An approach to large-scale field stress determination.pdf

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An approach to large-scale field stress determination

A.N. GALYBIN*, A.V. DYSKIN, B.G. TARASOV and R.J. JEWELL

The University of Western Australia, Nedlands, WA, 6907, Australia

(Received 15 February 1999; revised and accepted 28 September 1999)

Abstract. The construction of stable structures in rock masses requires knowledge of the in situ

stresses at the scale of excavations. However, the measurements obtained by the conventional overcoring

technique are related to a small scale (centimetres). To extrapolate them to the scales of interest to rock

mechanics (from meters to kilometres) requires a large number of individual stress measurements,followed by statistical analysis to avoid a considerable scatter of the measured values. In this paper, a

method is proposed based on (a) large-scale surface stress and modulus measurements using the

cylindrical jack method complemented by a special measuring scheme and then (b) back analysis for a

given excavation shape. The method allows the simultaneous reconstruction of the stress components at

the scale of excavation. A numerical simulation for a cylindrical excavation in an isotropic rock mass

demonstrates the high accuracy and robustness of the method. The presence of a fractured zone

surrounding the excavation can hamper the stress reconstruction, hence special measures should be taken

to conduct the measurements in competent rock.

Key words: in situ stress, in situ moduli, cylindrical jack, cylindrical excavation, back analysis

Introduction

Knowledge of the stress fields acting in the rock mass corresponding to both the

original (pre-mined) regional stress state and the local stress concentrations caused

by mining is necessary for the control of the fracture processes in existing

excavations and for the design of new ones.A number of direct and indirect methods of stress determination exist; however,

currently the most popular method of in situ stress measurements is the overcoring

method. The method is well established and (in its best variant) allows the

determination of the complete 3-D stress tensor at a single point remote from an

excavation. However, any one particular stress measurement is unlikely to be

representative due to a considerable scatter obtained from different measuring

locations (e.g., Cuisiat and Haimson, 1992). Thus a major drawback in the existing

overcoring technique can be identified as follows: the irrelevance of the resultsobtained by a small measuring device (tens of centimetres) to the scales of interest

for mining operations (from meters to kilometres).

* To whom correspondence should be addressed at: Tel. 61 8 9380 2631; fax: 61 8 9380 1044;e-mail: [email protected]

Geotechnical and Geological Engineering 17: 267289, 1999.

2000 Kluwer Academic Publishers. Printed in the Netherlands.267


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There are two ways to overcome this problem. The first is to undertake a number

of measurements spread over the area in question from which the mean value of the

stress components can be determined. However the number of measurements

required can be rather large. Indeed, if is the range of the stress variation within the

area and N measurements are conducted at random and at independently chosen

points, then the standard error of the stress reconstruction will be N1/2. In

particular, if the local stresses vary by 100% (which is not unusual given that, at a

contour of a circular hole under uniaxial compressionp, the stress concentration may

vary from p to 3p, i.e., by 400%), measurements atN 100 points are required to

reach an accuracy of 10%. Thus, reaching the proper scale requires a large number

of point measurements and becomes a very expensive exercise.

The second way is to use larger measuring devices. Cuisiat and Haimson (1992)

presented an illustration of how the scatter reduces when the diameter of the drilling

holes in the overcoring method increases. However the possibility of enlarging the

overcoring diameter is restricted. Measurements at the larger scale can, in principle,

be obtained by the flat-jack (slot) method in which the scale of the measurements is

determined by the slot dimensions; the volume of rock involved in the measurement

is of the order of 12 m3 (e.g. Goodman, 1989). However the method is laborious

since in its conventional form each single slot allows the determination of only one

stress component.

A potentially attractive method for obtaining a representative value for stresses ata larger scale is the under-excavation technique (Wiles and Kaiser 1994), in which

the driven excavation itself is used as a measuring device. In this situation the scale

of stress determination corresponds to the dimensions of the excavation. The method

is based on displacement monitoring as the excavation advances and then the

recalculation of the actual in situ stresses by back analysis based on a 3-D elastic

solution.

However, the under-excavation method also has some shortcomings. First, it

requires the knowledge of the rock mass deformation characteristics which areobviously different from the ones obtained in the laboratory. Thus, the determination

of the deformation moduli at the relevant scale remains a problem. This could be

overcome by complementing the displacement monitoring with stress measurements

around the excavation. However, if they were to be obtained by the overcoring

technique, the stresses determined would be at a scale much less than the one

involved in the elastic solution. This would still necessitate a considerable number of

measurements. In addition, the displacement measurements themselves would

require drilling of measuring boreholes ahead of the excavation face which mayinterfere with the mining operations.

A method of large-scale near surface stress measurements has been proposed by

Galybin et al. (1997). The method is based on the cylindrical jack method (Dean and

Beatty, 1968) complemented with a special measuring scheme ensuring the relevant

size of the measurements. In the original variant employed by Dean and Beatty only

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radial displacements were measured and the measuring pins were located relatively

close to the borehole. Shortcomings with the chosen system of measurement have

been identified and include:

1. The relatively close positioning of the measuring pins to each other and to thehole restricted the scale of stress measurements and affected accuracy.

2. The measurement system interfered with the drilling operations and therefore had

to be performed by removable deformometers.

3. Poissons ratio could not be determined and had to be obtained from separate

laboratory tests.

The scheme proposed by Galybin et al. (1997) eliminates these shortcomings and

allows the reconstruction of the deformation characteristics of rock at the scale ofexcavation.

