- Email: [email protected]
Analysis of borehole expansion and gallery tests in anisotropic rock masses
0148~9062/91 $3.00 + 0.00
ht. 1. Rock Med. Min. Sci. & Geomech. Abstr. Vol. 28, No. 5, pp. 383-396, 1991 F’rinted in Great Britain. All rights re~ervcd
Copyright 0 1991 Pergamon Press pk
Analysis of Borehole Expansion and Gallery Tests in Anisotropic Rock Masses B. AMADEIt W. 2. SAVAGES Closed-form solutions are used to show how rock anisotropy affects the variation of the modulus of deformation around the walls of a hole in which expansion tests are conducted. These tests include dilatometer and NX-jack tests in boreholes and gallery tests in tunnels. The effects of rock anisotropy on the modulus of deformation are shown for transversely isotropic and regularly jointed rock masses with planes of transverse isotropy or joint planes parallel or normal to the hole longitudinal axis for plane strain or plane stress condition. The closed-form solutions can also be used when determining the elastic properties of anisotropic rock masses (intact or regularly jointed)
in situ.
INTRODUCTION
Because of scale effects, laboratory tests on small rock samples are usually inadequate to measure the deformability of rock masses. This has led to the development of several static and dynamic field testing methods over the past 20yr. In the static methods, loads are applied on selected rock surfaces in boreholes, underground galleries or at the surface of rock outcrops and the resulting deformations are measured. These methods include plate bearing tests, flat jack tests, radial jacking tests, gallery tests and borehole expansion tests, all of which are well-documented in the literature. Testing procedures, case studies and reviews of their advantages and limitations can be found in Goodman et al. [1], Bieniawski [2], Goodman [3], Brown [4], Swolfs and Kibler [5] and Heuze and Amadei [6], among others. This paper focusses on the analysis of expansion tests in boreholes and gallery tests in tunnels. A number of testing devices are available which can be inserted into a borehole to apply a load and measure the direct response of the walls of the borehole. These devices either: (1) supply a uniform internal pressure in the borehole such as the dilatometers; or (2) supply a unidirectional pressure to a portion of the circumference of a borehole by forcing apart circular plates such as the NX-borehole jack of Goodman et al. [l]. The results of these tests are often presented in the form of curves of applied pressure vs diametral deformation. The diametral deformation is measured either in the direction of loading, as with the borehole jack, or at one or several locations around the borehole circumference as with some dilatometers. Gallery tests are similar to dilatomeTDepartrncntof Civil Engineering, University of Colorado, Boulder, CO 803094428, U.S.A. jU.S. Geological Survey, Box 25046, MS%, Denver, CO 80225, U.S.A.
Denver Federal Center,
ter tests in that a sealed section of a tunnel is pressurized and the rock mass deformation is measured at several points around the excavation circumference. Usually, the measurements of diametral deformation in expansion tests are analyzed by modelling the rock mass as a linearly elastic, isotropic and homogeneous continuum with Young’s modulus E and Poisson’s ratio v. A value for the Poisson’s ratio is assumed (often equal to 0.25) and the modulus E is determined from measured values of applied pressure q and hole diametral deformation U,, through equations borrowed from the theory of linear elasticity for isotropic media; e.g. 1 u, --
(l+v)
q2a=E’
for dilatometer
(l)
and gallery tests, and 1 Ud K(v9 m -q2a=E’
(2)
for the NX-borehole jack test. In these equations, 2a is the diameter of the hole, and 28, is an estimated value of the contact angle between the jack and the rock [5,6]. Equations (1) and (2) are also used in terms of increments of diametral deformation AU,, and increments of applied pressure Aq when pressure vs hole diametral deformation response curves are non-linear. The analysis of expansion test results described above is commonly followed in practice regardless of the anisotropic, discontinuous and heterogeneous nature of the rock mass of interest. The coefficient E would be the true elastic Young’s modulus of the rock mass if the latter actually behaved as a linearly elastic, isotropic, continuous and homogeneous medium. Any departure from these idealized characteristics causes the coefficient E to become, instead, an apparent modulus or modulus of &formation of the rock mass in the direction of loading and/or diametral measurement. The magnitude of this modulus can vary along the circumference of the
383
AMADEI and SAVAGE:
384
TESTS IN ANISOTROPIC
ANALYTICAL
hole if the rock mass shows anisotropic, discontinuous or heterogeneous behaviour. For dilatometer tests, such variations when plotted in polar form can give an insight into the anisotropy of the rock mass being studied [I. This can also be obtained with the borehole jack by loading, for instance, a jointed rock mass in directions parallel and normal to the joint planes. In a recent paper, Amadei and Savage [8] show that the directional character of the modulus of deformation of an anisotropic rock mass E, measured along the circumference of a hole during in situ expansion tests can be explained analytically by using closed-form solutions based on the theory of elasticity for anisotropic media. By replacing a regularly jointed rock mass by an anisotropic continuum, it is also shown that the modulus of deformation can be related to joint properties such as spacing, orientation and stiffnesses.
1
E,
--
VIU
E,,
ROCK
BACKGROUND
Consider the equilibrium of a rock mass that is modelled as an infinite, linearly elastic, anisotropic, continuous and homogeneous medium. The medium is bounded internally by a cylindrical surface of circular cross section of radius u that represents a hole (borehole or tunnel). Consider the geometry of Fig. 1 and let x, y, z be a Cartesian coordinate system with the z-axis defining the longitudinal axis of the hole. The rock mass is assumed to be orthotropic in a n, s, t Cartesian coordinate system attached to clearly defined planes of anisotropy. The orientation of these planes with respect to the x, y, z coordinate system is assumed to be known. The constitutive equation for the orthotropic medium in the n, s, t coordinate system is given by the following equation:
-2 V Es
-- 6VI”
0
0
0
1 E,
v, -E,
0
0
0
(3) 0
0
ok00 I,
0
0
0
0
&
0
“,
This paper first reviews the closed-form solutions of Amadei and Savage [8]. Then, these solutions are used to show how different degrees of rock anisotropy affect the variation of the modulus of deformation around the walls of a hole in which expansion tests are conducted. This is shown for transversely isotropic and regularly jointed rock masses with planes of transverse isotropy, or joint planes, parallel or normal to the hole longitudinal axis and for plane strain or plane stress conditions. It is also shown how the closed-form solutions can be used when determining the elastic properties of anisotropic rock masses (intact or regularly jointed) in situ.
S A
Fig. 1. Geometry
Ill
of the problem. Hole intersecting anisotropy in a rock mass.
planes of
or in a more compact matrix form: (s),, = W) (c)#W
(4)
Nine independent elastic constants are needed to describe the deformability of the medium in the n, s, t coordinate system. E,, ES and E, are the Young’s moduli in the n, s and t (or 1, 2 and 3) directions, respectively. G,, G,, and G,, are the shear moduli in planes parallel to the ns, nt and rt planes, respectively. Finally, = n, s, t) are the Poisson’s ratios that characterize vij (6 j the normal strains in the symmetry directions j when a stress is applied in the symmetry directions i. Because of symmetry of the compliance matrix (H), Poisson’s ratios Vii and vji are such that VijlEi =:VjilEj. Equations (3) and (4) still apply if the medium is transversely isotropic in one of the three M-, nt- or st -planes. In that case, only five independent elastic constants are needed to describe the deformability of the medium in the n, s, t coordinate system. In this paper, these constants will be called E, E’, v, v’ and G’ with the following definitions: (i) E and E’ are Young’s moduli in the plane of transverse isotropy and in direction normal to it, respectively;
AMADEI
and SAVAGE:
TESTS IN ANISOTROPIC
(ii) v and v’ are Poisson’s ratios characterizing the lateral strain response in the plane of transverse isotropy to a stress acting parallel or normal to it, respectively; and (iii) G’ is the shear modulus in planes normal to the plane of transverse isotropy.
ROCK
385
(4
Relations exist between E, E’, v, v’, G and G’ and the coefficients of matrix (H) in equation (3). For instance, for transverse isotropy in the St-plane: 1 -=-. E.
1 E”
1 1 -c-z-;
1
1 1 -_=-=-
Es
E
G,
E,
VII.3 V,I -=__=-.
v’
v*r vu -c-c_;
V
1 -=-
E.
E”
Es
E
Gs,
E,,
Et
G,,
1 G”
G
For known orientation of the planes of anisotropy with respect to the hole, the constitutive relation of the medium in the x, y, z coordinate system can be obtained by using second-order tensor coordinate transformation rules (see for instance [9]). This gives:
where
and
(A) is a (6 x 6) compliance matrix whose components uij (i, j = 1,6) depend on the elastic properties of the medium in the n, s, t coordinate system and the orientation of its planes of anisotropy with respect to the hole. Consider the loading conditions shown in Figs 2a and b. Figure 2a corresponds to the uniform radial loading of a hole (dilatometer and gallery tests) with a pressure q. In Fig. 2b the hole is loaded in the y-direction with a constant surface force per unit area q applied over two opposed arcs of angle 28, (NX-borehole jack test). Closed-form solutions for the displacement components along the hole circumference induced by the loads of Figs 2a and b are given by Amadei and Savage [fl] for an anisotropic medium. It is assumed in these solutions that the anisotropic medium deforms under a condition of generalized plane strain in the xy -plane [lo]. This assumption holds if the hole is loaded over a length that is much longer than its diameter. Also, displacements induced by the loading must be calculated at hole cross-sections located some distance inwards from the ends of the loaded zone (at least one hole diameter as a rule of thumb). Finally, body forces were neglected. The condition of generalized plane strain allows planes of anisotropy in the medium to be inclined with respect to the hole axis. All planes parallel to the xy-plane of Figs 2a and b are assumed to warp in the same way. In other words, longitudinal deformation induced by the loads in Figs 2a and b does not vanish and depends only on the x and y coordinates. For certain types and orientations of the anisotropy, generalized
Fig. 2. (a) Radial loading of a hole during dilatometer and gallery tests. (b) Loading of a hole over two opposed arcs of angle 2& with NX-borehole jack.
plane strain reduces to plane strain and the longitudinal deformation always vanishes. This occurs when the hole axis is at right angles to a plane of elastic symmetry such as: (1) when the medium is orthotropic in the x, y, z coordinate system (Fig. 3a) or has one of its three planes of symmetry normal to the z-axis (Fig. 3b); when the medium is transversely isotropic and the hole axis is either parallel (Fig. 3c) or perpendicular (Fig. 3d) to the planes of transverse isotropy; or (3) when the medium is isotropic. For all those cases, the compliances a&, us6, o.,~, usi (i = 1,2,3) in matrix (A) of equation (6) vanish. For dilatometer or gallery tests, Amadei and Savage [8] have shown that the hole diametral deformation U, measured at an angle 0 to the x-axis of Fig. 2a can be expressed as follows: ~~=A+Bcos20+Csin28 4 2a where A, B and C depend on the elastic properties of the anisotropic medium in the n, s, t coordinate system and the orientation of its planes of anisotropy with respect to the hole. For generalized plane strain conditions, A, B and C do not vanish. Conversely, if the medium is isotropic with Young’s modulus E and Poisson’s ratio v, B and C vanish and equation (7) reduces to equation (1).
386
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC
(4
(b)
(c)
(4
ROCK
Fig. 3. Four cases of anisotropy where the hole axis is normal to a plane of elastic symmetry: (a), (b) orthotropic medium; and (c), (d) transversely isotropic medium.
For the borehole jack tests, Amadei and Savage [8] have shown that the hole diametral deformation U,, in the direction of load application is such that: s-7 . tu,, t -=-
u:,
Similarly, the borehole jack test with loading in the y-direction gives:
K(v, A 1
1 ud --
(12)
q2a=E,’
(8) with
where /3, is expressed in radians and IJ: depends on the elastic properties of the anisotropic medium in the n, s, t coordinate system and the orientation of its planes of anisotropy with respect to the hole. For an isotropic medium with Young’s modulus E and Poisson’s ratio v, equation (8) reduces to equation (2) with: K(v,8.)=&(l-V2)
2 --$[l-(-l)“]sidm/I.. m-l
(9)
VARIATION IN MAGNITUDE OF THE MODULUS OF DEFORMATION
The variation in the magnitude of the modulus of deformation of an anisotropic rock mass measured at different points along the walls of a hole during expansion tests can be explained quantitatively with the analytical solutions presented above. To do so, equations (7) and (8) are rewritten as if the anisotropic medium were “isotropic” in the direction of loading and/or measurement, with an assumed Poisson’s ratio v and a modulus E,. Then, each diametral measurement in the dilatometer and gallery tests at an angle 8 from the x-axis can be interpreted from: 1 u, (l+v) -q2a=E,’
(10)
with E, = (1 + v)/(A + B cos 28 + C sin 20).
