3.60 >," l0 s psi k,)~ -

1"98 :e 1():i psi

and for Pads 1 and 2 : E~, -:- 3.95 x, 105 psi Eh

1'78 ~ l0 s psi.

2.3.2 Moduli derived from displacement of ground points--(a) Vertical moduli. Typical curves of vertical movement for ground points are shown in Fig. 14. Secant vertical moduli computed for various type A ground points are given in Table 1 assuming isotropic rock. These values correlate reasonably well with those calculated from the loading pad displacements. It is noticeable, however, that ground stations adjacent to the loading pads

.._

Peg 7

Peg8

i /1-//

2O0

g o )<

]0

20

30

40

0

I0

Verfi¢ol displocemenf,

20

30

40

50

m ×10 ~

Fio. 14. Vertical displacement of ground pegs (type A).

give slightly lower values of modulus, and ones farther removed higher values, compared with those derived from loading pad displacements. This can be correlated with fissures in the rock causing discontinuities and giving the ground a tendency to behave as separate loaded columns. The shape of the curves generally corresponds to those from the loading pads. Figures 15 and 16 show the vertical displacement profiles along the measuring lines. Derived vertical modulus values from stations type B are given in Table 2; again the results show good agreement with the previous ones. Profiles of horizontal movement are shown in Figs. 17 and 18. The movements are somewhat erratic though again their shape is similar to the vertical displacement distributions and the movement appears to be concentrated at the joints. No modulus values were obtained from these readings.

289

CABLE METHOD OF I N S I T U ROCK TESTING TABLE 1. VERTICAL

MODULI

FROM A-TYPE GROUND POINTS

DERIVED

E x 10 -5

Peg

A4 A5 A6 A7 A8 A9 A10 All A12 A13 A14 A15 A16 A17 A18 A25 A26 A27 A28 A29 A30 A31 A32 A33 A34

o x 100

-

(,~ x

3"95 3.41 3"30 2"86 2.99 3.04 3-10 3.61 3.25 3.16 3.00 3.24 3.76 4'03 5"14 4"86 3.64 3.41 3'00 2"86 3'01 3'23 3"38 4"74 5"06

2 4 0 Tons final loading 7 Tons final loading

z~

80

4

5

Tons firsl loading

E o ~. so

I

..

2

3

6

7

8

I

J

I

I

J

P

9

I0

II

t2

15

14

I

I

15 16

I -'~ "~--d 17

t8

19

Peg position

FIG. 15. Vertical displacement along pegs through pads 1 and 2.

20

290

O. C. ZIENKIEWICZ AND K. G. STAGG

o 190 Tons final loading x ~0

I0 Tons final loading

I00 i

~'- I00 Tons first loading

x

z= '~ a

50

X...--X~X 30 3t

32 33

3,4 35

36

3;"

Peg position

FIG. 16. Vertical displacement along peg lines through pads 3 and 4.

TABLE 2. VERTICAL MODULI DERIVED FROM B--TYPE GROUND POINTS

%

Peg

Ex 10 s

B4 B5 B6 B7 B8

4.98 4'24 3"57 3-16 2'84

o

6 0 Tons final loading

x

2 Tons final loading

A 2 0 Tons first loading

X ° 25 -o

._== X~X

2

$

4

5

6

?

8

9

I0

II

12

13

14

15

16

17

18

i

19 20

Peg position

FIG. 17. Horizontal displacement along peg lines through pads 1 and 2.

i

21

291

CABLE METHOD OF IN S I T U ROCK TESTING

x

o

60

Tons finolloodinq

~:

2

Tons finol Iooding

A 20

Tons first Ioeding

g fi

8o 25 i5

22 23 2,~ 25

26 2/' 28

29

30 51

32 33 34 35 36 57

Peg position

Fie. 18. Horizontal displacement along peg lines through pads 3 and 4.