This method is still restricted by the scale of measurement (possibly 12 m) and

also because only near surface measurements may be undertaken. The latter short-

coming is common for all types of near-surface measurements which can recover the

2-D stress tensor related to the surface of measurements. Hence, several measure-

ments from differently oriented surfaces with further calculations are required to

determine all stress components from the back analysis. This will allow a true large-

scale stress reconstruction.

In general, a consistent approach to the determination of the stress state at the large

scale should contain the following stages:

The development of more informative and less expensive methods of stress

measurements in excavations.

The performance of numerical (statistical) and laboratory testing of the proposed

methods of single measurements for determining their accuracy.

The development of the back analysis methods for the determination of the 3-D

stress state at an appropriate scale.

The present study attempts to address all these stages and to outline the problems

that have yet to be solved. It is proposed to use the modified cylindrical jack method

(Galybin et al., 1997) as a basic method for the determination ofin situ stresses and

deformation characteristics of rocks. This method enables the reconstruction of 2-D

stress tensors at the surface of an excavation simultaneously with elastic moduli at

the corresponding scale. The next two sections are devoted to careful analysis of this

method from statistical and instrumental points of view. The theory of stress and

moduli recovery is developed for isotropic and anisotropic rocks. A test on an

aluminium sample modelling a solid isotropic rock is performed and analysed.

Feasibility and accuracy of the stress state reconstruction at the scale of a mine is

then investigated on the basis of back analysis. The necessary corrections due to the

presence of a fracturing zone around the excavations are also discussed.

269LARGE-SCALE FIELD STRESS DETERMINATION


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Measuring scheme for the cylindrical jack method

Surface stress and modulus measurements

Three characteristics of an in situ stress field can be determined by measurements

near a free excavation surface. They are two principal stresses, 1, 2 (3 0 on the

free surface) and their orientation at the surface. There are also deformation

characteristics of the rock which generally need to be determined. For isotropic

elastic rock there are only two of them: Youngs modulus,E, and Poissons ratio, .

The problem of orthotropic rocks will also be considered in this paper for a particular

case when the surface of measurement is parallel to a plane of symmetry. In this case

one has a 2-D problem in which the number of independent moduli is four(Lekhnitskii, 1968). Therefore, in the case of isotropy, five parameters characterizing

the stress-strain state have to be determined: three dimensional parameters (1, 2and E) and two dimensionless parameters and . This means, that in an ideal

situation five independent measurements should be performed. The minimum num-

ber of independent measurements in the case of orthotropic rocks should obviously

be equal to seven. In both cases at least one of them must have the units of stress,

which provides reconstruction of stress magnitudes and Youngs moduli.

The cylindrical jack method indeed provides an independent measurement ofstress: the jack pressure. However, the variant employed in the 1960s by Dean and

Beatty (1968) had some shortcomings. These mainly arose from the measuring

scheme adopted which was (1) based on monitoring only radial displacements, and

not sufficient to recover Poissons ratio (this would require at least a measurement in

a perpendicular direction); and (2) the measuring pins were located close to the bore-

hole contour, hence the scale of measurement was only of the order of the borehole

diameter.

A new measurement system proposed by Galybin et al. (1997) has been developed

to overcome these shortcomings and thus enables in situ stress and full moduli

determination over a larger scale. The key innovation is the proposed arrangement of

measuring points. It will be described in the next section.

Since a single installation can recover only a 2-D stress tensor related to the

surface of measurements, a number of near surface measurements should be

performed for reconstruction of the complete 3-D stress tensor around an excavation.

For elastic rocks at least two installations on differently oriented surfaces are

required to compute all stress components. Locations of these measurements have to

be chosen to ensure applicability of the correspondent mathematical model to the

stresses recovered. For instance, in rocks consisting of thin horizontal layers it might

be necessary to use both the isotropic and orthotropic models of rocks depending on

the location of the measuring device. On all planes perpendicular to the layers, the

relationships for an orthotropic medium should be applied for calculation of the 2-D

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stress tensor, while on the planes parallel to the layers it is sufficient to employ an

isotropic model.

Existing scheme

Dean and Beatty (1968) used the following scheme:

1. Holes were drilled for pins at the vertices of an octagon.

2. Pins were inserted and deformometers located to measure distances in four radial

directions.

3. The deformometers were removed and a central hole drilled.

4. The cylindrical jack was inserted into the hole and the deformometers reinstalled.

5. Contractions between the radially-situated pins were measured.

6. Changes in radial distances were monitored during pressurizing of the hole.

7. The values of stresses and the shear modulus were obtained from a least squares

analysis of the measurements.

Mathematically the method was based on the Kirsch solution for a circular hole in

an elastic isotropic plane under plane-strain conditions. The changes in displacement

due first to drilling the hole of radius R and then pressurizing it to a pressure Q can

be presented in the following combined form using complex representation:

ur(r, ) iu(r, )R2

2Gr12

2Q

12

2 e2i()1R2

r2e2i() (1)where (r, are the polar coordinates with the origin situated at the hole centre, ur iuis the complex expression for the components of the vector of displacements in polar

coordinates, G is the shear modulus, 34, is Poissons ratio, 1 and 2 are

the remote principal stresses, is the angle between the first principal direction and the

x-axis of the reference coordinate frame, (x,y) as shown in Figure 1.

Equation (1) contains five unknown parameters: moduli of deformability (G, ) andthe characteristics of the in situ stress, (1, 2, ). In order to recover these parameters it

is sufficient, in principle, to perform four independent displacement measurements in

different directions after drilling the hole (Q 0) and at least one measurement (one

displacement component) after pressurizing the hole.

