Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 28, No. 6, pp. 477-488, 1991 0148-9062/91 $3.00 + 0.00 Printed in Great Britain. All rights reserved Copyright 1991 Pergamon Press pie

Time-dependent Tunnel Convergence II. Advance Rate and Tunnel-Support Interaction Y.-W. PANt J.-J. DONGt

A time-dependent model of tunnel convergence for tunnelling in a viscoelastic rock mass is proposed in the accompanying paper (this issue, pp. 469-475) to model the excavation-construction process during tunnelling in a rock mass with rheological properties. In this paper, a parametric study based on this model is presented to investigate the effects of the tunnelling advancing and support installation, respectively, on the tunnel convergence and on the support-pressure. Furthermore, a non-linear optimization procedure is suggested to calibrate the required model parameters from the in situ measured tunnel-convergence data. Discussion on the proper application of the proposed model and the optimization calibration are also included.

INTRODUCTION A time-dependent model of tunnel convergence for tunnelling in a viscoelastic rock mass is proposed in an accompanying paper [1]. In this hierarchical model, various important factors including: (1) the rheological properties of the rock mass; (2) the tunnelling advance- ment; and (3) the tunnel-support interaction, are taken into account to model the excavation--construction pro- cess during tunnelling in a rock mass with rheological properties.

In general, the creep-rate for an opening in a rock mass decreases with age and increases with the depth of the opening. The creep behaviour of a rock mass around a tunnel depends not only on the rheological properties of rock mass but also on the existing stress state [2]. As a result, the convergence of tunnel can be considered as a function of the rheological properties and the stress state of rock mass. However, the stress state of rock mass around a tunnel is strongly affected by the excavation/ support process of the tunnel construction, The resulting effects of the excavation/support process of tunnelling, in general, can be divided into the tunnelling advance- ment effect and the support installation effect.

The tunnel advancement effect mainly derives from the restraint of the convergence of the already excavated opening due to the unexcavated rock core ahead of the tunnel face, which provides a natural internal support pressure. This natural internal support decreases with increasing distance from the considered tunnel cross- section to the tunnel face. The radial displacement of the tunnel wall at any location in a circular tunnel can be described by the following equation:

U~(x) = F ( x ) U r ( x --, oo); (1) tlnstitute of Civil Engineering, National Chiao-Tung University,

Hsinchu, Taiwan 30050, Republic of China.

in which Ur(x "" oo) represents the radial displacement of the tunnel wall at an infinite distance from the tunnel face; it is equivalent to the radial displacement of a tunnel in a plane-strain condition. F(x) is a normalized displacement function relating to x, the distance from the considered tunnel cross-section to the tunnel face.

Hocking [3] made use of the method of boundary integral equations to evaluate the normalized displace- ment function F(x) under condition of an isotropic stress field in an elastic medium. Based on the study of the Frejus and Las Planas tunnels, Sulem et al. [4] proposed the following empirical equation for the con- vergence of a rock tunnel when the creep effect of the rock mass is excluded:

X 2

in which X is a parameter of the rock-tunnel and C~x is the tunnel convergence at x = ~ . From equations (1) and (2), one can extract the following empirical function to represent of the normalized displacement function F(x):

The ground response curve (or the characteristic line of a rock mass) portrays the relation between the tunnel displacement and support pressure. Many analytical and numerical solutions of the ground response curve consid- ering elasto-plastic behaviour of a rock mass are avail- able [5-10]. When the viscous behaviour of the rock mass is considered, the ground reaction curve also depends on the excavation sequence and the installation time of the support [11]. Thus, the tunnel-support interaction is also time-dependent.

477

478 PAN and DONG: TIME-DEPENDENT TUNNEL CONVERGENCE II

Some conclusions were drawn by Ward [12] on the support pressure developing with the tunnel advance- ment. First, the closer to the face the support is installed, the greater the load will be carried by the installed support. Second, the load carried by the installed sup- port is higher when the support stiffness is raised. A formula for calculating the support pressure is pro- posed by Ward [12] as follows:

P~ Ks[1 -F (x ) ] - (4) Po Ks + 2G ' in which Ps is the support pressure, Ks is the support stiffness and G is the shear modulus of the rock mass.

