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A model of the time-dependent interaction between rock and shotcrete support in a tunnel
Int. J. Rock Mech. Min. Sci. & Geomech. Abstr. Vol. 31, No. 3, pp. 213-219, 1994
Pergamon
Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/94 $7.00 + 0.00
0148-9062(93)E0009-D
A Model of the Time-dependent Interaction Between Rock and Shotcrete Support in a Tunnel YII-WEN PAN1" ZENG-LIN H U A N G t A viscoelastic model is developed to simulate the tunnel-support interaction of a circular tunnel in a viscoelsatic rock mass. This model accounts for the time-dependent stiffness and the yielding strain of shotcrete support. This model is formulated to analyze the effects of the time-dependence of support characteristics on tunnel convergence, support pressure, and tunnel-support interaction. Results of a parametric study reveal that the time-dependence of shotcrete stiffness has a strong influence on tunnel convergence and support pressure. Ignoring the yield strain of shotcrete may lead to the underestimation of tunnel convergence and the overestimation of the ultimate capacity of shotcrete support.
INTRODUCTION
The convergence of a tunnel often indirectly indicates the stability of the tunnel. The tunnel convergence in a competent rock, such as granite or marble, can quickly reach a stable state after excavation, even without man-made support. In some rock conditions, however, the tunnel convergence may be time-dependent due to a variety of reasons. The time-dependent tunnel convergence can result from the rheological properties of rock mass, the process of tunnel excavation, the timing of tunnel-support installation, and the time-dependent properties of the installed tunnel-support [1, 2]. In addition, the size of tunnel, gravity, and the method of tunnel excavation, also have some influence on the convergence of tunnel. Empirical approaches are adopted from time to time to estimate tunnel convergence and support pressure, and to design various types of tunnel-support. The in situ measurement of tunnel convergence often provides the means to monitor stability; and may be used as feedback for design modification of tunnel support. Analytical models correlating tunnel convergence and support pressure will be useful for interpreting the in situ measurement and for predicting tunnel convergence and support pressure. Although some elastic solutions, such as the Kirsh solution, are often used to analyze deformation and stress distribution in the rock mass around an opening, they cannot take the time-dependence of tlnstitute of Civil Engineering, National Chiao-Tung University, Hsinehu, Taiwan, R.O.C. 213
tunnel deformation and support pressure into account. Therefore, elastic solutions fail to model the tunnel mechanics for a tunnel excavated in a time-dependent rock mass or supported by a time-dependent material, such as shotcrete. Rock masses are essentially time-dependent materials with rheological properties. The creep behavior of rock masses has been studied by many researchers [3-6], either by laboratory tests or by in situ tests. Some [7-10] also make use of rheological models to simulate the time dependent deformation of rock material. Factors affecting the creep of rock materials may include stress type, stress level, confining pressure, and the loading environment of the loaded material. The creep behavior of rock material depends not only on the rheological properties but also on the subjected stress condition. Hence, the tunnel convergence is a function of the time-dependent properties of rock masses and the level of support pressure. Since the support pressure also depends on the excavation process and the timing of support installation, the tunnel convergence is largely affected by the same factors. Many researchers utilized rbeological models to describe the time-dependent tunnel convergence of rock tunnel before support installation [11-15]. In general, the tunnel convergence at time t can be described by a certain constant (either the initial or the ultimate tunnel convergence) and a creep function of the rock mass via the following expressions: Ur(t) = Ur(t--'O)Fc(t)
(1)
U,(t) = U,(t --*oo)Fc(t)
(2)
214
PAN and HUANG: INTERACTIONBETWEEN ROCK AND SHOTCRETE
in which Ur(t): tunnel convergence at a time t after tunnel excavation U~(t-+O): the initial tunnel convergence Ur(tOO0): the ultimate tunnel convergence Fc(t): the creep function of rock mass. Lombardi [11] indicated the importance of the inclusion of the three-dimensional effect due to tunnel excavation in developing reasonable analytical model of tunnel convergence using a plane-strain or plane-stress approach. The three-dimensional effect mainly arises from the natural self-support of rock near and in front of the excavation face against the already excavated part of the tunnel. This rock self-support tends to attenuate quickly with a distance x from the excavation face. As the distance x approaches infinity, an equivalent planestrain condition is satisfied. Thus distribution function F(x) can be used to describe the attenuation of rock stress. Ur(X ) = F(x)Ur(x ---~o0).
(3)
In the above expression, Ur(x--,m) represents the tunnel convergence at x = oo in a plane-strain condition which can be regarded as a reference value; F(x) is a normalized distribution function of x. F(x) has the properties that F(x) = 1 as x ~ o o and F(x) = 0 as x--*0. Ur(x--.oo) corresponds to the case when rock pressure disappears. If P0 is the initial stress in an isotropic condition, then the stress [1 - F ( x ) ] P o can be regarded as the natural support pressure against the already excavated tunnel provided by the rock in front of the excavation face. The tunnel-support interaction plays a very important role in determining the tunnel convergence and the support pressure (lining pressure) during the process of the tunnel excavation. This interaction is complicated. In general, it depends on the properties of both the rock mass and the installed supports, the process of tunnel excavation, and the timing of support installation [2, 16-18]. Proper design and installation of the tunnel-support are essential for limiting tunnel convergence and ensuring tunnel stability. Pan and Dong [19] developed a rheological model to simulate the time-dependent tunnel-support interaction problem during the excavation and construction process of a circular tunnel. The model takes into account various time-factors, including the creep effect, tunnel advancing effect, and the support effect. It distinguishes the tunnel advancing effect and the support effect from the creep behavior of the rock mass. Pan and Dong [20] also appraised important time-dependent factors of tunnel convergence. In their model, the tunnel-support is assumed to be made of an elastic material. It is, however, important to note that the support may be made of time-dependent material (such as shotcrete and some types of rock bolts) as well [2]. Shotcrete can possess time-dependent deformabilities (e.g. the shear modulus G and the Young's modulus E) [21] and can develop creep under a constant loading [22]. Besides,
shotcrete may yiel6 when its strain exceeds the yield strain. Any of these properties may result in the time-dependence of support stiffness, and hence affect the long-term tunnel convergence and support pressure. In this paper, the authors extent a previous viscoelastic model [19] to account for the time-dependent stiffness of shotcrete support. An empirical equation of the time-dependent Young's modulus of shotcrete [21] is adopted in the model. In addition, the yielding strain of shotcrete is taken into account. Following the formulation of the model, the effects of the time-dependence of support characteristics on tunnel convergence, support pressure, and tunnel-support interaction are examined. MODEL FORMULATION
Assumptions The following assumptions are made in developing the proposed model. 1. The rock mass is a homogeneous material subjected to an isotropic initial stress. 2. The tunnel is circular. 3. The effects of normal stress in the direction of tunnel axis and the shear stress near the tunnel-face are neglected. 4. The advancing effect and the support effect, together, can be treated as a change in the internal pressure, and can be approximated by the integration of infinite step functions. 5. As the distance from the excavation-face approaches infinity, an equivalent plane-strain condition is satisfied. The rock pressure is a function of the distance from the considered cross-section to the excavation-face. 6. The support is a closed ring with a constant thickness. The support contacts the rock mass as soon as the support is installed. 7. The change in the internal pressure is independent of the creep function. The rock pressure is independent of the support pressure.