It should also be noted that the difference in the displacements before and after the

pressurizing has the form

[u

r(r, )

iu(

r, )]before [

ur(

r, )

iu(

r, )]after

R2Q

2r (2)

Therefore by knowing the pressure magnitude,p, and the difference of the radial

component of displacement, it is possible to determine the shear modulus directly (see

also Dean and Beatty, 1968), while the tangential component should remain unchanged

which could be used for controlling the quality of measurements.



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Proposed measuring scheme

As one can see from equation (1), if other displacements in addition to radial displace-

ments were measured then it would be possible to determine the second deformation

modulus in isotropic rocks. It will be seen further that this gives a possibility to

determine all elastic moduli in the case of orthotropic rock as well. However, the

measurement of both radial and tangential components of displacement at the excava-

tion surface requires special equipment and is therefore not practical in the current

mining environment.

A more realistic operation is to measure the displacement (elongation/contraction)along a line. The sensitivity and resolution of these measurements should provide

sufficient accuracy. At the moment there are few devices for conducting such measure-

ments in field experiments: optical measurements, vibrating wire strain gauges or

LVDT transducers. In laboratory tests (described later) gauges in the form of pre-bent

steel strips were used. Applicability of all these or other methods to the field conditions

should be carefully investigated.

The minimum number of measurements performed depends upon rock character-

istics. For isotropic rock at least four extensometers oriented in different directions arerequired. This will provide four independent measurements of the deformation change

after the hole is drilled and another four after pressurizing (to provide a four-

fold redundancy to determine the shear modulus). For the determination of the shear

modulus the best orientation of the extensometers would be in radial directions accord-

ing to equation (2). However, as may be seen from equation (1), the radial displacement

Figure 1. Proposed measuring scheme.

272 GALYBIN ET AL.


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measurements ( const) alone are not sufficient to separate 12 and from their

product. Therefore some measurements are required in directions other than radial.

Equation (1) also shows that if the 2-D in situ stress field is isotropic (12) then

the parameter and, hence, the Poissons ratio will not affect the displacements and

therefore these parameters will not be determinable. This also means that for all stress

fields close to isotropic stress states, the accuracy of the determination of Poissons

ratio will be low. Nevertheless, this does not pose a considerable problem, since in this

case the influence of Poissons ratio on the stress field is minor.

A similar situation can be observed for anisotropic rock as well. The only difference

is that the number of measurements should correspond to the number of elastic moduli.

For orthotropic rock it should be not less than six (performed in different directions

after drilling the hole) plus an extra measurement after pressurizing.

For this reason and because of natural errors, the number of measurements should be

greater than the required minimum in order to provide some redundancy. The measure-

ment system should also not interfere with the drilling and pressurising operations. To

satisfy these constraints, the symmetrical arrangement shown in Figure 1 has been

suggested. In this arrangementNmeasurements are obtained in radial directions (lines

A1C1, A2C2, . . ., ANCN in Figure 1(a)) andNmeasurements in the circumferential

directions (lines B1B2, B2B3, . . ., BNB1 in the figure). The number of measuring

pairs,N, is chosen to satisfy the redundancy requirement and to provide the desirable

accuracy of the stress and moduli determination. The choice of the extensometer baselength is a compromise between the necessity to ensure an appreciable scale of meas-

urements and the sensitivity of the extensometers. The lengths of radial and circum-

ferential gauges may be different. It is anticipated that the best accuracy can be

achieved if the radial gauges are located closer to the hole and hence have their lengths

less than the radial gauges. However the difference in length should not be great to

ensure the same scale level of measured data.

The proposed measuring procedure mainly follows the Dean and Beatty scheme

described above but with the essential difference that extensometers do not have to beremoved during the drilling operations and pressurizing of the hole. This obviously

increases the accuracy.

To avoid the influence of the fractured zone around the excavation, pins should be

inserted deep enough to be located in competent rock, Figure 1(b), otherwise the results

of measurements will reflect properties of the fractured zone only and, thus, cannot be

used directly for recovering the complete 3-D stress tensor.

Procedure of the stress and moduli reconstruction

Variation of the distance between the pins

Let Pm(xm, ym) be coordinates of pins inserted into the rock before drilling the hole

and lmnPmPn be a vector connecting any two pins. Let the displacement vector



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V(x,y) {u(x,y), v(x,y)} after drilling the hole be known at any point (x,y). Then

small length variations, mn, of the vectors lmn can be expressed through the scalar

product

mnVmnlmn (3)

Here

lmn {cosmn, sinmn}, VmnV(Pm)V(Pn) (4)

where ij is the angle between the vector lmn and the positive direction of thex-axis.

