Time-independent and time-dependent deformation of small tunnels—III pre-failure behaviour

Time-independent and time-dependent deformation of small tunnels—III pre-failure behaviour

Int. J. Rock Mech. Min. Sci.& Geomech. Abstr. Vol. 19, pp. 307 to 324, 1982 0148-9062/82/060307-18503.00/0 Copyright © 1982 Pergamon Press Lid Print...

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Int. J. Rock Mech. Min. Sci.& Geomech. Abstr. Vol. 19, pp. 307 to 324, 1982

0148-9062/82/060307-18503.00/0 Copyright © 1982 Pergamon Press Lid

Printed in Great Britain. All rights reserved

Time-independent and Time-dependent Deformation of Small Tunnels III Pre-failure Behaviour P. K. KAISER*

N. R. MORGENSTERN*

Process simulation tests have been conducted to investigate the time-dependent deformation processes near tunnels at large depth or in weak rock. The test equipment has been described in Part I together with a presentation of the loading history aaopted for the tests. In Part II typical test results have been presented to document the behaviour of small tunnels in a jointed rock mass with time-dependent strength and deformation properties. A qualitative interpretation of the observed displacement pattern has been given. In this third part the measured tunnel closure and radial strain patterns are compared with those predicted from simple material models. The time-independent behaviour is compared with predictions assuming linear elastic material properties and the time-dependent tunnel response is compared with solutions assuming linear visco-elastic material properties. It is concluded that excessive deformations can be attributed to softening processes near the tunnel wall. The practical implications of this interpretation are discussed.

INTRODUCTION The success of modern tunnelling technologies such as mechanized excavation by tunnel boring machines or the New Austrian Tunnelling Method (NATM) depends, particularly when excavating in incompetent rock, largely on the time-convergence relationship which is controlled b y the time-dependent rock mass behaviour and the excavation and support history. The stand-up-time and the rate of tunnel closure are dominated by the excavation method and sequence, the support system and the installation procedure. Of equal concern is the rock mass response to tunnelling which is difficult to evaluate and utilize in the design procedure. For the purpose of studying the time-dependent processes controlling the performance of tunnels in highly stressed or overstressed rock, process simulation tests were conducted on a naturally jointed material. During this investigation small tunnels pre-excavated in a brittle, jointed coal representing a typical rock mass with time-dependent strength and deformation properties were loaded incrementally for periods of several weeks. The experimental facility, the properties of the coal and the loading history are described in Part I [1]. The deformation and stress redistribution processes observed during these tests will be described in detail in * D e p a r t m e n t of Civil Engineering, University of Alberta, E d m o n ton, Alberta, C a n a d a T6G 2G7.

the following. Monitoring of tunnel convergence and internal rock mass strains provided information about the instantaneous and the time-dependent rock mass response. The terminology, the data processing and presentation, and some typical data have already been presented in Part II [2] together with a brief qualitative interpretation of the data. The total accumulated tunnel closure or rock mass strain can seldom be determined accurately in the field and the total time-dependent displacements are difficult to determine because the initial strain rates are high compared with the loading or excavation rate. Moreover, access is limited at the outset. In practice, it has been found that it is convenient to use the deformation rate rather than the displacements as an indicator to evaluate tunnel performance. For example, at the Arlberg Tunnel [3] the tunnel wall displacement rate was used to determine when additional rock bolting was required and what the optimum bolt length should be. Additional or longer bolts were installed when the deformation rate exceeded 50 mm/day, and reasonable development of loads on the final lining was observed if the residual tunnel wall deformation rate in a tunnel with a diameter of 11 m was equal or less than 10 ram/month [4]. These values correspond approximately to tunnel closure rates of 37 x 10-3%/hr and 0.25 x 10-3%/hr. Because of the practical significance of these closure or strain rates most of the time-dependent observations will be presented in double logarith-

307

308

P.K. Kaiser and N. R. Morgenstern

mic deformation rate vs time plots. This relationship was found to be nearly linear for the test material (coal) [5] and a similar, linear relationship was observed from the tunnel closure and the radial rock mass strain observations [2]. This Part III contains a detailed interpretation of the pre-failure behaviour of these small tunnels. The term 'pre-failure' is used here to indicate that the unsupported tunnel remained stable without significant visible distress at the tunnel wall. Local yielding inside the rock mass and related stress redistribution may however occur at this stage, well before the tunnel fails. It is the main objective of this paper to illustrate experimentally how local overstressing with associated material softening can lead to increased tunnel convergence without the creation of a dilatant, strain-weakening yield zone. Much of the tunnel convergence measured in excess of that predicted from elasticity can be explained by non-linear deformation properties rather than by strength loss. The loading history adopted during the described test program does not correspond to the loading history observed in reality (see Part I). Particularly, the repeated reloading history is never observed in reality and the change in field stress ratio occurs only in few situations near multiple mine openings. Because of two factors limiting repeatability of test results, namely the complexity of the natural material used for testing and the natural variability in property distribution within each sample, the repeated loading history was chosen to show how the performance of a tunnel changes as the rock mass is altered due to time-independent processes, such as local overstressing during loading to high stress levels, and due to time-dependent processes, such as material creep with associated stress redistribution. The results presented in the following show how the rock mass properties and their distribution are affected by stress concentration and time, and how these alterations, which occur in reality during tunnel excavation, may affect the tunnel performance. Part IV will deal with the behaviour of tunnels during failure when the rock mass strength is exceeded. TIME-INDEPENDENT TUNNEL

PERFORMANCE The response of a rock mass with time-dependent strength and deformation properties to the excavation of a tunnel, or in the laboratory tests to the external loading of the specimen, is complex. Even the timeindependent or instantaneous deformations are influenced by time-dependent factors such as loading rate and loading history. Considering the time-dependent material properties of the highly fractured coal [5,6] and the stage-loading sequence described in Part I it must be expected that only very elaborate models of the constitutive relationship can accurately describe all observations and their development with time. For the purpose of detection and explanation of deformation processes occurring near tunnels it is, however, more

useful to compare the actual behaviour with those predicted for tunnels in materials with simple constitutive relation, i.e. linear elastic or linear visco-elastic materials. Such a comparison is presented in the following.

Interpretation of data by comparison with linear elastic material model The solution for the stress distribution around a circular hole in a fiat plate consisting of an elastic material is attributed to Kirsch and the derivation of these equations can be found in numerous texts. During a typical in situ monitoring program as well as during the laboratory testing, the tunnel wall displacements (called 'tunnel closure (u/a)' if divided by the tunnel radius a) and the difference in radial displacement between two points inside the rock mass (called 'average radial strain' if divided by the radial distance between the two measuring points) are measured. The corresponding equations for radial strain and displacements for a circular tunnel of radius (a) in a linear elastic material with a Young's Modulus E and a Poisson's ratio v (or a shear modulus G = E/J2(1 + v)] and a bulk modulus K = E/J3(1 - 2 v ) ] are summarized below for the condition of plane strain (e.g. [7-9]). While the stresses are independent of the loading history, the strains and displacements during the testing can be found by direct superposition of the pre-excavation deformations and the deformation occurring during excavation of the prestressed material. The radial strains and displacements occurring during the excavation of the prestressed plate corresponding more closely to the actual tunnelling situation (called 'REALITY') are:

--Radial strain (REALITY) er -

- G t l + v) [(1 -+- Nj(¢t 2) -I- (l - N)(cos 20) 2E X {40~ 2 -- 30( 4" - - 4 V U . Z l ]

where: a~... N = r... = 0...

