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# Analysis of couple problem of ground stress and temperature stress by alicejenny

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```									World Tunnel Congress 2008 - Underground Facilities for Better Environment and Safety - India

Analysis of couple problem of ground stress and temperature stress fields in
tunnel of fissured surrounding rock in cold region

QIU Wenge, SUN Bing, ZHANG Huijian & WU Mingfang
School of Civil Engineering, Southwest Jiaotong University, China

SYNOPSIS: The mathematical mechanical model and governing differential equation of the coupled
problem of stress and temperature fileds with and without phase change are presented base on plastoelasticity
and heat transfer theory, and the corresponding finite element formulae are derived. Take Queershan Tunnel
of Sichuan-Tibet Highway as the project background, the value, distribution regularities and influencing
factors of frost heaving stress in bending wall lining of fissured surrounding rock tunnel under ground stress
and temperature stress together action are studied by finite element method. The results show that frost
heaving stress is related to the embedded depth, fissure’s distance, fissure’s width, fissure’s angle, freezing
depth and distance between fissure and lining. Fissure’s angle is a main effect on the distribution of frost
heaving stress. In the same case, the embedded depth, fissure’s width and freezing depth are larger, and the
fissure’s distance and distance between fissure and lining are smaller while the frost heaving stress is larger,
and the largest frost heaving stress is the most when fissure’s angle is 45°, and it is the least when fissure’s
angle is 90°.
Key words: tunnel in cold region; fissured surrounding rock; couple; frost heaving stresses; frost heaving
vector; finite element method;

1.   INTRODUCTION                                              And take the bending wall lining for example, the
value, distribution regularities and influencing
Tunnel freeze injure has been a major problem to be            factors of frost heaving stress are studied by finite
resolved of tunnel projects in cold region, the freeze         element method. It will provide theoretical basis for
injure will threat the safety of tunnel structure and          the tunnels support principle and structure design in
operations,     produce     great     difficulties  in         cold and high altitude region, so it has strong
maintenance, and result in serious economic losses.            engineering and academic significance.
In recent years, there are a large number of
scientific research results of tunnel in cold region.          2.    CALCULATION PRINCIPLE
Lai Yuanming[1], Zhang Xuefu[2] have done the
nonlinear analysis of two fields and three fields              2.1 Governing differential equation and the finite
coupling problems of tunnel in cold region, and                    element formulae of stress field
researched the influence of seepage field and stress           When materials (surrounding rock and lining) are in
field on temperature field. Zhang Dehua[3], Lai                the state of flexibility, their constitutive relations are
Yuanming[4] researched elasticity solution and                 generalized Hooke law, using strain to represent
viscoelastic solution of frost heaving stress in               stress as follows:
perfect state. Wang Jianyu[5] analysed the frost
...(1)
heaving stress of tunnel liner. Zhang Xuefu[6], Yan            σ ij = 2Gε ij + 3λ ε mδ ij
Qixiang[7] studied the temperature field and heat
insulation layer of tunnel in cold region.                          In order to simulate rock excavation sequence,
using constitutive relations in the form of
The use of groundwater drainage and heat                  incremental:
insulation layer can’t eliminate frost heaving stress
completely, so it must be considered in the design.            d σ ij = 2G d ε ij + 3λ d ε mδ ij                   ...(2)
In this paper, the mathematical mechanical models
of the coupled problem of stress and temperature                   In this formula, G is shearing modulus and
fields with and without phase change are presented             equals to E/2(1 + v) λ s Lame constants which equal
base on plastoelasticity and heat transfer theory.

699
to Eν (1 + ν )(1 − 2ν ) ; ε m is                    volumetric          varying with temperature. The expression of
enthalpy is:
strain; δ ij is Kronecker symbol.
H = ∫ ρ C (T )dT                       ...(8)
Constitutive relation is expressed in the form                      and,
of matrix as follows:                                                   ∂H         ∂H ∂T              ∂T
=     ⋅   = ρ C (T ) ⋅                     ...(9)
∂t        ∂T ∂t              ∂t
{dσ } = ⎡ D e⎤ {d ε }
⎣ ⎦
...(3)
So we can get the governing differential
In this formula, [De] is elastic stiffness matrix.                 equation of transient temperature field with phase
laws, it usually only can build stress and strain                              ⎛ 2T   2   2 ⎞
incremental relations in the plastic state. The matrix                  ∂H
= λ⎜∂ + ∂        ⎟
T ∂ T                             ...(10)
+
expression of elastic-plasticity stress-strain relations                 ∂t    ⎜∂ 2 ∂ 2 ∂ 2⎟
are as follows:                                                                ⎝ x    y   z ⎠