3. ANALYSIS OF PLATE-BEARING TEST ASSUMING THE STRATA TO BEHAVE AS A TRANSVERSELY ISOTROPIC MATERIAL In such a material it is assumed that the x-axis is oriented in a direction perpendicular to the strata and that the z- and y-axes lie in the plane o f the strata, the relations between stress and strain can be written in a general form [4, 46] --

.----1 I

I

C l l -~- C12 -[- C13

~x

cry I

C12 -[- Ci1 mc C13

Ey

°'x

i I

Cla + C13 + Ca3 TXZ [

Yyz l "rxy [

EZ )' XZ

C44

/

(4)

C44

Yyz

~(Cn

-

Ct2)

where Cll; C12; C33; C44 are the five independent constants. These five constants may be expressed in terms o f engineering constants in the following form :

Clt = ¢-

nE

1 --

292

O. C. ZIENKIEWICZ AND K. G. STAGG

E ,p

C33=- ~ ( 1 - - v l 2) C44 -=~ G

where E ---- the elastic modulus normal to the plane of isotropy n × E = the elastic modulus in the plane of isotropy vl = the Poisson's ratio in the plane of isotropy vz = the Poisson's ratio expressing the effect of strains in the plane of isotropy on strains normal to this plane G = the shear modulus for between the plane of isotropy and directions normal to this plane. Two solutions can be obtained for the problem of a point load on the boundary of a semi-infinite isotropic elastic solid. The first, the Boussinesq solution, as already shown, is for normal loads and results in normal surface displacements of the form P wo =: A -

(6)

I'

where w0 = the normal surface displacement at a radius r from the point of application of a concentrated load P. The second solution is due to C~RUTTI, see Refs. [4, 53]. This gives tangential displacements u due to a concentrated tangential load Q and for the surface displacement may be expressed in the form u0==B

+C~

Q

(7)

where u0 is the tangential surface displacement in the x-direction at a radius r and co-ordinate distance x from the point of application of a concentrated tangential load Q. The constants A, B and C are dependent on the elastic constants E and v and the relation may be expressed as follows: A--

(1 - v~) E

B=

--~E---

(1 - - ~2)

I/

C--

(1 - ~)"

(8)

CABLE METttOD OF IN SIT U ROCK TESTING

293

A similar solution, due to M1CHELL[47], exists for the problem of a point load acting on the surface of a semi-infinite transversely isotropic elastic solid, with the surface parallel to the plane of isotropy. As before the results may be expressed in the form

wo -- A1 P~ ?uo -- B1

(!

q- Cl ~

(9)

Q

(10)

where w0, uo, P and Q have the same meaning as before and A~, B1 and C1 are constants dependent on the five elastic constants. The functional representation of the theory and of the relations between Aa; B1; C1 and the five elastic constants is rather complex. In the Appendix the extension of the original work by Michell is described and the necessary information for deriving the constants At; B1 and C1 from appropriate elastic material constants is given in general terms. For a more special case assuming an interdependence of elastic constants graphs permitting this evaluation to be done simply are given later in this section. The Michell results for point loads in the anisotropic case may be integrated in the same way as the corresponding expressions for the isotropic case to give the displacements due to a load distributed over a given area of the surface. The resulting expressions will have a form similar to those in the isotropic case since the variables under the integration signs have the same form in both cases, but will differ in magnitude by factors of A1/A; B1/B and C1/C in the relevant components. The constants A, B and C are special cases of the more general constants A1, B1 and C1. The equations (9) and (10) for w0 and u0 are not sufficient to determine the five elastic constants and for a complete determination three more independent relations would be required. In fact three more independent relations could be found in terms of displacements in directions normal to the direction of the loads producing them. For practical purposes, however, these additional relations are not very useful since the displacements involved are small compared with the displacements previously discussed and would be difficult to measure with any accuracy in a rock mass under field conditions. Furthermore the relations involve the five engineering constants in complex expressions and it would in any case be difficult to evaluate accurately all five engineering constants from the results of the five sets of measurements. The alternative approach is to assume values for certain of the engineering constants and evaluate the remaining ones from a limited set of measurements. This method is generally applied in the case of assumed isotropy where the elastic modulus is determined from a measurement of normal surface displacement due to a known normal load using an assumed value for Poisson's ratio. This is a particularly favourable situation since the value of A is very little affected by variations of Poisson's ratio within the range of values usually expected in rock masses, but if the same approach was used in conjunction with tangential loading experiments the calculated value of elastic modulus would be more dependent on the assumed value of Poisson's ratio since the constant C is sensitive to the value of v. In the case of a rock mass assumed to be transversely isotropic we have available two practicable experimental systems (a) the measurement of surface displacements in the xy plane due to a tangential surface load and (b) the measurement of normal displacements