(11)
(13) Equations (11) and (13) provide analytical expressions for the modulus of deformation of an anisotropic rock mass modelled as a linearly elastic continuum. The modulus depends on the anisotropic properties of the rock mass and the orientation of its planes of rock anisotropy with respect to the directions of measurement and/or loading in the hole. Because of the linear relations existing between coefficients hii and a,j of matrices (H) and (A) in equations (4) and (6), respectively, it can be shown that the ratio between the modulus of deformation E, and one of the Young’s moduli, say E,, of an orthotropic medium depends on the following eight dimensionless quantities:
5. E”’
Es
E,;
E v,; vm; vu; $
E,. G,,’
Es G,’
(14)
If the medium is transversely isotropic with, for instance, transverse isotropy in the St-plane, and using equation (5), the ratio E./E, = E./E is found to depend only on four dimensionless terms: E E’;
v;
v’;
G 1. G
(15)
Note that E./E, and EJE also depend on the orientation of the planes of anisotropy with respect to the hole in which the expansion tests are conducted. In addition, for
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC
ROCK
Table I. Expressions of coefficients qj for three orientafions of the planes of transverse isotropy
the dilatometer and gallery tests, these ratios also depend on the angle 0 of Fig. 2a.
Transverse isotropy parallel to yz-Plane xz-Plane xy -Plane
a,j
l/E’ -v’lE’ l/E
alI 013 %I
EFFECT OF INTACT ROCK ANISOTROPY ON THJ3 MODULUS OF DEFORMATION
l/E -v/E l/E l/E’ -v’iE’ - v’/E’ l/G’
I/E
*22
-v/E -v’/E’ I/G’
a23
For known values of the elastic properties of an orthotropic or transversely isotropic rock mass and known orientation of its planes of anisotropy, the solutions summarized above can be used to predict how the modulus of deformation of an anisotropic rock mass varies around the circumference of a borehole or a tunnel in which expansion tests are conducted. These solutions can also be used to solve the inverse problem, that is, in interpreting expansion tests to obtain rock mass deformability properties. The inverse problem will be discussed later in this paper. The effect of intact rock anisotropy on the modulus of deformation of a rock mass is shown below for the special case when the hole deforms in plane strain for which coefficients A, B, C and U: in equations (7) and (8) take simpler analytical forms. Consider, for instance, the anisotropy orientations of Figs 3a, 3d and 3c with the dip of the planes of transverse isotropy, JI, either 0 or 90”. In all three cases, the hole axes x, y, z are perpendicular to planes of elastic symmetry in the rock mass. For these orientations, Amadei and Savage [8] have shown that C always vanishes whereas A, B and U: are equal to:
381
42
as
l/E -v’/E’ l/E’ l/E - v’/E’ -v/E l/G
earlier in this paper. Substituting these coefficients into equation (16) and the resulting expressions of A, B and U: into equations (11) and (13) gives closed-form expressions for the ratio EJE in terms of the four dimensionless quantities of equation (15). For the dilatometer and gallery tests, EJE is equal to:
4 -= E
(1 + v)
(17)
A*+ B*cos26’
where v is assumed to be equal to the Poisson’s ratio in the planes of transverse isotropy and A* = AE and B* = BE are such that: A*=;;[2v.(l
+v)+(l
-$v’~
+J_)
-2v’(l
X
+v)+2;$(1
+v) +2
1+
:
(18)
-2/m],
(16)
where jIij = aij - U&ljJU~3 (i, j = l-6) and a,, are the coefficients of matrix (A) in equation (6). For a transversely isotropic rock mass with planes of transverse isotropy parallel to the st -plane of Fig. 1, Tables 1 and 2 give the expressions of coefficients ai, and fiij when the planes of transverse isotropy are also parallel to the yz-plane (Fig. 3c with $ = go”), the xzplane (Fig. 3c with JI = 0”) or parallel to the xy-plane (Fig. 3d). These coefficients are expressed in terms of the five elastic rock properties E, E’, v, v’ and G’ defined
1E B*=ZF
Table 2. Expressions of coefficients Bij for three orientations of the planes of transverse isotropy Transverse isotropy parallel to
A,
yz -Plane
xz -Plane
xy -Plane
B,, Al
(1 - v’*E/E’)/E’ (1 - v*)/E -v’(l + v)lE’ l/G’
(1 - v’)/E (1 - v’=EIE’)/E’ -v’(l + v)lE’ l/G’
(1 - v’*E/E’)/E (1 - v’~E/E’)/E -(v + ;;$/E’)lE
AMADEI and SAVAGE:
388 (a)
TESTS IN ANISOTROPIC
1’2r
ROCK
(W 1.0
0.6
Fig. 4. Polar variation of EJE with the angle 8 (0 d 0 Q n/2) for gallery or dilatometer tests in a medium with planes of transverse isotropy parallel to the yz-plane and for E/E’ = 1, 1.5,2, 3 and 4, Y = v’ = 0.25 and: (a) G/G’ = 1; and (b) G/G’ = 3. T’he isotropic case corresponds to E/E’ = G/G’ = 1 or when the planes of transverse isotropy are parallel to the xy-plane.
when the planes of transverse isotropy are parallel to the yz-plane (Fig. 3c with JI = 90’) and
for planes of transverse isotropy parallel to the xz-plane (Fig. 3c with J/ = O”). Finally, for planes of transverse isotropy parallel to the xy-plane (normal to the hole axis, Fig. 3d), A* = (1 f v), B* = 0 and E./E = 1, i.e. the rock behaves as if it were isotropic. As a numerical example, Figs 4 and 5 show variations of E,/E with the polar angle 8 in Fig. 2a (0 < 8 < x/2) for several degrees of intact rock anisotropy with E/E’ = 1, 1.5, 2, 3 and 4, G/G’ = 1 and 3, v = v’ = 0.25, and when the planes of transverse isotropy are parallel to the yz- and xz-planes, respectively. The isotropic case corresponds to E/E’ = G/G’ = 1 or when the planes of transverse isotropy are parallel to the xy-plane. The chosen domains of variation for E/E’ and G/G’ are reasonable in view of the literature surveys on elastic properties of anisotropic rocks conducted by Gerrard [1 1] and Amadei et al. [12]. Note that because of the symmetry associated with the dependence of EJE on cos 28 in equation (17), the variations of E./E with the angle 8 in Figs 4 and 5 are only shown in the first quadrant (0 s 8 < n/2). Figures 4 and 5 show that as E/E’ increases and for a fixed value of the ratio G/G’, the rock mass becomes more deformable in directions normal to the planes of transverse isotropy and the variation of EJE with the
0.0
0.2
0.4
0.6
0.6
1.0
Fig. 5. Polar variation of EJE with the angle 0 (0 C 0 6 x/2) for gallery or dilatometer tests in a medium with planes of transverse isotropy parallel to the xz-plane and for E/E’ = 1, 1.5,2,3 and 4, v = v’ = 0.25 an& (a) G/G’ - 1; and(b) G/G’ = 3. The isotropic case corresponds to E/E’ = G/G’ = 1 or when the planes of transverse isotropy are parallel to the xy-plane.
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC
389
ROCK
-&‘UII ---(Ea% 1.4
\ ---_ I ‘. -
$0.81 \ ‘y 0.6
0.4
\
'. . .
G/G'=2
---“GIG'_,
.. .
--__ --*_
GIG-3
- ---GK;'-2 --_ as-3
Fig. 6. Polar variation of EJE with the angle 0 (0 d 0 d n/2) for gallery or dilatometer tests in a medium with planes of transverse isotropy parallel to the yz-plane and for E/E'= 3,G/G'= 1,v = 0.25 and v’ = 0.15, 0.25 and 0.35. The isotropic case is also shown for Fig. 7. Variation of (E./E),, (EJE),and (E./E), with E/E' for comparison. G/G’ = 1, 2, 3 and v = v’ = 0.25(NX-borehole jack tests). The isotropic case corresponds to E/E'= G/G'= 1 (point I).
angle 8 becomes more non-uniform as expected. Note that the value of the modulus of deformation in directions parallel to the planes of transverse isotropy is not much affected by the value of E/E’ unlike the value of the modulus in directions normal to the planes of transverse isotropy. For a fixed value of E/E’, the polar variation of EJE with the angle 8 depends strongly on the value of G/G’. As G/G’ increases, the rock mass becomes more deformable in shear in planes normal to the planes of transverse isotropy which results in lower values for E,/E in all directions around the hole. Figure 6 shows the polar variations of E,/E with the angle 8 for v’ = 0.15, 0.25 and 0.35, E/E’ = 3, G/G’ = 1 and when the planes of transverse isotropy are parallel to the yz-plane. The value of the modulus of deformation decreases with increasing v’ in directions parallel to the planes of transverse isotropy and increases slightly with v’ in directions normal to the planes of transverse isotropy. For the NX-borehole jack test in which E, is determined in the direction of load application, equation (17) is replaced by the following:
when the planes of transverse isotropy are parallel to the xz-plane and at right-angles to the jack loading direction in Fig. 2b. Finally, for planes of transverse isotropy parallel to the xy-plane (normal to the hole axis), E./E is equal to:
E,
0
i? XY=
l-9
(22)
142;
Note that the Poisson’s ratio v in the numerators of equations (20-22) is assumed to be equal to the Poisson’s ratio in the planes of transverse isotropy. As a numerical example, Fig. 7 shows the variations of (E./E),, , (E./E), and (EJE), with E/E’ for G/G’ = 1, 2 and 3, and v =v’= 0.25. For a fixed value of E/E’, (E./E),, and (EJE), decrease as G/G’ increases. On the other hand, for a fixed value of G/G’, (EJE), is not much affected by the value of E/E’ whereas (E./E), decreases as E/E’ increases. According to equation (22), (E./E), is independent of G/G’ but increases with E/E’. Table 3 gives the values of (EJE),, , (E./E), and (EJE),
(20)
when the planes of transverse isotropy are parallel to the yz-plane, that is, the jack loading direction in Fig. 2b and
AMADEI and SAVAGE:
390
TESTS IN ANISOTROPIC
Table 3. Value of (EJE),, , (EJE), and (EJE),,. for three different values of Poisson’s ratio 1”’ v’
0.15 0.25 0.35
b’W), 0.901 1.014 1.214
intact rock and joints are assumed to undergo equal strains in directions parallel to the contact planes. This assumption makes all non-zero diagonal terms in equation (3) equal to the same value, -v/E. It also results in reducing the number of elastic constants necessary to describe the deformability of the rock mass when cut by a single joint set to four, namely, E, v, k,S and k,S.
WE),
WE), 0.522 0.629 0.854
1.005 1.154 1.482
for G/G’ = 1, E/E’ = 3, v = 0.25 and v’ = 0.15,0.25 and 0.35. All three modulus ratios increase with v’.
