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Analytical elasto-plastic solution for stress and plastic zone of surrounding rock in cold region tunnels G.Y. Gao a, b,⁎, Q.S. Chen a, b, Q.S. Zhang c, G.Q. Chen a, b a b c

Department of Geotechnical Engineering, Tongji University, Shanghai 200092, China Key Laboratory of Geotechnical and Underground Engineering of Ministry of Education, Tongji University, Shanghai 200092, China Bureau of Housing and Urban-Rural Development of Kunshan, Kunshan, Jiangsu Province 215300, China

a r t i c l e

i n f o

Article history: Received 29 August 2011 Accepted 22 November 2011 Keywords: Cold region tunnel Frost heaving force Stress in plastic zone Boundary of plastic zone Analytical elasto-plastic solution Semi-analytical method

a b s t r a c t To solve the engineering problem and put forward new practical theoretical insights to analysis and design, in this paper, an analytical solution for the frost heaving force in cold-region tunnel has been derived ﬁrstly using the continuity equation. Thereafter, a semi-analytical method to calculate the stress in the plastic zone of rock surrounding in cold-region tunnels is proposed by assuming that the frost rock was initially in the elastic state before the plastic zone was developed. Additionally, based on the approximate method for determining the plastic zone proposed by Kastner, the method for calculating the plastic zone in cold region tunnel is presented. Finally, the main results and conclusions are presented from a case study. The results show that frost heaving force results in a remarkable expansion of the plastic zone in the surrounding rock, and the rock in the plastic zone experiences hardening after the yield criterion is reached. Hardening effect can be taken into account in the proposed method for studying cold-region tunnels. © 2011 Published by Elsevier B.V.

1. Introduction With the dramatic development of economy in China, a great deal of infrastructures has been built in the past decades. High-speed transportation system, such as high-speed way and railway, is one of the most important projects by the central government. Tunnels have been inevitably constructed due to many mountains and plateaus. Until now, there have been proximately 40 tunnels in North China, Northeast, and Northwest and other places, such as Da Ban mountain tunnel, Feng Huo mountain tunnel on Qing–Tibet railway. However, frozen soils are widely distributed in such regions, which have raised a series of problems for the tunnels and greatly affected their normal operation. Although considerable studies have been conducted on a cold-regional tunnel in past decades, it is reported that all of the tunnels have shown different degrees of frost damage. Due to the inﬂuence of frost damage, some tunnels cannot be open to trafﬁc for half a year. Therefore, it is of great necessity for researchers to analyze coldregion tunnel in a further way. Studies on the issues of tunnel in cold regions have been carried out by many researchers in the past decades. Fritz (1984) presented an analytical solution for the time-dependent stresses and displacements in plane strain around a circular hole which it is loaded by an axisymmetric internal and far-ﬁeld pressure, assuming that material is elasto-viscoplastic with dilatant plastic deformations according to a ⁎ Corresponding author at: School of Civil Engineering, Tongji University, Shanghai, 200092, China. Tel./fax: + 86 21 55058685. E-mail address: [email protected] (G.Y. Gao). 0165-232X/$ – see front matter © 2011 Published by Elsevier B.V. doi:10.1016/j.coldregions.2011.11.007

non-associated ﬂow rule. Klein (1981) proposed that the timedependent material characteristics in the form of creep control the bearing capacity of any frozen earth support system. Lai et al. (2000b) established a mathematical mechanical model for earthquake response of the coupled problem of temperature, seepage, and stress ﬁelds with phase change, and derived the ﬁnite element formulae of the coupled problems from Galerkin's method combining with heat transfer, the theory of seepage and frozen soil mechanics. Zhang et al. (2002a) conducted non-linear analysis for the freezing–thawing situation of the rock surrounding the tunnel in cold regions under the conditions of different construction seasons, initial temperatures and insulations, and investigated the three-dimensional (3-D) temperature ﬁeld derived from Galerkin's method. Zhang et al. (2002b) also presented the ﬁnite element formulae of 3-D temperature ﬁelds based on Galerkin's method and the governing differential equations of the problem on temperature ﬁeld with phase change. Lai et al. (2002) worked out the approximate analytical solution of circular tunnels for temperature ﬁelds in coldregion by dimensionless and perturbative method. Zhang et al. (2004b) studied the coupled effects of moisture transfer and heat conduction with phase change, and the ﬁnite element formulae of this problem were derived from the governing differential equations and moisture transfer equations using Galerkin's method and the software for computers was edited, based on Biot's theory combining with Fourier transformation. Zhang et al. (2004a) discussed the damage propagation of rock from cold region tunnel under different conditions by CT; their achievements can provide a referenced basis for numerical computation and the safe running of cold-region tunnels. Lai et al. (2005) carried out 3-D nonlinear analysis for the coupled problem of the heat

