Analytical viscoelastic solution for frost force in cold-region tunnels

Analytical viscoelastic solution for frost force in cold-region tunnels

Cold Regions Science and Technology 31 Ž2000. 227–234 www.elsevier.comrlocatercoldregions Analytical viscoelastic solution for frost force in cold-re...

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Cold Regions Science and Technology 31 Ž2000. 227–234 www.elsevier.comrlocatercoldregions

Analytical viscoelastic solution for frost force in cold-region tunnels Yuanming Lai a,b,) , Wu Hui c , Wu Ziwang a , Liu Songyu b, Den Xuejun b a

State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions EnÕironmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China b Transportation College, Southeast UniÕersity, Nanjing 210096, People’s Republic of China c Department of Mechanics, Lanzhou UniÕersity, Lanzhou 730000, People’s Republic of China Accepted 11 September 2000

Abstract In this paper, the formulae of Laplace transform with respect to time for frost forces of the lining-frozen surrounding rock-unfrozen surrounding rock system are derived from the elastic–viscoelastic corresponding principle. And then, the frost forces and lining stresses in cold-region tunnels are obtained by the numerical inversion of the Laplace transform. Finally, an example is given. This example indicates that the lining stress including the frost factor is much larger than that not including the frost factor. q 2000 Elsevier Science B.V. All rights reserved. Keywords: Cold-region tunnels; Frost force; Viscoelastic solution

1. Introduction When a tunnel in the cold region has been excavated, the original stable thermodynamic condition in the tunnel is destroyed and replaced by a new thermodynamic system with natural convection and away from the solar thermal radiation and, thereby, conditions for the formation of seasonal or permanently frozen soil are created. When the fissure and pore water become frozen, an expansion in volume will occur. This volumetric expansion is restrained by the tunnel lining and the unfrozen surrounding rock; ) Corresponding author. State Key Laboratory of Frozen Soil Engineering, Cold and Arid Regions Environmental and Engineering Research Institute, Chinese Academy of Sciences, Lanzhou 730000, People’s Republic of China. E-mail address: [email protected] ŽY. Lai..

therefore, the frozen rock will result in forces on the lining, known as the frost force of the surrounding rock. The frost force and other forces acted on the tunnel lining may result in cracking and flaking in the tunnel and lead to some kind of frost damages, such as water-leaking, freezing and hanging-ice, threatening the normal transportation in tunnels. In the 1970s and 1980s, the surveys conducted in Japan’s northeastern cold-region tunnels indicated that frost force is the source of external forces causing the damage of cold-region tunnels. Moreover, the freezing–thawing process will accelerate the weathering of the surrounding rock, and the increase of the degree of rock-crushing provides more favorable conditions for the development of frost force, thus, increases the frost force. This vicious circle represents a serious threat to the lining stability. There are 33 railway tunnels in the north-

0165-232Xr00r$ - see front matter q 2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 2 3 2 X Ž 0 0 . 0 0 0 1 7 - 3

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Y. Lai et al.r Cold Regions Science and Technology 31 (2000) 227–234

eastern and northwestern cold regions of China. All of them have shown different degrees of frost damage. Due to the influence of frost damage, some tunnels cannot be open to traffic for 8–9 monthryear. Under the action of frost force, the crack width of some lining even reached 5 cm, and the normal traffic operations were seriously affected. Fritz Ž1984. presented an analytical solution for the time-dependent stresses and displacements in plane strain around a circular hole when it is loaded by an axisymmetric internal and far-field pressure. Klein Ž1981. proposed finite element method for time-dependent problems of frozen soils and investigated the mechanical behavior of temporary frozen earth support systems for tunnels using this kind finite element. Ladanyi Ž1980. has worked out the methods for direct determination of ground pressure on tunnel lining in a nonlinear viscoelastic rock, and Ladanyi Ž1984. presented a closed form solution of the time-dependent closure displacements, both free and restrained by the lining, as well as the lining pressure build-up with time for any moment of its installation, for a tunnel of circular cross-section. Lai et al. Ž1999. have made nonlinear analysis for the coupled problem of temperature and seepage fields in cold-region tunnels. Up to now, no research work on the computational method for frost force in cold-region tunnels has been reported in the literature. To solve these practical engineering problems and to provide some theoretical insights to design computations, the frost force in cold-region tunnels, based on the theory of viscoelasticity, is studied in this paper.