294

O. C. ZIENKIEWICZ AND K, G, STAGG

due to normal surface loads. This means that values must be assumed for ihree of the liw.~ engineering constants, and in general values of E and nE will be calcuJated assuming suitable values for G, vl and v2. It is reasonable to assume values for ~,~ and ~,e in the range 0-0-2 as in the case of isotropy, but since the expression for tangential loading is now a ts,, involved we can expect the calculated values of E and n to be more sensitive t¢3 the ehoser~ values of vl and v2 since, by analogy with the isotropic case, we ¢aa expcc~ C~ t~ be ~:,~n siderably dependent on the values of ~'1 and v 2. The choice of a suitable value for G is more difficult. An expression [i)~ (, in term~ oF E, n, vl, and us has been derived by Barden in which

G ....

nE (1 + n ÷ 2v2)

(Jl)

but the derivation of this expression contains some important fallacies. The most important of these is the assumption that the relationship between shear stress on any plane and the shear strain remains governed by the same constant G. This is an arbitrary statement which is not capable of rigorous proof; but nevertheless it may well represent a reasonable assumption of physical behaviour. An alternative assumption is introduced by F6PPL [49] which results in the following expression:

I/G == lIE +l/nE (12) I1

G' :: ( ~ + l) E for the particular case when vl - v2 = 0. This is clearly a special case of equation (11). Taking now the above restrictions and also assuming that the Poisson's ratios are equal to zero permits a determination of the more important elastic constants to be made for the measurements proposed. In particular we can now explicitly express the deformation constants as: //

vl =- v~ : : 0;

G

,(

A1---2-~

(n + 1) E

, ;)

I + ,~ +

(13)

1 ~/[2(1 -+- n)] Ol

2~rE

n

C1 ==0.

Vogt [1] has derived the expressions for the average displacements of rectangular plates due to uniformly distributed normal and tangential loads. For the particular case of a

C A B L E M E T H O D OF IN SITU R O C K T E S T I N G

295

square-loading plate of side length a, the required expressions, using our notation with the assumption (13), can be represented as

x/2)}P/a

= A14{log(1 r- ~/2) + ½(1 --

(14)

x/2)}Q/a

~ : : B14{log(l -}- y'2) [- ~(1

where P and Q are the respective total loads. Simplifying these expressions further gives equations (1) and (2). Curves of values of A 1 and B, plotted against the corresponding values of n are shown in Figs. 19-22.

!3"G!

I

c>sL . . . . . . . . . 0

L .........

0"5

L ...................

I'0

:

1"5

2"C

log ~n

Flu. 19. Values of A1 for n> 1.

1 [

o

log

I 2 i, n

Flu. 20. Values of A1 For n 4 1. 1.0

0'8

0"6

0"4

R

+n/]

B

o-.~P2(I +o)]

0'2

L 0

J

L_

Ol 5

I "0

Iog

l lI}

iOn

Flu. 21. Values of BI for n > I.

"~

El 0

296

*/"

O.C.

Z I E N K I E W I ( ' Z A N D K. G . S T A G G

l

0'5 [42(14n}]

6O

"n'nE ~

4O 3O

B~ 0 5142(1+n)1

2O ~0

o

i lOg ion

FIG. 22. Values of B1 for n< i.