G.Y. Gao et al. / Cold Regions Science and Technology 72 (2012) 50–57

transfer of surrounding rock and the heat convection between the air and the surrounding rock in cold-region tunnel. Lu and Jeng (2006) presented a semi-analytical solution of a circular tunnel surrounded by a poro-elastic medium and subjected to a moving axisymmetric ring load. Zhang et al. (2006) ﬁnished 3-D nonlinear analysis of coupled problem of heat transfer in the surrounding rock and heat convection between the air and the surrounding rock in the Feng-huo mountain tunnel. Yang et al. (2006) provided analysis model for frost heave with coupled water freezing, temperature and stress ﬁelds in coldregion tunnel excavation. Wu et al. (2010) established an elastic–plastic model for the mechanical behaviors of pipeline–soil system, using this model to calculate the stresses and deformations of the pipeline in permafrost regions. In short, the existing researches mainly focus on the cooling technology and numerical calculation. Zhu et al. (2010) presented a constitutive model of frozen soil with damage for the coupled problem, and Bronfenbrener and Bronfenbrener (2010) proposed a generalized model for secondary frost heave in freezing ﬁne-grained soils. Yang et al. (2010) did a mass of research for the mechanical characteristic of artiﬁcial frozen soils. Lu et al. (2011) obtained analytic stress solutions for a circular pressure tunnel at pressure and great depth including support delay. However, in the literatures, no research work on computational method of determining the plastic zone in the rock around the coldregion tunnels and the stress distribution in the plastic zone has been found yet. The instability of tunnels is closely associated with the development of the plastic zones around tunnels. In cold regions, after tunnels have been excavated, once the ﬁssure and pore water in rock become frozen, an expansion in volume will then occur. Because this volume expansion is restricted by the lining and unfrozen rock surrounding the tunnel, it will result in forces on the lining, called the frost force of surrounding rock. On the one hand, the mechanical properties of surrounding rock and the lining are severely affected due to frost heaving, and the original equilibrium can be destroyed. On the other hand, as the application of frost heaving force on the surrounding rock, the stress in the rock increase correspondingly. The plastic zones begin to develop when the stress exceeds yield stress of the rock. Once the plastic zones increase to a critical extent, tunnels will become unstable. Therefore, determination of the plastic zone and stress in the plastic zone are greatly signiﬁcant for the stability analyses of cold-region tunnels. Additionally, up to now, to the best knowledge of the authors, studies concerning the investigation of frost force are rather limited. In the 1970s and 1980s, the surveys conducted in Japan's northeastern cold-region tunnels indicated that frost force is the source of eternal forces causing the damage of cold-region tunnels (Lai et al., 2000a). In cold regions, tunnels that endured subzero temperatures such as in Tibetan Plateau pose a lot of problems related to the degradation of mechanical properties of the lining. One of the critical reasons is frost heaving force. Lai et al. (1998, 1999) made nonlinear analysis for the coupled problem of temperature, seepage and stress ﬁelds in coldregion tunnels. They derived the ﬁnite element formulae of the coupled problems from Galerkin's method based on heat transfer and the theory of seepage. Their work may offer reference to design computation for frost force of cold region tunnels. Thereafter, Lai et al. (2000a) proposed an analytical visco-elastic solution to study the frost force for cold region tunnels using Laplace transform with respect to time for frost forces and visco-elastic corresponding principle. It should be noted that, due to the complexity of the issues, simpliﬁed assumptions have been made in the previous study. Hence, an increased knowledge of frost forces is required. To solve the engineering problem and put forward new practical theoretical insights to analysis and design, in this paper, an analytical solution for the frost heaving force in cold-region tunnels have been derived ﬁrstly using the continuity equation. Thereafter, a semi-analytical method to calculate the stress in the plastic zone in the rock surrounding cold-region tunnels has been proposed. Also, based on the

51

approximate method for determining the plastic zone proposed by Kastner (1971), method for calculating the plastic zone in cold region tunnel is presented. Finally, the main results and conclusions are presented from one illustrated example. 2. Analytical solution for the frost heaving force in cold-region tunnel The volumetric expansion of frozen surrounding rock is restricted by the lining and unfrozen rock surrounding tunnel; thus, the frozen rock will result in forces σf on the tunnel lining and forces σH on the unfrozen surrounding-rock. When the tunnel becomes frozen, the original equilibrium condition is destroyed. The frost heaving forces and displacements analyzed in this paper are generated by the volumetric expansion of frozen surrounding rock. In the following equations, r0 is the inner radius of the tunnel lining; r1 is the outer radius of the tunnel lining; and r2 is the outer radius of the frozen rock, namely the inner radius of the unfrozen, as shown in Fig. 1. From the theory of elasticity, the radial displacement of tunnel lining is given by the following expression (Xu, 1990): uðr Þ ¼ −

ð1 þ μ l Þð1−2μ l Þσ f r 21 r ð1 þ μ l Þr 21 r 20 σ f − El r 21 −r 20 El r 21 −r20 r

ð1Þ

where El and μl are the elastic modulus and Poisson's ratio of the lining, respectively. The displacements of the outer surface of the tunnel lining (r = r1) is δ1 ¼ −

ð1 þ μ l Þð1−2μ l Þσ f r 31 ð1 þ μ l Þr 1 r 20 σ f − El r 21 −r 20 El r 21 −r 20

i ð1 þ μ l Þσ f r 1 h 2 2 ¼ − 2 2 ð1−2μ l Þr1 þ r 0 : El r 1 −r 0

ð2Þ

Let r1 and r2 be the inner radius and outer radius of the frozen surrounding rocks, respectively. The frozen surrounding rocks are subjected to the frost heaving force σf from lining inside and the expressive force σH from unfrozen surrounding rocks outside. And the radial displacements of the frozen surrounding rocks can be given by ωr 22 r 21 σ H −σ f 2 2 uðr Þ ¼ ω 1−2μ f σ f r 1 −σ H r 2 r− r 2 2 ω ¼ 1 þ μ f =Ef r 2 −r 1

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

ð3aÞ ð3bÞ

52

G.Y. Gao et al. / Cold Regions Science and Technology 72 (2012) 50–57

where Ef and μf are the elastic modulus and Poisson's ratio of the frozen surrounding rocks, respectively. Then we can get the displacements of the inner face of the frozen surrounding rocks: δf 1

r¼r 1

2 2 2 ¼ ω 1−2μ f σ f r 1 −σ H r 2 r 1 −ω σ H −σ f r 2 r 1 :