(               )
{dσ } = ⎡ D ep ⎤ {dε } = ⎡ D e⎤ − ⎡ D p⎤ {d ε}
⎣      ⎦         ⎣ ⎦ ⎣ ⎦                  ...(4)                    On the interface of surrounding rock, liner,
water and ice, they should satisfy the continuity

{}
T                                                       condition:
∂Φ
⎡ D e⎤ ⎡ D e⎤ ⎧ ⎫
∂Q
⎣ ⎦ ⎣ ⎦ ⎨∂σ ⎬ ⎩ ⎭
T I = T II = T m ,
⎛ λ ∂T ⎞ = ⎛ λ ∂T ⎞   ...(11)
⎡ ⎤ = ∂σ                                                                                           ⎜ ⎟ ⎜ ⎟
{}
...(5)
⎣D p⎦
⎝ ∂n ⎠ I ⎝ ∂n ⎠ II
T
∂Φ
⎡ De⎤ ⎧ ⎫
∂Q
A+              ⎣ ⎦ ⎨∂σ ⎬
⎩ ⎭                                          Where, Tm is interface temperature, as a
∂σ
function of time.
Substitute formula (5) into formula (4):                                     The initial temperature condition is:

{}
⎛            T                   ⎞
⎜         ∂Φ                  ∂Q ⎟                                                    = T 0 ( x, y, z )          ...(12)
⎜              ⎡ D e⎤ ⎡ D e⎤ ⎧ ⎫ ⎟
⎣ ⎦ ⎣ ⎦ ⎩∂σ ⎭ ⎨ ⎬
T
t =0
{dσ } = ⎜ ⎡ De⎤ −                        ⎟
∂σ                                                           The first boundary condition is:
⎟ {d ε }
...(6)

{}
⎜⎣ ⎦              T
⎜                                                                      T =Ta                                     ...(13)
∂Φ            ⎧∂Q ⎫ ⎟
⎡ ⎤⎨ ⎬ ⎟
⎜         A+         ⎣ De⎦ ⎩∂σ ⎭ ⎟
⎜                                                                      The second boundary condition is:
⎝            ∂σ                  ⎠
∂T
In this formula,         [D ]
p
is plasticity stiffness                λ
∂n
=0                             (14)
matrix; [Dep ] is elastic-plasticity stiffness matrix;                         The third boundary condition is:
Φ is yield function; Q is plastic potential; A is
harden parameter of strain.
∂n
λ
∂T
= h T a −T     (              )  ...(15)

2.2 Governing differential equation and the finite                           Here, Ta is atmosphere temperature; h is the
element formulae of temperature field                               film coefficient of convection.
For the transient temperature fields without
The thermal conduction governing differential                     and with phase change, the finite element
equation of three-dimensional transient temperature                     calculation formula can be got respectively by using
fields is as follow:                                                    the weighted residual and Galerkin methods as
⎛ 2T     2      2 ⎞                                         follow:
∂T
= λ⎜∂                  ⎟
T ∂ T
ρC               +∂     +                      ...(7)
[C ] ⎧
⎜∂  2      2     2
∂z ⎟                                                             ∂T ⎫
⎬ + [ K ]{T } = {F}
∂t            ∂y
⎝ x                ⎠                                                    ⎨                                    ...(16)
Here, T, ρ, C, λ indicate material temperature,                                    ⎩ ∂t ⎭
density, mass specific heat capacity and coefficient
[C ] ⎧
∂H ⎫
of thermal conductivity respectively.                                               ⎨        ⎬ + [ K ]{T } = {F}         ...(17)
After tunnel excavated, the fissure water froze                                   ⎩ ∂t ⎭
with process of phase change, and it released or
absorbed latent heat. The latent heat will be                               Here, [C] is transient temperature matrix; [K]
considered by defining the value of enthalpy                            is temperature stiffness matrix; {T} is node

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temperature vector; {F} is external load / boundary             freezing depth is greater than 150 cm, the tunnel site
conditions matrix .                                             belongs to typical arctic-alpine climate. According
to the geological drilling, the surrounding rocks are
2.3 Analysis of frost heaving stresses                          mainly Yanshan period granite, and they are great
thickness, hard rock.
Supposing that there exist N cracks in the
surrounding rock, take fissure Ni for study, the                3.2 Finite element model, calculation condition
increment strain of jointer plane is:                               and material parameters

{d ε i} = {d ε i'} + {d ε i''}
The finite element model is shown as Figure 1, the
...(18)         boundary of left, right and bottom are all 40m. Left
and right directions are restricted by horizontal

{ ε}
restraint, the base fixed by the vertical restraint, and
'                                               the surface is free. Tunnel lining and stratum layer
Here,    d i is the strain increment of fissure
use PLANE13 coupling unit to simulate, which has