Viscoelastic model for tunnel-support interaction In the Pan-Dong model [19], the tunnel convergence of a circular tunnel can be evaluated from the history of internal support pressure Pi(t). The internal support pressure just before tunnel excavation is the initial in situ pressure P0 (i.e. (Pi(O)=P0)). The internal support pressure is contributed from the support pressure P~(t) provided by the installed support and the rock pressure Pc(x(t)) provided by the unexcavated rock in front of the excavation face. As a result, Pi(t) = Ps(t) + Pc(x(t)).
(4)
P(t) denotes the change in the internal support-pressure; as a result, P(t) = Po - P~(t). From assumption (5), the rock pressure Pc(x) is a function of the distance x(t) from the considered location to the excavation face. Suppose Pc(x) can be described by a distribution func-
PAN and H U A N G :
INTERACTION BETWEEN ROCK A N D SHOTCRETE
tion F(x) such that Pc(x(t))= [1 - F ( x ( t ) ) ] P o , we can express
P(t) = F[x(t)]Po - P,(t).
F(x)=l--
~
.
(6)
Based on a viscoelastic formulation [19], the time-dependent tunnel convergence can be evaluated by the following convolution relation: U r ( t ) - gve {[--/5c - a
n ( t - q)/5~] * F¢(Q}.
(7)
in which the viscoelastic compliance function of rock; F~: the creep function of rock; H(t - h): the unit step function which satisfies H(t - h) = 0 for 0 ~< t < h and H(t-h)=l for t t > h ; ts: the installation time of support; gve :
/5c = dP¢(r_.) _ dr
outer boundary of the ring can be found as follows (Appendix):
(5)
The following empirical equation of the function F(x) can be extracted from the work of Sulem et al. [23]:
pod{F[x(Q]}" dr '
U~(r) = ~l + v ~ a2 b -aP~ -_- ~ 2 [ (1 - 2v¢)r + b_f1
(9)
in which E¢: re: a: b: r:
the the the the the
Young's modulus of the ring material Poisson's ratio of the ring material outer radius of the ring inner radius of the ring radius at which the deformation is concerned.
Using the above equation, the relation between the support pressure and the tunnel convergence can be expressed as follows: Or(t)--
Ur(/s)
a
(1
+re)
a
E~(t) a 2 - b 2 b2 xI(1-2vc)a+alPs(t
). (10)
For 0 ~< t < ts, the support pressure does not exist since the support is not yet installed. For t > q, we can find the following relation:
gve, F ftpod{Ftx(Q]} F¢(t - z)dz kJo dz
p _ dPs(z) dr
Modeling time-dependent support-stiffness
-
A constitutive relation of the installed support (support-stiffness) is required to evaluate the change in the support pressure P~(t) and the tunnel convergence Ur(t). The support-stiffness depends on the type, size, and material of the installed support. In the past, most studies assumed time-independent support-stiffness. For a shotcrete support, however, the stiffness may change with time due to the time-dependent nature of shotcrete material [21, 24]. Like concrete material, the strength and Young's modulus of shotcrete usually increase with time [21, 24]. In general, the modulus of shot crete gradually increases until it converges to a stable value. Weber [21] proposed the following empirical formula for the time-dependent Young's modulus of shotcrete based on experimental data: E¢(/) = E2s" exp[-- c(t -°'6 -- 28-0"6)]
215
(8)
in which E2s: 28-day Young's modulus of shotcrete c: a material parameter of shotcrete t: the elapsed time. The Poisson's ratio v¢ of shotcrete is assumed time-independent (constant). To model the time-dependent stiffness of shotcrete support on a circular tunnel, consider a hollow cylindrical ring. The elastic solution of the deformation of a cylindrical ring Ur(r) subjected to a pressure P, on the
f;
P ~ ( r ) F , ( t - ~) d r
s
]
Ur(t3 a
]
(!+___~) a F¢I -- 2v¢)a + b2 E¢(t) a 2 - b 2 L ' a ] Ps(t)"
(11)
Since a numerical integral can be utilized by means of the Gauss integration, the above integral equation can be written in a summation form. Subsequently, it is possible to solve for the tunnel convergence Ur (t) and the support pressure P,(t) at any time t using a time-marching algorithm.
"~0.007
~O.OOe
No Ittl~p~rt
...... E==5*10~lm'al - - - Era= t~{liPa)
~ 0.005 ~ O.OIN, d 0.005 ~ 0.002 ~ 0.00!
o Z o.ooo
0.0
5.b
to'.o
ts'.o
zo'.o
Time (days)
2d.o
30.0
Fig. 1. Normalized tunnel convergence against elapsed time for various E2s.
216
PAN and HUANG:
INTERACTION BETWEEN ROCK AND SHOTCRETE
o
When ~ exceeds ~f, the yielding of the shotcrete support occurs. Then
!.2
E u = 5 * 104.~MPa]
°•ol.0 ¸
...... z== lff~¥P!)
ca.
dr
E a = 2 * 10'(MPa)
~f~ v 0.8 • q) t* m
...... . FI
We thus can find the radius at the occurence of failure rf according to
0.6
0.4
J1~////~
dr
.-'""
~
rf=
-
-
(13)
.
(f
.obo
o.obz
o.obl
Normalized
0 . 0 '0 3
0.004
convergence
o.oo~
(U~/a)
Fig. 2. Tunnel-support interaction curves for various E28.