Ward [12] also found that in a weak rock, the displace- ment at the tunnel wall during tunnel advancement depends not only on the tunnel face position, but also on the elapsed time. This can explain why both the tunnel displacement and the loading on a support near the tunnel face tend to increase, no matter whether the tunnel face advances or remains unchanged. This phenomenon clearly demonstrates that creep behaviour of a rock mass should be reasonably modelled in the analysis of tunnel-support interaction.

In this paper, a parametric study of time-dependent tunnel convergence and tunnel-support interaction is presented. Furthermore, a non-linear optimization procedure is proposed to calibrate the required model parameters from the in situ measured tunnel- convergence data.

T H E T I M E - D E P E N D E N T T U N N E L - C O N V E R G E N C E M O D E L

A time-dependent model of tunnel convergence is proposed in the accompanying paper [I]. Based on this model, the time-dependent radial displacement Ur(t) in a circular opening with a radius a in a viscoelastic medium at time t can be evaluated by the following equation:

U~(t) a - g ~ [ h c ( t ) x Fc(t)]

-g~eK~Hs(t - ts)[D(t) Fc(t)], (5)

in which g v e K s =

Fc(t) = he(t) =

Po, P c ( t ) =

Hs(t - ts) =

O( t ) =

compliance function, support stiffness, creep function, d[P0 - Pc(t)]

dt initial pressure and the rock-core support pressure, respectively, unit step function such that Hs(t - q) = 1 for t i> t,, and Hs(t - is) = 0 for t < ts, the changing rate of the radial displace- ment:

d[Ur(t)- Ur(q)] D(t ) = dt

Six influence functions [gve, F~(t), D(t), he(t), K, and H ( t - t s ) ] are included in equation (5) to consider: (a) the creep effect; (b) the tunnel advancement effect; and (c) the support effect, respectively:

1. The creep effect results from F~(t) and gve. 2. The tunnel advancement effect results from h~(t)

and D(t) . 3. The support effect results from K,, H~(t - t~) and

D(t) , namely: (a) the effect of support stiffness results from Ks; (b) the effect of installation time results from

Hs(t - t s ) and O(t) . The creep Fc(t) and the compliance functions gw

depend on the assumed viscoelastic model. The function he(t) depends on the tunnel advancement rate; Hs(t - ts) and D(t ) depend on the installation-time of the support and the assumed viscoelastic model; and D(t ) depends on the support characteristics and the assumed visco- elastic model. In the next section, a parametric study is performed to investigate the effects of the tunnelling advancement and support installation, respectively, on the tunnel convergence and support pressure. The time- dependent tunnel-support interaction is also examined.

PARAMETRIC STUDY Tunnel advancement effect

The effect of tunnel advancement arises from the changing rate of the unexcavated rock-core pressure. Suppose the tunnel face advances at a constant speed V, then the unexcavated rock-core pressure can be related to the tunnel advancement rate V through a normalized displacement function F(x) . Assume that equation (3) is valid and Pc[x(t)] = PoE(x) , then:

o P c( t )= P o 1 - (x + X ) =P0 1-- (Vt + X (6)

It can be observed from the above equation that the unexcavated rock-core pressure Pc(t) depends on the elapsed time and tunnel advancement rate. As a conse- quence, the function he(t) also depends on the elapsed time and the tunnel advancement rate. The effects of the tunnel advancement rate can be noted in Figs 1-3 for different viscoelastic models. Figures 1-3 show the influ- ence of the tunnel advancement rate on the normalized tunnel convergence (Ur/a) for the Kelvin, Maxwell and generalized Kelvin models, respectively. For an identical viscoelastic model (based on the same creep function), the tunnel convergence approaches an identical final value, but the rate of tunnel convergence apparently varies with tunnel advancement rate. It reveals that the final tunnel convergence does not depend on tunnel advancement rate; however, a tunnel will take a longer time to reach the final tunnel convergence for a slower tunnel advancement rate than for a higher one. It can also be noted that the initial convergence rate depends on the tunnel advancement. Rate: the greater the tunnel