For the arrangement shown in Figure 1(a), the relative elongation (contractions)

between pins can be expressed by a single index, k 1 . . .N. Complex coordinates of

the pins are

Ak r1 exp2ikN

, Bk r2 exp2ikN

, Ck r3 exp2ikN

(5)Hence, for circumferential extensometers, Equation (4) takes the form

lk i expi 2k 1N

, VkV(Bk1)V(Bk) (6)Similarly for radial extensometers one has

lkN expi 2kN, VkNV(Ck)V(Ak) (7)

The elongations/contractions between the pins can now be expressed by only one

formula containing conjugated displacement vectors Vk u(xk,yk) iv(xk,yk)

kRe(lkVk), k 1 . . . 2N (8)

The next step is to express the displacement vectors Vk through stress components

and deformation characteristics of rocks. This requires some assumptions about rock

behaviour under loading. It will be assumed further that rocks are linear elastic and

homogeneous and may be either isotropic or orthotropic. This assumption essentially

simplifies the problem of stress reconstruction. In these cases the displacement vector

linearly depends on stress components, hence the deformation properties can be deter-

mined by a certain number of elastic moduli only, but not by a stress-strain curve. The

applicability of these assumptions to a real situation can be justified from the data

obtained during pressurizing the hole. Equation (2) demonstrates this for an isotropic

rock. If the dependence of measured elongations upon the jack pressure is linear then

the shear modulus is obviously constant, otherwise it is a function of the load and,

hence, the assumption of the linear rock behaviour should be discarded.

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Isotropic rocks

For isotropic rock the stress-displacement dependence is given by equation (1). When

new parameters given below are introduced

P21

2,

21

2cos2,

21

2sin2 (8)

the displacement vector in Cartesian coordinates can be expressed as follows

u(x,y) iv(x,y)R2

2Grei P Qe2i( i) 1R2r2e2i( i) (9)

Reconstruction of the shear modulus

First, the shear modulus G can be recovered on the basis ofequation (2). If pressure Q

has been applied to the hole the difference in the extensometers readings would be in

accordance with the displacement field

urQ

2G

R

r, u 0 (10)

whereR is the hole radius. Radial displacements create elongations of radial extens-

ometers (vectorr) and contractions of the circumferential extensometers (vector c).Then the shear modulus can be expressed separately for radial and circumferential data

by the following equation

GrQ

2

R

r3 r1

D

r, Gc

Q

2

R2

r22

d

c(11)

Here D and d are the lengths of the radial and circumferential extensometers

correspondingly, r and c are the mean values of the vectors r and ccorrespondingly.

Equation (11) can be generalized if displacements are continuously monitored dur-

ing pressurizing the hole up to the pressure Q. This can be done by the standard method

of linear regression. In this case, due to a large volume of data involved the shear

modulus is obtained with the greatest accuracy.

Reconstruction of in situ stresses and the Poissons ratio

It is assumed that shear modulus has already been found. In this case the jack

pressure can be assumed to be zero, Q 0. Then the problem contains four unknownparameters: P, , , . The number of readings available is equal to 2N. If the

number of independent readings is much greater than four then the readings are

known with sufficient redundancy, and the problem of simultaneous determination of

the moduli and stresses can be reduced to the problem of minimization of a proper

functional in the space of four variables.



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Let be a vector of readings and W(P, , , ) be the vector of displacements

given by the right-hand side of equation (9) then the problem becomes

F(P, , , ) W(P, , , ) min (12)

Here . . . stands for a norm.

Orthotropic rocks

Orthotropic rocks (foliated rocks or rocks with orthogonal sets of fractures) will now

be considered. In this 2-D consideration, it is assumed that the surface of measure-

ment is parallel to a plane of symmetry (i.e. a fracture or foliation plane). The

reference coordinate system can be introduced such that the x-axis is directed along

the foliation.Displacements in an orthotropic plane with a circular hole or radius r0 can be

expressed through complex potentials 1, 2 as follows (Lekhnitskii, 1968)

u(x, y) 2 Re[p11(x, y)p22(x, y)]

v(x, y) 2 Re[q11(x, y) q22(x, y)](13)

The following combination of the elastic moduli are introduced in equation (13)

pk (2k1)E

11 , qkk1E

11

2k E

12

k 1E12G (1)k 1E12G2

E1

E2

(14)

whereE1,E2 are Youngs moduli along the principal directions of elasticity, 1 is the

Poissons ratio in the x-direction, and G is the shear modulus which characterizes

changes of angles between principal directions.

If the plane is subjected to biaxial compression at infinity (1, 2) and internal

pressure Q inside the hole, the potentials have the following form

k(x, y)Akk(x, y)1, k 1, 2 (15)

Here A1, A2 are constants as follows

A1R

2

(P Q)(i2) ( i)(i2)

12

A2R

2

(PQ)(i1) ( i)(i 1)

12

(16)

Parameters P, and are determined in equation (9). Functions k have the

following form

k(x, y)xky (xky)2 r20(1k)2

r0(1 ik)(17)

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Parameters P, , and any four independent moduli in Hookes law (or their

combinations), form the full set of unknown parameters in the orthotropic case. The

following moduli 1, 2, a11E11 , a121E

11 will be used as independent

parameters in the example later.

Similar to the isotropic case one can minimize the following functional

F(1,2, a11, a12, P,, t) j 2 Re 2

k1

(pkcosj qksinj)Akk,j (18)where j is the angle betweenj-extensometer and the positive direction of thex-axis

of the reference coordinate system, functions k,j are defined as follows

k,jk(xj1, yj1)1k (xj, yj)

1 (19)

It should be noted that for orthotropic rocks, in contrast to the isotropic case, noneof the deformation moduli can be found from a separate test.

Examples of the reconstruction stresses and moduli

Numerical analysis

General

In order to illustrate the proposed technique the following examples are consideredfor both types of rocks (isotropic and orthotropic). In these examples the moduli and

the stress states have been specified, the ideal readings of the extensometers

computed and then distorted by introducing random errors to check how well the

original stress state and the Poissons ratio can be reconstructed using the proposed

method.