Modeling progressive failure of support Mortar mixed material, such as shotcrete, can yield under large compressive strain. A yield strain at Er= 0.003 is usually considered for concrete design. Under large compressive strain, shotcrete is likely to yield and result in progressive failure of the shotcrete support. To model the progressive failure of shotcrete, assume that (a) the radial deformation on the ring support is uniform, and (b) the shotcrete yields when the circumferential strain reaches Er. Suppose a circle with a initial radius r deforms to one with the final radius r'. The initial perimeter of the circle is l~ = 2hr. The final perimeter is
It = 2nr' = 21t(r - dr) = 2rrr - 2n dr in which dr = U~(t)-U~(t~) is the change in the radius. Then, the circumferential strain can be evaluated by E.
li
- -
. li
if
.
2rr d r
. 2nr
dr
r
.
(12)
The largest compressive strain on a ring support always occurs at the inner boundary of the ring. As a result, yielding will alway begin at the inner boundary. A perfect plastic condition after yielding is assumed; consequently, no increase in stress at the yielding point is allowed with any further increase in strain. As a consequence, the effective thickness of the ring support can be regarded as reduced. The yielding zone (in the manner of a yielding ring) can propagate outward along with further increase in the support pressure. The expansion of the yielding ring hence results in the progressive failure of the shotcrete support. When the yielding ring becomes as large as the whole ring support, the support cannot take any more pressure and finally reaches an ultimate state. A PARAMETRIC STUDY A parametric study based on the proposed model is performed to investigate the effects of time-dependent modulus Ec(t) and yielding strain Er of shotcrete on tunnel convergence and support pressure. A hypothetical circular tunnel with r = 5.0 and a shotcrete support with a = 5.0m, and b = 4.8 m are considered in the study. The mechanical behavior of rock is assumed to follow the Kelvin model, with P0 = Pv = Ph = 27 MPa, E r = 5 x 10 3 M P a , qr = 5 × 104 ( M P a . see), and vr = 0.25. The common parameters and variables used are as follows: E2s= 1 x 105MPa, c =0.596, v~= 0.17,
"~0.007
t~ 1.2 ~.~0.008 .-.--. -. ._.1 cN°:su-~p°rt: c_ = -.596
- - c = -2 . . . . . . c = -1 --c = -.596 - - e = 0
~1.0
-- - - c = 0
~ 0.005
~
0.5
~ 0.004
m 0.6
0.003
Q) ;@7/
~ 0.002
0.4 Q)
~ 0.001 Iq
'~
o Z o.ooo 0.0
V 5.0
I 10.0
U 15.0
Time
I 20.0
U 25.0
30.0
(days)
0.2
0.0 0.000
, 0.001
Normalized
Fig. 3. Normalized tunnel convergence against elapsed time for various c.
~ 0.002
convergence
, 0.003
(Ur/a)
Fig. 4. Tunnel-support interaction curves for various c.
0.004
PAN and HUANG:
INTERACTION BETWEEN ROCK AND SHOTCRETE
"~0.007
½ ~.0.000 ~ 0.005 t~ ~ 0.004
p,
0.003
~ 0.002 (
~ 0.001
d
a
y
/
--"- --- ts UI = 20 (d~2~)
0
Z o.ooo 0.0
~.b
~o'.o
t~'.o ~o'.o Time (days)
~'.o
3o.o
Fig. 5. Normalized tunnel convergence against elapsed time for various timing of support-installation.
t,=0 day, and vation = 2 m/day.
the
advancing
rate
of
exca-
Effects of E~(t) and the timing of support In this subsection, the effects of time-dependent E¢(t) and timing of shotcreting on tunnel-support interaction are examined; no yielding strain Ef is considered. Figure 1 shows the plot of the normalized tunnel convergence Ur/a against the elapsed time for various E2s. Shotcrete support with a higher 28-day Young's modulus can provide higher support stiffness and allow less tunnel convergence. Figure 2 shows the tunnelsupport interaction curves which display the relations of support pressure and tunnel convergence for different E2s. It can be observed that shotcrete support with a higher E28 causes a higher pressure and a lower deformation. The effect of the material parameter c on the tunnel-support interaction is shown in Figs 3 and 4. A greater value of c means that E~(t) takes a longer time
%
217
to reach E~s; in other words, the modulus of shotcrete in the earlier stage is smaller. The minimum value of c is zero; when c is zero, E¢(t) is always a constant (equal to E2s). From a regression analysis based on collected data [24], the range of the parameter c is found within 0.4 and 0.9. Figures 3 and 4 reveal that a higher value of c can result in a smaller support pressure and a larger tunnel convergence. Figures 5 and 6 compare the normalized convergence with the tunnel-support interaction for different timing of support installation, t~. Results in the figures show that earlier support installation assures a lower tunnel convergence accompanied with a higher support pressure.
Effects of EI The yielding strain Er can have significant influence on the tunnel-support interaction. Figure 7 shows the timedependent tunnel convergence calculated based on various Er. Figure 8 presents the tunnel-support interaction curves corresponding to various values of Er. As mentioned in the above context, yielding always begins at the inner boundary, then progressively expands outward. A perfect plastic condition is assumed in the yielded zone. When the strain at the outer boundary finally reaches the yielding strain, the support pressure cannot increase any more with further tunnel convergence and reaches the ultimate capacity. Ultimate support capacity appears to be a function of Er. From these figures, it can be seen that a smaller Er corresponds to a larger tunnel convergence and a smaller ultimate capacity of support pressure. Ignoring the importance of yielding strain may therefore underestimate tunnel convergence and overestimate support capacity. SUMMARY AND CONCLUSIONS A viscoelastic model has been developed to simulate the tunnel-support interaction of a circular tunnel with shotcrete support in a viscoelastic rock mass. It was
~'0.007
1.2
. . . . . .
It,, --
: =
days I,,...I
=
o.oo0
o
~ 0.005 Q)
"--" 0.8
~ 0.004 0.8
O 0.003 0.002 ~ 0.2
~ 0.001 -
-
~.o.o~s
O
~ 0.0
o.ooo
o.0bt o.doz o.ob3 Normalized convergence
o.oo4' (U~/a)
o.oo5
Fig. 6. Tunnel-support interaction curves for various timing of support-installation.
Z o.ooo
o.o
5.'o
io'.o
15'.o
zo'.o
25'.o
3o.o
Time(days)
Fig. 7. Normalized tunnel convergence against elapsed time for various Er.