EFFECT OF JOINT INDUCED ANISOTROPY ON THE MODULUS OF DEFORMATION
ROCK
If a regularly jointed mass is modelled as an equivalent continuum, equations (11) and (13) can also be used to relate the modulus of deformation in the direction of diametral measurement and/or loading in a hole to joint properties such as spacing, orientation and stiffnesses. For instance, for a rock mass cut by a single joint set parallel to the yz -, xz - or xy -plane of Fig. 1, substituting the expressions for l/E’, l/G’ and v//E’ of equation (23) into equations (18) and (19), the ratio EJE for the dilatometer and gallery tests is again given by equation (17) where A * and B* now depend on the Poisson’s ratio v and two dimensionless ratios a, and a, with:
Rock mass anisotropy can also be created by systems of regularly spaced rock joints also called joint sets in a rock that is otherwise isotropic. Since it is not possible to account for each joint plane when assessing the overall deformability of a regularly jointed rock mass, it is convenient to replace the rock mass by an equivalent anisotropic continuum [ 13-161. The rock mass is modelled as transversely isotropic if cut by a single joint set and orthotropic if cut by two or three orthogonal joint sets. E Consider for instance a rock mass cut by a single joint a’ = (1 - v*)k,S’ set oriented parallel to the St-plane of the n, s, t coordinate system of Fig. 1. The intact rock between the joints E is assumed to be linearly elastic and isotropic with a’ = (1 v*)k,S’ Young’s modulus E, Poisson’s ratio v and shear modulus G = E/2( 1 + v). If the joint set has spacing Sand normal For joints parallel to the yz-plane, A* and B* are now: and shear stiffnesses k, , k,, Duncan and Goodman [ 131 show that the regularly jointed rock mass can be reA*,(l - -v2) G 2v +(l +a,+JGi) placed by an equivalent transversely isotropic contin2 [ uum whose constitutive relation in the n, s, t coordinate system is given by equations (3) and (4). Furthermore, 2+a X the elastic constants in equation (3) satisfy equation (5) 1 with:
J~+g===&-uTi
1 _=L+i. E’ E
k,S’
i=L+i. G’ G
k,S’
vI,11. E’ E’
(23)
Hence, all non-zero off-diagonal terms in equation (3) are now equal to - v/E. Since k, and k, have units of stress/length or force/length3, the quantities k,S and k,S in equation (23) can be seen as normal and shear joint moduli. Note that the constitutive relation for a regularly jointed rock mass converges to that for an intact isotropic medium when the joint spacing or joint stiffness values in equation (23) approach infinity. Although convenient, the equivalent continuum formulation has certain limitations. For instance, Duncan and Goodman [13] make three main assumptions in the equivalent continuum approach: (i) the normal and shear joint stiffnesses are assumed constant and independent of the stress level acting across the joints; (ii) the joint response is assumed to remain in the elastic domain (pre-slip condition); (iii) the joints are assumed to have negligible thicknesses and not to create any Poisson’s effect upon loading of the rock mass. In other words,
B*=
9
+a,,-,/-)
y[(l
(25)
and for joints parallel to the xz-plane:
-v2)
A*_u --
2
2v
[
T--y+(l+Ji=n)
X J2+as+2JCr,-2J1+G(,
1 ,
J
B*=v[(l
(26)
-Jx)
Finally, for joints parallel to the xy-plane (normal to the hole axis), A * = (1 + v), B* = 0 and E,]E = 1, i.e. the jointed rock mass behaves as if it were isotropic. As numerical examples, Figs 8 and 9 show polar variations of E,/E with the angle 19(0 < 0 < n/2) for
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC
ROCK
391
of the modulus of deformation around the hole circumference approaches that of a circle. For the NX borehole jack test and for joints parallel to the yz-plane, i.e. the jack loading direction, equation (20) is replaced by the following: 2
E,
0 i?
,=J
(27)
2+=,+2Jx’
For joints parallel to the xz-plane, i.e. at right-angles to the jack loading direction, equation (21) becomes: 2
E,
0
(28) 2+a,+2JK’ Finally, for joints parallel to the xy-plane (normal to the hole axis), equation (22) is replaced by: 2
I=&%&
4
( >. z
Fig. 8. Polar variation of E./E with the angle 0 (0 < 6
eight cases of rock anisotropy when E/k,S ranges between 0 and 20, EIk,S ranges between 0 and 40 and v = 0.25 for joints parallel to the yz- and xz-plane, respectively. The isotropic case corresponds to E/k,S = E/k,S = 0 or when the joints are parallel to the xy-plane. As the rock mass becomes more isotropic, that is, as joint spacing and stiffnesses increase, the variation
“‘=l_
1 -vz v2 * E ’ +k,S
(29)
Table 4 gives values of (EJE),,, (EJE), and (EJE), for the eight cases of joint anisotropy considered above. For this example, the ratio of moduli of deformation measured in directions parallel and normal to the joint set, J% I&,, varies between 4.8 for case 1 and 1.Ofor case 8. Note that the value of (E,/E), is not much affected by the degree of rock mass fracturing. PLANE
STRAIN
VS PLANE
STRESS
Equations (17-29) were derived when the hole in which expansion tests are conducted deforms in plane strain and when the hole axes x, y and z are all perpendicular to planes of elastic symmetry in the rock mass. In general, the plane strain assumption holds as long as the hole is loaded over a length that is several times its diameter. Furthermore, the displacements induced by the loads in the hole must also be measured in crosssections located well within the loaded zone, say at least one hole diameter from the ends of the loaded zone. Conversely, if the loads shown in Figs 2a and b are applied over a smaller length, say of the order of one or two hole diameters or less, the plane strain assumption can no longer be used and must be replaced by plane stress. As discussed by Amadei [9], plane stress also implies that the hole be at right-angles to planes of elastic Table 4. Values of (E./E),,, (EJE), and (E./E), for eight different cases of joint induced anisotropy Case No.
Fig. 9. Polar variation of EJE with the angle 8 (0 d 0 d x/2) for gallery or dilatometer tests in a medium with a joint set parallel to the xz-plane and for eight cases of joint anisotropy. The isotropic case corresponds to E/k,S = E/k,S = 0 or when the joint planes are parallel to the xy-plane.
1 2 3 4 : I 8
El&S 20 10 5 2 0.4 0.2 0.02 0.0
EIkS 40 20 10 4 0.4 0.8 0.04 0.0
(WE), 0.212 0.364 0.475 0.638
(WE), 0.057 0.107 0.188 0.361
(WE), 0.940 0.943 0.947 0.957
0.873 0.929 0.992 1.0
0.731 0.844 0.981 1.0
0.989 0.981 0.998 1.0
AMADE
392
ant’ SAVAGE:
TESTS 1’N ANEOTROPTC
symmetry and therefore can only be used for the anisotropy types and orientations shown m Figs 3a-d. Under plane stress, it can be shown [ 17-201 that mean values of stress, strain and displacement in the xq‘-plane induced by the loads shown in Figs 2a and b and calculated over the length of the loaded section of the hole satisfy the same basic elasticity equations that govern the plane strain formulation. The difference is that all coefficients /Iii such as those appearing in equation (16) must now be replaced by the corresponding coefficients aij (i, j = l-6) of matrix (A) in equation (6). For plane stress, the expression of the modulus of deformation for a transversely isotropic rock mass with planes of transverse isotropy parallel to the yz-, xz- or xy-planes of Fig. 1 can be obtained by substituting coefficients aij of Table 1 for the corresponding p,, coefficients in equation (16). After algebraic manipulations, for the dilatometer and gallery tests, equation (17) still applies but A * and B* are now equal to:
X
/-2vJ+2$$(,+“)+2&2&],
ROCK
For the borehole jack rests, equations (20) and (21) are replaced by the following:
when the planes of transverse isotropy are parallel to the yz-plane and the jack loading direction and
9 (33)
when the planes of transverse isotropy are parallel to the xz-plane and at right-angles to the jack loading direction. Finally, for planes of transverse isotropy parallel to the xy-plane, (E,/E),v, = 1. For joint induced anisotropy, substituting equations (23) into equations (32) and (33) leads to equations similar to equations (27) and (28) with a, = E/k,S and a, = EIk,S. Note that for plane stress, K(v, /I,) in equations (12) and (13) is now equal to: 4 N K(v9Be)=~~~,;;;j[l
X
-2v’+2;$(1+“)+2
;,
1 -(-l)m]sinZm~c.
(34)
(30)
Jwhen the planes of transverse isotropy are parallel to the yz -plane and
(31)
for planes of transverse isotropy parallel to the xz-plane. As for plane strain, when the planes of transverse isotropy are parallel to the xy-plane, A * = (1 + v), B+=O and EJE= 1. For joint induced anisotropy, substituting equation (23) into equations (30) and (31) leads to expressions similar to equations (25) and (26) where (1 - v2) and v/(1 - v) are now replaced by 1 and v, respectively and u, = EIk,S, a, = E/k,S.
DETERMINING ELASTIC PROPERTIES OF ANISOTROPIC ROCK MASSES FROM EXPANSION TESTS
The variation in the modulus of deformation measured during expansion tests along the circumference of a borehole or tunnel in an anisotropic (intact or jointed) rock mass can be used to determine its elastic properties. This is shown below for transversely isotropic rock masses for which the unknown elastic constants are E, E’, v, v’ and G’ for intact rock and E, v, k,S and k,S if the rock is regularly jointed. In what follows, values for the Poisson’s ratios v and v’ are assumed as in the conventional isotropic analysis of expansion tests and the other three elastic constants E, E’ and G’ or E, k,S and k,S are determined from the test data. Applied pressure q vs hole diametral deformation U, response curves measured in situ are assumed to be non-linear and are analyzed in terms of increments of pressure and deformation Aq and AU,,. Thus, the elastic constants can be calculated over each increment of applied pressure and their stress dependency can be assessed. Dilatometer and gallery tests The three elastic constants E, E’, G’ or E, k,S and k,S can be determined by conducting dilatometer or gallery
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC 3.0
393
ROCK
r
WE’ 1
2.5 -2
_
44
3,
2.5 __ II8 +j
2*o _
(a) (W Fig. 10. (a) Dilatometer or gallery test in a hole with transverse. isotropy or joint planes normal to the hole axis. (b) Corresponding q vs U,/20 curve.
tests in two holes; one hole normal to the transverse isotropy or joints planes and another parallel to those planes. For these hole orientations, the closed-form solutions presented earlier in this paper apply. First, expansion tests are conducted in a hole having an axis normal to the planes of transverse isotropy or joint planes. For this anisotropy orientation, the rock mass behaves as if it were isotropic and all hole diametral deformations are equal (Fig. 10a). Over each increment of pressure Aq, the slope of the q vs Ud/2a curve is then equal to E/(1 + v) (Fig. lob). Assuming a value for v, the in-plane Young’s modulus E and shear modulus G = E/2(1 + v) can then be determined. Next, expansion tests are conducted in a hole parallel to the planes of transverse isotropy or joint planes. Let x, y, z be an arbitrary coordinate system attached to those planes. Since for this geometry C vanishes in equation (7), there are at most two independent diametral measurements that can be carried out in the hole. For each direction of diametral measurement, a curve q vs U,/2a is obtained. Consider for instance the case when the planes of transverse isotropy or joint planes are parallel to the yz-plane (Fig. 1la) and two diametral deformations U,, and U, are measured in directions normal (0, = 0’) and parallel (0, = n/2) to those planes, respectively (Curves 1 and 2 in Fig. 1lb). The respective slopes of those two curves over the same pressure increment Aq are equal to E,,/( 1 + v) and E&l + v). Dividing these slopes by E/(1 + v) measured in the first hole over the same pressure increment, gives values for e, = E,,/E and e, = Ed/E. Substituting these values into
q
Aq
----___
(a) (W Fig. Il. (a) Dilatometer or gallery test in a hole with transverse isotropy or joint planes parallel to the hole axis and yz-plane. Hole diametral deformations are measured at 0, = 0 (Curve 1) and 0, = n/2 (Curve 2). (b) Corresponding q vs U&a curves.
RMMS W-D
-2
/
GIG’
.5
1.5 1 t 1.01 0.0
fl I I
@ I
I
I
4.0
2.0
I
I
6.0
E.&l 2aQ Fig. 12. Variation of (E/2o)(AU,/Aq) with (E/2u)(AU,,/Aq) for different values of E/E’ and G/G’ ranging between I and 4, v = v’ = 0.25 and for plane strain conditions. Point I represents the isotropic case with E/E’ = G/G’ = 1.
equation (17) with 0, = 0 and Bz= n/2 leads to the following equations: A*+B*=_=_.-,
1+ v el
E AU.,, 2a Aq (35)
Af_B*_l+v --=-.__ E Au, e2
2a
Aq
For intact rock anisotropy, substituting the expressions of A * and B* given in equation (18) for plane strain or equation (30) for plane stress into equation (35) gives a system of two non-linear equations that can be solved for the two unknowns E/E’ and G/G’. Knowing E and G from the first hole, E’ and G’ can be calculated over the same pressure increment Aq. The modulus ratios E/E’ and G/G’ can be determined from charts similar to that shown in Fig. 12. In this figure, (E/2a) (AU,/Aq) and (E/2a) (AU,,/Aq) defined in equation (35) have been plotted using A* and B* defined by equation (18) for different values of E/E’ and G/G’ ranging between 1 and 4, v = v’ = 0.25 and for plane strain conditions. The isotropic case corresponds to E/E’ = G/G’ = 1 and is represented by point I with coordinates (1 + v, 1 + v). Knowing the values of AU.,, , AU, and Aq from the field tests and E from the first hole, Fig. 12 can be used to determine the two ratios E/E’ and G/G’ and therefore E’ and G’. Charts similar to Fig. 12 can be generated for other values of v and v’ and for plane stress conditions. For joint-induced anisotropy, substituting A l and B* given in equation (25) for plane strain and similar equations for plane stress into equation (35) gives a system of two non-linear equations that can be solved for E/k,S and E/k& Again knowing the values of AU,,, , AU, and Aq from the field tests and E from the first hole, allows the determinations of k,S and k,S and the corresponding joint stiffnesses k,, k, over each pressure increment Aq once S is measured in the field. Again, E/k,S and EIk,S can be determined from charts similar
q 3
AMADEI and SAVAGE:
394
TESTS IN ANISOTROPIC
ROCK
Y
x
2
(a)
Fig. 14. (a) Borehole jack test in a borehole with transverse isotropy or joint planes normal to the hole axis. (b) Corresponding q vs U,/2a curve.