ð4Þ

And the displacements of the outer face can be obtained similarly: 2 2 2 δf 2 r¼r ¼ ω 1−2μ f σ f r 1 −σ H r 2 r 2 −ω σ H −σ f r 1 r 2 :

ð5Þ

2

The displacements of the inner face of the unfrozen rocks are: δ2 ¼

r2 ð1 þ μ s Þσ H Es

ð6Þ

−δ1 þ δf 1 ¼ Δh1 as r ¼ r 1

δ2 −δf 2 ¼ Δh2 as r ¼ r 2

ð8Þ

where Δh2 is the displacements of the outer face of the frozen surrounding rocks caused by freezing. Δh1, Δh2 are given as follows: h i h i r 1 þr 2 2 2 2 −r 21 nα t r1 þr −r 21 2 2 ΔV 1 nα t π Δh1 ¼ ¼ ¼ 2πr 1 2πr 1 2r 1 ΔV 2 nα t π ¼ Δh2 ¼ 2πr 2

h

2 2 i r 22 − r1 þr 2 2πr 2

¼

h 2 2 i nα t r 22 − r1 þr 2 2r 2

ð9aÞ

ð9bÞ

where n is porosity of the surrounding rock, αt is the volumetric expansion per unit volume when water is frozen to ice. Because the radial thickness of frozen surrounding rock is small, one can take porosity of the surrounding rock as a constant in the following analysis for the sake of convenience. In the analysis, the centers of the inner and outer radii of the frozen rock were assumed to be ﬁxed, i.e., the displacements at these points are zero. This assumption was to facilitate the analytical solution. From Eqs. (1) to (9a) and (9b), the frost force can be obtained as follows: i nα hr þr 2 2 i 2 2 2 þ 2r t 1 2 2 −r 1 r2 − r1 þr 2 1 σf ¼ : 2 h i 4ω2 1−μ f r31 r32 ð 1þμ Þr l 1 þ E r2 −r ð1−2μ l Þr 21 þ r 20 ωr 1 1−2μ f r 21 þ r22 − 2 m lð 1 0Þ nα t r 1 r 2 ωð1−μ f Þ m

h

ð10aÞ r2 3 2 ð1 þ μ s Þ þ ω 1−2μ f r 2 þ ωr 1 r 2 Es

s

:

ð12aÞ

Because of freezing, an expansion of volume will occur. Therefore, the stresses acting on the inner face of unfrozen rocks (i.e., r ≥ r2) will increase. The increment of stresses can be given as follows: σ H r2 r2 2 σ H r2 σ sθ ¼ − 2 r τsrθ ¼ 0 σ sr ¼

)

:

ð12bÞ

3. Stresses in the plastic zone of surrounding rock For tunnels located in normal region instead of cold region, the plastic zones of surrounding rocks keep stable; forces acting on the lining are thus stable too. But in cold region, tunnels are subjected to subzero temperatures, which lead to the frozen pore water and therefore its volumetric expansion. Tremendous frost heaving forces accelerate the development of the plastic zones, and forces acting on the lining increase substantially, which may ﬁnally destroy the tunnels. Methods for calculating the stresses and the boundary of the plastic zones in surrounding rocks are presented herein. The ﬁrst step to obtain the plastic solution of stresses in surrounding rock is to establish its corresponding elastic solution before the tunnel becomes frozen. From the complex variable function solution of theory of elasticity, to solve the plane problem may come down to seek two analytic functions of ϕ(Z) and Φ(Z), which need to satisfy the given boundary conditions. For round tunnels, supposing the lining is closed in and its outer radius is equal to the radius of tunnel excavation, the key to solve the plane strain problem is to seek the analytic functions of ϕ(Z) and Φ(Z). Let ψ(Z) = ϕ '(Z), Φ(Z) = χ '(Z), Ψ(Z) = Φ '(Z) = χ " (Z), then stresses and displacements in polar coordinates can be expressed as: 2Gðu þ ivÞ ¼ kϕðZ Þ−Zϕ′ ðZ Þ−ΨðZ Þ

ð14Þ

Neglecting the friction between the lining and surrounding rocks, the boundary conditions are as follows (Yu, et al., 1983): r ¼ r0 : r ¼ r1 : r¼∞:

ð11Þ

ð13Þ

) h i σ r þ σ θ ¼ 2 ψðZ Þ þ ψðZ Þ h i : σ θ −σ r þ 2iτrθ ¼ 2 Z ψ′ ðZ Þ þ ΨðZ Þ e2ia

σ lr ¼ 0; τ rθ ¼ 0 σ r ¼ σ lr ; u ¼ ul τrθ ¼ τlrθ ¼ 0

ð10bÞ

nα t r2 −ð Þ þ 2ω 1−μ f σ f r 21 r 2 2r 2 h i σH ¼ r 2 ð1þus Þ þ ω 1−2μ f r 32 þ r 21 r2 E r 1 þr 2 2 2

)

ð7Þ

where Δh1 is the displacements of the inner face of the frozen surrounding rocks caused by freezing. Similarly, on the interface of frozen rocks and unfrozen rocks,

2

r 21 r 22 r2 r2 2 2 −r 1 r2 − 122 2 σ fr ¼ r 2 2 σ f þ 2 r2 σ H r2 −r 1 r 2 −r 1 2 2 2 2 r r r 2 2 1 r2 r1 − 12 2 −r2 2 r σ fθ ¼ 2 2 σ f þ r 2 2 σ H r2 −r 1 r 2 −r 1 τfrθ ¼ 0