(1)

vertical field stress a(horizontal)/~r(verticall radial distance from center of tunnel air with a = tunnel radius angle measured clockwise from vertical line through center of tunnel,

and f o r N =

1:

G-

--

O't,

2G (call

(2)

--Radial displacement (REALITY) u -

c%(1 + v)r [(1 + N)(~ z) + (1 - Nt(cos20) 2E x {4~2 - ~4 _ 4w21]

(3)

and for N = 1 at the tunnel wall r -~ a: oza

u. = 2 ~ ;

(4)

Deformation of Small Tunnels--III. Pre-failure Behaviour Stresses associated with the geological history of the rock mass are normally in equilibrium before tunnel excavation and the related deformations have terminated. In the process simulation tests, however, these deformations are measured during the loading of the block without the tunnel (called 'PLATE') and are part of the deformations measured while loading the specimen with the pre-excavated tunnel (see Part I). Under olane strain condition, the radial strains and displacements (for 'PLATE') are:

--Radial strain (PLATE) ~'-

a~(1 + v) [(1 - 2v)(1 + N) + (1 - N l c o s 2 0 ] 2E

and f o r N =

(5)

1:

avE ~ r - 6KG

(6)

--Radial displacement (PLATE) [9] U

or(1 + v)r [(1 - 2v)(1 + N) + (1 - N ) c o s 2 0 ] (7) 2E

I

and for N = 1 at a distance r = a (future tunnel wall): Ua

--

aoEa 6KG"

(8)

Superposition by adding these two sets of equations provides the equations for the radial strains and displacements observed during the process simulation tests while loading the pre-excavated specimen under plane strain conditions (called 'TEST'):

--Radial strain (TEST) ~(1 E r

--

_

_

-

v 2)

2E

[(1 + N ) ( 1 -

+ (1 - N)(1 + 3~ 4 - 4~t2)cos20] avv(1 + v) 2E [(1 + N)(1 + ~2) - (1 - N)(1 + 3~4)cos20]

(9)

and f o r N = 1: .

(10)

---Radial displacement (TEST) u-

ov(l - v2)r 2E

[(1 + N)(1 + c t 2)

+ (1 -- N)(1 - ct4 + 4~2)cos 20] o~v(l + v) [(1 + N)(1 - ctz) 2E - (1 - N)(1 - ~4)cos20]

(11)

309

and at the tunnel wall r = a: u. -

ov(l - v2)

and f o r N =

E

a[(1 + N) + 2(1 - N)cos 20]

(12)

1: u, = ~

+ 1 .

(13)

By comparison of equations (2), (6) and (10) or (4), (8) and (13) it can be seen that the strains and displacements measured in the 'TEST' depend on both the bulk and shear modulus of the test material whereas the deformations during excavation of a circular tunnel in a uniform stress field (N = 1) depend on the shear modulus only. Coal is a compressible material, particularly at low stress levels where crack closure dominates, and this complicates the application of the test results to describe the behaviour of real tunnels. The effect of compressibility must be separated to evaluate the real tunnel behaviour. This will be pursued further as appropriate in the following.

Observed stress-closure relationship Figure 1 summarizes the tunnel closure measurements of four tunnel diameters made during a series of 12 tests on one specimen. The stress ratio N and the loading history were varied as indicated. The average loading rate above 10 M P a is significantly reduced in some tests because of several one-day creep stages. After a dominant non-linear load-displacement response at stress levels below approx. 3 MPa, the stresstunnel closure records can be approximated by a linear relationship by determining the tangent at 10 M P a as indicated by the dashed line for Test 1.10. F r o m equation (12) and one linearized convergence curve it is possible to predict the other three if the Poisson's ratio is known and if the material is assumed to be homogeneous, isotropic and linear elastic. Such a prediction is shown on Fig. 1 (Part II) and reasons for deviation have also been noted. The coal is highly fractured and often non-homogeneous. A fair variation in material properties must be expected. As summarized in Part I the Young's Modulus determined by triaxial tests at confinement pressures between zero and 10MPa, varied between 900 and 2050MPa. Higher values of 1900-2050MPa were determined from rapid reloading tests of block samples whereas similar tests at slow loading rates revealed moduli of 1000-1700 MPa. The average deformation modulus, back-calculated from the linearized tunnel closure curves of a test with 9 one-day creep stages shown in Fig. 1 (Part II) is 1300MPa in the direction of the major principal field stress and 1100 and 1450 MPa in the directions parallel and perpendicular to the major joint set. These values correspond well with the results from slow loading tests on block samples. The adequacy of the assumed linear elastic material behaviour is further supported by the small displace-

310

P . K . Kaiser and N. R. Morgenstern

N0.980.9-1.01.06 0.57 1.08 TEST1.8 1.9 1.10 1.11 1.12

1.11 1.13

1.08 1.14

1.09 1.15

0.38 1.16

1.0 1.17

0.2 1.18

1,08 1.19

Q. I

o3

)

~

~

U,

-

L I '

l

t

[ ~ ~ ~Reached iofi LI'inmstrumen{116 it

ILl 13:: I-O3 O ._1

US~

,

eac

.

5:

:

ii~,

: . . . . . . .

TUNNEL CLOSURE (%)

Limit of Instrument] Rea~hed . 8

Scale:,

' 2.0%

Scale:'

,A.0%

Fig. I. Field stress vs tunnel closure diagrams for four tunnel diameters (see Fig. 2) of twelve reloading tests at field stress ratios between N = 0.2 and 1.08 (Test MC-1.8 to MC-I.19, Sargent [12]): (a) parallel to maximum field stress (03; (b) parallel to jointing (45~); (c) perpendicular to maximum field stress (90°); (d) perpendicular to jointing (1353.