P A N and D A N G : T I M E - D E P E N D E N T T U N N E L C O N V E R G E N C E - - I I 479

0.020

A .3 I

UJ O

" - ' 0 .015 O

0 c" ~ 0.010 > c- O

(0

-0

.N_ 0.005 O E 12t

Z

c c c c o V = 1 ( m / d a y ) = = = = = V = 2 ( m / d a y ) . . . . . V= 5 ( m / d a y ) = = . = = V = l O ( m / d a y )

the Kelvin's Model (C1=-1, ~xI=-5, (x2=-5,

g,,,,=O.O0001 ./MPa, X~3.6 m, K,=0.0 MPo/m, t,=0

0.000 '1' , , 0.0 5.0 10.0

Time Fig. 1. The influence of tunnel a d v a n c e m e n t ra te on the

I I 15.0 20.0 25.0

(days) normalized tunnel convergence for the Kelvin model.

advancement rate is, the greater the initial convergence rate will be.

The support effects The effects of stiffness and installation time of the

support on the radial tunnel displacement will be dis- cussed subsequently.

Figures 4--6 show the influence of support stiffness on the normalized tunnel convergence (U,/a) for the Kelvin, Maxwell and generalized Kelvin models, respectively. In

these figures, the curve for K, = 0 is equivalent to a condition when no support is installed. For each model (with the assumed parameters), the support is assumed to be erected immediately after the tunnel is excavated. It can be observed that the convergence of the tunnel becomes saturated within a short time; the tunnel is obviously stabilized from the contribution of the sup- port. The tunnel convergence is largely repressed due to the installation of a support; the higher is the stiffness, the smaller the tunnel convergence that will occur. Take

0.030

t o I

uJ 0.025 0 v

0

"C--~ 0.020

d 0 C ~ 0.015 0 > C 0 o 0.010 "O O N "6

0.005 O

Z

V : 1 ( m / d a y ) V= 2 (m/day, )

,- -'- ..:, = ,- V= 5 ( m / d a y ) V = l O ( m / d a y ) the Mexwell's Model (C~=0.00001, C2=1.5, a = - 5 ,

g,,.=O.O0001 ./MPQ, X-3.6 m K , = O . O M P o / m , t , = O )

0.000 ~ , , , , 0.0 5.0 I 0.0 15.0 20.0 25.0

Time (days) Fig. 2. The influence of tunnel advancement rate on the normalized tunnel convergence for the Maxwell model.

480 PAN and DANG: TIME-DEPENDENT TUNNEL CONVERGENCE--II

0 .020

r ~ I

h J 0

~-" 0 .015 0

d 0 E So.o o %_

> C 0

(.3

._N 0 .005 O E t - O

Z

0 .000

. . . . . V=10 ( m / d a y ) the Generalized Kelvin's Model (CI=1, cx1=-5, C2=-1, ~2=-5,

g~,=O.O0001 /MPo, X=3.6 m K,=O.O MPo/m, t,=O )

I I 1 I 0.0 5.0 10.0 15.0 20.0 25.0 Time (days)

Fig. 3. The influence of tunnel advancement rate on the normalized tunnel convergence for the generalized Kelvin model.

the case of K, = 5000 MPa/m for example, the installed support reduces the tunnel convergence by about 35% for the Kelvin model, and about 45% for the Maxwell model.

Figures 7-9 show the influence of support stiffness on the normalized support pressure and the normalized radial pressure of the rock mass, P/Po, for the Kelvin, Maxwell and generalized Kelvin models, respectively. The support is assumed to be erected immediately after the tunnel is excavated (i.e. ts = 0). Ground response

curves corresponding to different values of K, are pre- sented in these figures. The difference in the support stiffness results in the variation of the ground response curves. This clearly demonstrates that the tunnel- support interaction during the tunnelling process should be taken into account carefully.