In both cases the same 2-D stress tensor has been used for calculation of ideal

readings. The principal compressive stresses at infinity, 132 MPa, 218

MPa, are inclined to the x-axis at an angle 20.The following geometrical parameters were used in both cases as independent

ones: hole radius, R 0.04 m, length of circumferential exensometers (i.e. B1B2,

B2B3 etc), d 0.2 m. The length r2 is to be determined in accordance with the

number of extensometers and their length. For the proposed scheme (Figure 1)r2 is

given by a simple formulae d 2r2sin(/N). Other distances shown in the figure are

r1 r2 d/2, r3 r2 d/2.

The reconstruction was performed with the number of measuring pairs N 5,

6, 7, 8.

Isotropic rocks

The accuracy of the determination of the shear modulus is not analysed here, as it

will obviously be the best one since it uses all 2Nmeasurements for determination of

this single parameter.



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Consider now modelling of the readings. The following moduli are chosen:

E 81.9 GPa, 33.5 GPa which yields 2.11 ( 0.222). These data corres-

pond to granite and were taken from Lama and Vituruki (1978).

The readings are modelled as the ideal ones computed for the above stress states

using equations (6) and (9) and then distorted by adding to each of them an

independently generated random error uniformly distributed within the range of

(10%, 10%). That is if the ideal readings form the vector Y with the components

Yj, then the readings used for the reconstruction will have the components

Yj(1 0.1j) where j, j 1, ..., 2N are independent random variables uniformly

distributed over the interval (1, 1).

For each number of the measuring pairs, 50 sets of readings were independently

generated, and the values of P, , and were reconstructed for every set ofreadings. The averages over 100 sets of readings and the variation coefficients (the

standard deviation over the average) were then computed.

The comparison of principal stresses and the principal direction computed from

the average values of the reconstructed P, , with the original ones are shown in

Table 1. Variation coefficients representing the relative errors of the reconstruction

versus the number of the measuring pairs Nare also shown in Table 1.

For this isotropic case the relative errors in the reconstruction of and , which

are proportional to the differences between the principal stresses, can be high, butreduce with an increase in the number of measurement pairs. On the other hand, the

hydrostatic component of the stress state is reconstructed with high accuracy in all

cases. An examination of the relative errors shows that a reasonable compromise

between the accuracy of reconstruction and the minimization of the number of

measurements can be achieved with N 6.

Orthotropic rock

Similar to the isotropic case, a numerical experiment has been undertaken to estimatethe accuracy of the stress and modulus reconstruction. The assumed moduli correspond

to a schist (Lama and Vituruki, 1978):E1 63.4 GPa, E2 20 GPa, G 7.9 GPa;

the Poissons ratio is 1 0.067.

In contrast to the isotropic case the data for pressurizing the hole has also been

modelled. Two values of the cylindrical jack pressure were used: Q 0 and 10 MPa.

Table 1. Average reconstructed stresses and modulus compared to the original ones.Variation coefficients (%) are given in brackets

N

Number of measurement pairs, N

5 6 7 8

Original

valuesP (MPa) 25.0 (2.0) 25.0 (2.4) 25.0 (1.6) 25.0 (1.5) 25.0 (MPa) 5.28 (8.9) 5.39 (8.0) 5.34 (5.1) 5.41 (4.5) 5.36 (MPa) 4.39 (11.5) 4.54 (6.7) 4.57 (6.9) 4.54 (4.5) 4.50 2.15 (6.9) 2.09 (5.7) 2.09 (5.0) 2.07 (4.7) 2.11

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The function (equation 12) written for the Euclidean norm was minimized using a

procedure built into MATHCAD 6 PLUS with the initial value obtained from the

exact values of the parameters randomly disturbed within 30%. It should be

emphasized that the minimization procedure used is very sensitive to the initial

value, which suggests that a search for a more robust minimization method is

necessary.

For each number of measuring pairs, 50 sets of readings were independently

generated, the values of P, , and 1, 2, a11E1

1 , a121E1

1 were recon-structed for every set of readings. The averages over 50 sets of readings and the

variation coefficients (the standard deviation over the average) were then computed.

The principal stresses and the principal direction computed from the average

values of the reconstructed parameters may be compared with the original ones from

the results presented in Table 2 for different numbers of measuring pairs N.

The variation coefficients representing the relative errors of the reconstruction are

shown in Table 2 in brackets. It can be seen that the errors do not vary much when

N changes from 5 to 8, which suggests that the minimum number of 2*2*5 20

measurements already gives sufficient redundancy to determine the 7 unknown

parameters. It should be noted that reconstruction of the parameter a12(, E11 ) is

subjected to large errors, although the average values are close to the exact ones.

Laboratory test

Aluminium sample

The main aim of laboratory testing was to check the reliability and stability of thegauges and the cylindrical jack during the test. Aluminium was chosen to manu-

facture the sample because its Youngs modulus is close to the moduli of strong solid

rocks. Thus, something similar to hard field conditions were modelled in the

experiment. An aluminium sample having dimensions 0.2 0.2 0.085 m was

tested. The sample had a hole in its centre of radius R 0.0125 m.