{ ε}''                    the multi-DOF of structure displacement and
water in course of freezing,            d i     is the          temperature. Figure 2 is the diagrammatic sketch of
the distribution of fissure. In the following text, we
compressive strain increment of ice pressed by the              use H, D, W, β, R, L and σf to represent the
surrounding rock in course of freezing. The strain              embedded depth, fissure’s distance, fissure’s width,
increment of contact area           {d ε ii}   between          fissure’s angle, freezing depth, distance between
fissure and lining and frost heaving stress
surrounding rock and lining can be obtained from                separately. In the course of calculating, the value of
strain increment of jointer plane      {d ε i} , so the         H are 6, 12, 36m, the value of D are 0.5, 1, 2 m, the
value of W are 0.001, 0.005, 0.01m, the value of β
total strain increment of contact area caused by frost          are 30, 45, 90°, so the calculation conditions are 81
heaving of all the fissure water is:                            kinds. The preliminary support use gunite concrete
C20, 15 cm thickness, use C30 reinforced concrete

{d ε k} = i∑1{d ε ii}                                           for second lining, thickness of 25 cm, material
N
...(19)         parameters are shown in Table 1. We consider the
=                                                    phase change latent heat through the definition of
And the stress increment is:                               material enthalpy which changes with temperature,
when water is at -1,0 and 1 oC, the value of enthalpy

{d σ k} = [ D]{d ε k}                           ...(20)
are 37.8e6, 79.8e6, 121.8e6J/m3 separately
compared to the value of -10 oC, and the enthalpy
value on other points will be got by automatic linear
When lining in a flexible admission stage, use             interpolation.
[De] to instead of [D], in the plastic stage use [Dep].
Then calculate the normal stress component in
lining of stress increment which is frost heaving
stress.

3.   EXAMPLE OF ANALYSIS

3.1 Project background

Queershan Tunnel of Sichuan-Tibet Highway on
State Road 317 line belongs to middle-low latitude
and high-altitude cold region, where the climate is
cold. On east and west side, the year average
temperature is 3.2 oC ~ -15.2 oC, when the elevation
vary from 3,800 to 5,200 meters. The largest                                Figure 1 The finite element model

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Figure 2 Diagrammatic sketch of the distribution of fissure

Table 1 Material parameters

parameter                 Surrounding rock         preliminary support       second lining   water    ice
Young’s modulus E (GPa)                  20                         21                       31              1.28
Poisson coefficient ν                 0.22                       0.2                      0.2               0.3
Friction angle
55
φ (°)
Cohesion c (MPa)                      1.8
Density ρ (kg·m-3)                   2600                      2300                      2500     1000    913
Frost heaving ratio η                                                                               9%
Thermal conductivity (W·m-
2.6                       1.51                     1.51     0.57    2.22
1·K-1)
Specific heat C (J·kg-1·K-1)             630                        840                      840      4200    2100

3.3 Calculated load and construction procedure                    3.4 The calculation results and analysis
In the construction stage, the main load is the                   The following basic conclusions have been got from
gravity of stratum layer and concrete lining. In the              the numerical simulation analysis:
phase of frost heaving, we define the initial                          (1) Fissure’s angle is a main effect on the
temperature of material on the basis of construction                       distribution of frost heaving stress. As
phase first, then impose temperature convective                            shown in Figure 3, it is a diagram of frost
load on the contact boundary between lining and                            heaving stress distribution under the
atmosphere. The initial temperature is 1oC, air                            condition of that H is 6m, D is 0.5m, W is
0.01m. When β is 45°, the larger frost
temperature is -10oC, and the convection heat
heaving stress occur near the foot of side
transfer coefficient of lining and atmospheric is                          wall and arch crown, and the smaller frost
15W·m-2·K-1. The calculation procedure: initial                            heaving stress occur near the inverted
ground stress field simulation → tunnel                                    arch and hance. When β is 60 and 90°, the
Excavation→ application of preliminary support                             frost heaving stress gradual increase from
→application of secondary lining →mass of rock                             arch crow to the foot of side wall, and the
smaller frost heaving stress occurs near
frost heaving process simulation.