2

where Es and μ s are the elastic modulus and Poisson's ratio of the unfrozen rocks, respectively. On the interface of lining and frozen rocks, the continuity conditions of displacements should be met:

m¼

Frost heaving force in frozen surrounding rocks (i.e., r b r2) can be expressed by:

σr ¼

P ½ð1 þ λÞ þ ð1−λÞ cos2θ 2

P σ θ ¼ ½ð1 þ λÞ−ð1−λÞ cos2θ 2 P τrθ ¼ − ð1−λÞ sin2θ 2

)

:

ð15Þ

G.Y. Gao et al. / Cold Regions Science and Technology 72 (2012) 50–57

The complex number form of the boundary condition Eq. (15) are expressed as: jzj ¼ r 0 : jzj ¼ r 1 :

jzj ¼ þ∞ :

i 0 zψ ðzÞ þ ΨðzÞ ¼ 0 h 0 i 2iα ψðzÞ þ ψðzÞ−e zψ ðzÞ þ ΨðzÞ ¼ σ lr h i 0 Re kϕðZ Þ−Zϕ ðZ Þ−ΨðZ Þ ¼ 2Gulr h i 2 ψðZ Þ þ ψðZ Þ ¼ P ð1 þ λÞ h 0 i 2ia 2 Z ψ ðZ Þ þ ΨðZ Þ e σ θ ¼ P ð1−λÞ cos2θ−i½P ð1−λÞ sin2θ ψðzÞ þ ψðzÞ−e

2iα

h

)

where P is the force acting on tunnel lining, λ is the coefﬁcient of lateral earth pressure, θ is the vertical angle, subscript l denotes the lining, σlr,ul,τlrθindicate the normal stress, displacement and shear stress of the lining, respectively. The analytic functions for surrounding rocks are:

)

i Ph 2 −1 ϕðZ Þ ¼ ð1 þ λÞZ þ ð1−λÞβr 1 Z 4 i : Ph 2 −1 4 −3 ΦðZ Þ ¼ − ð1−λÞZ þ ð1 þ λÞγr 1 Z þ ð1−λÞδr1 Z 2

ð16Þ

Substituting the expression of complex variable function in Eq. (16) into Eqs. (13) and (14), and combining boundary conditions Eq. (15) or Eq. (15a), stresses and displacements of surrounding rocks are obtained (Yu, et al., 1983): "

2

!

2

4

!

P γr 2βr 3δr ð1 þ λÞ 1− 21 þ ð1−λÞ 1− 2 1 − 4 1 cos2θ 2 r r r " ! ! # 2 4 P γr 1 3δr 1 ð1 þ λÞ 1 þ 2 −ð1−λÞ 1− 4 cos2θ σθ ¼ 2 r r ! P βr 2 3δr 4 τrθ ¼ − ð1−λÞ 1 þ 21 þ 4 1 sin2θ 2 r r ( " # ) 2 Pr 1 2δr 21 2γ ð1 þ λÞ þ ð1−λÞ βðk þ 1Þ þ 2 cos2θ u¼ 8Gr r " # Pr 2 2δr 2 v ¼ − 1 ð1−λÞ βðk−1Þ− 2 1 sin2θ 8Gr r

σr ¼

#

)

)

In Eqs. (15) to (19), variables of both surrounding rock and tunnel lining are deﬁned as follows: k ¼ 3−4μ; kl ¼ 3−4μ l E El ;G ¼ 2ð1 þ μ Þ l 2ð1 þ μ l Þ h i G ðkl −1Þr 21 þ 2r 20 γ¼ 2Gl r 21 −r20 þ G ðkl −1Þr 21 þ 2r 20 3 GH þ Gl r 21 −r 20 β¼2 3 GH þ Gl ð3k þ 1Þ r 21 −r 20 3 GH þ Gl ðk þ 1Þ r 21 −r 20 δ¼− 3 GH þ Gl ð3k þ 1Þ r 21 −r 20 G¼

ð15aÞ

53

6

4 2

H ¼ r 1 ðkl þ 3Þ þ 3r1 r 0 ð3kl þ 1Þþ 4 2

6

þ3r 0 r 1 ðkl þ 3Þ þ r 1 ð3kl þ 1Þ 2

A1 ¼

P r ð1 þ λÞð1−γ Þ 2 1 2 4 r 1 −r 0

P r2 r2 A2 ¼ − ð1 þ λÞð1−γÞ 2 0 1 2 2 r 1 −r 0 2 2 4 2 2 4 r r 2r 0 1 1 þ r0 r1 þ r0 3P ð1−λÞð1 þ δÞ A3 ¼ 2 2 3 4 r −r 1 0 2 2 2 r 1 r 1 þ 3r 0 P A4 ¼ ð1−λÞð1 þ δÞ 3 4 r 2 −r 2 1 0 2 4 2 2 r 1 r 1 þ r 0 r 1 þ 2r 20 3P A5 ¼ − ð1−λÞð1 þ δÞ 2 2 3 2 r −r 1 0 4 2 2 2 2 r 0 r 1 3r 1 þ r 0 r1 P A6 ¼ ð1−λÞð1 þ δÞ 2 2 3 2 r −r 1

ð20Þ

) :

ð21Þ

0

By summing the stresses in Eqs. (12a) and (17), stress ﬁeld in frozen surrounding rocks (i.e., r b r2) can be derived by assuming that the frost rock was initially in the elastic state before the plastic zone was developed:

ð17Þ

σ r ¼ σ r þ σ fr σ θ ¼ σ θ þ σ fθ τ rθ ¼ τ rθ þ τfrθ

) :