Figures 10-12 show the influence of installed time of a support on the normalized tunnel convergence (Ur/a) for the Kelvin, Maxwell and generalized Kelvin models, respectively. For each model (with the assumed

0.020

I Ld O ~ ' 0 . 0 1 5

0

d 0

~ 0 . 0 1 0 &._

> C 0

(.9

- 0 G) .N 0.005 O E 0

Z

~ i . 0 C. ^ cooco K,= 0 . 0 (MPa/.rn) ===== K,=IO00.O (MPa/.m) ~ ' - K,--5000.O (MPa/m)

K, =5000.0 (MPa/m) the Kelvin's Model (C,=-1 , ~ = - 5 , a ~ = - ~

g,,=O.OqO01 /MPa,. X 3.6 m, V=2 m/day, t,=O )

0.000 , , l I 0.0 5.0 10.0 15.0 20.0 25.0

Time (days) Fig. 4. The influence of support stiffness on the normalized tunnel convergence for the Kelvin model.

PAN and DONG: TIME-DEPENDENT TUNNEL CONVERGENCE--II 481

0 . 0 3 0

r-) I w 0 . 0 2 5 o

v

O

"C 0.020

E ~ 0 . 0 1 5 > C O

c) 0.010 ~3

N

(3

E 0 . 0 0 5 O

Z

c.~co K,= O.0 (MPa /m) ===== K,=IO00.O (MPa /m) = = = = = K , = 3 0 0 0 . O (MPo/m) . . . . . K , = 5 0 0 0 . O ( M P o / m )

the Maxwell's Model (CI=O.OO001, C~=1.5, = = - 5 ,

gv,=0.Oq001 / M P a , X=3.6 m V=2 m/day, t,=O )

0 . 0 0 0 ~ i I , , 0.0 5.0 10.0 15.0 20.0 25.0

T ime (days ) Fig. 5. The influence of support stiffness on the normalized tunnel convergence for the Maxwell model.

parameters), the support stiffness is assumed to be 3000 MPa/m. In these figures, the curve for t, = oo corresponds to a condition when no support is installed. From these figures, it can be observed that the sooner the support is installed, the lower the final tunnel conver- gence that will be reached. It can also be seen that the final tunnel convergence may be reduced significantly if support is installed too late.

Figures 13-15 show the effect of support timing on the support pressure and the normalized radial pressure of

the rock mass, P/Po, for the Kelvin, Maxwell and generalized Kelvin models, respectively. The support stiffness is assumed to be 3000 MPa/m. Ground response curves corresponding to different support installation times t,, are presented in these figures. The difference in the support installation time results in the variation of the ground response curve, These results evidently reveal that the timing of the tunnel installation may greatly influence the tunnel convergence as well as the support pressure.

0 . 0 2 0

e,3 I uJ

c )

" J 0 . 0 1 5 O

D 6 0 E ~ 0 . 0 1 0 > C O

o

"O G) ._N 0 . 0 0 5

0 Z

hi , : : n

cocoa K,= O.O (MPo/.m) ===== K,=IO00.O (MPa/m) ===== K,='3000.O (MPa/m) . . . . . . K,=5000.O (MPo/m)

the Generalized Kelvin's Model (C~=1, cx,=-5, .C=-1, a~=-5 ,

g, ,=0 .0QO01 /MPo, . X = 3 . 6 m V=2 m/day, t,---O )

0.000 I , , , , 0.0 5.0 I 0.0 15.0 20.0 25.0

T i m e ( d a y s ) Fig. 6. The influence of support stiffness on the normalized tunnd convergence for the generalized Kelvin model.

482 PAN and DANG: TIME-DEPENDENT TUNNEL CONVERGENCE I1

1.10

1.00

0.90

0_0.80

.0.70

0.60 (0 (1) o_ 0.50 -0 N ~ 0.40 O

0.30 0

Z 0.20

0.10

o .oo 0.000

ccc :o K,= 0.0 (MPa/m) c : = = : K,=IO00.O (MPo/.m) -~-~-~-~ K,=3000.O (MPa/m) . . . . . K,=5000.O (MPa/m)

the Kelvin's Model (C1=- -1 , ~X,=--5, (Xz=--5,

gv,=O.O0001 /MPa, X=3.6 m. V=2 m/day, t,=O )

0.005 0.010 0.015 0.020 Normalized Convergence, Ur /o ( 10E-3 )

Fig. 7. The influence of support stiffness on the normalized pressure for the Kelvin model.