ment observed in the direction of the minimum principal stress (Fig. 1, Part II) which should be zero for an ideal linear elastic material under a field stress ratio of N = 0.33. Because of these observations it seems appropriate to compare the measured displacements with the deformations predicted from a linear, homogeneous, isotropic, elastic material model. Several considerations may cause deviations from this idealized behaviour and must be recognized during such a comparison: (a) Non-linear elastic material. Rock elements close to the unsupported tunnels are stressed to high stress levels at reduced confining pressure. The secant deformation modulus for non-linear elastic elements stressed to high stress levels is lower than for elements at lower stress levels far from the tunnel wall. On average the deformation moduli decrease as the field stress increases and this reflects in increased tunnel wall displacements. During testing of coal this effect may partially be compensated by simultaneous stiffening due to compaction. Zones of significantly higher stress concentrations are created on opposite sides of a circular tunnel during non-hydrostatic loading and stress reduction occurs in areas between these zones. For N < l, the highly stressed areas are found near the springline. Because of non-linear deformation properties these areas will soften, deform more and the displacement of the roof will increase beyond the value predicted from linear elastic material models. Non-linear deformation properties constitute one cause of axisymmetric softening if N = 1 or mirror-

symmetric softening if N < 1. Such softening leads to tunnel wall convergence in excess of that predicated by linear elasticity. A second reason having the same net effect is: (b) Yielding due to global or local overstressin#. In accordance with the elasto-plastic, strain-weakening model introduced by Egger [10] yielding may occur in highly stressed areas and the related straining may be accelerated by the associated strength loss and dilation within the plastic zone. This process may be concentrated in local zones of weak material or local areas of stress concentration created, for example, by corners, irregular opening geometry or non-hydrostatic field stress conditions as well as in global zones of more or less constant width around the opening. The overall effect of the yielding processes is the same as discussed earlier. It causes a local or global softening which, in the case of overstressing, is partially irreversible. During most tests little evidence could be found in support of the global yielding concept [10]. For example, the tunnel in specimen MC-2 remained stable as described in Part II and only local cracking was observed along the springline, roof and floor. The internal strains recorded close to the tunnel (see Fig. 4, Part I) were in the order of 0.30-0.9~o which is just sufficient to reach the yield point but much less than is required to reach the failure strain of coal. Shear along local planes of weakness and crushing in local areas occurred in the later stages of Test MC-I and the effects on tunnel performance will be discussed later in more detail. (c) Time-dependent rock deformation. Under sustained

Deformation of Small Tunnels--III. Pre-failure Behaviour load rock deforms which reduces the current secant deformation modulus. These deformations may be recoverable (i.e. if visco-elastic), partially recoverable, or irreversible (i.e. if visco-plastic). This time-dependent softening may occur globally, or locally in areas of high creep potential and causes a uniform or non-uniform time-dependent deformation pattern. Unless the rock mass properties are loading history dependent, the instantaneous response to loading should remain unchanged. The rock mass response will, however, deviate if permanent deformations and related stress redistributions result from differential creep under sustained loads. Each one of these processes causes a redistribution of stresses from highly stressed areas to areas where, because of increased confinement or material stiffness and strength, higher stresses can be tolerated. This stress redistribution is associated with deformations as long as there is an areal variation in rockmass deformation modulus. It was one purpose of these process simulation tests to find evidence for the hypothesis of time-dependent stress-redistribution because of timedependent strength loss. However it was found, as will be shown later, that the other processes described above or a combination of two or more of them are of greater importance. The test data show that the timedependent and time-independent post-failure behaviour only dominates when a relatively large zone of rock has failed near the opening. Furthermore, this stress redistribution is likely to occur in a non-uniform and non-symmetric fashion even under conditions where the opening geometry and the stress field are symmetric. The data presented later provide an excellent example in support of this statement. In practice, stress redistribution due to repeated loading is seldom encountered with the exceptiofi of special conditions which may be found during mining operations. However, the excavation procedure, the excavation sequence and layout, and the support

311

method and procedure do affect the stress redistribution processes and hence the tunnel performance. It is therefore important to understand these stress redistribution processes and to evaluate how strongly they influence the deformation pattern near tunnels.

Behaviour under hydrostatic field stress (N = 1) Sample MC-1 with a 15cm tunnel was loaded 12 times under various stress ratios varying between 1.11 and 0.2 as shown in Fig. 1. Nine of these tests were executed with a nearly hydrostatic field stress and variations detected between these tests must be attributed to changes which occurred inside the sample due to stress redistribution or material property modification. Table 1 summarizes the normalized tunnel closure ratios for the maximum, and minimum deformation direction as well as for two sets of perpendicular axes (for orientations see Fig. 2). Both maximum and minimum closure ratios generally increase during repeated testing. The minimum deformation axis (at 135°) remains unchanged up to Test MC-I.13 but the maximum closure seldom occurs in the direction perpendicular to the minimum deformation axis. No consistent pattern of anisotropic response can be detected. Both closure ratios parallel and perpendicular to the jointing are on average about 1.08. The average maximum closure ratio u(max)/u(min) is higher at 1.21 but the orientation varies. There is a slight trend for larger movements parallel to the jointing and less perpendicular to it. Because of this nearly isotropic response it is of interest to compare the average tunnel closure normalized to the predicted tunnel closure with results from a number of sequential tests (Fig. 2). Initially repeated loading increases the material stiffness, probably because of compaction of the coal, and then after a drastic change during the non-hydrostatic test at N = 0.57 repeated loading causes a gradual softening which results in a total increase of the total average tunnel closure by about 42~o. An additional 15-22~o closure was observed during tests with reduced loading

TABLE 1. NORMALIZED TUNNEL CLOSURE DURING TESTS WITH HYDROSTATIC STRESS FIELD Test number

1.8 1.9 1.9 slow 1.10 1.11 1,12 1.13 1.14 1.14 slow 1.15 1.16 1.17 1.18 1.19

a~ (max) MPa

10.5 15 15 15 15 15 15 15 15 15 15 15 10 12

u(max)/uo

u(min)/uo

(orientation)

(orientation)

1.05 (0) 1.05 (90) 1.50 (90) 1.11 (0) 1.52 (45) 1.48(0) 1.45 (45; 135) 1.64 (90) 1.45 (0) 1.49 (45) 1.5 (0; 45)

0.99 (45; 135) 0.93 (135) 0.96 (135) 0.83 (135)

u(O)/u(90)

1.04 1.12 1.33 1.32 non-symmetric loading 1.20 (135) 0.98 1,14(135) 1.07 1.27 (90) 1.06 1.30 (45) 0.89 1.40 (45; 90; 135) 1.03 non-symmetric loading 1.35 (90) 1.06 non-symmetric loading 1.27 (90) 1.18

u(45)/u(135)

u(max)/u(min)

1.00 1.07 1.25 1.12

1.06 1.13 1.56 1.33

1.26 1,17 1.0 0.9 1.00

1,26 1,30 1,14 1,18 1.03

1.06

1.10

1.03

1.18

Note: u(max) and u(min) have been adjusted to account for slight deviation of N from unity. Orientation: 0 . . . . . . parallel to maximum field stress; 45 . . . . . . parallel to jointing; 90 . . . . . . perpendicular to maximum field stress; 135 . . . . . . perpendicular to jointing. u0 . . . . . . predicted tunnel closure for N = l, Eo = 1.5 GPa, v = 0.2, homogeneous, isotropic, linear elastic material,

312

P . K . Kaiser and N. R. Morgenstern o

1.6 z

1.5

o

1.4

F<[ rr

1.3

LJJ

1.2

U') 0 .._1 o

1.1

J LIJ Z Z I--

1.0

0.9 0.8

"' (.9

0.7

o:

0.6

<[

LIJ >

90

90

co

LO

i

(D

d 45

i

~ r

0

=I

r

IE

',

0

Several observations are of practical importance: (a) The rate of loading (or excavation) q[]~,cts the magnitude of the tunnel wall convergence hut it does not significantly alter the instantaneous rock mass response. A comparison of test MC-I.9 or 1.14 (Fig. 2) with the subsequent rapid loading tests MC-I.10 or 1.15 shows that the slow loading history did not affect the rock mass behaviour during the subsequent test. The timedependent movements vary over a wider range than the instantaneous tunnel wall displacements giving the rock mass an apparent anisotropic response. (b) Local overstressitlg q/ the rock ahers the overall rock mass response by a~l apparent softening qf the rock. Such local overstressing was achieved to a large degree during Test M C - I . l l at N = 0 . 5 7 resulting in an increased average tunnel closure by 35°~,, and a nonsymmetric closure increase varying between 25 and o/ During Test MC-I.16 and 1.18 at N = 0.38 and 57~o. 0.20 less additional local overstressing occurred as indicated by the minimal increase in average tunnel closure. However, the increasing range of closure measurements shows that more non-symmetric responsc resulted from local material property modification. This is also indicated by the gradual rotation of the diameter with minimum tunnel closure from 135 ~ to 90L The 90°-dia showed maximum closure during the earlier parts of the test series. It should be noted that the rock mass behaved like a continuum during these tests because no structural instability was visible until Test MC-I.13 and because little damage occurred before Test MC-I.18 (N = 0.20) when a shear zone developed along a major inclined discontinuity near the tunnel wall (Fig. 3). Furthermore,

04

Z 0

z.45~.45

145 0 ' ~ ]- 'V' T

I

So~te°~

,' o/ l/ 9

0 ~I~9Q,J. 135 4"5 .135~

Jointing 0 ol 4 ~ / / /

~£t/~eo/X_l~q~

///

o~?~oo

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Loading Rate: Rapid Q 5 MPa/hr. Slow 0 0.04 MPa/hr. I

I

I

I

I

I

I

i

MC- 1.8 1.9 1.10 1.12 1.13 1.14 1.15 1.17 1.19 TEST NUMBER

Fig. 2. Average tunnel closure ratio for nine tests with nearly hydrostatic field stress ratio N = 0.9~1.09. Range of measurements indicated by length of bar. Numbers give orientation of measuring diameter.

rates. The range between the smallest closure (u/a = 1.06%) observed perpendicular to jointing during Test MC-I.10 and the largest closure (u/a = 2.09%) measured along the diameter parallel to the minimum stress axis during Test MC-I.14 (slow loading rate) exceeds 80% of the tunnel closure uo/a of 1.28% at 10 M P a field stress.

Maximum field stress

/

4,

"~

, /

. GI" /

/

~=. 'I0

E

E

....... - "w~u~xxJa

Joints Crocks [open) S h e a r zone Crushed rock

~

Surficially grouted crock o . m . . , Surficiol depressions Scale: Approx. 1 : 4 0

Fig. 3. Top view of Specimen MC-1 after Test 1.19 (photo reduced map after Sargent [12]).

Deformation of Small Tunnels--III. Pre-failure Behaviour

G

5.0

Q

313

CirculaOpeni r ng

EllipticalOpening

----

--.....

I E=Eo; o=0.2 II E=Eo/1.5 19=0.2

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/

°

III b/a = 1.5; E = E o IV b/a = 2: E = E o 19= 0.2

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0.7 0.6

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FIELD STRESS RATIO N Fig. 4. Predicted tunnel closure ratio as a function of the field stress ratio N for: (a) (I) a circular opening in rock with modulus E0; (II) a circular opening in rock with a reduced modulus E = Eo/1.5; (b) (III) an elliptical opening with axis ratio b/a = 1.5, E = Eo; (IV) an elliptical opening with axis ratio b/a = 2.0, E = Eo.

the performance of the tunnel as recorded by convergence measurements during Test MC-I.19 is almost unaffected (N = 1) by this major shear zone passing near the tunnel at a depth of about 30% of a tunnel diameter. Two conclusions can be drawn from this observation: First, the tunnel wall convergence does not reflect the failure process and hence it is not a good indicator to evaluate opening stability. Second, major shear zones may not affect the tunnel performance if the stress field is favourable and prevents movements and instability by confinement. The same observations were made by G u e n o t [14] on another test specimen. The tunnel behaves like an opening in an elastic medium as long as continuity is maintained and no major shear displacement occurs•

(c) A significant increase in tunnel convergence, beyond the closure predicted by linear elasticity, may be observed because of stress redistribution processes. These are caused by local overstressing, time-dependent rock deformation or other mechanisms which result in an apparent softening of the rock mass. The conventional concept of a global plastic zone or post-failure weakening behaviour in an extensive zone near the tunnel was not confirmed by these test results. These comments will be discussed further following the presentation of the tunnel response under nonhydrostatic field stresses.

Behaviour under non-hydrostatic field stress (N 4: 1) Theoretical behaviour. Under non-hydrostatic field stresses the convergence varies along the circumference of the circular tunnel (with radius ao and decreases from a maximum in the major principal stress direction to a minimum in the direction perpendicular to it. The predicted tunnel closure, normalized to the closure under a hydrostatic stress field, is plotted in Fig. 4 for a linear elastic material with Young's Modulus Eo as a function of the stress ratio N. For a circular opening this closure ratio varies linearly as indicated by the full lines for four tunnel diameters (I). In case of a reduction in the deformation modulus of the rock mass in a circular zone near the tunnel opening, it behaves like a larger opening of radius a ~> a0 or an opening of same size but in a rock mass with reduced average deformation modulus E ~< E0. Such a modulus decrease may result from non-linear material properties or, if timedependent, from creep in the highly stressed zone near the tunnel. This causes an upward shifting of the three closure ratio curves as indicated by the dashed lines (II) for an average modulus decrease to E = Eo/1.5 (or an apparent opening size increase from a o to a = 1.5 ao). A naturally jointed material is a non-homogeneous continuum with a non-uniform material property distribution. Non-uniform stress distributions with local stress concentrations due to high stiffness zones and

314

P . K . Kaiser and N. R. Morgenstern properties, all observed closure ratios should fall onto the full lines (I) (Fig. 4). However, if axisymmetric and mirror-symmetric softening occurs, a gradual shifting toward the behaviour predicted for the cases II to IV will be observed. Observed behaviour. The convergence data from 12 subsequent tests are summarized in Figs 5a-d for the four tunnel diameters including the relationship predicted from linear elasticity. The sequence of testing is indicated by the arrows connecting the data points. The measured tunnel closure is in most cases larger than predicted and the deviation increases generally during subsequent tests. This deviation can be explained by the processes of softening and related stress redistribution. During a particular test, stress concentrations near the opening wall are redistributed because of overstressing and softening. The response of the rock mass during the following test will then reflect the changes which have been caused during the preceding tests. The influence of

softening due to localized overstressing or local creep property variations must be expected. If such localized softening occurs simultaneously on each side of the circular tunnel it will deform in a manner similar to an elliptical opening with an axis ratio b/a. The corresponding closure ratio curves, for the 0 :~and 90 ° direction only, are shown by the dash-doited lines on Fig. 4 for two b/a-ratios of 1.5 (III) and 2.0 (IV) with the shorter axis in the major principal stress direction. Such a shape change due to non-uniform stiffness distribution increases the tunnel convergence in the major stress direction significantly while it reduces the convergence in the minor stress direction only slightly. During Test MC-1 the specimen with a 15 cm tunnel was repeatedly loaded under various stress ratios (N = 1, 0.57, 0.38 and 0.2) and between the non-hydrostatic tests loaded hydrostatically at two loading rates (0.5 and 0.04 MPa/hr). If the rock mass behaves in a linear elastic manner and maintains its deformation