ð22aÞ

Analytic functions for tunnel lining are: ϕl ðZ Þ ¼ A1 Z þ A4 Z 3 þ A3 Z −1 Φl ðZ Þ ¼ A5 Z þ A2 Z −1 þ A6 Z −3

) :

ð18Þ

Substituting the expression of complex variable function in Eq. (18) into Eqs. (13) and (14), and combining boundary conditions in Eq. (15), stresses and displacements of the lining are obtained (Yu, et al., 1983): σ lr ¼ 2A1 þ A2 r −2 − A5 þ 4A3 r −2 −3A6 r −4 cos2θ σ lθ ¼ 2A1 −A2 r −2 þ A5 þ 12A4 r 2 −3A6 r −4 cos2θ τ lrθ ¼ A5 þ 6A4 r 2 −2A3 r −2 þ 3A6 r −4 sin2θ i h i o 1 nh −1 3 −1 −3 ul ¼ þ ðkl −3ÞA4 r −A5 r þ ðkl þ 1ÞA3 r −A6 r cos2θ ðkl −1ÞA1 r−A2 r 2Gl i 1 h 3 −1 −3 sin2θ ðkl þ 3ÞA4 r þ A5 r−ðkl −1ÞA3 r −A6 r vl ¼ 2Gl

)

ð19Þ where subscript l denotes the lining.

By summing the stresses in Eqs. (12b) and (17), stress ﬁeld in unfrozen surrounding rocks (i.e., r ≥ r2) can be derived by assuming that the frost rock was in the elastic state at the time when the plastic zone began to develop: σ r ¼ σ r þ σ sr σ θ ¼ σ θ þ σ sθ τ rθ ¼ τ rθ þ τsrθ

) :

ð22bÞ

The plastic stress solution can be obtained based on the following assumptions: (1) The distribution of radial stresses σ pr in the plastic zone has the same formation as that of the elastic solution σ er at the time when the plastic zone began to develop; superscript p and e are respectively the plastic and elastic states. (2) The orientation of the principal stress in the plastic zone is the same as at the elastic state for surrounding rocks at the time when the plastic zone began to develop.

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G.Y. Gao et al. / Cold Regions Science and Technology 72 (2012) 50–57

With the above assumptions, using the Mohr–Coulomb criterion, the tangent stresses σ pθ and shear stresses τprθ can be derived. The plastic condition expressed by the principal stresses can be given by:

In the plastic zone, the plastic yield criterion should be met. The Mohr–Coulomb criterion is quite suitable for geotechnical material, and it can be given as: σ pr þ cctgφ

2c cosφ 1 þ sinφ p þ σ p1 ¼ σ : 1− sinφ 1− sinφ 3

ð23Þ

According to the above assumptions, the directions of the principal stresses in the plastic zone can be obtained by equation of the elastic stresses at the time when the plastic zone began to develop: tg2ϕ ¼

2τerθ e σ θ −σ er

ð24Þ

where ϕ is the angle between the axis and direction of minor principal stress. Also in light of the mentioned assumptions, radial stresses in the plastic zone σ pr are equal to those at the elastic state σ er at the time when the plastic zone began to develop, which gives:

)

i Ph 2 2 4 ð1 þ λÞ 1−γα þ ð1−λÞ 1−2βα −3δα cos2θ þ 2 r 21 −α 2 r 22 α 2 r 2 −r 2 σf þ 2 2 2 2 σH 2 2 r 2 −r 1 r2 −r 1 i Ph 2 2 4 σ pr ¼ σ er ¼ ð1 þ λÞ 1−γα þ ð1−λÞ 1−2βα −3δα cos2θ þ 2 r2 σ H 22 r σ pr ¼ σ er ¼

rbr 2

r≥r 2

σ pθ

þ cctgφ

¼

1− sinφ : 1 þ sinφ

ð29Þ

Combining Eqs. (28) and (29) gives: 2 sinφ lnr þ C 1 ln σ pr þ cctgφ ¼ 1− sinφ

ð30Þ

where, C1 is a constant, and it can be determined by boundary conditions. Before the surrounding rocks become frozen, the force acting on tunnel lining is P(1 − γ), and after the rocks becomes frozen, the force acting on the lining is Pi = P(1 − γ) + σf. The constant C1 can be given as: C 1 ¼ lnðP i þ cctgφÞ−

2 sinφ lnr 1 : 1− sinφ

ð31Þ

Substituting Eq. (31) into Eq. (30), radial stresses in frozen the plastic zone (i.e., r b r2) can be obtained: 2 sinφ h i r 1− sinφ σ pr ¼ P ð1−γ Þ þ σ f þ cctgφ −cctgφ: r1

ð25Þ

ð32aÞ

Radial stresses in unfrozen the plastic zone (i.e., r ≥ r2) are given as:

where α = r1/r. Since the values and directions of σ pr have been known, the stresses in the plastic zone can be obtained from the Mohr's stress circle, which is given by the equations: 3 1− cos2 ϕ c cosφ 1− sinφ 5 σ p3 ¼ 4σ pr − 1− sinφ cos2ϕ 1− sinφ 2 3 1− cos2 ϕ c cosφ 2c cosφ 4 p p 5 1 þ sinφ þ σr − σ1 ¼ 1− sinφ 1− sinφ cos2ϕ 1− sinφ 2

σ pθ ¼

)

:

ð26Þ

σ p1 þ σ p3 σ p1 −σ p3 þ cos2ϕ ¼ σ p1 þ σ p3 −σ pr 2 2 τ prθ ¼

σ p1 −σ p3 sin2ϕ 2

)

:

# 2 sinφ r 1− sinφ −cctgφ: r2

! þ σ H þ cctgφ

ð32bÞ

Substituting Eq. (32a) and (32b) into Eq. (26) gives whole stresses in the plastic zone. When the coefﬁcient of lateral earth pressure, λ, is much greater or much lesser than 1, radial stresses in the plastic zone can be calculated from Eq. (25). When λ is equal to 1, radial stresses in the plastic zone can be calculated from Eq. (32a) and (32b), then substituting the results into Eq. (26) gives all other stresses.