4. CONCLUSIONS The pilot tests carried out on the new 'double-cable' method of in situ testing of rock deformability have proved that it is practicable and that it has important advantages relative to other procedures. Although the tests were not carried out under ideal conditions, consistent values of both normal and tangential moduli of rock were obtained and with a 'participating mass' of rock much larger than in conventional tests. The experimental values obtained in the field averaged 3.7 > 105 and 1.9 :- 105 psi for vertical and horizontal moduli respectively, compared with the modulus of 3.3:2:106 psi obtained from samples of the rock tested in the laboratory, which latter, moreover, did not show marked anisotropy. This brings out the importance of testing representative masses of rock in situ, so that the influence of cracks and fissures can be manifested. At the present stage of development further pilot tests appear unnecessary. The next step is the application of the new technique to an actual site investigation, preferably to a case where alternative methods of test are also specified; only thus can the necessary experience of the method and its further development be achieved. In further tests, improvements of a minor nature can readily be envisaged from the experience so far gained. The use of more rigid loading pads will reduce the errors due to pad compression, for instance. The use of dial gauges appears preferable to that of DEMEC gauges. Particular care must be taken over the rigidity of the bridge from which measurements are taken. The possible economic advantages of the new technique seem clear; an assessment of actual savings compared with other test procedures cannot be attempted here, as it will depend on many factors of the particular situation concerned. Amongst such factors are: the amount of experience the contractor has in stressing and anchoring cables and the existence or not of galleries or boreholes already made for other purposes. An exploration of the site by borehole methods is advantageous geologically as, for a given cost, a much more thorough investigation of the rock formation and its singularities can be undertaken. Under such conditions the use of cables for rock testing is clearly advantageous, as the additional cost is only that of the cables and their anchorages. If, as appears feasible, a detachable cable anchorage can be developed, the same cables can be re-used on a number of test occasions with further economy.

CABLE METHOD OF IN SITU ROCK TESTING

297

Some of the salient advantages of the cable procedure are summarized below: (a) Deformability can be determined in the same directions as the actual proposed loading and in the same location where the actual loads will subsequently occur. (b) With a large size of pad feasible and the simultaneous loadings of two cables, a large 'participating mass' of rock can be tested. (c) Tests can be repeated at various levels of excavation either by removing or protecting the cables during blasting. (d) With the 'double-cable' arrangement, h~ situ shear tests can be carried out with little additional cost. (e) Any method of drilling boreholes can be adopted, such as percussion, since the test is not sensitive to minor inaccuracies in the borehole. (f) Boreholes can be also used for geological exploration if suitably drilled, and with expanding borehole plugs the variation of'local' moduli with depth can be investigated.