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these changes on convergence will depend mainly on the magnitude of the preceding stress level, the stress ratio N which largely controls the magnitude of the change in closure ratio for two cases: (I) axisymmetric stress concentration and the rate of loading or the time softening has occurred before a test at reduced field allowed for stress redistribution. The influence of these stress ratio N and (II) axisymmetric softening occurs gradually during loading for a test at reduced field factors is discussed separately in the following: Effect of loading rate. Materials with time-dependent stress ratio. The second case (II) was observed during deformation properties deform under sustained load or loading at N = 0.58 where the average deformation exhibit decreasing deformation moduli with decreasing modulus decreased by about 20~o but the behaviour on loading rate [11]. The modulus may be time-dependent the subsequent test can be predicted assuming reduced but not stress level-dependent as in linear visco-elastic linear elastic material properties (see Fig. 5a and b: materials or it may vary with the stress level too. essentially parallel lines to those predicted from elastiHence, the accumulated tunnel wall displacement, or city). Further softening to a total closure ratio of 1.45 the closure ratio, will increase with time or stress level occurred during the following tests at field stress ratios and the tunnel will respond as if it was excavated in a near unity. material of lower deformation modulus. The effect of Non-axisymmetric softening. When loading at field such a softening is illustrated in Fig. 4 and this behav- stress ratios significantly different from unity high stress iour was observed during both tests performed at low concentrations are created in areas on opposite sides of loading rates (Fig. 2 or Fig. 5) and under hydrostatic the opening perpendicular to the maximum field stress stress field. The change in closure ratio varies between direction. Increased straining in these areas may lead to 10 and 50% for the various tunnel diameters. During mirror-axisymmetric softening. Figure 7 presents a both Test MC-1.9 and 1.14 most of the time-dependent schematic diagram that illustrates the path corretunnel closure was recovered and little effect on the sponding to two cases where mirror-symmetric softenbehaviour of the following tests can be detected; i.e. on ing has occurred: (I) softening has occurred before a Fig. 5b both tests after the slow loading tests, labeled test at reduced stress ratio and (II) softening occurs (s), behave almost identically with the tests executed gradually during loading. The type II behaviour can before the slow loading tests. According to the theoreti- again be observed from the test results plotted in Fig. cal prediction presented in Fig. 4 non-axisymmetric clo- 5a and b. However, this mirror-symmetric response is sure ratios are expected only if non-axisymmetric superimposed on the axisymmetric behaviour described softening occurs. The higher than average closure ratios in Fig. 6. The combination of both cases is shown in the observed in the 90°-direction (Fig. 5b) indicate an schematic diagram in Fig. 8 for a test sequence with enhanced time-dependent softening at 'roof and floor' N = 0.75, 0.5 and 0.33. Excellent agreement with this resulting in a behaviour like an opening of elliptical predicted behaviour can be found by comparison of shape with the longer axis in the vertical (0 °) direction. Figs 5b and 8b (where N = 0.57, 0.38 and 0.2). Axisymmetric softening. Figure 6 presents schematic So far it was assumed that the variation in deformadiagrams which illustrate the theoretically predicted tion modulus is stress level-independent (linear elastic

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material). This is not correct for the test material coal which follows a pronounced non-linear stress-strain relationship at high stress levels near failure. During tests with low field stress ratios (N = 0.38 and 0.2) higher stress levels were reached locally on opposite sides of the tunnel. The average deformation modulus at these elevated stress levels therefore is locally much lower. This is reflected in a higher b/a-ratio and hence higher tunnel closure ratios. The corresponding behaviour is shown in Fig. 8a (Fig. 8b is not affected by stress leveldependent moduli). Again good agreement between Fig. 8a (dashed lines) and Fig. 5a can be observed. A quantitative comparison with Fig. 4 leads to the conclusion that the average deformation modulus E is about Eo/1.45 and the b/a-ratio must have exceeded 2 at N = 0.2 In summary, it was found that axisymmetric stress redistribution reflected an apparent softening of the rock mass by an average modulus reduction of about 30~o and resulted in an increase in tunnel closure by about 45~o beyond the initial elastic response. Nonaxisymmetric stress redistribution due to additional softening in highly stressed regions resulted in a further tunnel closure increase by 125~o (at N = 0.2) to a total of more than three-times the elastic response predicted from the initial modulus. It must be repeated that yielding along a shear zone had occurred at this stage. However the subsequent hydrostatic reloading test (MC-I.19) showed little deviation from the previous tests at N = 1. The interpretation presented so far assumed that the principal convergence axes correspond with the princi-

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pal field stress axes. A comparison of the 45 ~ and 135 ° tunnel convergence data presented in Fig. 5c and 5d show however that this is not fully justified. An average axisymmetric softening to cause an increased closure ratio of about 1.45 can also be derived from these figures. In the direction parallel to the jointing this ratio is reached much more rapidly and the total closure ratio during tests with reduced field stress ratios is on average smaller by about 0.3. This indicates that the rock structure caused a slight rotation of the non-axisymmetric softening zones to areas at about 0 -- 100-120 ° and 0 = 280-300 °. Only more sophisticated analytical methods considering the regional deformation modulus distribution and using more complex numerical models will allow for more accurate fitting of the data presented. It should however be noted that the strain measurements from extensometers inside the rock mass (data from MC-1 not presented here [12]) further support these conclusions.

Comparison with test MC-2 Almost identical behaviour was observed (Fig. 9) during another test series on Sample MC-2 [15] which was loaded in five repeated loading tests with N = 1.0/ 0.5/0.75/0.33 (see also Fig. 1; Part II) and 0.20 to the following respective stress levels of 7/14/3/10/15 and 15 MPa. The jointing in this test was inclined at 135 ° to the major principal field stress direction and the deformation modulus, back calculated from the first test, was slightly lower at 1370MPa. The tunnel showed little distress at the end of the test series as described in Part II (Fig. 7 and Plate 1). This sample was loaded to a lower stress level relative to its capacity. Consequently,