2.:: 0
10
5
15
20
X~L.!%b
.-_v 1
25
30
2a Aq l-V2 1-v Fig. 13. Variation of X and Y defined in equation (36) for different values of a, ranging between 0 and 20 and a, ranging between 0 and 40 and plane strain. Point I represents the isotropic case with a, = a, = 0. For plane stress, E and v must be substituted for E/(1 - v2) and v/(1 -v) in X, Y, a, and a,.
to that shown in Fig. 12. After substituting equation (25) into (35), and after rearrangement, two quantities X and Y can be defined as follows: E AU,, 2a Aq
1
_y=-.-._--
1-v’
V
l-v
(36) y
EA& =2a’F’l-v2
---
1
V
l-v
=J3+&&$TZ ”
X and Y have been plotted in Fig. 13 for different values of a, ranging between 0 and 20 and a, ranging between 0 and 40. The isotropic case corresponds to a, = a, = 0 and is represented by point I with coordinates (1, 1). For plane stress, Fig. 13 can still be used by substituting respectively E and v for E/( 1 - v2) and Y,a,anda,. v/(1 -v)inX, Note that for regularly jointed rock, all three moduli E, k,S and k,S can be determined from expansion tests in the hole parallel to the joint planes by first loading the jointed rock by an amount Aq followed by a cycle of unloading and reloading over the same pressure increment. If the slope of the curve q vs UJ2a during the unloading-reloading cycle can be used as an estimate to E/(1 + v) for intact rock, as suggested by Goodman [3] for static tests on fractured rock, then e, 5: E,,/E and e2 = Ed/E can be obtained directly as the ratios between the slopes of the loading and unloading-reloading curves over the same pressure increment Aq. Borehole jack tests The three elastic constants E, E’, G’ or E, k,S and k,S can be determined by conducting borehole jack tests in
two boreholes; one borehole normal to the planes of transverse isotropy or joint planes and another parallel to those planes. First, consider the test in a borehole normal to the planes of transverse isotropy or joint planes (Fig. 14a). According to equation (12), over each increment of pressure Aq, the slope of the q vs U,,/2a curve recorded during borehole jack tests is equal to E,/K(v, 8,) where K(v, /I,) is given by equation (9) for plane strain and equation (34) for plane stress. The measured slope (see Fig. 14b) gives an estimate of E, = (E,), which is also related to E and E/E’ through equation (22) or E and E/k,S through equation (29). For plane stress, E, = (E,), gives a direct measurement of the intact rock modulus E. On the other hand, for a borehole parallel to the planes of transverse isotropy or joint planes (Fig. 15a), the slopes of the q vs U,, curves obtained by conducting jack tests in directions parallel and normal to those planes give estimates of (E,),, and (El)l, respectively (see Fig. 15b). These moduli are related to E, E/E’ and G/G’ through equations (20), (21) for intact rock or to E, E/k,S and E/k,S through equations (27), (28) for jointed rock and plane strain. For plane stress, equations (20) and (21) are replaced by equations (32) and (33). Hence, by measuring (El)_, (E,), and (E,), , a system of three non-linear equations can be constructed that can be solved for E, E/E’ and G/G’ or E, k,S and k,S. Another way of solving for E/E’ and G/G’ or E/k,S and E/k,S is to use charts similar to those of Figs 12 and 13. Let U,, and U, be the borehole diametral deformations measured in loading directions normal and parallel to the planes of transverse isotropy or joint
(a)
04 Fig. 15. (a) Borehole jack test in a borehole with transverse isotropy or joint planes parallel to the hole axis and yz-plane. Hole diametral deformations are measured by conducting jack tests in directions normal (0, = 0, Curve 1) and parallel (6, = n/2, Curve 2) to those planes. (b) Corresponding q vs U&a curves.
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC ROCK
‘i
0
5
_---
-- 1-J
30
c-
C-
A? 3.0 7-L Y
-'90
0-0
-
c-
20
--$8 I w 4 i
2.0-
Q-*
10
s
ti* *O
/f , 1.0
t
/ /
11
0
15
10
5
E A&jr _.-2a Q
20
1 WI%)
Fig. 16. Variation of (E/2a)(AU,,/Aq)/K(v,8,) with (E/2a) (AU,, /Aq)/K(v, &) for different values of a,, ranging between 0 and 20 and K, ranging between 0 and 40 and plane strain. Point I represents the isotropic case with K, = 01, = 0. For plane stress, E must be substituted for E/(1 - v2) in K, and 01,.
planes in the second borehole. For those two loading directions, equation (12) can be rewritten as follows: E AU,, a’F’K(v,B,)= E
1
E E, ,/) 0
1
E E, 1’ 0
(37)
AU,, zdq’K(v,B,)=
Using the expressions for (EJE),, and (EJE), derived in this paper for plane stress and plane strain, (E/2a) (AU,,lAq)lWv, 8,) and (E/2a) (AWAq)/ K(v, /I,) can be plotted for different values of E/E’ and G/G’ or E/k,S and EIk,S. A numerical example is shown in Fig. 16 for regularly jointed rock and for different values of a, ranging between 0 and 20 and a, ranging between 0 and 40 and plane strain. This figure can also be used for plane stress conditions with u, = EIk,S and u, = E/k& From a practical point of view, the problem remains that although AU,, , AU,, and Aq are measured in the field and although K(v, 8,) can be calculated once v is estimated and /3, is known, the in-plane modulus E cannot be determined from the first borehole as easily as with the dilatometer and gallery tests. This limitation can be overcome: (1) if the rock deforms in plane stress for which E is measured directly in the first borehole; or (2) if the rock is regularly jointed and E is determined during cycles of unloading and reloading as suggested previously for the dilatometer and gallery tests. CONCLUSION
The directional character of the modulus of deformation E, of an anisotropic rock mass measured along the walls of a borehole or tunnel during in situ expansion tests can be explained analytically by using closed-form
395
solutions based on the theory of elasticity for anisotropic media. The modulus depends on the anisotropic properties of the rock mass, its type of anisotropy (transversely isotropic or orthotropic) and the orientation of the rock mass planes of anisotropy with respect to the directions of measurements and/or loading in the hole in which the expansion tests are conducted. By replacing a regularly jointed rock mass by an equivalent anisotropic continuum, the modulus of deformation can also be related to joint properties such as spacing and stiffness. The effect of the degree of rock anisotropy on the variation of the modulus of deformation around the circumference of a hole in which expansion tests are conducted is shown for intact transversely isotropic and regularly jointed rock masses with transverse isotropy or joint planes parallel or normal to the hole. For intact rock, the ratio between the modulus of deformation Ea and the Young’s modulus E in the planes of transverse isotropy of the rock depends on four dimensionless terms: two Poisson’s ratios v and v’, and two ratios of Young’s moduli E/E’ and shear moduli G/G’ defined in planes parallel and normal to the planes of transverse isotropy. For jointed rock, EJE depends on the intact rock Poisson’s ratio v and the ratios between the intact rock Young’s modulus E and the normal and shear joint moduli k,S and k,S. The analysis presented in this paper is limited to anisotropic rock masses with structural features (layers, beddings, schistosity, etc.) that are much smaller than the diameter of the hole in which expansion tests are conducted. For regularly jointed rock, the equivalent continuum approach is limited to geometries for which joint spacing is small compared to the size of the problem of interest. For instance, for borehole expansion tests, and for most commonly-used borehole diameters, joint spacing must be small in order to use the equivalent continuum approach. On the other hand, for gallery tests that usually involve larger volumes of rocks than the borehole tests, this approach seems to be more appropriate for a much wider range of joint spacings. The variation of the modulus of deformation observed along the circumference of a borehole or tunnel during expansion tests not only provides a qualitative assessment of the anisotropic character of the rock mass of interest but can also be used to determine its anisotropic properties. As shown in this paper, assuming values for the Poisson’s ratios, expansion tests in two holes, one normal to transverse isotropy or joint planes and another parallel to those planes can be used to determine E, E/E’ and G/G’ for intact rock and E, E/k,S and E/k&3 for jointed rock. For the hole normal to the planes of transverse isotropy or joint planes, all diametral measurements are the same and therefore only one independent diametral measurement can be obtained. For the other hole parallel to the transverse isotropy or joint planes and in a coordinate system attached to those planes, there are at most two independent diametral measurements. Hence, with two holes, a total of three diametral measurements can be obtained that lead to a system of three non-linear equations and three un-
AMADEI and SAVAGE:
396
TESTS IN ANISOTROPIC
knows:
E, E/E’ and G/G’ or E, E/k,,S and EIk,S. This system can be solved directly or charts can be used to determine E/E’ and G/G’ in the second hole. The stress dependency of the elastic constants can be assessed by analyzing pressure vs diametral deformation curves in an incremental form. Note that the elastic constants v, v’, E, E’, G and G’ must also satisfy certain inequalities associated with the thermodynamics constraint that the strain energy of the rock must always remain positivedefinite [12,21]. Accepted
for publication 5 February 1991. REFERENCES
I. Goodman R. E., Van T. K. and Heuze F. E. Measurement of rock deformability in boreholes. Proc. 10th U.S. Symp. on Rock Mech., Univ. of Texas. Austin. DD. 523-555 (1972). 2. Bieniawski 2. T. Deter&&g rock mass deformability: experience from case histories. ht. J. Rock Mech. Min. Sci. & Geomech. Abstr. IS, 237-248 (1978). 3. Goodman R. E. Introduction to Rock Mechanics, 2nd Edn. Wiley,
New York. (1989). 4. Brown E. T. Rock characterization testing and monitoring. ISRM Suggested Metho&. Pergamon Press, Oxford (1980). 5. Swolfs H. S. and Kiblcr J. D. A note on the Goodman jack. Rock Mech. 15, 57-66 (1982). 6. Heuze F. E. and Amadei B. The NX-borehole jack: a lesson in trial and error. Int. J. Rock Mech. Min. Sri. & Geochem. Abstr. 22, 105-I I2 (1985). 7. Rocha M., Da Silveira A., Rodrigues F. P., Silverio A. and
Ferreira A. Characterization
of the deformability of rock masses
8. 9. 10. 11.
12.
ROCK
by dilatometer tests. Proc. 2nd. Congr. Int. Sot. Rock Mech. (ISRM), Belgrade, 2-32, Vol. 1, pp. 509-516 (1970). Amadei B. and Savaae W. Z. Comprehemive Rock Etwineerinx (Edited by J. A. Hudson). Pergamon Press, Oxford (1991). Amadei B. Rock Anisotropy and the Theory of Stress Measurements. Springer-Verlag, New York (1983). Lekhnitskii S. G. Theory of Elasticity of anAniPotropic Body. Mir Publishers, Moscow (1977). Gerrard C. M. Background to mathematical modelling in geomechanics: the roles of fabric and stress history. Proc. Int. Symp. on Numerical Methodr, Karlsruhe, pp. 33-120 (1975). Amadei B., Savage W. Z. and Swolfs H. S. Gravitational stresses in anisotropic rock masses. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 24, 5-14 (1987).
13. Duncan J. M. and Goodman R. E. Finite element analyses of slopes in jointed rock. Final Report to U.S. Army Corps of Engineers, Vicksburg, MI, Report S-68-3 (1968). 14. Morland L. W. Elastic anisotropy of regulatory jointed media. Rock Mech. 8, 35-48
(1976).
15. Gerrard C. M. Equivalent elastic moduh of a rock mass consisting of orthorhombic layers. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 19, 9-14 (1982).
16. Gerrard C. M. Elastic modulus of rock masses having one, two or three sets of joints. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 19, 15-23 (1982). 17. Filon L. N. G. On an approximate solution for the bending of a
18. 19. 20.