Acknowledgement--The authors wish to acknowledge with thanks the assistance and financial support of the Civil Engineering Research Association and the Cementation Co. Ltd. REFERENCES 1. VOGT F. Uber die Berechnung der Fundament Deformation, Det. Norske Videnskops Akademi, Oslo (1925). 2. Trial Load Method of Analysing Arch Dams, Boulder Canyon Project Final Report; Part V--Technical Investigations, Bulletin 1, U.S. Bureau of Reclamation, Department of the Interior, Washington, D.C. 3. BousslyESQ J. Application des Potentials, Paris (1885). 4. TIMOSHENKO S. and GOODIER J. N. Theory o f Elasticity, 2nd edn., McGraw-Hill, New York (1951). 5. WALDORF W. A., VELTROP J. and CURTIS J. J. Foundation modulus tests for Karadj arch dam, J. Soil Mech. Fdns. Div. Am. Soe. Cir. Engrs July (1963). 6. RoCIJA M., SERAHM J. L. and DA SILVEIRA A. F. Deformability of Foundation Rocks, Proceedings o f the Fifth Congress on Large Dams, Vol. III, p. 531, Paris (1955). 7. STUCKV A. Le Centre de recherches pour l'6tude des barrages, Centenaire a l'Ecole Polytechniqtw de Lausanne, p. 106, Lausanne, Switzerland (1953), 8. NONVEILLER E. The determination of the deformation of loaded rock in tunnels, Proc. Yugoslav Soc. Soil Mech. Found. Engng No. 8, 43 (1954). 9. TOURNON G, Sulla determinazione della deformabilita delle roccie in posto, Atti Rass. tee. Soc. In~,. Archit. Torino, NS 8, 15 (1954). 10. SERAHM J. L. Rock Mechanics Considerations in the Design of Concrete Dams, Procee~hngs ~f the International Conference on State o f Stress in the Earth's Crust, Paper 14, Santa Monica, California, May (1963). 11. GlCoT H. Measure de la Deformabilit6 du Sol de Fondation du Barrage de Rosscns, Proceedings o f the Third Congress on Large Dams, Vol. II, Paper R56, Stockholm, Sweden (1948). 12. Foundation Tests at Davis Dam, Saint]Research Lab. Report SPI8 and SP18A, U.S. Bureau of Reclamation, Department of the Interior, Denver, Colorado, (1948); (1951). 13. DELARCrE M. J. and MARXOTTIM. M. Quelques probl6mes de m6canique des sols au Maroc, Annls Inst. tech. Bdtim p. 37, Sept. (1950). 14. KUJUNDZIG B. Survey of the methods of experimental investigation of mechanical characteristics of the rock masses in Yugoslavia, Institute for Development of water resources, Belgrade, Jaroslav Cerni Transl. No. 26 (1963). 15. JOVANOVICt . Ispitivanje elasticnik osobina stene uredajem S. Dva jastuka Saopstenja Hidroteknikog Inst. Jaroslav Cerni Cr. 2 (1955). 16. JOVANOW¢ L. Ispitivanje elasticnik osobina stene uredajem S. Dva jastuke Saopstenja Hidroteknikog Inst. VIII Cong. Jugoslovenskog drustva za mechaniku tha i fitndivanje, Cr. 2 (1959). 17. ZmNKXEWlCZO. C. and STAGGK. G. In-situ Testing o f Rock Deformability, Report to the Civil Engineering Research Association, London (1965). 18. KUJUNDZIC B. Anisotropie des Massifs Rocheux, Proceedings o f the Fourth Institute Conference on Soil Mechanics and Foundation Engineering, London (1957). 19. KUJUNDZlC B. Mesure de caracteristique des roches en place, Annls Inst. tech. Bdtim, No. 125 (1958); Revue Ind. mindr. No. 10 (1957).

298

o.c.