Deformation of Small Tunnels--III. Pre-failure Behaviour

317

expands non-symmetrically toward the left side. It can be shown [13] and [15] (or from equation 10) that the radial strain at extensometer 1 is compressive while it is extensional at extensometer 3 for case (a) in a compressible material like coal. Extensometer 2 will measure an Circulor tunnel intermediate strain depending on the relative contriEr bution due to compressibility. After propagation of a (bl mirror-symmetric softened zone near the spring lines t"'O~-~ 3 3 I this behaviour will be more pronounced at extensI ometers 1 and 3 while extensometer 2 will tend toward Elliptical zero strain depending on the exact shape of the sofequivo lent opening tened zone. During Test MC-2 at N = 0.2 the softened zone expanded non-symmetrically toward extensometer 6 and 7 at higher stress levels. This propagation was t¢ 8 caused only at stress levels in excess of the ones induced tel 7\q:-f I--~- 3 ~8 "~.'T'j'~/ 6 3 2 4 I. 5 during earlier tests and was non-symmetric because of material property variations. Extensometer 6 is I expected to be influenced first and strongest as it is Elliptical, equivalent opening included in the soft zone. The other extensometers propagating towards ,, instruments 6 and 7 ~r should respond less drastically and only where further Fig. 10. Schematic diagrams to illustrate radial strain development softening occurs at the respective locations. The during propagation of a non-symmetric zone of softening. expected response is presented in Fig. 10c for all eight extensometers. The measured average radial strains are less local overstressing occurred and a lower closure summarized in Fig. 11 (see also Fig. 5, Part II). During ratio was observed (Fig. 9). a comparison of Figs 10 and 11 it must be considered The average radial strains measured during this test that non-linear behaviour due to crack closure affects near the tunnel wall support the interpretation the real behaviour (Fig. 11) at low stress levels by shiftpresented earlier extremely well. Figure 10 illustrates in ing the stress-strain curve. All extensometers perform a schematic diagram the expected radial strain behav- as predicted (Fig. 10). Again, it should be noted that no iour for (a) a circular tunnel in a linear elastic material distress was visible at the tunnel surface (Plate 1, Part under a non-hydrostatic field stress (e.g. N = 0.2), (b) a II shows the tunnel after unloading) even though more circular tunnel where mirror-symmetric softening than 2% extensional, radial strain was recorded locally. occurred near the spring lines, and (c) for a case, corre- The maximum tunnel closure u/a recorded during this sponding to Test MC-2, where the softening zone test is more than 6% (see Fig. 6, Part II). N • small (e~ 0.2)

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P.K. Kaiser and N. R. Morgenstern

Practical implications Several practical conclusions can be drawn from these test results and their interpretation. First, the elastic material surrounding the opening seems to dominate the opening performance but the magnitude and distribution of the displacements and strains is affected by stress redistribution or softening processes near the opening. The actual opening behaves like an equivalent opening of different size and shape in a material with unchanged properties [13]. Local material modification near the opening due to overstressing, during loading with N 4; 1 in the test or during excavation in reality, increases the convergence beyond the one predicted from the measured deformation properties without creating a large strain-weakened, dilatant zone. Second, the non-symmetric distribution of zones of overstressing does affect the orientation of the principal closure directions. Hence, maximum closure may be observed in the direction of the minor field stress simply due to a change in local stiffness and strength. Similar behaviour must be expected in the field where local overstressing due to many reasons may occur. Local geological features are of prime importance in this regard. Third, tunnel wall displacements several times those predicted from the initially measured rock mass properties may be observed because of global or local softening in areas near the opening wall. Local strength loss and dilation may not be required to explain these movements. The apparent success of the strain-weakening model (often called strain-softening) may simply be attributed to the fact that the brittle plastic model sensibly predicts the actual stress distribution and hence the related displacement field. Our experience from this testing program shows that the processes assumed for modelling and matching observed behaviour often do not correspond with the physical processes observed [14], [-16]. There is ample evidence that the strainweakening model has been misused by simply matching convergence measurements without proving the validity of the model by comparing it with the actual deformation processes or at least by evaluation of the internal strains measured by extensometers [17]. This view needs to be pursued further by an investigation of the strain distribution inside the rock mass near tunnel openings. TIME-DEPENDENT T U N N E L PERFORMANCE A higher level of complexity is introduced when one attempts to simulate analytically the time-dependent response of an opening in a naturally jointed material with time-dependent peak strength, post-peak strengthstress-strain relationship [6] and time-dependent as well as stress level-dependent properties. For the loading history adopted for these process simulation tests (Fig. 3, Part 1), this problem is further complicated by two factors: the external load application and the stage-

loading sequence. The real loading condition of internal unloading by tunnel excavation is presently under experimental investigation but these results are not yet available for publication. Hence, it is necessary to clearly separate, during the analysis, the time-dependent deformations due to loading of the specimen without tunnel ('Plate') and due to internal unloading during excavation of the tunnel. This is of greatest importance because it can be seen from equations (4) and (13} that the radial displacements in the tests depend on the compressibility of the material whereas they are independent of it in reality. Both Kaiser [15] and da Fontoura [18] have shown in independent data analyses that the time-dependent tunnel behaviour is affected by a time-dependent compression of the plate, a hydrostatic creep component, and by a time-dependent deformation under deviatoric loading, a deviatoric creep component. At high stress levels, where local stresses approach the peak strength of the rock mass near the tunnel, a third component associated with local softening (not weakening) or stress redistribution must be considered. In the following these three components will be evaluated by comparison with tunnels in materials with simple constitutive relationships. On the basis of this comparison it will be possible to draw practical conclusions from the results of the tests even though the loading history in reality differs from the one adopted in the test.

Interpretation of data by comparison with linear viscoelastic material model The derivation of the visco-elastic solutions for radial strains and tunnel wall displacements by application of the LaPlace Transformation has been summarized by Kaiser [15]. Both, the hydrostatic and the deviatoric, creep components were modelled by a three-parameter solid composed of a spring (with stiffness E21 or Ell) in series with a Kelvin element with a spring (of stiffness E22 or El2) and a dash pot (with a coefficient of viscosity N22 or N12). For the examples presented later, no attempt was made to determine the rheological parameters for the selected model by curvefitting material test data even though this would be possible by comparison of creep and recovery data. The parameters for the calculations were estimated from test results presented by Kidybifiski [19] and Morlier [20] and selected in such a way that calculated creep rates and retardation times compared well with observations from our tests. These parameters (Fig. 12) result in a strain-time relationship where the deviatoric creep portion has a short retardation time and the hydrostatic creep component has a relatively long retardation time. The results of this model are shown in Fig. 12 for one particular set of parameters, for a stress increment of I MPa and a unit field stress ratio. The displacements at the tunnel wall (u(a)) and at the sample boundary (u(4a)) as wcll as the radial strains at the tunnct surface (1.Oa), at a radius of 1.5 and 2.0 times thc tunnel radius (a) are shown. While a continuously decreasing tunnel

Deformation of Small Tunnels--III. Pre-failure Behaviour

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by the same process that controls the time-dependent deformations of a real tunnel. Because of several orders of magnitude difference in strain rate immediately after loading, the effect of compressibility can be neglected at this stage. Hence, observations from this early stage can be used to compare with or predict the behaviour of real tunnels. Before discussing the third aspect, the stress redistribution, it is beneficial to further consider the effect of compressibility on the behaviour of the small tunnels tested. The compressibility in a highly fractured, stressrelieved body is highly non-linear and the bulk modulus K increases rapidly with increasing confining pressure or stress level. The schematic diagrams in Fig. 14a presents the radial strain distribution for the test condition and for three levels of compressibility: (1) incompressible material where the strain distribution of the test corresponds with reality; (2) bulk modulus K = E / 3 ; and (3) an extremely low bulk modulus such that the hydrostatic compression dominates. If we make the simple assumption that during three sequential load increments the conditions change from (3) to