The forces acting on domain I are shown in Fig. 2. From the theory of elasticity written by Xu Ž1982., the stresses and radial displacements in domain I under axisymmetric plane strain state are given by the following expressions:

I

sr s y I

su s y

u Ir s

1 y a2rr 2 2

1 y a rb

2

1 q a2rr 2 1 y a2rb 2

¶ Pb

•,

Ž 1.

ß

Pb

Ž 1 q m 1 . 2 m 1 y Ž 1 q a2rr 2 . Pb E1 Ž 1 y a 2rb 2 .

r,

Ž 2.

where E1 and m 1 are the elastic modulus and Poisson’s ratio of the lining, respectively; and Pb is the uniformly distributed forces acting on the lining. The forces acting on domain II are shown in Fig. 3. Because of freezing, an expansion in volume will occur. Assume that the water content and the unfrozen water content of the surrounding rock are W and Wu , respectively, and the volumetric expansion

2. Mechanical model and solution in the Laplacian image space A freezing tunnel in cold region can be divided into three domains, as shown in Fig. 1. Domain I, domain II and domain III represent the lining, the frozen surrounding rock and the unfrozen surrounding rock, respectively. Domain I and domain III are elastic domains, while domain II is a viscoelastic domain. According to the corresponding principle of elasticity and viscoelasticity, the first step to obtain the viscoelastic solution for the system shown in Fig. 1 is to establish its corresponding elastic solution.

Fig. 1. The mechanical model of frost heaving of rocks surrounding tunnel.

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The forces acting on domain III are shown in Fig. 4. The Lami’s solution to this case is given by:

sr III s III

su s Fig. 2. The diagram of tunnel lining subjected to forces.

per unit volume is a when water is frozen to ice. And then, the linear strain induced by frost will be: 1 ´ 0 s a Ž W y Wu . . Ž 3. 3 Under the action of uniformly distributed forces Pb and Pc , and subjected to a linear frost strain ´ 0 , the elastic solutions of stresses and displacements for the axisymmetric plane strain problem in the frozen domain are as follows:

sr II s su II s

1 y c 2rr 2 c2 y b2 1 q c 2rr 2 2

c yb

2 2

u II r s

b 2 Pb q b 2 Pb y 2

b 2rr 2 y 1 c2 y b2 1 q b 2rr 2 2

c yb 2

•,

2

2

2

d rc y 1 d rr q 1 d 2rc 2 y 1

Pc y

1 y c 2rr 2 2

1 y c rd

2

1 q c 2rr 2 1 y c 2rd 2



sr III s y III

su s

¶ P0

•,

Ž 6.

ß

P0

c2 r

c2 r2

Pq 2 c

ž

c2 r2

ž

Pc y 1 q



/

y 1 P0 c2 r2

/

P0

•.

Ž 7.

ß

By using Eq. Ž7., the constitutive equations and the displacement–strain relations, the displacements in domain III can be obtained: u )r s

Ž 4.

2

Pc y

The outer radius d of the surrounding rock is by far larger than c; therefore, it can be assumed as infinite. Let d `, Eq. Ž6. becomes:

¶ c 2 Pc

d 2rr 2 y 1

r Ž 1 q m3 .

c2

E3

r

ž

P y 1q 2 c

c2 r2

/

y 2 m 3 P0 .

Ž 8.

ß

2

c 2 Pc

2

2

The displacements given by the Eq. Ž8. are the absolute displacements of the surrounding rock, in

2

r Ž 1 q m 2 . w Ž 1 q c rr y 2 m 2 . b Pb y Ž 1 q b rr y 2 m 2 . c Pc x E2 Ž c 2 y b 2 . q r´ 0 Ž1 q m 2 . ,

Ž 5.

where E2 and m 2 denote the elastic modulus and Poisson’s ratio of domain II, respectively. In the following discussions, substitutions to the parameters E2 and m 2 will be made by using the corresponding principle of elasticity and viscoelasticity.