ZIENKIEWICZ AND K. G. STAGG

20. WESTERGAARDB. On the elastic distortion of a cylindrical hole by a localis,~d hydrostatic prc,~t~c Theodore yon Karman Anniversary Volume, California Institute of Technology (1941). 21. IVKOVICH M. Deformation of cylindrical tube of infinite length by a radially symmetrical loat[ ~)~ ~ finite section, Sl. Radova No. 2, Belgrade (1953). 22. IONOVV. N. Equilibrium of an elastic thick-walled tube under internal pressure applied to a sectio~ o|' its length, Yestnofk M G U No. 5, (1965). 23. FREY-BARO. Die 7 Dehnungs messungen in Druckstollen des Kraftwerkes Lucend~y Schweiz. t~a,:1. 65, (41) 557 (1947). 24. OBERTIE. and VERDUCCIE. La galleria forzata dell impian to di lovero, Energia Elett. p. 95, Feb. (1949). 25. BERNARDP. Mesure des Modules l~lastiques et Application au Calcul des Gal6ries en Charge, Proceedings of the Third Institute Conference on Soil Mechanisms and Foundation Engineering, Vol 11, p. 145, Zurich (1964). 26. OBERTI G. Richerche sperimentali sulla deformabilita della roccia di fondazione della diga del Piave, G. Genio cir. No. 1l, 607 (1948). 27. LE~RRmR G. Le comportement du rochcr dans les gal6ries blind6es, Houille blanche |iA, March/April, 144 (1956). 28. KUJUNDZlC B. Metode eksperimentalnog odredivanja modula etasticnosti stene, Na.~c Grader. Cr. 3 (1956). 29. MULLERW. Un appareil pour mesurer les d6formations des tunnels et pour determiner I'elasticit6 de !a roche, lngenieria, B. Aires March/April, 79 (1947). 30. LAZEREVICD. and KUJUNDZICB. Mechanical characteristics of mountain masses. P~oc. Yugoslav Soc. Soil Mech. Found. Engng No. 7 (1954). 31. KUJUNDZICB. M6thodes de D6termination l~xp6rimentale du Module d'l~la'.;ticit6 des Roches, ~vmpo;ium sur l'observation des Ouvrages, No. 17, Rilem, Lisbon (1955). 32. LAUVERH. Ein Gerat zur Ermittlung der Flesnach-giebigkeit fur die Bemessung on Druckstollen-und Druckschachtauskleidungen, Geologic Bauwes. 2/3 (1960). 33. MAYERA. et al. Mesure des Modules de D6formation des Massifs Rocheux dans les Sondages, Proceedings of the Eighth Congress on Large Dams, Edinburgh, Paper R.16, May (1964). 34. KUJUNDZICB. Eksperimentalus ispiosobina stene uredajem sdva justuka, V1H Jugoslovenskog Drustua Mekaniku Fundiranje Cr. 2 (t959). 35. Fox P. P., MAYERA. A. and TALOBREJ. A. Foundations of the Pablavi Dam on Dex River, Proceedings of the Eighth Congress on Large Dams, Edinburgh, Report R.1 (1964). 36. MASONW. P. Piezo Electric Crystals and their Application to Ultrasonics, Van Nostrand N.Y. (1950) 37. JONESR. Nondestructive Testing of Concrete, Cambridge University Press (1962). 38. EvISON F. F. The seismic determination of Young's modulus and Poisson's ratio for rocks ht-situ, Gdoteehnique 6, (3) (1956). 39. JANOD A. and MERMIN P. La mesure de characteristiques du rocher en place a l'aide du dilatometre a verim cylindrique, Travaux 610-612 (1954). 40. ONODERAT. F. Dynamic Investigation of Foundation Rocks ht-situ, Proceedings of the l~Tfth Symposh#n on Rock Mechanics, Minnesota, Pergamon Press (1962). 41. SUTHERLANDR. B. Some Dynamic and Static Properties of Rock, Proceedings of the [-iJi'h .S),mposium on Rock Mechanics, Minnesota, Pergamon Press (1962). 42. GREC~ORYA. R. Shear Wave Measurements of Sedimentary Rock Samples under Compression, Ptvcecdings of the Fifth Symposium on Rock Mechanics, Minnesota, Pergamon Press (1962). 43. RINEHARTJ. S., FORTIN J. P. and BURG1N L. Propagation Velocity of Longitudinal Waves in Rocks. Effect of Stress, Stress Level of Waves, Water Content, Porosity, Temperature, Stratification and Texture, Proceedings of the Fourth Symposium on Rock Mechanics, Pennsylvania State University, U.S.A. (1961 ). 44. GRINE D. R. and FOWLESD. H. The Attenuation of Shock Waves in Solid Materials with Seismic Applications, Proceedings of the Third Symposium on Rock Mechanics, Colorado School of Mines, Golden, Colorado, U.S.A. (1959). 45. FO(3ELSOND. E., ArrcmsoN T. C. and DUVALL W. 1. Propagation of Peak Strain and Strata Energy from Explosion Generated Strain Pulses in Rock, Proceedings of the Third Symposium on Rock Mechanics, Colorado School of Mines, Golden, Colorado, U.S.A. (1959). 46. JAEGERL. C. Elasticity, Fracture and Flow, Methuen, London (1956). 47. MICHELL J. H. The stress in an aeotropic elastic solid with an infinite plane boundary. Proc. Lond. math. Soc. 247, June (1900). 48. BARDENL. Stresses and displacements in a cross-anisotropic soil, G~otechnique 198, Sept. (1963). 49. F~JPPL L. Drang undZwang 2, (1944). 50. KUJUNDZXCB. Prilog eksperimentalnom odredivanju modula elasticnosti stene in situ, Nase Grader. Cr. 8 (1954). 51. TALOBREJ. La Mecanique des Roches, Paris (1957). 52. JAEgeR C. Rock mechanics for hydro power engineering, Wat. Pwr Sept./Oct. (1961). See also FER~USON F. F. Proceedings of the Eighth Congress on Large Dams, Edinburgh, May (1964).