218
PAN and HUANG:
•,•
INTERACTION BETWEEN ROCK AND SHOTCRETE
1.20
o
1.00-
...... ~ . ~ - - - ev,o.oozs
I:~ 0.80, O)
U) O)
0.60
I~ 0.40
0.20-
O
z
0.80
o.ooo
o.ool
o.oo2
o.oos
o.oo4
o.0o5
Normalized convergence ( U r / a ) Fig. 8. Tunnel-support interaction curves for various Er.
formulated to consider the time-dependent stiffness and the yielding strain of shotcrete support. An empirical equation is adopted in the model to describe the Young's modulus of shotcrete. The yield strain of shotcrete is also included to take the ultimate capacity of support into account. Following the formulation of the model, the effect of the time-dependence of support chracteristics on tunnel convergence, support pressure, and tunnel-support interaction are appraised. A parametric study reveals that the time dependent modulus of shotcrete has a strong influence on tunnel convergence and support pressure. Moreover, ignoring the yield strain of shotcrete may underestimate tunnel convergence and overestimate the ultimate capacity of shotcrete support. Acknowledgement--Tiffs work was financially supported by the National Science Council of the Republic of China under contract No. NSC 81-0410-E009-517. This support is gratefully acknowledged. Accepted for publication 8 October 1993.
9. Crter N. L. et al. Creep and creep rupturc of granitic r•ck. Mechanical Behavior q/Crustal Rocks, Geophysical Monograph 24, pp. 61-82. Amer. Geophys. Union (19811. 10. Hrseman S. and Passaris E. Creep tests for storage cavity closure prediction. The Mechanical Behavior of Salt, Proc. First Con]~ (Hardy and Langer, Eds), pp. 119 157. Trans Tech. Publ., Clausthal-Zellerfeld (1984). 11. Lombardi G. Some comments on the convergence continerment method. Underground Space 4, 249 258 (1980). 12. Gnirk P. F. and Johnson R. E. The deformational behavior of a circular mine shaft situated in a viscoelastic medium under hydrostatic stress. Proc. 6th Syrup. On Rock Mech., pp. 231-259 (1964). 13. Jumikis A. R. Rock Mechanics. Trans Tech PUN., ClausthalZellerfeld (1979). 14. Massier D. and Cristescu N. In situ creep of rock. Rev. Roumaine Sci. Tech. Set. Mec. Appl. 26, 687-702 (1981). 15. Wang S. et al. The back-analysis method from displacement for viscoelastic rock mass. 2nd Int. Symp. on Field Measurement in Geomechanics (Sakurai, Ed.), pp. 1059-1068. (1988). 16. Daemen J. J. K. Problems in tunnel support mechanics. Underground Space, I, pp. 163-172 (1977). 17. Hoek E. and Brown E. T. Underground Excavation in Rock, Chap. 8. Institution of Mining Metallurgy, London, England (1980). 18. Brady B. H. G. and Brown E. T. Rock Mechanics for Underground Mining, Chap. 11. George Allen & Unwin Ltd, London, U K . (1985). 19. Pan Y. W. and Dong J. J. Time-dependent tunnel convergence I. Formulation of the model. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 28, 469-475 (199la). 20. Pan Y. W. and Dong J. J. Time-dependent tunnel convergence II. Advance rate and tunnel-support interaction. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 28, 477-488 (1991b). 21. Weber J. W. Empirische Formeln zur Beschreibung der Festigkeitsentwicklung und der Entwicklung des E-Moduls von Beton Betonwerk und Fertigteiltechnik (1979). 22. Bazant Z. P. Limitations of strain-hardening model for concrete creep. Cement Concrete Res. 17, 505-509 (1987). 23. Sulem J., Panet M. and Guenot A. Closure analysis in deep tunnels. Int. J. Rock. Mech. Min. Sei. Geomech. Abstr. 24, 145 154 (1987). 24. Mahar J. W. Shotcrete Practice in Underground Construction: Final Report. Dept of Civil Engrg, Univ. of Illinois at Urbana-Champaign, Springfield, VA (1975).
APPENDIX Deformation of a Circular Ring Subjected to Pressure To model the time-dependent stiffness of shotcrete support on a circular tunnel, consider a hollow cylindrical ring subjected to an isotropic pressure Ps on the outer boundary of the ring. The following equations can be found from the elastic solutions:
C ar= A + ~
REFERENCES I. Kerisel J. Commentary on the general report. Underground Space 4, 233-239 (1980). 2. Duddeek H. On the basic requirements for applying the convergence-confinement method. Underground Space 4, 241-247 (1980). 3. Lamma R. D. and Vutukuri V. S. Handbook on Mechanic Properties of Rocks, Vol. 3. Trans Tech Publications, Rockport, MA (1978). 4. Hunsche U. Measurement of creep in rock salt at small strain rate. Proc. 2nd Conf. on the Mechanical Behavior of Salt, Hannover, September (1984). 5. Hardy H. R. et al. Creep and microseismic activity in geologic materials. Proc. llth U.S. Rock Mech. Syrup., Berkeley, pp. 377-413 (1969). 6. Reynold T. D. and Gloyna E. F. Creep measurement in salt mines. Proc. 4th Syrup. Rock mech., University Park, PA, pp. 11-17 (1961). 7. Langer M. Rheological behavior of rock masses. Proc. Fourth Int. Congr. on Rock Mechanics, Montreux, Balkema, Rotterdam, pp. 29-62 (1979). 8. Langer M. The rheological of rock salt. The Mechanical Behavior of Salt, Proc. First Conf. Trans Tech Publ., Clausthal-Zellerfeld (1984).
(AI)
r =
and C ao = A -- -5.
(A2)
r"
Apply the following boundary conditions to equations (A l) and (A2): r = a =~ P~
C
=
A -l-a2, --
and c
r =b =~O= A + -Q5.
One obtains a2
_a2b 2
As a result, (A3)
PAN and HUANG:
INTERACTION BETWEEN ROCK AND SHOTCRETE
and
219
we have
ao=(~)p~+(
a2b2---~P--2 ~a 2-- b2] r 2"
Under a plane-strain condition, *z = V(ar + ao) since 1 ~z = ~ [O'z -- V(O'r "q" O'0)] = 0.
Now that
(A4)
l+v q = ~ - [(1 -- v)er -- vao].
(A5)
Substitute equations (A3) and (A4) into (A5) to obtain equation (A6) l + v . a2P' [ :ib2] (A6) Er= E a2_b 2 ( I - 2 v ) - _ . Integrate the above equation, we can obtain the following
1
l+va2PsF Ur=~-.a~--b~L(l-2v)r + ~f] .