21.
beam of rectangular cross section under any system of load with special reference to points of concentrated or discontinuous loading. Proc. R. Sot. Land. Ser. A. 201, 65-154. Love A. E. H. A Treatise on The Mathematical Theory of Elasticity. Cambridge University Press (1920). Lekhnitskii S. G. Anisotropic Plates. Gordon & Breach, London (1968). Green A. E. and Zema W. Theoretical Elasticity. Clarendon Press, Oxford (1968). Pickering D. J. Anisotropic parameters for soils. Geotechnique 2Q, 271-276 (1970).
ht. 1. Rock Med. Min. Sci. & Geomech. Abstr. Vol. 28, No. 5, pp. 383-396, 1991 F’rinted in Great Britain. All rights re~ervcd
Copyright 0 1991 Pergamon Press pk
Analysis of Borehole Expansion and Gallery Tests in Anisotropic Rock Masses B. AMADEIt W. 2. SAVAGES Closed-form solutions are used to show how rock anisotropy affects the variation of the modulus of deformation around the walls of a hole in which expansion tests are conducted. These tests include dilatometer and NX-jack tests in boreholes and gallery tests in tunnels. The effects of rock anisotropy on the modulus of deformation are shown for transversely isotropic and regularly jointed rock masses with planes of transverse isotropy or joint planes parallel or normal to the hole longitudinal axis for plane strain or plane stress condition. The closed-form solutions can also be used when determining the elastic properties of anisotropic rock masses (intact or regularly jointed)
in situ.
INTRODUCTION
Because of scale effects, laboratory tests on small rock samples are usually inadequate to measure the deformability of rock masses. This has led to the development of several static and dynamic field testing methods over the past 20yr. In the static methods, loads are applied on selected rock surfaces in boreholes, underground galleries or at the surface of rock outcrops and the resulting deformations are measured. These methods include plate bearing tests, flat jack tests, radial jacking tests, gallery tests and borehole expansion tests, all of which are well-documented in the literature. Testing procedures, case studies and reviews of their advantages and limitations can be found in Goodman et al. [1], Bieniawski [2], Goodman [3], Brown [4], Swolfs and Kibler [5] and Heuze and Amadei [6], among others. This paper focusses on the analysis of expansion tests in boreholes and gallery tests in tunnels. A number of testing devices are available which can be inserted into a borehole to apply a load and measure the direct response of the walls of the borehole. These devices either: (1) supply a uniform internal pressure in the borehole such as the dilatometers; or (2) supply a unidirectional pressure to a portion of the circumference of a borehole by forcing apart circular plates such as the NX-borehole jack of Goodman et al. [l]. The results of these tests are often presented in the form of curves of applied pressure vs diametral deformation. The diametral deformation is measured either in the direction of loading, as with the borehole jack, or at one or several locations around the borehole circumference as with some dilatometers. Gallery tests are similar to dilatomeTDepartrncntof Civil Engineering, University of Colorado, Boulder, CO 803094428, U.S.A. jU.S. Geological Survey, Box 25046, MS%, Denver, CO 80225, U.S.A.
Denver Federal Center,
ter tests in that a sealed section of a tunnel is pressurized and the rock mass deformation is measured at several points around the excavation circumference. Usually, the measurements of diametral deformation in expansion tests are analyzed by modelling the rock mass as a linearly elastic, isotropic and homogeneous continuum with Young’s modulus E and Poisson’s ratio v. A value for the Poisson’s ratio is assumed (often equal to 0.25) and the modulus E is determined from measured values of applied pressure q and hole diametral deformation U,, through equations borrowed from the theory of linear elasticity for isotropic media; e.g. 1 u, --
(l+v)
q2a=E’
for dilatometer
(l)
and gallery tests, and 1 Ud K(v9 m -q2a=E’
(2)
for the NX-borehole jack test. In these equations, 2a is the diameter of the hole, and 28, is an estimated value of the contact angle between the jack and the rock [5,6]. Equations (1) and (2) are also used in terms of increments of diametral deformation AU,, and increments of applied pressure Aq when pressure vs hole diametral deformation response curves are non-linear. The analysis of expansion test results described above is commonly followed in practice regardless of the anisotropic, discontinuous and heterogeneous nature of the rock mass of interest. The coefficient E would be the true elastic Young’s modulus of the rock mass if the latter actually behaved as a linearly elastic, isotropic, continuous and homogeneous medium. Any departure from these idealized characteristics causes the coefficient E to become, instead, an apparent modulus or modulus of &formation of the rock mass in the direction of loading and/or diametral measurement. The magnitude of this modulus can vary along the circumference of the
383
AMADEI and SAVAGE:
384
TESTS IN ANISOTROPIC
ANALYTICAL
hole if the rock mass shows anisotropic, discontinuous or heterogeneous behaviour. For dilatometer tests, such variations when plotted in polar form can give an insight into the anisotropy of the rock mass being studied [I. This can also be obtained with the borehole jack by loading, for instance, a jointed rock mass in directions parallel and normal to the joint planes. In a recent paper, Amadei and Savage [8] show that the directional character of the modulus of deformation of an anisotropic rock mass E, measured along the circumference of a hole during in situ expansion tests can be explained analytically by using closed-form solutions based on the theory of elasticity for anisotropic media. By replacing a regularly jointed rock mass by an anisotropic continuum, it is also shown that the modulus of deformation can be related to joint properties such as spacing, orientation and stiffnesses.
1
E,
--
VIU
E,,
ROCK
BACKGROUND
Consider the equilibrium of a rock mass that is modelled as an infinite, linearly elastic, anisotropic, continuous and homogeneous medium. The medium is bounded internally by a cylindrical surface of circular cross section of radius u that represents a hole (borehole or tunnel). Consider the geometry of Fig. 1 and let x, y, z be a Cartesian coordinate system with the z-axis defining the longitudinal axis of the hole. The rock mass is assumed to be orthotropic in a n, s, t Cartesian coordinate system attached to clearly defined planes of anisotropy. The orientation of these planes with respect to the x, y, z coordinate system is assumed to be known. The constitutive equation for the orthotropic medium in the n, s, t coordinate system is given by the following equation:
-2 V Es
-- 6VI”
0
0
0
1 E,
v, -E,
0
0
0
(3) 0
0
ok00 I,
0
0
0
0
&
0
“,
This paper first reviews the closed-form solutions of Amadei and Savage [8]. Then, these solutions are used to show how different degrees of rock anisotropy affect the variation of the modulus of deformation around the walls of a hole in which expansion tests are conducted. This is shown for transversely isotropic and regularly jointed rock masses with planes of transverse isotropy, or joint planes, parallel or normal to the hole longitudinal axis and for plane strain or plane stress conditions. It is also shown how the closed-form solutions can be used when determining the elastic properties of anisotropic rock masses (intact or regularly jointed) in situ.
S A
Fig. 1. Geometry
Ill
of the problem. Hole intersecting anisotropy in a rock mass.
planes of
or in a more compact matrix form: (s),, = W) (c)#W
(4)
Nine independent elastic constants are needed to describe the deformability of the medium in the n, s, t coordinate system. E,, ES and E, are the Young’s moduli in the n, s and t (or 1, 2 and 3) directions, respectively. G,, G,, and G,, are the shear moduli in planes parallel to the ns, nt and rt planes, respectively. Finally, = n, s, t) are the Poisson’s ratios that characterize vij (6 j the normal strains in the symmetry directions j when a stress is applied in the symmetry directions i. Because of symmetry of the compliance matrix (H), Poisson’s ratios Vii and vji are such that VijlEi =:VjilEj. Equations (3) and (4) still apply if the medium is transversely isotropic in one of the three M-, nt- or st -planes. In that case, only five independent elastic constants are needed to describe the deformability of the medium in the n, s, t coordinate system. In this paper, these constants will be called E, E’, v, v’ and G’ with the following definitions: (i) E and E’ are Young’s moduli in the plane of transverse isotropy and in direction normal to it, respectively;
AMADEI
and SAVAGE:
TESTS IN ANISOTROPIC
(ii) v and v’ are Poisson’s ratios characterizing the lateral strain response in the plane of transverse isotropy to a stress acting parallel or normal to it, respectively; and (iii) G’ is the shear modulus in planes normal to the plane of transverse isotropy.
ROCK
385
(4
Relations exist between E, E’, v, v’, G and G’ and the coefficients of matrix (H) in equation (3). For instance, for transverse isotropy in the St-plane: 1 -=-. E.
1 E”
1 1 -c-z-;
1
1 1 -_=-=-
Es
E
G,
E,
VII.3 V,I -=__=-.
v’
v*r vu -c-c_;
V
1 -=-
E.
E”
Es
E
Gs,
E,,
Et
G,,
1 G”
G
For known orientation of the planes of anisotropy with respect to the hole, the constitutive relation of the medium in the x, y, z coordinate system can be obtained by using second-order tensor coordinate transformation rules (see for instance [9]). This gives:
where
and
(A) is a (6 x 6) compliance matrix whose components uij (i, j = 1,6) depend on the elastic properties of the medium in the n, s, t coordinate system and the orientation of its planes of anisotropy with respect to the hole. Consider the loading conditions shown in Figs 2a and b. Figure 2a corresponds to the uniform radial loading of a hole (dilatometer and gallery tests) with a pressure q. In Fig. 2b the hole is loaded in the y-direction with a constant surface force per unit area q applied over two opposed arcs of angle 28, (NX-borehole jack test). Closed-form solutions for the displacement components along the hole circumference induced by the loads of Figs 2a and b are given by Amadei and Savage [fl] for an anisotropic medium. It is assumed in these solutions that the anisotropic medium deforms under a condition of generalized plane strain in the xy -plane [lo]. This assumption holds if the hole is loaded over a length that is much longer than its diameter. Also, displacements induced by the loading must be calculated at hole cross-sections located some distance inwards from the ends of the loaded zone (at least one hole diameter as a rule of thumb). Finally, body forces were neglected. The condition of generalized plane strain allows planes of anisotropy in the medium to be inclined with respect to the hole axis. All planes parallel to the xy-plane of Figs 2a and b are assumed to warp in the same way. In other words, longitudinal deformation induced by the loads in Figs 2a and b does not vanish and depends only on the x and y coordinates. For certain types and orientations of the anisotropy, generalized
Fig. 2. (a) Radial loading of a hole during dilatometer and gallery tests. (b) Loading of a hole over two opposed arcs of angle 2& with NX-borehole jack.
plane strain reduces to plane strain and the longitudinal deformation always vanishes. This occurs when the hole axis is at right angles to a plane of elastic symmetry such as: (1) when the medium is orthotropic in the x, y, z coordinate system (Fig. 3a) or has one of its three planes of symmetry normal to the z-axis (Fig. 3b); when the medium is transversely isotropic and the hole axis is either parallel (Fig. 3c) or perpendicular (Fig. 3d) to the planes of transverse isotropy; or (3) when the medium is isotropic. For all those cases, the compliances a&, us6, o.,~, usi (i = 1,2,3) in matrix (A) of equation (6) vanish. For dilatometer or gallery tests, Amadei and Savage [8] have shown that the hole diametral deformation U, measured at an angle 0 to the x-axis of Fig. 2a can be expressed as follows: ~~=A+Bcos20+Csin28 4 2a where A, B and C depend on the elastic properties of the anisotropic medium in the n, s, t coordinate system and the orientation of its planes of anisotropy with respect to the hole. For generalized plane strain conditions, A, B and C do not vanish. Conversely, if the medium is isotropic with Young’s modulus E and Poisson’s ratio v, B and C vanish and equation (7) reduces to equation (1).
386
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC
(4
(b)
(c)
(4
ROCK
Fig. 3. Four cases of anisotropy where the hole axis is normal to a plane of elastic symmetry: (a), (b) orthotropic medium; and (c), (d) transversely isotropic medium.
For the borehole jack tests, Amadei and Savage [8] have shown that the hole diametral deformation U,, in the direction of load application is such that: s-7 . tu,, t -=-
u:,
Similarly, the borehole jack test with loading in the y-direction gives:
K(v, A 1
1 ud --
(12)
q2a=E,’
(8) with
where /3, is expressed in radians and IJ: depends on the elastic properties of the anisotropic medium in the n, s, t coordinate system and the orientation of its planes of anisotropy with respect to the hole. For an isotropic medium with Young’s modulus E and Poisson’s ratio v, equation (8) reduces to equation (2) with: K(v,8.)=&(l-V2)
2 --$[l-(-l)“]sidm/I.. m-l
(9)
VARIATION IN MAGNITUDE OF THE MODULUS OF DEFORMATION
The variation in the magnitude of the modulus of deformation of an anisotropic rock mass measured at different points along the walls of a hole during expansion tests can be explained quantitatively with the analytical solutions presented above. To do so, equations (7) and (8) are rewritten as if the anisotropic medium were “isotropic” in the direction of loading and/or measurement, with an assumed Poisson’s ratio v and a modulus E,. Then, each diametral measurement in the dilatometer and gallery tests at an angle 8 from the x-axis can be interpreted from: 1 u, (l+v) -q2a=E,’
(10)
with E, = (1 + v)/(A + B cos 28 + C sin 20).