CABLE M E T H O D OF IN SITU ROCK TESTING

299

53. LOVE A. E. H. Mathematical Theory o f Elasticity, 4th edn. 54. CHAPMAN E. J. K. Pressure Tests in Rock Galleries for the Ffestiniog Pumped-Storage Plant, Proceedings o f the Seventh Congress on Large Dams, Rome, Paper R.22 (196l). 55. TALOBR~ J. La D6termination ExpSrimentale de la R6sistance des Roches d'appui des Barrages et des Patois de Souterrains, Proceedings o f the Seventh Congress on Large Dams, R o m e Paper R.37 (1961). 56. MULLER L. Die Geomechanik in der Praxis des lngenieur und Bergbaues, Geolgie Bauwes. 25, 2/3 (1960), 57. LANE R. G. T. and ROFF J. W. Kariba Underground Works. Design and Construction Methods, Proceedings o f the Seventh Congress on Large Dams, Paper R.16 (1961).

APPENDIX Michell's Solution f o r Transverse lsotropy

Michell uses five elastic constants denoted by A, C, F, L and N which are related to the constants defined in the present text such that : A -- Cll ~ tie 1 -- v2f_, ') ~9 It

C--

C:~.= E ( I -- vl ~)

F :- Ch~ = E v,,(1 -k ,'1)

L = C44 - - G N : I(CI1 -- C ] 2 ) -

where

~=

(I -~- vl) 1 - , ' 1 -

nE 2( 1 -f ,q)

V22,1 /" n

'

Intermediate constants are also introduced defined as: ql,',_=

F + L × A -k pl, ~L A

"~1, 2 ~

where

L -f pl, ~(F q- L) A

p~-L(F q L) + p{(F-[ L) 2 ~ L - A C } + L ( F q- L) :~ 0

and

a+f3=F ~q~ ~ flq,,. = C -- F.

Taking the surface as the plane y = 0 with the x y plane being tile plane of isotropy and a surface load whose c o m p o n e n t s are az. rz., rzu the displacement w in the y direction is given by

td+

(q',- -- ql) d z = q 2 \ d x - b

where

dy -k dz J - - q 1 \ d.r + dy q-

2,~d ~/7~ V~,

¢"~r, dx'dy'

300

O. C. ZIENKIEWICZ AND K. G. STAGG

¢1 : :

1

II

~/'yly2

1

dx'dv"

r~

"

7; u

2rra V'Y'-'- "~/yl

Xt .... I "v/YtY2 [ ~ e ] d x ' d v ' 2rra ~/72 -- V'71 J J "r~ " and

rl e -=

(x

~ O'

x') e

-

y3 e -~ re'

¢2, ¢2, )(2 and r2 are obtained by interchanging the suffixes 1, 2, and writing/3 for a. Michell shows that for a point load ,rz = 14/at the origin with r~. = r~u = 0 the surface displacement

W0 ~

~/A {(v/AC + L)'-' - (F + L)2}~ W . . . . . . . . . . . 2rr~/L ( A C - - FZ) r

•

The lateral displacement u in the x-direction in the case of general surface loading is given as:

, d td¢~'" d¢~"] ~-~tZ q- dx ]

~,- ( q z -- q , ) d y

where .

. . . 2~ra ~/72 - - ~/Yl

rzx

] z

z

ra) - - r j } dx'dy"

¢1% )(1" are found by substituting r ~ and ~ for ~-~ and fro", @e", )(2" are found as above and 1

~4', being obtained by replacing ~..~ by r.y y a ==:

L N

rz 2 = (x + x ' ) 2 - t - ( y ~ y,)2 I - 2 " .

This expression for u cannot generally be conveniently expressed in terms o f the basic constants. A n exception to this is the special case discussed in the main text.