Table 2. Average reconstructed stresses and moduli for orthotropic rock compared to the original

ones. Variation coefficients (%) are given in brackets

N

Number of measurement pairs, N

5 6 7 8

Original

valuesi1 2.78 (2.2) 2.75 (1.3) 2.70 (1.8) 2.74 (1.9) 2.7310i2 0.65 (2.9) 0.64 (1.9) 0.65 (2.9) 0.65 (2.6) 0.65

E11 104 (MPa) 1.52 (3.1) 1.60 (3.9) 1.63 (3.1) 1.58 (3.0) 1.58

1E11 105 (MPa) 1.4 (122) 0.85 (186) 0.87 (143) 0.93 (133) 1.08

P (MPa) 25.0 (3.1) 24.8 (2.3) 24.8 (2.4) 25.1 (2.0) 25.0 (MPa) 5.5 (7.5) 5.13 (5.3) 5.24 (7.2) 5.36 (5.4) 5.36 (MPa) 4.7 (8.9) 4.47 (6.5) 4.45 (5.3) 4.47 (5.7) 4.50



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The measuring pins were 2 mm steel bolts screwed into the sample at a depth of

approximately 0.01 m. Twelve pins were inserted in the manner shown in Figure 2.

Six of them were located at a distance r1 r2 0.03 m (inner pins) from the centre

of the hole, another six (outer pins) at the distance r3 0.08 m.

Six circumferential gauges were installed between the inner pins, they had the

length d 0.03 m. Six gauges of length D 0.05 m were installed in the radial

direction between the inner and outer pins.

Each gauge consisted of a bent strip made of spring steel with four tensometersglued on the opposite surfaces (two on each side). One of the strip edges was

sharpened and another had an adjustable screw fixed to it. Each gauge was calibrated

to allow for measurement of strain with an accuracy of 106.

The cylindrical jack was represented by a steel tube with outer diameter of 2 cm

covered by a rubber tube, Figure 2. One end of the steel tube was closed, while

machine oil was pumped through the other end. The steel tube had a number of small

holes through which the oil escaped into the space between the steel and the rubber

tubes. After securing the jack into the hole followed by pumping of oil into the jack,the rubber expanded which created the pressure inside the hole. The pressure used

during the test was up to Q 40 MPa.

The readings of gauges are shown in Figure 3. All circumferential gauges show

negative (compressive) strains with the mean value of1.13*104. Radial gauges

were under tension with the average strain of 0.34*104.

Figure 2. Aluminum sample with cylindrical jack, pins and gauges.

280 GALYBIN ET AL.


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Determination of the shear modulus and Poissons ratio

An attempt to obtain the shear modulus on the basis of equation (11) leads to the

following values: Gc 3.06*104 MPa and Gr 3.87*10

4 MPa. As one can see, the

difference between the values obtained is 26%. Since the gauges demonstrated quite

stable behaviour during the test, this difference cannot be related to experimental

errors. Rather, it reflects the influence of external boundaries of the sample. Thus a

corresponding correction should be introduced for analysing the response of the

aluminium sample.

For simplicity, instead of the real geometry, a circular ring has been considered. Its

internal radius corresponds to the radius of the hole (0.0125 m). The external radius

has been calculated from the condition that the volumes of the real sample and thecircular ring coincide. Thus, the ratio of the external radius to the radius of the hole

has been found to be equal to 9. This value was used for the further calculation

described below.

For the case of a circular ring with the internal surface subjected to a pressure Q

the displacement field is determined as follows (Muskhelishvili, 1953)

ur

Q

E

1

2 1

r2

R2

r

r2

R2

r

, u

0 (20)

It should be noted that in contrast to the displacement field given in equation 10,

equation (20) depends upon Poissons ratio. This fact enables its reconstruction

directly from the pressurization test alone. The corresponding formulae for the shear

modulus and Poissons ratio assume the form

Figure 3. Deformations of circumferential (d 0.03 m) and radial (D 0.05 m) extensometersafter pressurizing the hole into the aluminum sample to the pressure of 40 MPa.

Numbers refer to microstrains.



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GQ

2

2

2 1

(r3 r1)R2

r3r21

Dd

rd cD1

GHGH

, HQ2

12 1

D2

drr3d cr1D

(21)

For 9 equation (21) give the following moduli of the sample: G 3.3*104

MPa, 0.3, E 2(1)G 8.5*104 MPa.

It should be noted that the shear modulus obtained by equation (21) weakly

depends upon the ratio . With respect to Poissons ratio this dependence becomes

more pronounced. For instance all possible values of: 0 0.5 can be obtained

if varies in the range 6.6 11.4.

Full in situ stress reconstruction

The technique proposed for measurements of surface stress and moduli values can

form the basis for complete, large scale in situ stress reconstruction. Since a single

installation can recover only a 2-D stress tensor related to the surface of measure-

ments, measurements from at least three installations on differently oriented surfaces

are required to compute all stress components. In order to extend these data andrecover the original stress state in the rock mass itself, further computations are

required.

This can be achieved by combining the results of the stress measurements at

different locations on the excavation walls by means of back analysis of the solution

for the excavation. In order to do this, in the case of an elastic rock mass, the

deformation characteristics of the rock mass are required (for an elastic isotropic rock

mass, only the Poissons ratio is needed) and they are obtained simultaneously with

the stress measurements when the proposed method is used. In the case of a non-elastic rock mass a number of additional moduli measurements might be necessary.

The method will be demonstrated for an isotropic rockmass with a cylindrical

tunnel arbitrarily oriented with respect to the principal directions of the original

(natural) stress. In this case the existence of an analytic solution simplifies the

analysis. More general cases of excavation geometry or/and anisotropic rocks will

obviously require numerical calculations. This will make the problem more laborious

because the number of elastic moduli to be determined increases (e.g., for orthotropic

rocks nine elastic moduli characterizing spatial deformations should be determined).The approach to the analysis will nevertheless stay the same.