(11)
(13) Equations (11) and (13) provide analytical expressions for the modulus of deformation of an anisotropic rock mass modelled as a linearly elastic continuum. The modulus depends on the anisotropic properties of the rock mass and the orientation of its planes of rock anisotropy with respect to the directions of measurement and/or loading in the hole. Because of the linear relations existing between coefficients hii and a,j of matrices (H) and (A) in equations (4) and (6), respectively, it can be shown that the ratio between the modulus of deformation E, and one of the Young’s moduli, say E,, of an orthotropic medium depends on the following eight dimensionless quantities:
5. E”’
Es
E,;
E v,; vm; vu; $
E,. G,,’
Es G,’
(14)
If the medium is transversely isotropic with, for instance, transverse isotropy in the St-plane, and using equation (5), the ratio E./E, = E./E is found to depend only on four dimensionless terms: E E’;
v;
v’;
G 1. G
(15)
Note that E./E, and EJE also depend on the orientation of the planes of anisotropy with respect to the hole in which the expansion tests are conducted. In addition, for
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC
ROCK
Table I. Expressions of coefficients qj for three orientafions of the planes of transverse isotropy
the dilatometer and gallery tests, these ratios also depend on the angle 0 of Fig. 2a.
Transverse isotropy parallel to yz-Plane xz-Plane xy -Plane
a,j
l/E’ -v’lE’ l/E
alI 013 %I
EFFECT OF INTACT ROCK ANISOTROPY ON THJ3 MODULUS OF DEFORMATION
l/E -v/E l/E l/E’ -v’iE’ - v’/E’ l/G’
I/E
*22
-v/E -v’/E’ I/G’
a23
For known values of the elastic properties of an orthotropic or transversely isotropic rock mass and known orientation of its planes of anisotropy, the solutions summarized above can be used to predict how the modulus of deformation of an anisotropic rock mass varies around the circumference of a borehole or a tunnel in which expansion tests are conducted. These solutions can also be used to solve the inverse problem, that is, in interpreting expansion tests to obtain rock mass deformability properties. The inverse problem will be discussed later in this paper. The effect of intact rock anisotropy on the modulus of deformation of a rock mass is shown below for the special case when the hole deforms in plane strain for which coefficients A, B, C and U: in equations (7) and (8) take simpler analytical forms. Consider, for instance, the anisotropy orientations of Figs 3a, 3d and 3c with the dip of the planes of transverse isotropy, JI, either 0 or 90”. In all three cases, the hole axes x, y, z are perpendicular to planes of elastic symmetry in the rock mass. For these orientations, Amadei and Savage [8] have shown that C always vanishes whereas A, B and U: are equal to:
381
42
as
l/E -v’/E’ l/E’ l/E - v’/E’ -v/E l/G
earlier in this paper. Substituting these coefficients into equation (16) and the resulting expressions of A, B and U: into equations (11) and (13) gives closed-form expressions for the ratio EJE in terms of the four dimensionless quantities of equation (15). For the dilatometer and gallery tests, EJE is equal to:
4 -= E
(1 + v)
(17)
A*+ B*cos26’
where v is assumed to be equal to the Poisson’s ratio in the planes of transverse isotropy and A* = AE and B* = BE are such that: A*=;;[2v.(l
+v)+(l
-$v’~
+J_)
-2v’(l
X
+v)+2;$(1
+v) +2
1+
:
(18)
-2/m],
(16)
where jIij = aij - U&ljJU~3 (i, j = l-6) and a,, are the coefficients of matrix (A) in equation (6). For a transversely isotropic rock mass with planes of transverse isotropy parallel to the st -plane of Fig. 1, Tables 1 and 2 give the expressions of coefficients ai, and fiij when the planes of transverse isotropy are also parallel to the yz-plane (Fig. 3c with $ = go”), the xzplane (Fig. 3c with JI = 0”) or parallel to the xy-plane (Fig. 3d). These coefficients are expressed in terms of the five elastic rock properties E, E’, v, v’ and G’ defined
1E B*=ZF
Table 2. Expressions of coefficients Bij for three orientations of the planes of transverse isotropy Transverse isotropy parallel to
A,
yz -Plane
xz -Plane
xy -Plane
B,, Al
(1 - v’*E/E’)/E’ (1 - v*)/E -v’(l + v)lE’ l/G’
(1 - v’)/E (1 - v’=EIE’)/E’ -v’(l + v)lE’ l/G’
(1 - v’*E/E’)/E (1 - v’~E/E’)/E -(v + ;;$/E’)lE
AMADEI and SAVAGE:
388 (a)
TESTS IN ANISOTROPIC
1’2r
ROCK
(W 1.0
0.6
Fig. 4. Polar variation of EJE with the angle 8 (0 d 0 Q n/2) for gallery or dilatometer tests in a medium with planes of transverse isotropy parallel to the yz-plane and for E/E’ = 1, 1.5,2, 3 and 4, Y = v’ = 0.25 and: (a) G/G’ = 1; and (b) G/G’ = 3. T’he isotropic case corresponds to E/E’ = G/G’ = 1 or when the planes of transverse isotropy are parallel to the xy-plane.
when the planes of transverse isotropy are parallel to the yz-plane (Fig. 3c with JI = 90’) and
for planes of transverse isotropy parallel to the xz-plane (Fig. 3c with J/ = O”). Finally, for planes of transverse isotropy parallel to the xy-plane (normal to the hole axis, Fig. 3d), A* = (1 f v), B* = 0 and E./E = 1, i.e. the rock behaves as if it were isotropic. As a numerical example, Figs 4 and 5 show variations of E,/E with the polar angle 8 in Fig. 2a (0 < 8 < x/2) for several degrees of intact rock anisotropy with E/E’ = 1, 1.5, 2, 3 and 4, G/G’ = 1 and 3, v = v’ = 0.25, and when the planes of transverse isotropy are parallel to the yz- and xz-planes, respectively. The isotropic case corresponds to E/E’ = G/G’ = 1 or when the planes of transverse isotropy are parallel to the xy-plane. The chosen domains of variation for E/E’ and G/G’ are reasonable in view of the literature surveys on elastic properties of anisotropic rocks conducted by Gerrard [1 1] and Amadei et al. [12]. Note that because of the symmetry associated with the dependence of EJE on cos 28 in equation (17), the variations of E./E with the angle 8 in Figs 4 and 5 are only shown in the first quadrant (0 s 8 < n/2). Figures 4 and 5 show that as E/E’ increases and for a fixed value of the ratio G/G’, the rock mass becomes more deformable in directions normal to the planes of transverse isotropy and the variation of EJE with the
0.0
0.2
0.4
0.6
0.6
1.0
Fig. 5. Polar variation of EJE with the angle 0 (0 C 0 6 x/2) for gallery or dilatometer tests in a medium with planes of transverse isotropy parallel to the xz-plane and for E/E’ = 1, 1.5,2,3 and 4, v = v’ = 0.25 an& (a) G/G’ - 1; and(b) G/G’ = 3. The isotropic case corresponds to E/E’ = G/G’ = 1 or when the planes of transverse isotropy are parallel to the xy-plane.
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC
389
ROCK
-&‘UII ---(Ea% 1.4
\ ---_ I ‘. -
$0.81 \ ‘y 0.6
0.4
\
'. . .
G/G'=2
---“GIG'_,
.. .
--__ --*_
GIG-3
- ---GK;'-2 --_ as-3
Fig. 6. Polar variation of EJE with the angle 0 (0 d 0 d n/2) for gallery or dilatometer tests in a medium with planes of transverse isotropy parallel to the yz-plane and for E/E'= 3,G/G'= 1,v = 0.25 and v’ = 0.15, 0.25 and 0.35. The isotropic case is also shown for Fig. 7. Variation of (E./E),, (EJE),and (E./E), with E/E' for comparison. G/G’ = 1, 2, 3 and v = v’ = 0.25(NX-borehole jack tests). The isotropic case corresponds to E/E'= G/G'= 1 (point I).
angle 8 becomes more non-uniform as expected. Note that the value of the modulus of deformation in directions parallel to the planes of transverse isotropy is not much affected by the value of E/E’ unlike the value of the modulus in directions normal to the planes of transverse isotropy. For a fixed value of E/E’, the polar variation of EJE with the angle 8 depends strongly on the value of G/G’. As G/G’ increases, the rock mass becomes more deformable in shear in planes normal to the planes of transverse isotropy which results in lower values for E,/E in all directions around the hole. Figure 6 shows the polar variations of E,/E with the angle 8 for v’ = 0.15, 0.25 and 0.35, E/E’ = 3, G/G’ = 1 and when the planes of transverse isotropy are parallel to the yz-plane. The value of the modulus of deformation decreases with increasing v’ in directions parallel to the planes of transverse isotropy and increases slightly with v’ in directions normal to the planes of transverse isotropy. For the NX-borehole jack test in which E, is determined in the direction of load application, equation (17) is replaced by the following:
when the planes of transverse isotropy are parallel to the xz-plane and at right-angles to the jack loading direction in Fig. 2b. Finally, for planes of transverse isotropy parallel to the xy-plane (normal to the hole axis), E./E is equal to:
E,
0
i? XY=
l-9
(22)
142;
Note that the Poisson’s ratio v in the numerators of equations (20-22) is assumed to be equal to the Poisson’s ratio in the planes of transverse isotropy. As a numerical example, Fig. 7 shows the variations of (E./E),, , (E./E), and (EJE), with E/E’ for G/G’ = 1, 2 and 3, and v =v’= 0.25. For a fixed value of E/E’, (E./E),, and (EJE), decrease as G/G’ increases. On the other hand, for a fixed value of G/G’, (EJE), is not much affected by the value of E/E’ whereas (E./E), decreases as E/E’ increases. According to equation (22), (E./E), is independent of G/G’ but increases with E/E’. Table 3 gives the values of (EJE),, , (E./E), and (EJE),
(20)
when the planes of transverse isotropy are parallel to the yz-plane, that is, the jack loading direction in Fig. 2b and
AMADEI and SAVAGE:
390
TESTS IN ANISOTROPIC
Table 3. Value of (EJE),, , (EJE), and (EJE),,. for three different values of Poisson’s ratio 1”’ v’
0.15 0.25 0.35
b’W), 0.901 1.014 1.214
intact rock and joints are assumed to undergo equal strains in directions parallel to the contact planes. This assumption makes all non-zero diagonal terms in equation (3) equal to the same value, -v/E. It also results in reducing the number of elastic constants necessary to describe the deformability of the rock mass when cut by a single joint set to four, namely, E, v, k,S and k,S.
WE),
WE), 0.522 0.629 0.854
1.005 1.154 1.482
for G/G’ = 1, E/E’ = 3, v = 0.25 and v’ = 0.15,0.25 and 0.35. All three modulus ratios increase with v’.