(A7)
Pergamon
Copyright © 1994 Elsevier Science Ltd Printed in Great Britain. All rights reserved 0148-9062/94 $7.00 + 0.00
0148-9062(93)E0009-D
A Model of the Time-dependent Interaction Between Rock and Shotcrete Support in a Tunnel YII-WEN PAN1" ZENG-LIN H U A N G t A viscoelastic model is developed to simulate the tunnel-support interaction of a circular tunnel in a viscoelsatic rock mass. This model accounts for the time-dependent stiffness and the yielding strain of shotcrete support. This model is formulated to analyze the effects of the time-dependence of support characteristics on tunnel convergence, support pressure, and tunnel-support interaction. Results of a parametric study reveal that the time-dependence of shotcrete stiffness has a strong influence on tunnel convergence and support pressure. Ignoring the yield strain of shotcrete may lead to the underestimation of tunnel convergence and the overestimation of the ultimate capacity of shotcrete support.
INTRODUCTION
The convergence of a tunnel often indirectly indicates the stability of the tunnel. The tunnel convergence in a competent rock, such as granite or marble, can quickly reach a stable state after excavation, even without man-made support. In some rock conditions, however, the tunnel convergence may be time-dependent due to a variety of reasons. The time-dependent tunnel convergence can result from the rheological properties of rock mass, the process of tunnel excavation, the timing of tunnel-support installation, and the time-dependent properties of the installed tunnel-support [1, 2]. In addition, the size of tunnel, gravity, and the method of tunnel excavation, also have some influence on the convergence of tunnel. Empirical approaches are adopted from time to time to estimate tunnel convergence and support pressure, and to design various types of tunnel-support. The in situ measurement of tunnel convergence often provides the means to monitor stability; and may be used as feedback for design modification of tunnel support. Analytical models correlating tunnel convergence and support pressure will be useful for interpreting the in situ measurement and for predicting tunnel convergence and support pressure. Although some elastic solutions, such as the Kirsh solution, are often used to analyze deformation and stress distribution in the rock mass around an opening, they cannot take the time-dependence of tlnstitute of Civil Engineering, National Chiao-Tung University, Hsinehu, Taiwan, R.O.C. 213
tunnel deformation and support pressure into account. Therefore, elastic solutions fail to model the tunnel mechanics for a tunnel excavated in a time-dependent rock mass or supported by a time-dependent material, such as shotcrete. Rock masses are essentially time-dependent materials with rheological properties. The creep behavior of rock masses has been studied by many researchers [3-6], either by laboratory tests or by in situ tests. Some [7-10] also make use of rheological models to simulate the time dependent deformation of rock material. Factors affecting the creep of rock materials may include stress type, stress level, confining pressure, and the loading environment of the loaded material. The creep behavior of rock material depends not only on the rheological properties but also on the subjected stress condition. Hence, the tunnel convergence is a function of the time-dependent properties of rock masses and the level of support pressure. Since the support pressure also depends on the excavation process and the timing of support installation, the tunnel convergence is largely affected by the same factors. Many researchers utilized rbeological models to describe the time-dependent tunnel convergence of rock tunnel before support installation [11-15]. In general, the tunnel convergence at time t can be described by a certain constant (either the initial or the ultimate tunnel convergence) and a creep function of the rock mass via the following expressions: Ur(t) = Ur(t--'O)Fc(t)
(1)
U,(t) = U,(t --*oo)Fc(t)
(2)
214
PAN and HUANG: INTERACTIONBETWEEN ROCK AND SHOTCRETE
in which Ur(t): tunnel convergence at a time t after tunnel excavation U~(t-+O): the initial tunnel convergence Ur(tOO0): the ultimate tunnel convergence Fc(t): the creep function of rock mass. Lombardi [11] indicated the importance of the inclusion of the three-dimensional effect due to tunnel excavation in developing reasonable analytical model of tunnel convergence using a plane-strain or plane-stress approach. The three-dimensional effect mainly arises from the natural self-support of rock near and in front of the excavation face against the already excavated part of the tunnel. This rock self-support tends to attenuate quickly with a distance x from the excavation face. As the distance x approaches infinity, an equivalent planestrain condition is satisfied. Thus distribution function F(x) can be used to describe the attenuation of rock stress. Ur(X ) = F(x)Ur(x ---~o0).
(3)
In the above expression, Ur(x--,m) represents the tunnel convergence at x = oo in a plane-strain condition which can be regarded as a reference value; F(x) is a normalized distribution function of x. F(x) has the properties that F(x) = 1 as x ~ o o and F(x) = 0 as x--*0. Ur(x--.oo) corresponds to the case when rock pressure disappears. If P0 is the initial stress in an isotropic condition, then the stress [1 - F ( x ) ] P o can be regarded as the natural support pressure against the already excavated tunnel provided by the rock in front of the excavation face. The tunnel-support interaction plays a very important role in determining the tunnel convergence and the support pressure (lining pressure) during the process of the tunnel excavation. This interaction is complicated. In general, it depends on the properties of both the rock mass and the installed supports, the process of tunnel excavation, and the timing of support installation [2, 16-18]. Proper design and installation of the tunnel-support are essential for limiting tunnel convergence and ensuring tunnel stability. Pan and Dong [19] developed a rheological model to simulate the time-dependent tunnel-support interaction problem during the excavation and construction process of a circular tunnel. The model takes into account various time-factors, including the creep effect, tunnel advancing effect, and the support effect. It distinguishes the tunnel advancing effect and the support effect from the creep behavior of the rock mass. Pan and Dong [20] also appraised important time-dependent factors of tunnel convergence. In their model, the tunnel-support is assumed to be made of an elastic material. It is, however, important to note that the support may be made of time-dependent material (such as shotcrete and some types of rock bolts) as well [2]. Shotcrete can possess time-dependent deformabilities (e.g. the shear modulus G and the Young's modulus E) [21] and can develop creep under a constant loading [22]. Besides,
shotcrete may yiel6 when its strain exceeds the yield strain. Any of these properties may result in the time-dependence of support stiffness, and hence affect the long-term tunnel convergence and support pressure. In this paper, the authors extent a previous viscoelastic model [19] to account for the time-dependent stiffness of shotcrete support. An empirical equation of the time-dependent Young's modulus of shotcrete [21] is adopted in the model. In addition, the yielding strain of shotcrete is taken into account. Following the formulation of the model, the effects of the time-dependence of support characteristics on tunnel convergence, support pressure, and tunnel-support interaction are examined. MODEL FORMULATION
Assumptions The following assumptions are made in developing the proposed model. 1. The rock mass is a homogeneous material subjected to an isotropic initial stress. 2. The tunnel is circular. 3. The effects of normal stress in the direction of tunnel axis and the shear stress near the tunnel-face are neglected. 4. The advancing effect and the support effect, together, can be treated as a change in the internal pressure, and can be approximated by the integration of infinite step functions. 5. As the distance from the excavation-face approaches infinity, an equivalent plane-strain condition is satisfied. The rock pressure is a function of the distance from the considered cross-section to the excavation-face. 6. The support is a closed ring with a constant thickness. The support contacts the rock mass as soon as the support is installed. 7. The change in the internal pressure is independent of the creep function. The rock pressure is independent of the support pressure.