More severe implications result from the presence of a damaged zone at the

excavation surfaces. The influence of this zone will be discussed in the next section.

It is convenient to introduce a Cartesian coordinate frame Ox1x2x3 with thex3-axis

directed along the generatrix of the tunnel and axes x1 and x2 coinciding with the

282 GALYBIN ET AL.


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secondary principal directions. Then the original stress tensor has the following

form

11 0 13

0 22 23 (22)

13 23 33

In this case the stress field around the excavation can be presented as a super-

position of the stress fields obtained in plane strain (the response to compressions 11and 22), anti-plane (out-of-plane) strain (the response to shear loads 13 and 23) and

the compression along the excavation axis, 33.

It is also convenient to introduce cylindrical coordinates, (r, , z) such that

x1 rcos,x2 rsin,x3z. Stress components on the excavation wall can then be

expressed as

1122

22211

2cos(2)

z 33z2(13 cos13 sin)

(23)

rrzr 0

The components r, , and, z, as well as Poissons ratio of the rock mass, can

be measured in different points on the excavation wall by the method described in theprevious sections. The components of the stress tensor, equation (22) as well as the

coordinate angle of, say, the first measuring point, 1 (this will serve as a refer-

ence point; the coordinate angles of other measuring points can then be determined

with respect to the first one) are then to be determined from equation (23).

Let non-zero stress components be determined atMdifferent locations given by k,

k 1, . . . M.

k1122

22211

2cos(2

k

)

zk33

k (24)

zk2(13 cosk13 sink)

The coordinate angles k are unknown, but their differences are known and can be

denoted

kk1. (25)

The reference angle 1 remains unknown.

It is now convenient to introduce new parameters characterizing the plane strain

part of the original stress tensor

P2211

2, D

2211

2(26)

System (24) then assumes the form



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kPD cos(2k 2)

zk33P D cos(2k 2) (27)

zk2(13 cos(k)13 sin(k))

Let the surface stresses be determined with sufficient redundancy, i.e. M 3. Theproblem of stress determination can then be reduced to the problem of minimization

of a proper functional in the space of 6 variables

F(P,D,13,23,33,) min (28)

In particular, one can use the following functional

F(P,D,33,13,23,)N

k1

(kPD cos(2k 2))

2

N

k1

(zk33PD cos(2k 2))

2 (29)

Nk1

(zk 2(13 cos(k)23 sin(k)))

2

A numerical experiment has been undertaken to estimate the accuracy of the stress

reconstruction. The following values for stresses were used: 112 MPa, 22

3 MPa, (corresponds to P2.5 MPa and D0.5 MPa), 331 MPa,

13 0.5 MPa, 230.2 MPa and the reference angle chosen was 22.The surface stresses at points k are modelled as the ideal ones computed for the

above stress state using the equations (24)(27) and then distorted by adding to each

of them an independently generated random error uniformly distributed within the

range of 20%. Poissons ratio was 0.3 and was supposed to be known in this

simulation. In real situations Poissons ratio can be determined as an average of the

values reconstructed simultaneously with the stresses at the measuring points.

Functional (29) was minimized using a procedure built into the package MATH-

CAD 6 PLUS. The initial guess necessary for the numerical minimization has beencomputed by using the first three sets of measurements (k 1, 2, 3) to find

parameters P, D, , and then the value obtained for the angle was used to find

parameters 33, 13 and 23 from the first two sets of stress measurements.

For each number of surface measurements, M 3 . . . 16, 200 combinations of

readings were independently generated, and the values ofP were reconstructed. The

averages over 200 sets of readings and the variation coefficients, , (the standard

deviation over the absolute value of the mean, e.g., for a parameter X, XVar(X)/

E(X)) were then computed.Figure 4 shows the mean values and variation coefficients representing the relativeerrors (%) of the reconstruction for all 6 unknown parameters for different numbers

of measuring points.

It can be seen from the plots that the accuracy of reconstruction is quite high given

that the error specified in the stress measurements was very large, 20%. Thus, 7

284 GALYBIN ET AL.


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Figure 4. Reconstruction of the initial stress state vs. the number of measurements. Each broken line

is the average value of the reconstructed parameter, while horizontal dashed line shows the exact

value. Solid lines show the coefficients of variation which represent the relative error of

reconstruction: (a) the hydrostatic pressure, P; (b) the deviatoric component, D; (c) normal

component, 33; (d) shear component, 13; (e) shear component, 23; (f); the reference angle, .



20/23

points of measurement are normally enough to bring the error of the stress reconstruc-

tion below 20%, while some components are restored with even higher accuracy.

Further increase in the number of measuring points can improve the accuracy even

beyond the accuracy of the original measurements.

Thus, the proposed combined method allows the accurate reconstruction of the full

stress state at the level of the excavation without the shortcomings characteristic of

the under-excavation technique, viz. the necessity to conduct displacement measure-

ments in front of the advancing excavation face.

Effect of the presence of a fractured zone around the excavation

The method has been tested under the assumption that the rock mass is homo-

geneous. However, in some cases the excavation is surrounded by a heavily fractured

zone created either due to the excavation process, or due to deterioration of the rock

adjacent to the excavation walls. In this case the non-homogeneity of the rock mass

can no longer be neglected and its effect should be investigated.