PARAMETER CALIBRATION FROM TUNNEL-CONVERGENCE DATA USING THE NON-LINEAR OPTIMIZATION TECHNIQUE

Non-linear optimization Back analyses are often performed to find the required

material parameters of a model from in situ measure- ment [e.g. 13-15]. For the purpose of back analysis, the

non-linear optimization technique can be applied to calibrate the required parameters of the proposed model in terms of one or more sets of data from in situ measurement. A non-linear optimization, in principle, is to search for the optimized value of a non-linear object function. Among many different non-linear optimization techniques, the Levenberg-Marquardt method has been well established, the required computer subroutines are

110

1.00

0.90

a_ 0.80 O_ 0.70

o 6 o (D k . n 0.50

o 4 o 0 E 0.30 O Z 0.20

0 . 1 0

0.00 0.000 0.005 0.010 0.015 0.020 0.025 0.030

Normalized Convergence, Ur /o (10E-3 ) Fig. 8. The influence of support stiffness on the normalized pressure for the Maxwell model.

PAN and DONG: TIME-DEPENDENT TUNNEL CONVERGENCE--II 483

1.10

1 . 0 0

0.90

~_ 0.80 (3._ 0.70

0.60

e_ 0.50 " O

0.40 0

E 0.30 0 z

0.20

0.10

o.oo - ~ 0.000

c e e e e K,= 0.0 (MPa/m) "= "= K,=IO00.O (MPo/m) .~=- '~ K,=3000.O {MPa/.m) . . . . . K,=5000.O (MPo/m) ceeee the Generolized Kelvin's Model c---=---(C1=1, o(1=-5 , C2=-1 , ~2=--5, -"-~-~':- g,,,=O.O0001 /MPo, X=3.6 m, ........ V=2 m/doy, t,=O )

0.005 0.010 0.015 0.020 Normolized Convergence, Ur /o ( l O E - 3 )

Fig. 9. The influence of support stiffness on the normalized pressure for the generalized Kelvin model.

also readily available in mathematic packages (e.g. IMSL [16]). This method is suitable for locating the global optimum [17-19]. Formulation of the non-linear optimization method is beyond the scope of this study; interested readers are referred to Fletcher [19].

The object function q~ {x } in this ease is defined by a non-linear "error square" function, which is the sum of the squares of the differences between n (model) calcu-

lated data, Ui({x}), (i = 1, n) and n (in si tu) measured tunnel-convergence data, V~ (i = 1, n):

(~{x} = ~ [Ui({x})- Vi] 2, (7) i=1

in which {x } is the matrix consisting of the undetermined model parameters. The involved non-linear optimiz- ation, then, is to search for a set of unknown model

0.020

A

I w o ~J0.015

O

Q; C ~ 0.010 > C 0 o "tO

.N 0.005 O

0 z

0.000 0.(

t A '

o o = ~ o

0 0 0 0 0

i (doys) (doys) ~ ~ -~ -" t ,= 4 (,days)

. . . . . . t ,= 8 (,doys) -'~-'.' '- t,=infinite (doys)

the Kelvin's Model (C~=-1, ~ = - 5 . , ~2=-5,

gv,=0.O0001 /MPo, X=3.6 m, V=2 m/doy, K,=3000 MPo/m )

I I I I 5.0 10.0 15.0 20.0 25.0

Time (doys) Fig. 10. The influence of support timing on the normalized tunnel convergence for the Kelvin model.

484 PAN and DONG: TIME-DEPENDENT TUNNEL CONVERGENCE- Ii

0.050

1 w 0.025 o

[3

"~C-~ 0.020

d 0 C ~ 0.015 0 > C 0

t) 0.01 0 -[3 (I) N

O

E 0.005 0 7

ooecot ,= 0 (doys) oc --- " = t,= 2 (`doys)

. . . . . t,= 8 .(dys) :.~''~" t,=infinite (,days)

the Moxwell's Model (C~=0.00001, C~=1.5, cx=-5,

g v,=O.O0.O01 /MPa, X=3.6 m. V=2 m/day, K,=3000 MPo/m )

0 . 0 0 0 ~ , , , 0.0 5.0 10.0 15.0 20.0 25.0

Time (days) Fig. 11. The influence of support timing on the normalized tunnel convergence for the Maxwell model.

parameters {x} corresponding to a series of Ui({x}), (i = 1, n) such that the object function (the error square function) is minimum. A computer program based on the Levenberg-Marquardt method and the time- dependent tunnel convergence model is developed for calibrating the required parameters of the proposed model in terms of data from in situ measured data. This program can be used to perform calibration based on

one set or several independent sets of measured data; the lower and upper bounds of each parameter can also be defined. Besides, various weight functions can be assigned to different sets of calibrated data in order to distinguish the various reliabilities of different data sets.