Based on the approximate method for calculating the boundary of the plastic zone proposed by Kastner (1971), method for calculating the boundary of the plastic zone in cold region tunnel is presented herein. The plastic condition is given by:

ð27Þ sinφ ¼

For axisymmetric problems, stresses and deformations are functions of radius r and independent of θ, either the elastic or plastic zones, and the thickness of the plastic zone are constant. The equilibrium equation of stress is: ∂σ r σ r þ σ θ ¼ 0: þ r ∂r

r21 γ r 22

4. Boundary of the plastic zone of surrounding rock

For axisymmetric problems, let λ = 1, from Eqs. (24) and (26), we get ϕ = 0 and τprθ ¼ 0. And Eq. (25) becomes: α 2 r 2 −r 2 r 2 −α 2 r 2 2 σ pr ¼ P 1−γα þ 2 2 2 1 σ f þ 2 2 2 2 σ H ; r b r 2 r2 −r 1 r 2 −r 1 2 r2 2 p σ r ¼ P 1−γα þ σ H 2 ; r ≥ r2 r

" σ pr ¼ P 1−

ð28Þ

rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 2ﬃ σ eθ −σ er þ 2τerθ σ eθ þ σ er þ 2cctgφ

ð33Þ

Substituting Eq. (22a) and (22b) into Eq. (33), we obtain: 2 2 2 2 sin φ σ eθ þ σ er þ 4c cosφ sinφ σ eθ þ σ er þ 4c cos φ 2 2 ¼ σ eθ −σ er þ 2τerθ

ð34Þ

G.Y. Gao et al. / Cold Regions Science and Technology 72 (2012) 50–57

λ ≠ 1, the distribution of the plastic zone is too complicated to use analytic method.

or 2

4c þ 4σ eθ σ er ¼

h i2 2 σ eθ þ σ er cosφ−2c sinφ þ 2τerθ :

ð35Þ

If the boundary of the plastic zone is located in the frozen domain (i.e., r b r2), substituting the elastic stresses obtained from Eq. (22a) into Eq. (35) gives the equation of the boundary: 2 4c þ 4 σ θ þ σ fθ σ r þ σ fr h i2 2 ¼ σ r þ σ fr þ σ θ þ σ fθ cosφ−2c sinφ þ ð2τ rθ Þ :

ð36aÞ

The solution of Eq. (36a) gives the expression of r. If r > r2, it means the boundary of the plastic zone is located in the unfrozen domain. Substituting the elastic stresses obtained from Eq. (22b) into Eq. (35) gives the equation of the boundary: 2

4c þ 4ðσ θ þ σ sθ Þðσ r þ σ sr Þ

2

2

¼ ½ðσ r þ σ sr þ σ θ þ σ sθ Þ cosφ−2c sinφ þ ð2τrθ Þ :

ð36bÞ

The redistribution of the plastic stresses is not taken into consideration in the previous method. For axisymmetric condition where λ = 1, the previous method can be modiﬁed. After the boundary of the plastic zone is determined, radial plastic stress σ pr at any point can be obtained from Eq. (25) or Eq. (32a) and (32b), and then solution of Eq. (26) gives shear stresses τ prθ in the plastic zone in polar coordinates. The calculated plastic zone can be simpliﬁed as an area equivalent circle. Let radial stresses σ pr and shear stresses τprθ be forces acting on the face of virtual circle, and we get the elastic stress ﬁeld of rocks at the elastic state (i.e., r ≥ r2): " # P 2a2 σ pr 2 2 4 ð1 þ λÞ 1−a þ ð1−λÞ 1−4a þ 3a cos2θ þ 2 P " # P 2a2 σ pr 2 4 ð1 þ λÞ 1 þ a −ð1−λÞ 1 þ 3a cos2θ− σ eθ ¼ 2 P " # 2 2a τ prθ P 2 4 ð1−λÞ 1 þ 2a −3a sin2θ þ τ erθ ¼ 2 P

σ er ¼

)

5. Example study Herein, the same example of a circular tunnel in cold region as used in the literature (Lai et al., 2000a) is studied. 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. The yearly average temperature is −3 °C, and the temperature of the atmosphere inside the tunnel in the coldest period is − 15 °C. 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 r0 = 4.5 m, r1 = 5.5 m, r2 = 9.0 m. The elastic modulus and Poisson's ratio of tunnel lining are El = 20000MPa and μl = 0.2, respectively. The elastic modulus and Poisson's ratio for the unfrozen surrounding rock are Es = 800MPa and μs = 0.33, respectively. The elastic modulus and Poisson's ratio for the frozen surrounding rock are Ef = 240.50MPa and μf = 0.35, respectively. Void ratio is 0.18; the volumetric expansion per unit volume is 9% when water is frozen to ice; cohesion is c = 1.2 MPa and angle of internal friction is φ = 45°. The coefﬁcient of lateral earth pressure is λ = 0.5; the density of surrounding rock is ρ = 2500kg/m 3. 5.1. Solution of the frost heaving force Substituting the inner radius, outer radius, elastic modulus and Poisson's ratio of frozen surrounding rock into Eqs. (3b) and (10b) give