Viscoelastic model for tunnel-support interaction In the Pan-Dong model [19], the tunnel convergence of a circular tunnel can be evaluated from the history of internal support pressure Pi(t). The internal support pressure just before tunnel excavation is the initial in situ pressure P0 (i.e. (Pi(O)=P0)). The internal support pressure is contributed from the support pressure P~(t) provided by the installed support and the rock pressure Pc(x(t)) provided by the unexcavated rock in front of the excavation face. As a result, Pi(t) = Ps(t) + Pc(x(t)).
(4)
P(t) denotes the change in the internal support-pressure; as a result, P(t) = Po - P~(t). From assumption (5), the rock pressure Pc(x) is a function of the distance x(t) from the considered location to the excavation face. Suppose Pc(x) can be described by a distribution func-
PAN and H U A N G :
INTERACTION BETWEEN ROCK A N D SHOTCRETE
tion F(x) such that Pc(x(t))= [1 - F ( x ( t ) ) ] P o , we can express
P(t) = F[x(t)]Po - P,(t).
F(x)=l--
~
.
(6)
Based on a viscoelastic formulation [19], the time-dependent tunnel convergence can be evaluated by the following convolution relation: U r ( t ) - gve {[--/5c - a
n ( t - q)/5~] * F¢(Q}.
(7)
in which the viscoelastic compliance function of rock; F~: the creep function of rock; H(t - h): the unit step function which satisfies H(t - h) = 0 for 0 ~< t < h and H(t-h)=l for t t > h ; ts: the installation time of support; gve :
/5c = dP¢(r_.) _ dr
outer boundary of the ring can be found as follows (Appendix):
(5)
The following empirical equation of the function F(x) can be extracted from the work of Sulem et al. [23]:
pod{F[x(Q]}" dr '
U~(r) = ~l + v ~ a2 b -aP~ -_- ~ 2 [ (1 - 2v¢)r + b_f1
(9)
in which E¢: re: a: b: r:
the the the the the
Young's modulus of the ring material Poisson's ratio of the ring material outer radius of the ring inner radius of the ring radius at which the deformation is concerned.
Using the above equation, the relation between the support pressure and the tunnel convergence can be expressed as follows: Or(t)--
Ur(/s)
a
(1
+re)
a
E~(t) a 2 - b 2 b2 xI(1-2vc)a+alPs(t
). (10)
For 0 ~< t < ts, the support pressure does not exist since the support is not yet installed. For t > q, we can find the following relation:
gve, F ftpod{Ftx(Q]} F¢(t - z)dz kJo dz
p _ dPs(z) dr
Modeling time-dependent support-stiffness
-
A constitutive relation of the installed support (support-stiffness) is required to evaluate the change in the support pressure P~(t) and the tunnel convergence Ur(t). The support-stiffness depends on the type, size, and material of the installed support. In the past, most studies assumed time-independent support-stiffness. For a shotcrete support, however, the stiffness may change with time due to the time-dependent nature of shotcrete material [21, 24]. Like concrete material, the strength and Young's modulus of shotcrete usually increase with time [21, 24]. In general, the modulus of shot crete gradually increases until it converges to a stable value. Weber [21] proposed the following empirical formula for the time-dependent Young's modulus of shotcrete based on experimental data: E¢(/) = E2s" exp[-- c(t -°'6 -- 28-0"6)]
215
(8)
in which E2s: 28-day Young's modulus of shotcrete c: a material parameter of shotcrete t: the elapsed time. The Poisson's ratio v¢ of shotcrete is assumed time-independent (constant). To model the time-dependent stiffness of shotcrete support on a circular tunnel, consider a hollow cylindrical ring. The elastic solution of the deformation of a cylindrical ring Ur(r) subjected to a pressure P, on the
f;
P ~ ( r ) F , ( t - ~) d r
s
]
Ur(t3 a
]
(!+___~) a F¢I -- 2v¢)a + b2 E¢(t) a 2 - b 2 L ' a ] Ps(t)"
(11)
Since a numerical integral can be utilized by means of the Gauss integration, the above integral equation can be written in a summation form. Subsequently, it is possible to solve for the tunnel convergence Ur (t) and the support pressure P,(t) at any time t using a time-marching algorithm.
"~0.007
~O.OOe
No Ittl~p~rt
...... E==5*10~lm'al - - - Era= t~{liPa)
~ 0.005 ~ O.OIN, d 0.005 ~ 0.002 ~ 0.00!
o Z o.ooo
0.0
5.b
to'.o
ts'.o
zo'.o
Time (days)
2d.o
30.0
Fig. 1. Normalized tunnel convergence against elapsed time for various E2s.
216
PAN and HUANG:
INTERACTION BETWEEN ROCK AND SHOTCRETE
o
When ~ exceeds ~f, the yielding of the shotcrete support occurs. Then
!.2
E u = 5 * 104.~MPa]
°•ol.0 ¸
...... z== lff~¥P!)
ca.
dr
E a = 2 * 10'(MPa)
~f~ v 0.8 • q) t* m
...... . FI
We thus can find the radius at the occurence of failure rf according to
0.6
0.4
J1~////~
dr
.-'""
~
rf=
-
-
(13)
.
(f
.obo
o.obz
o.obl
Normalized
0 . 0 '0 3
0.004
convergence
o.oo~
(U~/a)
Fig. 2. Tunnel-support interaction curves for various E28.