In the simplest case the fractured zone can be modelled by an elastic ring

possessing moduli lower than those of the rock mass. In order to analyse the

possibility of using the method for stress reconstruction in this case, a simpleexample will be considered in which the ambient stress field satisfies the plane strain

conditions and is purely isotropic (i.e., 1122P, 121323 0). The

excavation radius is taken as equal to unity and the external radius of the fracture

zone is taken to be Rd 1 as illustrated in Figure 5. Then by taking the axial

symmetry into account and introducing the unknown contact pressure, pd on the

boundary between the fractured zone and the rock mass, the corresponding complex

potentials can be expressed in the form (e.g., Muskhelishvili, 1953):

(z)P2

, (z)(Ppd)Rd

2

z2(30)

(z)P

2zC1, (z)(Ppd)

Rd2

z2C2 (31)

outside the fractured zone, and

(z)pd

2

Rd2

Rd2 1, (z)p

d

Rd2

Rd2 1

1

z2(32)

(z)pd

2

Rd2

Rd2 1

zC3, (z)pdRd

2

Rd2 1

1

zC4 (33)

inside the fractured zone.

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The displacements outside the fractured zone can be obtained from equation (31)

by acknowledging that the constants C1 and C2 vanish after neglecting the rigid body

movements; i.e.

ur iu1

2

z

r(1) P

2z (Ppd)

R2d

z (34)

The displacements inside the fractured zone are obtained from equation (33) by

taking into account that the terms containing constants C3 and C4 should be zero dueto the axial symmetry; i.e.

ur iu1

2d

R2d

R2d 1

z

r(1d) P

2z (Ppd)

R2d

z (35)

Here d 34d, d and d are the shear modulus and Poissons ratio of the fractured

zone respectively.

Continuity of the displacements allows the determination of the dimensionless

interface pressure qdpd/P

qd(1)(R2d 1)

2R2d 2mR2dd 2m

(36)

where m/d 1.

Figure 5. An example: the elastic model of the fractured zone.



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By substituting equation (36) into equation (32) and equation (35) and using the

Kolosov formulae, one can determine the non-zero stress and displacement compo-

nents at the excavation wall:

contourP 1

2md 2(m 1)R2d

, urcontour 1

d

4d

contour (37)

It can be seen that the proposed method based on the surface stress and modulus

measurements can only recover a certain combination of the unknown ambient stress

P, the fractured zone radius,Rd and the moduli of the rock mass (the surface measure-

ments can only provide the moduli of the fractured zone). The surface displacement is

proportional to the stress, hence measuring it will not contribute to the separation of

the unknown parameters. Therefore, in the case considered the under-excavation

method will not permit stress reconstruction, so using it in combination with the

proposed method can only increase the accuracy of the determination of the fractured

zone moduli.

This example demonstrates that the fractured zone becomes a serious obstacle for

stress reconstruction, which is why it is essential to install the measuring pins and the

cylindrical jack at a depth sufficient to reach the competent rock(Figure 1b).

Conclusions

A method is proposed which is based on (a) large-scale surface stress and modulus

measurements at a number of points in the excavation; and (b) the back analysis for

a given excavation shape. It allows the simultaneous reconstruction of the stress

components at the scale of excavation.

The proposed measuring scheme for near surface reconstruction of 2-D stress

tensor allows the simultaneous determination of the stress components and the elastic

moduli of the rock mass at a scale considerably larger than conventional methods.The accuracy of the reconstruction is not uniform and depends on the number of

pairs of deformation measurement. The analysis suggests that six measuring pairs are

sufficient for reasonable accuracy.

Laboratory test on an aluminium sample demonstrates that resolution of the gauges

used is sufficient for accurate recovery of the small deformations expected in field

conditions, even for the case of strong solid rocks. However, due to the relatively

small dimensions of the laboratory samples the influence of their boundaries

becomes considerable and has to be taken into account.The numerical simulation of the under-excavation method for the case of a

cylindrical excavation in an isotropic rock mass has shown the high accuracy and

robustness of the method. The presence of a fractured zone surrounding the

excavation can hamper the stress reconstruction, hence special measures should be

taken to conduct the measurements in competent rock.

288 GALYBIN ET AL.


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Acknowledgements

The authors acknowledge the support of the Large Australian Research Council

Grant A89701124. The authors thank Mr Frederic Verheyde for his help in con-

ducting the laboratory test. The third author acknowledges the support of the

Gledden Trust.

References

Cuisiat, F.D. and Haimson, B.C. (1992) Scale effects in rock mass stress measurements. Int. J. Rock

Mech. Min. Sci. & Geomech. Abstr., 29, 99117.

Dean, A.H. and Beatty, R.A. (1968) Rock stress measurements using cylindrical jacks and flat jacks at

North Broken Hill Limited. Broken Hill Mines Monograph No 3, 18. Australian Inst. Min. Metal,

Melbourne.

Galybin, A.N., Dyskin, A.V. and Jewell, R.J. (1997) A measuring scheme for determining in situ stresses

and moduli at large scale. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 34, 157162.

Goodman, R.E. (1989) Introduction to Rock Mechanics. John Wiley & Sons, New York.

Lama, R.D and Vutukuri, V.S. (1978)Handbook of Mechanical Properties of Rocks, vol. II. Trans Tech

Publ., Switzerland.

Lekhnitskii, S.G. (1968)Anisotropic plates . Gordon and Breach, Science Publishers, Inc., New York.

Muskhelishvili, N.I. (1953) Some Basic Problems of the Mathematical Theory of Elasticity . P. Noordhoff

Ltd, Groningen-Holland.

Wiles, T.D. and Kaiser, P.K. (1994) In situ stress determination using the Under-excavation Technique.Parts I, II. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr., 31, 439446, 447456.


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