For purposes of illustration, a set of in situ measured tunnel-convergence data are taken from Sulem et al. [4] to demonstrate the calibration method using a

0.020

r~ I

UJ 0

"J0.015 [9

d 0 C ~0.010 k- (D > C 0

-3

.~ 0.005 0

0 z

0.000

Z Z Z J. ~ T - o c -- =

ys = o -o

iii?i :L:,,o the General ized Kelvin's Model (CI=1, (x~=-5, .Cz=- l , a2=-5,

gv,=O.O0001 /MPo, X=3.6 .m. V=2 m/day, K,=3000 MPo/m )

I I I I 0.0 5.0 10.0 15.0 20.0 25.0

Time (days) Fig. 12. The influence of support timing on the normalized tunnel convergence for the generalized Kelvin model.

PAN and DONG: TIME-DEPENDENT TUNNEL CONVERGENCE--II 485

1.10

1 .00

0.90

o_0.80 o_

0 . 7 0

0.60 O3 , ~- 0.50 "o

o 4 o . ~ 0 E 0.30 0 Z

0.20

0.10

e : : : o t , = 0 (days) c==== t,= 2 (days) * = = =" t , = 4 (days) . . . . . t , = 8 (days)

the Kelvin's Model (C t= -1 , a1=--5 , e~=-5,

gv,=O.O0001 /MPa, X=3.6 m, V=2 m/day. K,=3000 MPa/m

0.00 0.000 0.005 0.010 0.015 0.020

Normal ized Convergence, Ur /o ( 1 0 E - 3 ) Fig. 13. The influence of support timing on the normalized pressure for the Kelvin model.

non-linear optimization technique. The adopted data come from the measurement of the Las Planas tunnel, Base A, section 7 [3].

The results of tunnel-convergence calibration The distortional stress-strain relation of the ground is

assumed to follow the Kelvin model. The creep function Fc(t) takes the following form for the Kelvin model [1]:

Fc(t) = 1 + C, e - ' , ' - (C, + 1)e-=:'. (8)

The conditions for the considered tunnel are: Pv= 1.84MPa and a = 10.5m. There are five par- ameters that need to be calibrated in this case:

X of equation (3), gve of equation (5), C1, ~t, ~2 of equation (8).

The parameter calibration is based on the measured data of tunnel convergence during the first 8 days. For lack of actual properties and installation time of the

1.10

1 .00

0.90

0_0.80 o_

.0 .70

0.60 O 3

n 0.50 "0

0.40 0 E 0.30 0 Z

0.20

c : : = o t , = 0 (days) a = = = = t,= 2 (days) = = = = ~ t,= 4 (days) =-- . . . . t ,= 8 (days) oeceo t.=-=infinite (days)

the Maxwell's Model (C~=0.00001, C~=1.5, (x=-5.

g,,=O.OqO01 /MPo, X=3.6 m, v=2 m/day, K,=3000 MPo/m )

0.10

0.00 0.000 0.005 0.010 0.015 0.'020 0.025 0.030

Normalized Convergence, Ur /o (10E-.3) Fig. 14. The influence of support timing on the normalized pressure for the Maxwell model.

486 PAN and DONG: TIME-DEPENDENT TUNNEL CONVERGENCE-- 1I

1.10

1 . 0 0

0.90

o_ 0.80 [3_

.0 .70 o

0.60 2 n 0.50 - 0

0.40 '6

0.30 o Z

0.20

0.10

o.oo 5 ~= o.ooo

c c c c o t,= 0 (days) u==== ts= 2 {days~

. . . . . . . t~= 8 {days) ooooo t,=infinite (days) =c==3 the Generalized Kelvin's Model ~ - = ~ (C1=1, ~x~=-5, C~=-1, ~x2=-5,

gv,=O.O0001 /MPo, X=3.6 m, V=2 m/day, Ks=3000 MPo/m )

0.005 0.010 0.015 0.020 Normal ized Convergence, U r / a ( 1 0 E - 3 )

Fig. 15. The influence of support timing on the normalized pressure for the generalized Kelvin model.

support, the support effect cannot be taken into account. As a result, no support is assumed in the parameter calibration illustrated.