Modeling progressive failure of support Mortar mixed material, such as shotcrete, can yield under large compressive strain. A yield strain at Er= 0.003 is usually considered for concrete design. Under large compressive strain, shotcrete is likely to yield and result in progressive failure of the shotcrete support. To model the progressive failure of shotcrete, assume that (a) the radial deformation on the ring support is uniform, and (b) the shotcrete yields when the circumferential strain reaches Er. Suppose a circle with a initial radius r deforms to one with the final radius r'. The initial perimeter of the circle is l~ = 2hr. The final perimeter is
It = 2nr' = 21t(r - dr) = 2rrr - 2n dr in which dr = U~(t)-U~(t~) is the change in the radius. Then, the circumferential strain can be evaluated by E.
li
- -
. li
if
.
2rr d r
. 2nr
dr
r
.
(12)
The largest compressive strain on a ring support always occurs at the inner boundary of the ring. As a result, yielding will alway begin at the inner boundary. A perfect plastic condition after yielding is assumed; consequently, no increase in stress at the yielding point is allowed with any further increase in strain. As a consequence, the effective thickness of the ring support can be regarded as reduced. The yielding zone (in the manner of a yielding ring) can propagate outward along with further increase in the support pressure. The expansion of the yielding ring hence results in the progressive failure of the shotcrete support. When the yielding ring becomes as large as the whole ring support, the support cannot take any more pressure and finally reaches an ultimate state. A PARAMETRIC STUDY A parametric study based on the proposed model is performed to investigate the effects of time-dependent modulus Ec(t) and yielding strain Er of shotcrete on tunnel convergence and support pressure. A hypothetical circular tunnel with r = 5.0 and a shotcrete support with a = 5.0m, and b = 4.8 m are considered in the study. The mechanical behavior of rock is assumed to follow the Kelvin model, with P0 = Pv = Ph = 27 MPa, E r = 5 x 10 3 M P a , qr = 5 × 104 ( M P a . see), and vr = 0.25. The common parameters and variables used are as follows: E2s= 1 x 105MPa, c =0.596, v~= 0.17,
"~0.007
t~ 1.2 ~.~0.008 .-.--. -. ._.1 cN°:su-~p°rt: c_ = -.596
- - c = -2 . . . . . . c = -1 --c = -.596 - - e = 0
~1.0
-- - - c = 0
~ 0.005
~
0.5
~ 0.004
m 0.6
0.003
Q) ;@7/
~ 0.002
0.4 Q)
~ 0.001 Iq
'~
o Z o.ooo 0.0
V 5.0
I 10.0
U 15.0
Time
I 20.0
U 25.0
30.0
(days)
0.2
0.0 0.000
, 0.001
Normalized
Fig. 3. Normalized tunnel convergence against elapsed time for various c.
~ 0.002
convergence
, 0.003
(Ur/a)
Fig. 4. Tunnel-support interaction curves for various c.
0.004
PAN and HUANG:
INTERACTION BETWEEN ROCK AND SHOTCRETE
"~0.007
½ ~.0.000 ~ 0.005 t~ ~ 0.004
p,
0.003
~ 0.002 (
~ 0.001
d
a
y
/
--"- --- ts UI = 20 (d~2~)
0
Z o.ooo 0.0
~.b
~o'.o
t~'.o ~o'.o Time (days)
~'.o
3o.o
Fig. 5. Normalized tunnel convergence against elapsed time for various timing of support-installation.
t,=0 day, and vation = 2 m/day.
the
advancing
rate
of
exca-
Effects of E~(t) and the timing of support In this subsection, the effects of time-dependent E¢(t) and timing of shotcreting on tunnel-support interaction are examined; no yielding strain Ef is considered. Figure 1 shows the plot of the normalized tunnel convergence Ur/a against the elapsed time for various E2s. Shotcrete support with a higher 28-day Young's modulus can provide higher support stiffness and allow less tunnel convergence. Figure 2 shows the tunnelsupport interaction curves which display the relations of support pressure and tunnel convergence for different E2s. It can be observed that shotcrete support with a higher E28 causes a higher pressure and a lower deformation. The effect of the material parameter c on the tunnel-support interaction is shown in Figs 3 and 4. A greater value of c means that E~(t) takes a longer time
%
217
to reach E~s; in other words, the modulus of shotcrete in the earlier stage is smaller. The minimum value of c is zero; when c is zero, E¢(t) is always a constant (equal to E2s). From a regression analysis based on collected data [24], the range of the parameter c is found within 0.4 and 0.9. Figures 3 and 4 reveal that a higher value of c can result in a smaller support pressure and a larger tunnel convergence. Figures 5 and 6 compare the normalized convergence with the tunnel-support interaction for different timing of support installation, t~. Results in the figures show that earlier support installation assures a lower tunnel convergence accompanied with a higher support pressure.
Effects of EI The yielding strain Er can have significant influence on the tunnel-support interaction. Figure 7 shows the timedependent tunnel convergence calculated based on various Er. Figure 8 presents the tunnel-support interaction curves corresponding to various values of Er. As mentioned in the above context, yielding always begins at the inner boundary, then progressively expands outward. A perfect plastic condition is assumed in the yielded zone. When the strain at the outer boundary finally reaches the yielding strain, the support pressure cannot increase any more with further tunnel convergence and reaches the ultimate capacity. Ultimate support capacity appears to be a function of Er. From these figures, it can be seen that a smaller Er corresponds to a larger tunnel convergence and a smaller ultimate capacity of support pressure. Ignoring the importance of yielding strain may therefore underestimate tunnel convergence and overestimate support capacity. SUMMARY AND CONCLUSIONS A viscoelastic model has been developed to simulate the tunnel-support interaction of a circular tunnel with shotcrete support in a viscoelastic rock mass. It was
~'0.007
1.2
. . . . . .
It,, --
: =
days I,,...I
=
o.oo0
o
~ 0.005 Q)
"--" 0.8
~ 0.004 0.8
O 0.003 0.002 ~ 0.2
~ 0.001 -
-
~.o.o~s
O
~ 0.0
o.ooo
o.0bt o.doz o.ob3 Normalized convergence
o.oo4' (U~/a)
o.oo5
Fig. 6. Tunnel-support interaction curves for various timing of support-installation.
Z o.ooo
o.o
5.'o
io'.o
15'.o
zo'.o
25'.o
3o.o
Time(days)
Fig. 7. Normalized tunnel convergence against elapsed time for various Er.