Fig. 3. The diagram of frozen surrounding rocks acted by forces.

Fig. 4. The diagram of domain III subjected to forces.

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230

which the primary displacements prior to the excavation of the tunnel are included. For practical engineering problems, only the relative displacements of the excavated tunnel, also called the additional displacements, are meaningful, i.e. the primary displacements caused by the primary stresses prior to excavation should be subtracted from the absolute displacements. Denoting the primary displacements by u 0 and the relative displacements by u III r , then we have: ) u III r s ur y u0 .

Pc s

y

u0 s y

E3

P0 .

u III r s

Ž 1 q m3 . c 2 rE3

Ž 11 .

In the above expressions, the forces Pb and Pc are unknown and have to be determined by the conditions of displacement continuity on the common boundary of adjacent domains, i.e.: u Ir < rs b s u IIr < rsb < u IIr < rs c s u III r rsc

5

.

Ž 12 .

The solutions of Pb and Pc can be obtained by substituting Eqs. Ž2., Ž5. and Ž11. into Eq. Ž12.. The expressions for Pb and Pc are given by: Pb s

1

D

q

½

Ž 1 q m3 .

y ´ 0 Ž 1 q m2 . q

E3 =

Ž 1 q m2 . Ž c 2 q b 2 y 2 m2 c 2 . E2 Ž c 2 y b 2 .

´ 0 Ž 1 q m2 .

2 1 y m22 c 2 E2 c 2 y b 2

Ž

.

Ž

.

5

,

Ž 1 q m3 . E3

P0

Ž 1 y m22 . b 2´ 0 Ž 1 q m 2 . D E2 Ž c 2 y b 2 .

,

Ž 14 .

E1 E2 Ž b 2 y a 2 . Ž c 2 y b 2 .

q

Ž 1 q m 2 . 2 Ž b 2 q c 2 y 2 m 2 b 2 .Ž c 2 q b 2 y 2 m 2 c 2 . E22 Ž c 2 y b 2 .

Ž 1 q m 1 . w2 m 1 b 2 y Ž b 2 q a 2 . x Ž 1 q m 3 . 2

2

2

q

4 Ž 1 y m 22 . c 2 b 2

E1 E3 Ž b y a .

y

E22 Ž c 2 y b 2 .

Ž1 q m 2 . Ž1 q m 3 . Ž b 2 q c 2 y 2 m 2 b 2 . E22 Ž c 2 y b 2 .

2

2

Ž 15 .

.

Eqs. Ž13. and Ž14. are the elastic solutions for calculating the frost force in tunnels. Based on the experiments conducted for surrounding rock samples with a freezing temperature ranging from y108C to y158C, it can be seen that the constitutive relation of the rock samples is close to the Poyting–Thomson model shown in Fig. 5. The constitutive equation for this model is:

s˙ q

E1)

h

s s Ž E1) q E2) . ´˙ q

E1) E2)

h

´.

Ž 16 .

Denoting: As

E1)

h

,

Bs

E1) E2)

h

,

C s E1) q E2) ,

then Eq. Ž16. can be written in the following concise form:

s˙ q A s s C´˙ q B´ .

Ž 13 .

P0

E3

Ž 1 q m 1 . Ž 1 q m 2 . w2 m 1 b 2 y Ž b 2 q a 2 . x Ž c 2 q b 2 y 2 m 2 c 2 .

y

Ž Pc y P0 . .

Ž 1 q m3 .

5

where:

Ž 10 .

Substituting Eqs. Ž8. and Ž10. into Eq. Ž9., the relative displacements in domain III are obtained:

Ž 1 q m2 . Ž b q c 2 . y 2 m2 b 2 E2 Ž c 2 y b 2 .

q2

Ds

r Ž 1 q m3 . Ž 1 y 2 m3 .

½

= ´ 0 Ž 1 q m2 . q

Ž 9.