EFFECT OF JOINT INDUCED ANISOTROPY ON THE MODULUS OF DEFORMATION
ROCK
If a regularly jointed mass is modelled as an equivalent continuum, equations (11) and (13) can also be used to relate the modulus of deformation in the direction of diametral measurement and/or loading in a hole to joint properties such as spacing, orientation and stiffnesses. For instance, for a rock mass cut by a single joint set parallel to the yz -, xz - or xy -plane of Fig. 1, substituting the expressions for l/E’, l/G’ and v//E’ of equation (23) into equations (18) and (19), the ratio EJE for the dilatometer and gallery tests is again given by equation (17) where A * and B* now depend on the Poisson’s ratio v and two dimensionless ratios a, and a, with:
Rock mass anisotropy can also be created by systems of regularly spaced rock joints also called joint sets in a rock that is otherwise isotropic. Since it is not possible to account for each joint plane when assessing the overall deformability of a regularly jointed rock mass, it is convenient to replace the rock mass by an equivalent anisotropic continuum [ 13-161. The rock mass is modelled as transversely isotropic if cut by a single joint set and orthotropic if cut by two or three orthogonal joint sets. E Consider for instance a rock mass cut by a single joint a’ = (1 - v*)k,S’ set oriented parallel to the St-plane of the n, s, t coordinate system of Fig. 1. The intact rock between the joints E is assumed to be linearly elastic and isotropic with a’ = (1 v*)k,S’ Young’s modulus E, Poisson’s ratio v and shear modulus G = E/2( 1 + v). If the joint set has spacing Sand normal For joints parallel to the yz-plane, A* and B* are now: and shear stiffnesses k, , k,, Duncan and Goodman [ 131 show that the regularly jointed rock mass can be reA*,(l - -v2) G 2v +(l +a,+JGi) placed by an equivalent transversely isotropic contin2 [ uum whose constitutive relation in the n, s, t coordinate system is given by equations (3) and (4). Furthermore, 2+a X the elastic constants in equation (3) satisfy equation (5) 1 with:
J~+g===&-uTi
1 _=L+i. E’ E
k,S’
i=L+i. G’ G
k,S’
vI,11. E’ E’
(23)
Hence, all non-zero off-diagonal terms in equation (3) are now equal to - v/E. Since k, and k, have units of stress/length or force/length3, the quantities k,S and k,S in equation (23) can be seen as normal and shear joint moduli. Note that the constitutive relation for a regularly jointed rock mass converges to that for an intact isotropic medium when the joint spacing or joint stiffness values in equation (23) approach infinity. Although convenient, the equivalent continuum formulation has certain limitations. For instance, Duncan and Goodman [13] make three main assumptions in the equivalent continuum approach: (i) the normal and shear joint stiffnesses are assumed constant and independent of the stress level acting across the joints; (ii) the joint response is assumed to remain in the elastic domain (pre-slip condition); (iii) the joints are assumed to have negligible thicknesses and not to create any Poisson’s effect upon loading of the rock mass. In other words,
B*=
9
+a,,-,/-)
y[(l
(25)
and for joints parallel to the xz-plane:
-v2)
A*_u --
2
2v
[
T--y+(l+Ji=n)
X J2+as+2JCr,-2J1+G(,
1 ,
J
B*=v[(l
(26)
-Jx)
Finally, for joints parallel to the xy-plane (normal to the hole axis), A * = (1 + v), B* = 0 and E,]E = 1, i.e. the jointed rock mass behaves as if it were isotropic. As numerical examples, Figs 8 and 9 show polar variations of E,/E with the angle 19(0 < 0 < n/2) for
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC
ROCK
391
of the modulus of deformation around the hole circumference approaches that of a circle. For the NX borehole jack test and for joints parallel to the yz-plane, i.e. the jack loading direction, equation (20) is replaced by the following: 2
E,
0 i?
,=J
(27)
2+=,+2Jx’
For joints parallel to the xz-plane, i.e. at right-angles to the jack loading direction, equation (21) becomes: 2
E,
0
(28) 2+a,+2JK’ Finally, for joints parallel to the xy-plane (normal to the hole axis), equation (22) is replaced by: 2
I=&%&
4
( >. z
Fig. 8. Polar variation of E./E with the angle 0 (0 < 6
eight cases of rock anisotropy when E/k,S ranges between 0 and 20, EIk,S ranges between 0 and 40 and v = 0.25 for joints parallel to the yz- and xz-plane, respectively. The isotropic case corresponds to E/k,S = E/k,S = 0 or when the joints are parallel to the xy-plane. As the rock mass becomes more isotropic, that is, as joint spacing and stiffnesses increase, the variation
“‘=l_
1 -vz v2 * E ’ +k,S
(29)
Table 4 gives values of (EJE),,, (EJE), and (EJE), for the eight cases of joint anisotropy considered above. For this example, the ratio of moduli of deformation measured in directions parallel and normal to the joint set, J% I&,, varies between 4.8 for case 1 and 1.Ofor case 8. Note that the value of (E,/E), is not much affected by the degree of rock mass fracturing. PLANE
STRAIN
VS PLANE
STRESS
Equations (17-29) were derived when the hole in which expansion tests are conducted deforms in plane strain and when the hole axes x, y and z are all perpendicular to planes of elastic symmetry in the rock mass. In general, the plane strain assumption holds as long as the hole is loaded over a length that is several times its diameter. Furthermore, the displacements induced by the loads in the hole must also be measured in crosssections located well within the loaded zone, say at least one hole diameter from the ends of the loaded zone. Conversely, if the loads shown in Figs 2a and b are applied over a smaller length, say of the order of one or two hole diameters or less, the plane strain assumption can no longer be used and must be replaced by plane stress. As discussed by Amadei [9], plane stress also implies that the hole be at right-angles to planes of elastic Table 4. Values of (E./E),,, (EJE), and (E./E), for eight different cases of joint induced anisotropy Case No.
Fig. 9. Polar variation of EJE with the angle 8 (0 d 0 d x/2) for gallery or dilatometer tests in a medium with a joint set parallel to the xz-plane and for eight cases of joint anisotropy. The isotropic case corresponds to E/k,S = E/k,S = 0 or when the joint planes are parallel to the xy-plane.
1 2 3 4 : I 8
El&S 20 10 5 2 0.4 0.2 0.02 0.0
EIkS 40 20 10 4 0.4 0.8 0.04 0.0
(WE), 0.212 0.364 0.475 0.638
(WE), 0.057 0.107 0.188 0.361
(WE), 0.940 0.943 0.947 0.957
0.873 0.929 0.992 1.0
0.731 0.844 0.981 1.0
0.989 0.981 0.998 1.0
AMADE
392
ant’ SAVAGE:
TESTS 1’N ANEOTROPTC
symmetry and therefore can only be used for the anisotropy types and orientations shown m Figs 3a-d. Under plane stress, it can be shown [ 17-201 that mean values of stress, strain and displacement in the xq‘-plane induced by the loads shown in Figs 2a and b and calculated over the length of the loaded section of the hole satisfy the same basic elasticity equations that govern the plane strain formulation. The difference is that all coefficients /Iii such as those appearing in equation (16) must now be replaced by the corresponding coefficients aij (i, j = l-6) of matrix (A) in equation (6). For plane stress, the expression of the modulus of deformation for a transversely isotropic rock mass with planes of transverse isotropy parallel to the yz-, xz- or xy-planes of Fig. 1 can be obtained by substituting coefficients aij of Table 1 for the corresponding p,, coefficients in equation (16). After algebraic manipulations, for the dilatometer and gallery tests, equation (17) still applies but A * and B* are now equal to:
X
/-2vJ+2$$(,+“)+2&2&],
ROCK
For the borehole jack rests, equations (20) and (21) are replaced by the following:
when the planes of transverse isotropy are parallel to the yz-plane and the jack loading direction and
9 (33)
when the planes of transverse isotropy are parallel to the xz-plane and at right-angles to the jack loading direction. Finally, for planes of transverse isotropy parallel to the xy-plane, (E,/E),v, = 1. For joint induced anisotropy, substituting equations (23) into equations (32) and (33) leads to equations similar to equations (27) and (28) with a, = E/k,S and a, = EIk,S. Note that for plane stress, K(v, /I,) in equations (12) and (13) is now equal to: 4 N K(v9Be)=~~~,;;;j[l
X
-2v’+2;$(1+“)+2
;,
1 -(-l)m]sinZm~c.
(34)
(30)
Jwhen the planes of transverse isotropy are parallel to the yz -plane and
(31)
for planes of transverse isotropy parallel to the xz-plane. As for plane strain, when the planes of transverse isotropy are parallel to the xy-plane, A * = (1 + v), B+=O and EJE= 1. For joint induced anisotropy, substituting equation (23) into equations (30) and (31) leads to expressions similar to equations (25) and (26) where (1 - v2) and v/(1 - v) are now replaced by 1 and v, respectively and u, = EIk,S, a, = E/k,S.
DETERMINING ELASTIC PROPERTIES OF ANISOTROPIC ROCK MASSES FROM EXPANSION TESTS
The variation in the modulus of deformation measured during expansion tests along the circumference of a borehole or tunnel in an anisotropic (intact or jointed) rock mass can be used to determine its elastic properties. This is shown below for transversely isotropic rock masses for which the unknown elastic constants are E, E’, v, v’ and G’ for intact rock and E, v, k,S and k,S if the rock is regularly jointed. In what follows, values for the Poisson’s ratios v and v’ are assumed as in the conventional isotropic analysis of expansion tests and the other three elastic constants E, E’ and G’ or E, k,S and k,S are determined from the test data. Applied pressure q vs hole diametral deformation U, response curves measured in situ are assumed to be non-linear and are analyzed in terms of increments of pressure and deformation Aq and AU,,. Thus, the elastic constants can be calculated over each increment of applied pressure and their stress dependency can be assessed. Dilatometer and gallery tests The three elastic constants E, E’, G’ or E, k,S and k,S can be determined by conducting dilatometer or gallery
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC 3.0
393
ROCK
r
WE’ 1
2.5 -2
_
44
3,
2.5 __ II8 +j
2*o _
(a) (W Fig. 10. (a) Dilatometer or gallery test in a hole with transverse. isotropy or joint planes normal to the hole axis. (b) Corresponding q vs U,/20 curve.
tests in two holes; one hole normal to the transverse isotropy or joints planes and another parallel to those planes. For these hole orientations, the closed-form solutions presented earlier in this paper apply. First, expansion tests are conducted in a hole having an axis normal to the planes of transverse isotropy or joint planes. For this anisotropy orientation, the rock mass behaves as if it were isotropic and all hole diametral deformations are equal (Fig. 10a). Over each increment of pressure Aq, the slope of the q vs Ud/2a curve is then equal to E/(1 + v) (Fig. lob). Assuming a value for v, the in-plane Young’s modulus E and shear modulus G = E/2(1 + v) can then be determined. Next, expansion tests are conducted in a hole parallel to the planes of transverse isotropy or joint planes. Let x, y, z be an arbitrary coordinate system attached to those planes. Since for this geometry C vanishes in equation (7), there are at most two independent diametral measurements that can be carried out in the hole. For each direction of diametral measurement, a curve q vs U,/2a is obtained. Consider for instance the case when the planes of transverse isotropy or joint planes are parallel to the yz-plane (Fig. 1la) and two diametral deformations U,, and U, are measured in directions normal (0, = 0’) and parallel (0, = n/2) to those planes, respectively (Curves 1 and 2 in Fig. 1lb). The respective slopes of those two curves over the same pressure increment Aq are equal to E,,/( 1 + v) and E&l + v). Dividing these slopes by E/(1 + v) measured in the first hole over the same pressure increment, gives values for e, = E,,/E and e, = Ed/E. Substituting these values into
q
Aq
----___
(a) (W Fig. Il. (a) Dilatometer or gallery test in a hole with transverse isotropy or joint planes parallel to the hole axis and yz-plane. Hole diametral deformations are measured at 0, = 0 (Curve 1) and 0, = n/2 (Curve 2). (b) Corresponding q vs U&a curves.
RMMS W-D
-2
/
GIG’
.5
1.5 1 t 1.01 0.0
fl I I
@ I
I
I
4.0
2.0
I
I
6.0
E.&l 2aQ Fig. 12. Variation of (E/2o)(AU,/Aq) with (E/2u)(AU,,/Aq) for different values of E/E’ and G/G’ ranging between I and 4, v = v’ = 0.25 and for plane strain conditions. Point I represents the isotropic case with E/E’ = G/G’ = 1.
equation (17) with 0, = 0 and Bz= n/2 leads to the following equations: A*+B*=_=_.-,
1+ v el
E AU.,, 2a Aq (35)
Af_B*_l+v --=-.__ E Au, e2
2a
Aq
For intact rock anisotropy, substituting the expressions of A * and B* given in equation (18) for plane strain or equation (30) for plane stress into equation (35) gives a system of two non-linear equations that can be solved for the two unknowns E/E’ and G/G’. Knowing E and G from the first hole, E’ and G’ can be calculated over the same pressure increment Aq. The modulus ratios E/E’ and G/G’ can be determined from charts similar to that shown in Fig. 12. In this figure, (E/2a) (AU,/Aq) and (E/2a) (AU,,/Aq) defined in equation (35) have been plotted using A* and B* defined by equation (18) for different values of E/E’ and G/G’ ranging between 1 and 4, v = v’ = 0.25 and for plane strain conditions. The isotropic case corresponds to E/E’ = G/G’ = 1 and is represented by point I with coordinates (1 + v, 1 + v). Knowing the values of AU.,, , AU, and Aq from the field tests and E from the first hole, Fig. 12 can be used to determine the two ratios E/E’ and G/G’ and therefore E’ and G’. Charts similar to Fig. 12 can be generated for other values of v and v’ and for plane stress conditions. For joint-induced anisotropy, substituting A l and B* given in equation (25) for plane strain and similar equations for plane stress into equation (35) gives a system of two non-linear equations that can be solved for E/k,S and E/k& Again knowing the values of AU,,, , AU, and Aq from the field tests and E from the first hole, allows the determinations of k,S and k,S and the corresponding joint stiffnesses k,, k, over each pressure increment Aq once S is measured in the field. Again, E/k,S and EIk,S can be determined from charts similar
q 3
AMADEI and SAVAGE:
394
TESTS IN ANISOTROPIC
ROCK
Y
x
2
(a)
Fig. 14. (a) Borehole jack test in a borehole with transverse isotropy or joint planes normal to the hole axis. (b) Corresponding q vs U,/2a curve.