ω¼

ð1 þ 0:35Þ −4 ¼ 1:106 10 ; m ¼ 0:0719968: 240:5 92 −5:52

Substituting ω, m and other parameters into Eq. (10a) and (10b) gives σf = 3.089MPa, σH = 2.036MPa; then substituting r1, r2,σf and σH into Eq. (12a) and (12b), the corresponding frost heaving force gives:

ð37Þ where a = r*/r, r* is the equivalent circle radius of the plastic zone area. Substituting Eq. (37) into Eq. (35) gives:

3 2cctgφ 2 1þλþ sin φ 2 4 p 6 7 ð1 þ λÞP−2σ r 1−2a þ 3a P 2 7 cos 2θ þ 6 4 2ð1−λÞP a2 sin2 φ þ 2−3a2 − ð1−λÞa2 sin2 φ þ 2−3a2 5 cos2θ− 2

2 2 2 4 2 ð1 þ λÞP−2σ pr a 1−2a þ 3a − 2 2 − 4ð1−λÞ2 P 2 a2 sin2 φ þ 2−3a2 4a a sin2 φ þ 2−3a2

55

2450:25 2450:25 −30:25 0:061 þ 81− 0:04 2 r2

r

2450:25 2450:25 0:061− 81− 0:04 σ fθ ¼ 30:25− r2 r2 τfrθ ¼ 0 164:92 σ sr ¼ r2 164:92 σ sθ ¼ − r2 τsrθ ¼ 0

σ fr ¼

)

)

:

2 2cctgφ 2 2 1þλþ sin φ a2 τ prθ P − þ 4ð1−λÞ2 a2 sin2 φ þ 2−3a2 a2 ð1−λÞ2 P 2 a2 sin2 φ þ 2−3a2

−

τ prθ

2

4

ð38Þ

1 þ 2a −3a sin2θ ¼ 0: P ð1−λÞ a2 sin2 φ þ 2−3a2

The solution of Eq. (38) gives the expression of a. Referring to a = r*/r, the expression of r can be obtained. From the above-mentioned, the boundary of the plastic zone is determined. Using iterative method presented previously, more precise solution can be obtained. For non-axisymmetric problems where

5.2. Solution of stresses in surrounding rock before the tunnel becomes frozen Substituting the corresponding values from the example into Eqs. (20) and (21) gives:

k ¼ 1:6667; kl ¼ 2:2 G ¼ 300; Gl ¼ 8333 γ ¼ 0:1152 β ¼ 1:72 δ ¼ −0:906 H ¼ 818076

)

A1 ¼ 0:9954P A2 ¼ −40:6P A3 ¼ 60:6P A4 ¼ 0:0323P A5 ¼ −4:1P A6 ¼ 197:4P

)

56

G.Y. Gao et al. / Cold Regions Science and Technology 72 (2012) 50–57

Substituting the above results into Eq. (17) gives the stresses in surrounding rock of tunnel:

2:5 3:4848 104:1 2486:9 þ 0:5 1− 2 þ cos2θ 1:5 1− 2 4 2 r r r

2:5 3:4848 2486:9 −0:5 1 þ cos2θ 1:5 1 þ σθ ¼ 2 r2 r4

2:5 52:1 2486:9 sin2θ τ rθ ¼ − 0:5 1 þ 2 − 2 r r4

σr ¼

) :

Then, substituting the above results into Eq. (19) gives the stresses in tunnel lining: σ lr ¼ 1:991−40:6r −2 2:5− −4:1 þ 242:4r −2 −592:2r −4 2:5 cos2θ −2 2 −4 σ lθ ¼ 1:991 þ 40:6r 2:5 þ −4:1 þ 0:3876r −592:2r 2:5 cos2θ 2 −2 −4 τlrθ ¼ −4:1 þ 0:1938r −121:2r þ 592:2r 2:5 sin2θ

) :

5.3. Solution of stresses in surrounding rock when the tunnel becomes frozen Summing the stresses in surrounding rock before tunnel becomes frozen and the frost heaving force after tunnel is frozen, based on Eq. (22a), one can get the stresses in frozen surrounding rock (i.e., r b r2):

104:1 2486:9 45 σ r ¼ 0:625 1− 2 þ cos2θ þ 2 þ 3:27 4 r r r

2486:9 6:55 σ θ ¼ −0:625 1 þ cos2θ þ 2 þ 0:48 r4 r

52:1 2486:9 τrθ ¼ −0:625 1 þ 2 − sin2θ r r4

)

Substituting the expression of stresses in the surrounding rock after tunnel is frozen and the values of c and φ into Eq. (36a), one can get:

6:534 65:1 1554:3 5:76 þ 4 1:875− 2 þ 0:625− 2 þ cos2θ r r r4

6:534 1554:3 1:875 þ 2 − 0:625 þ cos2θ r r4

2

46:026 65:1 3108:6 2 þ 1:25 þ − sin2 2θ: ¼ 0:954− r2 r2 r4 From the above equation, the values of r for different values of θ can be obtained, which are given in Table 1. For θ = 0 0 (i.e., around tunnel vault and tunnel invert), the plastic zone is the largest and its boundary of the plastic zone is r = 8.739 m, which indicates that the largest boundary is 3.239 m from the elastic to plastic zones after surrounding rock is frozen. While for θ ≥ 34.1°, the radius of the plastic zone is less than 5.500 m, which indicates that the surrounding rock is still at the elastic state. Similarly, the plastic zone before tunnel becomes frozen can be obtained from Eq. (36a), which is shown in Table 2; the radius of the plastic zone is less than 5.500 m, which means no rock reaches the plastic state. Comparing Table 1 with Table 2, the results show that around tunnel vault and tunnel invert come into the plastic zone after the tunnel is frozen. 5.5. Solution of stresses in the plastic zone From Eq. (24), one can get: − 1 þ 52:1 − 2486:9 sin2θ r2 r4 i tg2ϕ ¼ h : 10:45 52:1 2486:9 − 1− r2 þ r4 cos2θ −2:23 r2