The primary displacements can be obtained by taking c s 0 in Eq. Ž8., hence:

Ž 1 q m 1 . 2 m 1 b 2 y Ž b 2 q a2 . D E1 Ž b 2 y a 2 . 1

Ž 17 .

For linear viscoelastic materials under complex stress state, the stress tensor can be decomposed to a deviatoric tensor, Si j , and a spherical tensor, si j , and

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231

Fig. 5. Poyting–Thomson’s model.

the general expression of the rheological constitutive equation in the differential form is: P1 Si j s Q1 e i j P2 s i i s Q 2 ´ i i

5

,

Ž 18 .

comparing Eq. Ž21. with Eq. Ž18., the moduli K and G of the linear viscoelastic material can be obtained: K Ž t. s

1 Q2 3 P2

,

Gs

1 Q1 2 P1

where P1 , Q1 , P2 and Q2 are polynomials of the operator ErEt. When the stress level is not high, the change in volume caused by the spherical tensor can be considered as elastic, while the deviatoric strains induced by the deviatoric tensor can be described, according to our foregoing explanation, by the Poyting–Thomson model. Under the conditions described in this paper, the embodied form of Eq. Ž18. will be:

From the relations between E, m , and K, G, the relaxation modulus and the Poisson’s ratio of relaxation can be obtained as the following:

S˙i j q ASi j s 2Ce˙i j q 2 Be i j

By taking Laplace transform of the expressions above, we have:

si i s 3 K ´ i i

5

,

Ž 19 .

where K is the volumetric modulus. From Eq. Ž19., the following relations can be derived: E P1 s

Et P2 s 1,

E q A,

Q1 s 2C Q2 s 3 K

Et

¶ • ß.

q2 B

si i s 3 K ´ i i

5

.

mŽ t . s

9KQ1 Q1 q 6 KP1 3 KP1 y Q1 Q1 q 6 KP1

,

.

9K Ž CS q B .

Es

Ž 22 .



Ž C q 3 K . S q B q 3 KA •. Ž 3 K y 2C . S q Ž 3 KA y 2 B . ms 2 Ž C q 3 K . S q 2 Ž B q 3 KA .

Ž 23 .

ß

Ž 20 .

For linear elastic materials, the constitutive equations are: Si j s 2Ge i j

EŽ t . s

Ž 21 .

According to the corresponding principle in the book, AViscoelasticityB, written by Flugge Ž1975. and

Let ´ 0 s ´ X0 H Ž t ., P0 s P0X H Ž t ., where H Ž t . is a step function, i.e.: HŽ t. s

½

1, 0,

t)0 t-0

Substituting the above expressions for ´ o and P0 into Eqs. Ž13. and Ž14. and then performing Laplace transform to the both sides of Eqs. Ž13. and Ž14., and noting that the elastic modulus E1 , E3 and Poisson’s

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ratio m 1 , m 3 in the lining and the unfrozen domain do not change vs. time, we have: Pb s

1

½

DS

Ž 1 q m3 .

q

Pc s

Ž

1 y m22

E2 Ž c 2 y b 2 .

y 2 ´ X0 Ž 1 q m 2 . q

E3 =

Ž1 q m2 . Ž c 2 q b 2 y 2 m 2 c 2 .

´ X0 Ž 1 q m 2 .

.c

2

2

E2 Ž c y b 2 .

5

,

`

Ž 1 q m3 . E3

FŽ t. s P0X

Ž 24 .

Ž 1 q m 1 . 2 m 1 b 2 y Ž b 2 q a2 . E1 Ž b 2 y a 2 . D 1

y

½

Ž1 q m2 . Ž b 2 q c 2 y 2 m2 b 2 . E2 Ž c 2 y b 2 .

