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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 Because loading and unloading under different change: 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 700 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 701 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. 702 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 703 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 construction principle of adjacent tunnel. 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 Wu Mingfang graduated in Civil 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. 704

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