218
PAN and HUANG:
•,•
INTERACTION BETWEEN ROCK AND SHOTCRETE
1.20
o
1.00-
...... ~ . ~ - - - ev,o.oozs
I:~ 0.80, O)
U) O)
0.60
I~ 0.40
0.20-
O
z
0.80
o.ooo
o.ool
o.oo2
o.oos
o.oo4
o.0o5
Normalized convergence ( U r / a ) Fig. 8. Tunnel-support interaction curves for various Er.
formulated to consider the time-dependent stiffness and the yielding strain of shotcrete support. An empirical equation is adopted in the model to describe the Young's modulus of shotcrete. The yield strain of shotcrete is also included to take the ultimate capacity of support into account. Following the formulation of the model, the effect of the time-dependence of support chracteristics on tunnel convergence, support pressure, and tunnel-support interaction are appraised. A parametric study reveals that the time dependent modulus of shotcrete has a strong influence on tunnel convergence and support pressure. Moreover, ignoring the yield strain of shotcrete may underestimate tunnel convergence and overestimate the ultimate capacity of shotcrete support. Acknowledgement--Tiffs work was financially supported by the National Science Council of the Republic of China under contract No. NSC 81-0410-E009-517. This support is gratefully acknowledged. Accepted for publication 8 October 1993.
9. Crter N. L. et al. Creep and creep rupturc of granitic r•ck. Mechanical Behavior q/Crustal Rocks, Geophysical Monograph 24, pp. 61-82. Amer. Geophys. Union (19811. 10. Hrseman S. and Passaris E. Creep tests for storage cavity closure prediction. The Mechanical Behavior of Salt, Proc. First Con]~ (Hardy and Langer, Eds), pp. 119 157. Trans Tech. Publ., Clausthal-Zellerfeld (1984). 11. Lombardi G. Some comments on the convergence continerment method. Underground Space 4, 249 258 (1980). 12. Gnirk P. F. and Johnson R. E. The deformational behavior of a circular mine shaft situated in a viscoelastic medium under hydrostatic stress. Proc. 6th Syrup. On Rock Mech., pp. 231-259 (1964). 13. Jumikis A. R. Rock Mechanics. Trans Tech PUN., ClausthalZellerfeld (1979). 14. Massier D. and Cristescu N. In situ creep of rock. Rev. Roumaine Sci. Tech. Set. Mec. Appl. 26, 687-702 (1981). 15. Wang S. et al. The back-analysis method from displacement for viscoelastic rock mass. 2nd Int. Symp. on Field Measurement in Geomechanics (Sakurai, Ed.), pp. 1059-1068. (1988). 16. Daemen J. J. K. Problems in tunnel support mechanics. Underground Space, I, pp. 163-172 (1977). 17. Hoek E. and Brown E. T. Underground Excavation in Rock, Chap. 8. Institution of Mining Metallurgy, London, England (1980). 18. Brady B. H. G. and Brown E. T. Rock Mechanics for Underground Mining, Chap. 11. George Allen & Unwin Ltd, London, U K . (1985). 19. Pan Y. W. and Dong J. J. Time-dependent tunnel convergence I. Formulation of the model. Int. J. Rock Mech. Mining Sci. Geomech. Abstr. 28, 469-475 (199la). 20. Pan Y. W. and Dong J. J. Time-dependent tunnel convergence II. Advance rate and tunnel-support interaction. Int. J. Rock Mech. Min. Sci. Geomech. Abstr. 28, 477-488 (1991b). 21. Weber J. W. Empirische Formeln zur Beschreibung der Festigkeitsentwicklung und der Entwicklung des E-Moduls von Beton Betonwerk und Fertigteiltechnik (1979). 22. Bazant Z. P. Limitations of strain-hardening model for concrete creep. Cement Concrete Res. 17, 505-509 (1987). 23. Sulem J., Panet M. and Guenot A. Closure analysis in deep tunnels. Int. J. Rock. Mech. Min. Sei. Geomech. Abstr. 24, 145 154 (1987). 24. Mahar J. W. Shotcrete Practice in Underground Construction: Final Report. Dept of Civil Engrg, Univ. of Illinois at Urbana-Champaign, Springfield, VA (1975).
APPENDIX Deformation of a Circular Ring Subjected to Pressure To model the time-dependent stiffness of shotcrete support on a circular tunnel, consider a hollow cylindrical ring subjected to an isotropic pressure Ps on the outer boundary of the ring. The following equations can be found from the elastic solutions:
C ar= A + ~
REFERENCES I. Kerisel J. Commentary on the general report. Underground Space 4, 233-239 (1980). 2. Duddeek H. On the basic requirements for applying the convergence-confinement method. Underground Space 4, 241-247 (1980). 3. Lamma R. D. and Vutukuri V. S. Handbook on Mechanic Properties of Rocks, Vol. 3. Trans Tech Publications, Rockport, MA (1978). 4. Hunsche U. Measurement of creep in rock salt at small strain rate. Proc. 2nd Conf. on the Mechanical Behavior of Salt, Hannover, September (1984). 5. Hardy H. R. et al. Creep and microseismic activity in geologic materials. Proc. llth U.S. Rock Mech. Syrup., Berkeley, pp. 377-413 (1969). 6. Reynold T. D. and Gloyna E. F. Creep measurement in salt mines. Proc. 4th Syrup. Rock mech., University Park, PA, pp. 11-17 (1961). 7. Langer M. Rheological behavior of rock masses. Proc. Fourth Int. Congr. on Rock Mechanics, Montreux, Balkema, Rotterdam, pp. 29-62 (1979). 8. Langer M. The rheological of rock salt. The Mechanical Behavior of Salt, Proc. First Conf. Trans Tech Publ., Clausthal-Zellerfeld (1984).
(AI)
r =
and C ao = A -- -5.
(A2)
r"
Apply the following boundary conditions to equations (A l) and (A2): r = a =~ P~
C
=
A -l-a2, --
and c
r =b =~O= A + -Q5.
One obtains a2
_a2b 2
As a result, (A3)
PAN and HUANG:
INTERACTION BETWEEN ROCK AND SHOTCRETE
and
219
we have
ao=(~)p~+(
a2b2---~P--2 ~a 2-- b2] r 2"
Under a plane-strain condition, *z = V(ar + ao) since 1 ~z = ~ [O'z -- V(O'r "q" O'0)] = 0.
Now that
(A4)
l+v q = ~ - [(1 -- v)er -- vao].
(A5)
Substitute equations (A3) and (A4) into (A5) to obtain equation (A6) l + v . a2P' [ :ib2] (A6) Er= E a2_b 2 ( I - 2 v ) - _ . Integrate the above equation, we can obtain the following
1
l+va2PsF Ur=~-.a~--b~L(l-2v)r + ~f] .
(A7)