= ´ X0 Ž 1 q m 2 . q

q

To obtain the primitive function of image F Ž t . from the function F Ž S ., the function F Ž t . is expanded into a series with some set of orthogonal functions, i.e.:

Ž 1 q m3 . E3

2 Ž 1 y m22 . b 2 Ž 1 q m 2 . ´ X0 2

2

D E2 Ž c y b . S

P0X

5 1 S

,

Ž 25 .

where Pb , Pc and D are the Laplace transforms of Pb , Pc and D, respectively. Replacing E2 and m 2 in Eqs. Ž24. and Ž25. by E and m given by Eq. Ž23., the viscoelastic solution in the Laplacian image space for the frost force in cold-region tunnels is obtained. Then, performing the inverse Laplace transforms to Eqs. Ž24. and Ž25., the behavior of variation of the forces Pb Ž t . and Pc Ž t . vs. time can be established. From the Eq. Ž1., the variation of frost stresses can be obtained. However, due to the complexity of Eqs. Ž24. and Ž25., it is very difficult to obtain the analytical expression of the inverse transform. Therefore, numerical method of inversion, presented by Papoulis Ž1956., will be used.

Ý Cl f l Ž t . ,

Ž 27 .

ls1

where f l Ž t . is a known set of orthogonal functions, which can be trigonometric series, Legendre polynomials or Jacobi polynomials, etc. Substituting Eq. Ž27. into Eq. Ž26. and using the orthogonality of the functions f l Ž t . and a set of known values of the image function F Ž S ., the coefficients Cl in Eq. Ž27. can be determined; thereby, the function F Ž t . will be obtained. The new method of the Laplace transform, proposed by Papoulis Ž1956., is adopted in this paper, and the sine functions are taken for f l Ž t .. A set of equally spaced points on the real axis in the image space is selected as following: S s Ž 2 l q 1. d ,

d ) 0, l s 0,1,2, PPP

A variable u is introduced by defining cos u s eyd t ; thus, the interval Ž0, q`. of the t axis is transformed to the interval Ž0, pr2. of the u axis. If F Ž0. s 0, then: N

FŽ t. s

Ý Cl sin Ž 2 l q 1. arccos eyd t

4

p 22



d F Žd . s C0 4

p PPP

d F Ž3 d . s C 0 q C 1 4

p

d F w Ž2 N q 1. d x s

• 2N

2N

ž / ž / ž / ž / y

N

2Ny1

q

`

yS t

H0 F Ž t . e

dt,

where S is the parameter of transform.

Ž 26 .

2N

y

i

FŽ S. s

Ž 29 .

where the coefficients Cl are determined by the following equations:

2N

The Laplace transform F Ž S . of function F Ž t . is:

,

ls1

22N

3. Numerical inverse transform

Ž 28 .

iy 1

C 0 q PPP

ß

C Ny i q PPP qC N

Ž 30 . If F Ž0. / 0, let F1Ž t . s F Ž t . y F Ž0.. Then, according to the theorem of initial value in the theory of Laplace transform, we have the relation F Ž0. s

Y. Lai et al.r Cold Regions Science and Technology 31 (2000) 227–234

233

lim S ™ ` SF Ž S .. Denoting the Laplace transform of F1Ž t . by F1Ž S ., then: F1 Ž S . s F Ž S . y

F Ž 0. S

.

Using the above technique of inverse transform, the approximate values of F1Ž t . can be determined from F1Ž S .; thereby, the approximate values of F Ž t . can be obtained from the relation F Ž t . s F1Ž t . q F Ž0..

4. Numerical example A circular tunnel in cold region is studied here. The elevation of the tunnel above sea level is 3800 m. The studied cross-section of the tunnel is 100 m below the mountain surface, i.e. H s 100 m. The yearly average temperature is y38C, and the temperature of the atmosphere inside the tunnel in the coldest period is y158C. From the thermal and mechanical parameters of the considered lining and the surrounding rock, the frost depth can be calculated and its value is 3.5 m. The geometrical dimensions of the tunnel are a s 4.5 m, b s 5.5 m and c s 9.0 m. The linear frost strain is ´ X0 s 0.0075, P0X s 1.533 MPa; the elastic modulus and the Poisson’s ratio of the lining are E1 s 20 000 MPa and m s 0.2, respectively. The elastic modulus and Poisson’s ration for the unfrozen surrounding rock are E3 s 800 MPa and m 3 s 0.333, respectively. From test results, the measured viscoelastic parameters of the frozen rock samples are: E1) s 208.4 MPa, E2) s 32.1 MPa, h s 18.3 MPa h. Substituting the above parameters into expressions of solution in the image space, i.e. into Eqs. Ž24. and Ž25., and using the method described in Section 3 to perform the numerical inverse transform by a small program, the variation of the frost force Pb Ž t . vs. time, acting on the lining, can be worked out. The parameter d can be chosen from the interval Ž0.04, 3., and the value of d s 0.05 has been used in this example, and calculations have been made for N s 10 and N s 20, respectively. Through these calculations, it can be seen that it is sufficient to take N s 10 for achieving enough convergence accuracy. The calculated results are shown in Fig. 6. From Fig. 6, it can be seen that the frost force Pb Ž t . will reach a maximum value of 4.2298 MPa when t s 0.9 h. Substituting this maxi-