The calculated results of the displacement time history based on the calibrated parameters and the measured displacement time history are compared in Fig. 16. It is seen that agreement is extremely good in the early part of the curve, but is relatively imperfect in the long-term deformation. This discrepancy can be explained as fol-

lows: since the support effect significantly affects the final deformation of a tunnel, the long-term calculated defor- mation which ignores the support effects tends to be overestimated.

Suggestions for model parameter calibration The result shown in Fig. 16 is simply a demonstration

of the application of the optimization technique on the model parameter calibration for the tunnel convergence.

6.50

6.00

5.50 0 " ~ - -

5.00

E 4.5O E

4.00 (1) 0 g 3.5o

3 0 0 C o o 2.50

2.00

1 . 5 0

eccco computed = = = = = measured

the Kelvin's Model (X=6.438 m, g=0.0000175 /MPa,

C~=5.65 e i - - - 1 2 . 3 6 e2=-140.96)

1 . 0 0

0.50 I I I " J i 0.0 5.0 10.0 15.0 20.0 25.0 30.0

Time (days) Fig. 16. The comparison of the calculated and measured time histories of tunnel convergence for the illustrated case.

PAN and DONG: TIME-DEPENDENT TUNNEL CONVERGENCE--II 487

To properly apply the proposed model and the optimiz- ation calibration, the following remarks are suggested:

1. The installation time and the properties of the support should be taken into account when the support conditions change from site to site.

2. The installation time of the convergence indicators should remain the same for every monitored tunnel cross-section. Otherwise, the installation time and the distance from the tunnel face should be recorded in order to eliminate the influence of the initial convergence.

3. The selection of the creep function to represent the rock behaviour should be very carefully made, especially when the long-term tunnel convergence is predicted based on a short-term convergence curve. The appropriate viscoelastic models for various types of rock can be found in the hand- book of Lama and Vutukuri [20].

4. The initial convergence largely affects the interpret- ation of the long-term convergence. The installation time of the convergence indicator and it's distance from the tunnel face should remain constant. Otherwise, the following items should be recorded along with the in situ tunnel-convergence measure- ment: (1) the time when convergence indicators are installed; and (2) the distance from the tunnel- face.

5. To predict the long-term tunnel convergence and support pressure, the time of support installation should be recorded, especially for a stiffer support system.

Acknowledgement--This work was financially supported by the National Science Council of the Republic of China under Contract No, NSC80-0410-E-009-10. This support is gratefully acknowledged.

Accepted for publication 19 March 1991.

S U M M A R Y AND C O N C L U S I O N S

In this paper, a parametric study of time-dependent tunnel convergence the tunnel-support interaction is presented. Moreover, a non-linear optimization pro- cedure is suggested to calibrate the required model parameters from the in situ measured tunnel-conver- gence data. The following conclusions can be drawn:

1. The proposed model in the accompanying paper [1] takes the time factors, including the creep effect, the tunnel advancement effect and the support effect, into account.

2. From the parametric study, the following are found:

(a) the tunnel advance rate does not affect the final tunnel convergence of a tunnel; how- ever, it affects the initial convergence and the required time for a tunnel to reach the final convergence;

(b) the tunnel convergence can be repressed due to the support installation; the higher is the support stiffness, the smaller the tunnel con- vergence will occur;

(c) the tunnel-support interaction during a tun- nelling process should be taken into account carefully;

(d) the sooner the support is installed, the lower the final tunnel convergence which will be reached. The final tunnel convergence may not be reduced significantly if support is installed too late;

(e) the timing of the tunnel installation may greatly influence the tunnel convergence as well as the support pressure.

3. An optimization technique is suggested to calibrate the required parameters of the proposed model. In using the suggested technique, the proper selection of a viscoelastic rock-mass model is essential.

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