2.:: 0
10
5
15
20
X~L.!%b
.-_v 1
25
30
2a Aq l-V2 1-v Fig. 13. Variation of X and Y defined in equation (36) for different values of a, ranging between 0 and 20 and a, ranging between 0 and 40 and plane strain. Point I represents the isotropic case with a, = a, = 0. For plane stress, E and v must be substituted for E/(1 - v2) and v/(1 -v) in X, Y, a, and a,.
to that shown in Fig. 12. After substituting equation (25) into (35), and after rearrangement, two quantities X and Y can be defined as follows: E AU,, 2a Aq
1
_y=-.-._--
1-v’
V
l-v
(36) y
EA& =2a’F’l-v2
---
1
V
l-v
=J3+&&$TZ ”
X and Y have been plotted in Fig. 13 for different values of a, ranging between 0 and 20 and a, ranging between 0 and 40. The isotropic case corresponds to a, = a, = 0 and is represented by point I with coordinates (1, 1). For plane stress, Fig. 13 can still be used by substituting respectively E and v for E/( 1 - v2) and Y,a,anda,. v/(1 -v)inX, Note that for regularly jointed rock, all three moduli E, k,S and k,S can be determined from expansion tests in the hole parallel to the joint planes by first loading the jointed rock by an amount Aq followed by a cycle of unloading and reloading over the same pressure increment. If the slope of the curve q vs UJ2a during the unloading-reloading cycle can be used as an estimate to E/(1 + v) for intact rock, as suggested by Goodman [3] for static tests on fractured rock, then e, 5: E,,/E and e2 = Ed/E can be obtained directly as the ratios between the slopes of the loading and unloading-reloading curves over the same pressure increment Aq. Borehole jack tests The three elastic constants E, E’, G’ or E, k,S and k,S can be determined by conducting borehole jack tests in
two boreholes; one borehole normal to the planes of transverse isotropy or joint planes and another parallel to those planes. First, consider the test in a borehole normal to the planes of transverse isotropy or joint planes (Fig. 14a). According to equation (12), over each increment of pressure Aq, the slope of the q vs U,,/2a curve recorded during borehole jack tests is equal to E,/K(v, 8,) where K(v, /I,) is given by equation (9) for plane strain and equation (34) for plane stress. The measured slope (see Fig. 14b) gives an estimate of E, = (E,), which is also related to E and E/E’ through equation (22) or E and E/k,S through equation (29). For plane stress, E, = (E,), gives a direct measurement of the intact rock modulus E. On the other hand, for a borehole parallel to the planes of transverse isotropy or joint planes (Fig. 15a), the slopes of the q vs U,, curves obtained by conducting jack tests in directions parallel and normal to those planes give estimates of (E,),, and (El)l, respectively (see Fig. 15b). These moduli are related to E, E/E’ and G/G’ through equations (20), (21) for intact rock or to E, E/k,S and E/k,S through equations (27), (28) for jointed rock and plane strain. For plane stress, equations (20) and (21) are replaced by equations (32) and (33). Hence, by measuring (El)_, (E,), and (E,), , a system of three non-linear equations can be constructed that can be solved for E, E/E’ and G/G’ or E, k,S and k,S. Another way of solving for E/E’ and G/G’ or E/k,S and E/k,S is to use charts similar to those of Figs 12 and 13. Let U,, and U, be the borehole diametral deformations measured in loading directions normal and parallel to the planes of transverse isotropy or joint
(a)
04 Fig. 15. (a) Borehole jack test in a borehole with transverse isotropy or joint planes parallel to the hole axis and yz-plane. Hole diametral deformations are measured by conducting jack tests in directions normal (0, = 0, Curve 1) and parallel (6, = n/2, Curve 2) to those planes. (b) Corresponding q vs U&a curves.
AMADEI and SAVAGE:
TESTS IN ANISOTROPIC ROCK
‘i
0
5
_---
-- 1-J
30
c-
C-
A? 3.0 7-L Y
-'90
0-0
-
c-
20
--$8 I w 4 i
2.0-
Q-*
10
s
ti* *O
/f , 1.0
t
/ /
11
0
15
10
5
E A&jr _.-2a Q
20
1 WI%)
Fig. 16. Variation of (E/2a)(AU,,/Aq)/K(v,8,) with (E/2a) (AU,, /Aq)/K(v, &) for different values of a,, ranging between 0 and 20 and K, ranging between 0 and 40 and plane strain. Point I represents the isotropic case with K, = 01, = 0. For plane stress, E must be substituted for E/(1 - v2) in K, and 01,.
planes in the second borehole. For those two loading directions, equation (12) can be rewritten as follows: E AU,, a’F’K(v,B,)= E
1
E E, ,/) 0
1
E E, 1’ 0
(37)
AU,, zdq’K(v,B,)=
Using the expressions for (EJE),, and (EJE), derived in this paper for plane stress and plane strain, (E/2a) (AU,,lAq)lWv, 8,) and (E/2a) (AWAq)/ K(v, /I,) can be plotted for different values of E/E’ and G/G’ or E/k,S and EIk,S. A numerical example is shown in Fig. 16 for regularly jointed rock and for different values of a, ranging between 0 and 20 and a, ranging between 0 and 40 and plane strain. This figure can also be used for plane stress conditions with u, = EIk,S and u, = E/k& From a practical point of view, the problem remains that although AU,, , AU,, and Aq are measured in the field and although K(v, 8,) can be calculated once v is estimated and /3, is known, the in-plane modulus E cannot be determined from the first borehole as easily as with the dilatometer and gallery tests. This limitation can be overcome: (1) if the rock deforms in plane stress for which E is measured directly in the first borehole; or (2) if the rock is regularly jointed and E is determined during cycles of unloading and reloading as suggested previously for the dilatometer and gallery tests. CONCLUSION
The directional character of the modulus of deformation E, of an anisotropic rock mass measured along the walls of a borehole or tunnel during in situ expansion tests can be explained analytically by using closed-form
395
solutions based on the theory of elasticity for anisotropic media. The modulus depends on the anisotropic properties of the rock mass, its type of anisotropy (transversely isotropic or orthotropic) and the orientation of the rock mass planes of anisotropy with respect to the directions of measurements and/or loading in the hole in which the expansion tests are conducted. By replacing a regularly jointed rock mass by an equivalent anisotropic continuum, the modulus of deformation can also be related to joint properties such as spacing and stiffness. The effect of the degree of rock anisotropy on the variation of the modulus of deformation around the circumference of a hole in which expansion tests are conducted is shown for intact transversely isotropic and regularly jointed rock masses with transverse isotropy or joint planes parallel or normal to the hole. For intact rock, the ratio between the modulus of deformation Ea and the Young’s modulus E in the planes of transverse isotropy of the rock depends on four dimensionless terms: two Poisson’s ratios v and v’, and two ratios of Young’s moduli E/E’ and shear moduli G/G’ defined in planes parallel and normal to the planes of transverse isotropy. For jointed rock, EJE depends on the intact rock Poisson’s ratio v and the ratios between the intact rock Young’s modulus E and the normal and shear joint moduli k,S and k,S. The analysis presented in this paper is limited to anisotropic rock masses with structural features (layers, beddings, schistosity, etc.) that are much smaller than the diameter of the hole in which expansion tests are conducted. For regularly jointed rock, the equivalent continuum approach is limited to geometries for which joint spacing is small compared to the size of the problem of interest. For instance, for borehole expansion tests, and for most commonly-used borehole diameters, joint spacing must be small in order to use the equivalent continuum approach. On the other hand, for gallery tests that usually involve larger volumes of rocks than the borehole tests, this approach seems to be more appropriate for a much wider range of joint spacings. The variation of the modulus of deformation observed along the circumference of a borehole or tunnel during expansion tests not only provides a qualitative assessment of the anisotropic character of the rock mass of interest but can also be used to determine its anisotropic properties. As shown in this paper, assuming values for the Poisson’s ratios, expansion tests in two holes, one normal to transverse isotropy or joint planes and another parallel to those planes can be used to determine E, E/E’ and G/G’ for intact rock and E, E/k,S and E/k&3 for jointed rock. For the hole normal to the planes of transverse isotropy or joint planes, all diametral measurements are the same and therefore only one independent diametral measurement can be obtained. For the other hole parallel to the transverse isotropy or joint planes and in a coordinate system attached to those planes, there are at most two independent diametral measurements. Hence, with two holes, a total of three diametral measurements can be obtained that lead to a system of three non-linear equations and three un-
AMADEI and SAVAGE:
396
TESTS IN ANISOTROPIC
knows:
E, E/E’ and G/G’ or E, E/k,,S and EIk,S. This system can be solved directly or charts can be used to determine E/E’ and G/G’ in the second hole. The stress dependency of the elastic constants can be assessed by analyzing pressure vs diametral deformation curves in an incremental form. Note that the elastic constants v, v’, E, E’, G and G’ must also satisfy certain inequalities associated with the thermodynamics constraint that the strain energy of the rock must always remain positivedefinite [12,21]. Accepted
for publication 5 February 1991. REFERENCES
I. Goodman R. E., Van T. K. and Heuze F. E. Measurement of rock deformability in boreholes. Proc. 10th U.S. Symp. on Rock Mech., Univ. of Texas. Austin. DD. 523-555 (1972). 2. Bieniawski 2. T. Deter&&g rock mass deformability: experience from case histories. ht. J. Rock Mech. Min. Sci. & Geomech. Abstr. IS, 237-248 (1978). 3. Goodman R. E. Introduction to Rock Mechanics, 2nd Edn. Wiley,
New York. (1989). 4. Brown E. T. Rock characterization testing and monitoring. ISRM Suggested Metho&. Pergamon Press, Oxford (1980). 5. Swolfs H. S. and Kiblcr J. D. A note on the Goodman jack. Rock Mech. 15, 57-66 (1982). 6. Heuze F. E. and Amadei B. The NX-borehole jack: a lesson in trial and error. Int. J. Rock Mech. Min. Sri. & Geochem. Abstr. 22, 105-I I2 (1985). 7. Rocha M., Da Silveira A., Rodrigues F. P., Silverio A. and
Ferreira A. Characterization
of the deformability of rock masses
8. 9. 10. 11.
12.
ROCK
by dilatometer tests. Proc. 2nd. Congr. Int. Sot. Rock Mech. (ISRM), Belgrade, 2-32, Vol. 1, pp. 509-516 (1970). Amadei B. and Savaae W. Z. Comprehemive Rock Etwineerinx (Edited by J. A. Hudson). Pergamon Press, Oxford (1991). Amadei B. Rock Anisotropy and the Theory of Stress Measurements. Springer-Verlag, New York (1983). Lekhnitskii S. G. Theory of Elasticity of anAniPotropic Body. Mir Publishers, Moscow (1977). Gerrard C. M. Background to mathematical modelling in geomechanics: the roles of fabric and stress history. Proc. Int. Symp. on Numerical Methodr, Karlsruhe, pp. 33-120 (1975). Amadei B., Savage W. Z. and Swolfs H. S. Gravitational stresses in anisotropic rock masses. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 24, 5-14 (1987).
13. Duncan J. M. and Goodman R. E. Finite element analyses of slopes in jointed rock. Final Report to U.S. Army Corps of Engineers, Vicksburg, MI, Report S-68-3 (1968). 14. Morland L. W. Elastic anisotropy of regulatory jointed media. Rock Mech. 8, 35-48
(1976).
15. Gerrard C. M. Equivalent elastic moduh of a rock mass consisting of orthorhombic layers. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 19, 9-14 (1982).
16. Gerrard C. M. Elastic modulus of rock masses having one, two or three sets of joints. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 19, 15-23 (1982). 17. Filon L. N. G. On an approximate solution for the bending of a
18. 19. 20.
21.
beam of rectangular cross section under any system of load with special reference to points of concentrated or discontinuous loading. Proc. R. Sot. Land. Ser. A. 201, 65-154. Love A. E. H. A Treatise on The Mathematical Theory of Elasticity. Cambridge University Press (1920). Lekhnitskii S. G. Anisotropic Plates. Gordon & Breach, London (1968). Green A. E. and Zema W. Theoretical Elasticity. Clarendon Press, Oxford (1968). Pickering D. J. Anisotropic parameters for soils. Geotechnique 2Q, 271-276 (1970).