Fig. 6. The time history of frost heaving force Pb Ž t ..

mum value into Eq. Ž1., the corresponding stresses on the surface of the lining are su I < rs a s 25.59 MPa, which already exceeds 21 MPa, the compressive strength of concrete 300 a in China. Therefore, cracks will occur in some parts of the lining. If the frost factor is ignored, the stresses on the surface of the lining will be su I s 11.80 MPa, showing a reduction of 53.9% with compared to the viscoelastic solution when the frost force is included in the analysis. This is the reason that cracks may still occur in cold-region tunnels, which are considered safe from the calculations given by the conventional theory. Therefore, the effect of frost force should be taken into account for cold-region tunnels. When the elastic modulus of the surrounding rock increases, the initial stresses in the lining caused by the self-weight of the surrounding rock will reduce, while the additional stresses caused by frost force will increase. On the other hand, when the elastic modulus of the surrounding rock reduces, the initial stresses in the lining will increase, while the additional stresses caused by frost force will decrease.

5. Conclusions The formulae of Laplace transform with respect to time for frost forces of the lining-frozen surrounding rock-unfrozen surrounding rock system is presented from the elastic–viscoelastic corresponding principle. The numerical inversion approach of the Laplace transform for the frost forces and lining stresses in

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cold-region tunnels are proposed. Numerical example illustrates that the lining stress including the frost factor is much larger than that not including the frost factor. So, the influence of this factor on the tunnel lining should be taken into account in cold regions’ tunnel design.

Acknowledgements This study was supported in part by The Foundation of AHundred People PlanB of Chinese Academy of Sciences Žto Dr. Y.M. Lai. and by a grant from the key research project of Transportation Ministry of China. The authors wish to thank two anonymous reviewers for suggesting improvements to the manuscript. Their comments are helpful. We sincerely appreciated Dr. Niu Fujun and doctoral student Yu Wenbing for their help during preparing this paper for publication.

References Flugge, W., 1975. Viscoelasticity. 2nd edn. pp. 46–50. Fritz, P., 1984. An analytical solution for axisymmetric tunnel problems in elasto–viscoplastic media. Int. J. Numer. Anal. Meth. Geomech. 8 Ž3., 325–342. Klein, J., 1981. Finite element method for time-dependent problems of frozen soils. Int. J. Numer. Anal. Meth. Geomech. 5 Ž2., 263–283. Ladanyi, B., 1980. Direct determination of ground pressure on tunnel lining in a non-linear viscoelastic rock. 13th Can. Rock Mech. Symp., Toronto. CIM Spec. Vol. vol. 22, 126–132. Ladanyi, B., 1984. Tunnel lining design in a creeping rock. Design and Performance of Underground Excavations. ISRMr BGS, Cambridge, pp. 19–26. Lai, Y.M., Wu, Z.W., Zhu, Y.L. et al., 1999. Nonlinear analysis for the coupled problem of temperature and seepage fields in cold region tunnels. Cold Reg. Sci. Technol. 29 Ž1., 89–96. Papoulis, A., 1956. A new method of the Laplace transform. Q. Appl. Math. 14 Ž4., 405–414. Xu, Z.L., 1982. Theory of Elasticity. 2nd edn. High Educational Press, Beijing, pp. 73–108.