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the inverted arch. When β is 45 and 90°,                             σ f ,max (90°).   We can also see that the
the distribution of frost heaving stress is
slope of the curves decrease with the
symmetrical, and when β is 60°, the
increase of R which indicates that L is
skewed pressure phenomenon occurs.
(2) As shown in Figures.4~7, they are the                                larger, while σ f is smaller.

variation of
σ f ,max with H, D, W and β.                  4.   CONCLUSIONS
We can find that, in the same case, H, W
Through the above analysis,              the    following
and R are larger, and D and L are smaller
conclusions are derived:
while σ f is larger, and the compared
(1) When fissure’s angle is 45°, the larger frost
result of σ      f , max   in different fissure’s              heaving stress occur near the foot of side wall
angle is     σ f ,max (45°)> σ f ,max (60°)>                   and arch crown, and the smaller frost heaving

a 45°                                       b 60°                                  c 90°
Figure 3 Diagram of frost heaving stress distribution

Figure 4 Variation of σf,max with H                           Figure 5 Variation of σf,max with D

Figure 6 Variation of σf,max with W                               Figure 7 Variation of σf,max with β

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stress occur near the inverted arch and hance.                 7.   YAN Qixiang, HE Chuan, ZENG Dongyang (2005),
When fissure’s angle is 60 and 90°, the frost                       “Study of temperature field and heat preservation
heaving stress gradual increase from arch crow                      and insulation layer for tunnel in cold area.” Journal
to the foot of side wall, and the smaller frost                     of Sichuan University, 37(3):24-27.
heaving stress occurs near the inverted arch.
When fissure’s angle is 45 and 90°, the                        BIOGRAPHICAL DETAILS OF THE AUTHORS
distribution of frost heaving stress is                                            Qiu Wenge graduated in Tunnel and
symmetrical, and when fissure’s angle is 60°,                                      Underground Railway Engineering
the skewed pressure phenomenon occurs.                                             from      the    Southwest    Jiaotong
(2) In the same case, the embedded depth, fissure’s                                    University in 1982, and obtained a
width and freezing depth are larger, and the                                       Bachelor. He graduated in Bridge and
Tunnel      Engineering    from     the
fissure’s distance and distance between fissure                                    Southwest Jiaotong University in
and lining are smaller while the frost heaving                                     1988, and obtained a Master. Then he
stress is larger.                                                                  obtained a PhD. in Bridge and Tunnel
(3) In the same case, the compared result of the                   Engineering at the University of Southwest Jiaotong in
largest frost heaving stress in different fissure’s            2003, and became a professor. From 1982 to now he
angle                                            is            worked for Southwest Jiaotong University. He specialize
in mechanism research and design of frozen soil tunnel,
σ f ,max (45°)> σ f ,max (60°)> σ f ,max (90°).                design method and application game of single lining and
REFERENCES                                                                          Sun Bing graduated in Civil
1.   LAI Yuanming, WU Ziwang, ZHU Yuanlin (1999),                                   Engineering at the University of
“Nonlinear analyses for the couple problem of                                  Southwest Jiaotong in 2003, and
temperature, seepage and stress fields in cold region                          obtained a Bachelor. He graduated in
Applied Mechanics Engineering from
tunnels.” Chinese Journal of Geotechnical
the Southwest Jiaotong University in
Engineering, 21(5):529-533.                                                    2006, and obtained a Master. Now he is
2.   Zhang Xuefu, Yu Wenbing, LIU Zhiqiang (2006),                                  studying for the PhD. in Bridge and
“Three dimensional nonlinear analyses for coupled                              Tunnel Engineering. He specialize in
problem of seepage field and temperature field of             mechanism research and design of frozen soil tunnel.
cold regions tunnels.” Chinese Journal of
Geotechnical Engineering, 28(9):1095-1100.                                       Zhang Huijian graduated in Civil
3.   ZHANG De hua, WANG Mengshu, TAN                                                  Engineering at the University of
Zhongshen (2003), “Effect of frost heaving on                                    Southwest Jiaotong in 2005, and
tunnel surpporting systems of Fenghuoshan railway                                obtained a Bachelor. Now he is
tunnel.” Chinese Journal of Geotechnical                                         studying for the PhD. in Bridge and
Engineering, 25(5):571-573.                                                      Tunnel Engineering. He specializes in
mechanism research and design of
4.   LAI Yuanming, WU Ziwang, ZHU Yuanlin (1999),                                     tunnel and underground engineering.
“Analytical Viscoelastic Solution for Frost Force of
Cold Regional Tunnels.” Journal of the China
railway society, 21(6):70-74.                                                    Engineering from the University of
5.   WANG Jianyu, HU Yuanfan (2004), “A Discussion                                    Southwest Jiaotong in 2007 and
on Frost –Heaving Force on Tunnel Lining.” Journal                               obtained a Bachelor. Now she is a
of Glaciology and Geocryology, 26(1): 112 119.                                   Postgraduate of Southwest Jiaotong
University. She is studying for a
6.   Zhang Xuefu, Su Xinmin, LAI Yuanming (2004),
master's degree and specialize in
“Nonlinear     analyses    for    three-dimensional
mechanism research and design of
temperature fields in cold-region tunnels.” CHINA                                frozen soil tunnel.
CIVIL ENGINEERING JOURNAL, 37(2):47-53.

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