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(2) to (1), the schematic stress-strain curves shown in Fig. 14b, for r -- a and r = 2a, result. A comparison of this diagram with the radial strains measured and presented in Fig. llc shows that high compressibility dominates at stress levels below 4-5 MPa (N = 0.2). The data from Test MC-2.6 (N = 0.75) in Fig. 13a and b indicate that compressibility still dominates at stress levels up to or slightly in excess of 9 MPa. This apparent discrepancy is to be explained by local variations in compressibility and, even more important, by the higher local stress level due to higher stress concentration during the test with a lower field stress ratio. It follows that observations from the intermediate stress range are of greatest value for comparison with real behaviour because the bulk modulus in this range is almost constant and has reached a relatively high value. The stress pattern near the tunnel, predicted from linear visco-elasticity in an isotropic, homogeneous body does not change with time under the boundary conditions of the tests. The effects of the third component, the stress redistribution, therefore cannot be

described adequately by this material model. However, during the interpretation of the time-dependent tunnel performance we have shown that softening occurs at high stress levels in areas of stress concentration near the opening wall. It seems reasonable to assume that this softening process is time-dependent and that a time-dependent propagation of these softened zones affects the strain and displacement rates. This process of time-dependent propagation of softening has not yet been simulated analytically, except in a very crude manner by Kaiser [13], but the following conceptual evaluations supports the hypothesis of such a process. Figure 15 schematically illustrates the propagation of a softened zone towards an extensometer measuring average radial strain between point A and C. For this diagram, it was assumed that the softened zone possesses an extremely low modulus and does not provide significant support to the surrounding undisturbed rock mass. Hence, it can be simulated by an equivalent opening which propagates in time to a larger opening of modified shape. Figure 15

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presents the radial strain distribution, for a test in a presented in Fig. 10 (Part II) where propagation at elematerial with some compressibility, immediately after vated stress levels can be observed. Furthermore, it can load application to, at some intermediate time tl and be seen from this figure that the opening during reloadafter time-dependent straining has stopped. Figure, 15a ing, after propagation, behaves as if the opening size illustrates the condition before propagation of a sof- and shape was changed during the earlier test and only tened zone at a low stress level and Figure 15b after further propagation changes the mode of behaviour as and while the equivalent opening propagates to an discussed in Part II. It should also be remembered that elliptical shape at a higher stress level. The propagation this propagation was not detectable from the tunnel of an equivalent opening increases the extensional closure measurements (see Part II) except at the highest radial strain between point A and C. Figure 15c shows stress level during the last test MC-2.9 (N = 0.2). the three radial strain components, both at a low and To further illustrate the insensitivity of tunnel closure high stress level, due to hydrostatic creep (compression) measurements to the physical process of time-depenand deviatoric creep (extension), and opening propa- dent rock mass deformation and in support of the gation (extension) only for the high stress level. Super- earlier conclusions Fig. 16a summarizes the creep cloposition of these three components results in the strain- sure of the tunnel measured in four directions after time relationship plotted by heavy lines in Fig. 15c. The 100 hr of sustained load. While Fig. 10b (Part II) clearly corresponding behaviour is plotted schematically in the indicates a propagation of a softened zone at stresses double logarithmic strain rate vs time diagram in Fig. above 9 MPa, no such evidence can be found from the 13a as curve (2). This curve has been generated by creep closure record. It must, however, be repeated that superposition of linear double logarithmic strain rate vs this creep closure measured in the test includes both time relationships for hydrostatic creep (I), deviatoric creep components due to deviatoric and hydrostatic creep (II) and opening propagation (III). Again good creep. It is larger than the one which would be encouncorrespondence with the measured strain rates tered during unloading by tunnel excavation. presented in Figs 13d, e and f for stress levels in excess The creep strains predicted from the 3-parameter of 13 MPa can be observed. This effect of time-depen- solid, linear visco-elastic model are shown in Fig. 16b dent propagation and stress redistribution is further for the incremental loading history of the test. It was supported by the radial strain vs time diagrams assumed that creep deformations terminate completely

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before application of the next increment. For Case A, it was assumed that the parameters are independent of the stress level, whereas it was assumed for Case B that the modulus of the spring in the Kelvin element decreases linearly with increasing stress level. A comparison with Fig. 16a shows that there is excellent agreement between measurements and Case B predictions. The predictions, based on the parameters used for Case B (Fig. 16b) and a retardation time t(ret) = 30 hr, for test MC-2.9 with N = 0.2 are plotted in Fig. 16c and can be compared with the measurements presented in the same figure. Between Test MC-2.6 and MC-2.9 the tunnel had been reloaded twice at stress ratios of N = 0.5 and N = 0.33 (see Fig. 9). During this repeated loading softening around the opening resulted in a significant increase in tunnel wall closure, particularly parallel to the major field stress. The measured time-dependent tunnel wall closure compares reasonably well with the one predicted from the simple visco-elastic model. It does not reflect the drastic changes which occurred between Test MC-2.6 and 2.9. However, the increasing discrepancy at the higher stress levels indicates that propagation of a softened zone occurs and that the

opening behaves as if its shape is gradually being changed. This is indicated by the increased creep deformation in all four directions. Figure 17 summarizes the tunnel wall closure rates, 5 and 50 hr after each loading increment, as a function of the applied field stress level. The stress-creep rate relationship predicted by visco-elasticity for a fixed time after loading would show the same patterns as the stress-creep closure relationship. Both, the creep closure and the creep rate are directly proportional to the stress increment. This similarity is evident from Figs 16 and 17. Figure 17 shows that the closure rate is also strongly influenced by the loading history. Low rates are observed after small loading increments and higher rates after larger loading increments. These observations are in general agreement with field measurements where often higher closure rates are observed if large excavatioq rounds follow rapidly and lower closure rates are recorded during smaller or delayed excavation rounds. The interaction between the tunnel support and the ground may, however, change this tunnel response pattern. At elevated stress levels the closure rate increases rapidly, particularly when propagation of a zone of softened rock occurs.