:

)

From Eq. (26), the stresses in plastic zone can be obtained:

Assuming the frost rock was initially in the elastic state before the plastic zone was developed, according to Eq. (22b), the stresses in unfrozen surrounding rock (i.e., r ≥ r2) is:

104:1 2486:9 158:4 σ r ¼ 0:625 1− 2 þ cos2θ þ 1:875 þ 2 4 r r r

2486:9 107 σ θ ¼ −0:625 1 þ cos2θ þ 1:875− 2 r4 r

52:1 2486:9 τrθ ¼ −0:625 1 þ 2 − sin2θ r r4

5.4. Solution of the plastic zone

)

:

104:1 2486:9 45 σ pr ¼ 0:625 1− 2 þ cos2θ þ 2 þ 3:27 r r4 r h i 0:293 2 p p σ 3 ¼ σ r −2:896 1− cos ϕ 1−0:707 cos2ϕ h i 1:707 2 p p σ 1 ¼ 5:791 þ σ r −2:896 1− cos ϕ 1−0:707 cos2ϕ σ p þ σ p3 σ p1 −σ p3 þ cos2ϕ ¼ σ p1 þ σ p3 −σ pr σ pθ ¼ 1 2 2 σ p −σ p3 sin2ϕ τprθ ¼ 1 2

:

The plastic stresses at tunnel vault where r = 6m, 7m, 8m before and after the tunnel becomes frozen are shown in Tables 3 and 4, re-

Table 1 Values of r and θ after freezing. θ (°)

0

10

20

30

34.1

40

50

60

70

80

90

r (m)

8.739

8.442

7.576

6.156

5.500

4.752

4.071

3.746

3.579

3.500

3.476

Table 2 Values of r and θ before freezing. θ (°)

0

10

20

30

34.1

40

50

60

70

80

90

r (m)

4.879

4.853

4.763

4.602

6.593

4.416

4.249

4.122

4.034

3.984

3.968

G.Y. Gao et al. / Cold Regions Science and Technology 72 (2012) 50–57

57

Acknowledgments

Table 3 Plastic stresses in surrounding rock of tunnel before freezing. r (m)

σr(MPa)

σθ(MPa)

τrθ(MPa)

σ1(MPa)

σ3(MPa)

6 7 8

1.711 1.686 1.761

0.232 0.736 0.9726

0 0 0

1.711 1.686 1.761

0.232 0.736 0.973

This work has been supported by the National Natural Science Foundation of China (project number: 51178342), Shanghai Leading Academic Discipline Project (project number: B308) and Kwang-Hua Fund for College of Civil Engineering, Tongji University. References

Table 4 Plastic stresses in surrounding rock of tunnel after freezing. r (m)

σr(MPa)

σθ(MPa)

τrθ(MPa)

σ1(MPa)

σ3(MPa)

6 7 8

4.54 4.13 3.99

32.24 29.85 29.04

0 0 0

32.241 29.85 29.04

4.54 4.13 3.99

spectively. From the comparison, it can be found that the plastic stresses in the surrounding rock of tunnel greatly increase due to the effect of frozen rock. Take the case of r = 6 m for instance, the plastic stress in the surrounding rock of tunnel after the rock becomes frozen is 2.6 times greater than that before the rock becomes frozen. Therefore, the frost heaving force should be greatly concerned in the stability analyses of tunnel in the cold regions. Additionally, the illustrated results show that the plastic stress at the point of 6 m above the tunnel can reach 4.54 MPa after frost heaving. This value is slightly larger than the biggest stress (4.23 MPa) reported in Lai et al. (2000a), in which the frost rock was regarded as a visco-elastic material. The difference between plastic solution and visco-elastic solution may come from that the rock in the plastic zone experiences hardening after the yield criterion is reached. The hardening effect can be taken into account in the proposed method for studying cold-region tunnels.

6. Conclusions An attempt has been made in this study to investigate the plastic zone and the stress distribution in the plastic zone in the rock surrounding the tunnels. Firstly, an analytical solution for the frost force in cold-region tunnels is derived using the continuity equation. Thereafter, a semi-analytical method to calculate the stress in plastic zone in the rock surrounding cold-region tunnels is proposed by assuming that the frost rock was initially in the elastic state before the plastic zone was developed. Also, based on the approximate method for determining plastic zone proposed by Kastner, method for calculating the plastic zone in cold region tunnel is presented. Finally, with one illustrated example, it indicates that tremendous frost heaving forces can be induced by the frozen rock, and such frost heaving force expands the plastic zone in the surrounding rock of tunnel remarkably, which exert a greatly unfavorable inﬂuence on the stability of tunnel in cold regions. Therefore, the frost force should be of great concern in the design of cold region tunnels. Also, determination of the stress in the plastic zone in the rock surrounding the tunnels and its corresponding plastic zone are important for the stability analyses of cold-region tunnel. The rock in the plastic zone has hardening effect after reaching yield criterion, hardening effect can be taken into account in the study of cold-region tunnel with proposed method in this study. It is hoped that the study reported here may provide some theoretical insight in design computation and solve the practical engineering problems.

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