Deformation of Small Tunnels--III. Pre-failure Behaviour 15

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Practical implications From our tests, it seems that closure rates in excess of 1-2 x 10-a%/hr are indicative of ongoing stress redistribution as a result of the propagation of softened zones. As indicated in the introduction to this paper, at the Arlberg Tunnel remedial anchoring was necessary when tunnel closure rates in the order of 30 x 10-a%/hr were measured and the final liner was only installed when the rates dropped below about 0.25 x 10-3%/hr. It is too early to establish specific guidelines, but the test results indicate that there may be threshold values which can be used to differentiate modes of opening behaviour. For example, for the small tunnels tested, below 1 x 10-3%/hr tunnel closure the opening is in a state of equilibrium and only time-dependent deformation due to creep of the rock mass are observed. Above 1 x 10-3%/hr tunnel closure, the equivalent opening propagates because of rock mass softening, or yielding as the case may be at higher stress levels. However this propagation may terminate at another state of equilibrium if the closure rates are not excessive. As will be discussed in Part IV, the threshold between stable and onset of instability propagation, where failure occurs, may be as high as or in excess of 10 x 10 3~o/hr, The closure rate may be a useful indicator of the global rock mass response to the tunnel excavation procedure. It may be possible to use it for the evaluation of the effectiveness of ground control measures.

323

However, it is of practical significance to note how sensitive the time-dependent closure is to the loading history. During Test MC-2.9 (Fig. 16) approximately twice the actually measured amount of creep closure would be observed if the tunnel was loaded directly to 10 MPa. About three times the actually measured creep closure would be observed under such loading conditions during Test MC-2.6. The incremental stress change per excavation round dominates the timedependent tunnel closure. Naturally, the actual loading rate during tunnel construction varies drastically because of inevitable delays caused by many construction related factors. Hence, the tunnel closure rate measurements must be related to the construction history. Furthermore, the comparison between time-dependent tunnel closure and radial strain measurements, observed by extensometers, illustrates that closure measurements are poor indicators of the physical processes which dominate the behaviour of the rock near the tunnel wall. They do not reflect the magnitude and direction of the propagation of zones of softened rock. The radial strain measurements provide more useful information about the size and shape of these softened zones and their rate of propagation. These radial strain measurements become of major importance in situations where the stresses near the opening approach the rock mass strength and softening is enhanced by the non-linear rock mass stress-strain behaviour. At this stage the opening behaves as if its size and shape had been changed and this change in apparent opening geo15

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324

P.K. Kaiser and N. R. Morgenstern

merry must be considered when evaluating tunnel wall closure data. Our tests have shown that extensometers are much better indicators of the local ground behaviour. It has been seen that softening occurs well before yielding and weakening is initiated and that the process of softening results in excessive movements, compared to the ones predicted using elastic material models, largely because of non-symmetric propagation. The complexity of the geological rock mass structure will almost always provide conditions susceptible for non-symmetric propagation of softened zones near the tunnel. The existence of such zones and the related effect on stress redistribution must be considered during the evaluation of performance records. It is our belief that this pre-failure softening process does not receive adequate recognition and that the strain-weakeninq model is often misused to match field observations where failure has not yet occurred.

Concludin9 remarks The tests of unsupported small tunnels in naturally jointed rock have shown that deviation from the elastic rock mass response may occur well before the rock mass is stressed beyond its peak strength and weakens because of excessive straining. The tunnel behaviour at this point is dominated by the time-independent and time-dependent properties of the undisturbed rock mass far from the opening and by the local stress redistribution process due to the propagation of softening near the tunnel wall. It will be necessary to further support these findings by field evidence. Examination of published measurements and our own observations reported here support these conclusions. Acknowledgements- The authors wish to acknowledge the continuing financial support by the National Sciences and Engineering Research Council of Canada. Received 30 March 1982.

REFERENCES I. Kaiser P. K. & Morgenstern N. R. Time-dependent deformation of small tunnels 1. Experimental facilities. Int. J. Rock. Mech. Min. Sci. & Geomech. Ahstr. 18, 129-140 (t981).

2. Kaiser P. K. & Morgenstern N. R. Time-dependent deformation of small tunnels- II. Typical Test data. Int..I. Rock. Mech. Mi,. Sci. & Geomech. Abstr. 18, 141-152 (1981). 3. John M. Application of the New Austrian "Funnelling Method under various rock conditions. Rapid Excavation and Tunnellim/ C o ~ Proc., Vot. 1, Chap. 26, pp. 409 426 (1981). 4. John M. Adjustment of programs of measurements based on the results of current evaluation. Proc. Int. Syrup. on Field Measurements in Rock Mechanics, Vol. 2, pp. 639 656 (1977). 5. Da Fontura S. & Morgenstern N. R. Stress strain-time relationship for jointed coal. D~t. Syrup. on Weak Rock Tokyo, Vol. 1~ pp. 105 110(1981). 6. Kaiser P. K. & Morgenstern N. R. Time-dependent deformation of jointed rock near failure. 4th Con qr. I S R M , Vol. 1, pp. 195-202 (1979). 7. Terzaghi K. & Richart F. E. Stresses in rock about cavities, Geotechnique 3, 57-90 (1952). 8. Deere D. U., Peck R. B., Monsees J. E. & Schmidt B. Design of tunnel liners and support systems. Rep. for t,LS. Dept of Transportation, Contr. No. 3-0152, 287 pp (1969). 9. Berry D. S. Deformation of a circular hole driven through a stressed viscoelastic material. Int. J. Rock Mech. Min. Sci. 4, 181--187 (1967). I0. Egger P. Einfluss des Post-Failure-Verhaltens yon Fels auf den Tunnel Ausbau. Veroeffentlichungen des Inst. fuer Bodenmech. und Felsmech., Univ. Fridericiana, Karlsruhe, Heft 57, 83 pp (1973). 11. Kaiser P. K. & Morgenstern N. R. Phenomenological model for rock with time-dependent strength. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 18, 153-166 (1981). 12. Sargent D. W. J. A process simulation test on a small inclined tunnel in jointed coal. M.Sc. thesis, Dept of Civil Engineering., Univ. of Alberta, 179 pp (1981). 13. Kaiser P. K. A new concept to evaluate tunnel performance-. Influence of excavation procedure. 22nd U.S. Rock Mechanics Syrup. Boston, pp. 264~271 (1981). 14. Guenot A. Investigation of tunnel stability by model tests. M.Sc. thesis, Dept of Civil Engineering, Univ. of Alberta, 217 pp (1979}. 15. Kaiser P. K. Time-dependent behaviour of tunnels in jointed rock masses, Ph.D. thesis, Dept of Civil Engineering, Univ. of Alberta, 395 pp (1979). 16. Kaiser P. K. Tunnel design---conclusions from long-term model tests. 4th Conyr. I S R M , Vol. 3, pp. 144-145 (1979). 17. Kaiser P. K. Monitoring for the evaluation of the stability of underground openings. Ist Ann. Conf. on Strata Control in Mininq, West Virginia Univ., pp. 90-97 (1981). 18. da Fontoura S. A. B. Time-dependent response of rock masses during tunnelling. Ph.D. thesis, Dept of Civil Engineering, Univ. of Alberta, 314 pp ([980). 19. Kidybifiski A. Rheological models of upper Silesian carboniferous rocks. Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. 3, 279 306 (1966). 20. Morlier P. Etude exp6rimentale de la d6formation des roches. Revue lnst.fr. P~trole, Octobre, Part 2, 1187 (1064). 21. Terry N B. The elastic properties of coal. British National Coal Board. Mining Research Establishment Rep. 2080, Part 6 (1956).