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Calculation of the surrounding rock pressure on a shallow buried tunnel using linear and nonlinear failure criteria
Automation in Construction 37 (2014) 191–195
Contents lists available at ScienceDirect
Automation in Construction journal homepage: www.elsevier.com/locate/autcon
Calculation of the surrounding rock pressure on a shallow buried tunnel using linear and nonlinear failure criteria Mingfeng Lei, Limin Peng, Chenghua Shi ⁎ School of Civil Engineering, Central South University, Changsha, Hunan 410075, China
a r t i c l e
i n f o
Article history: Accepted 8 August 2013 Available online 11 September 2013 Keywords: Shallow buried tunnel Surrounding rock pressure Terzaghi theory Linear and nonlinear Complex strata Limit equilibrium method
a b s t r a c t A method to calculate the surrounding rock pressure on a shallow buried tunnel was established using linear and nonlinear failure criteria based on Terzaghi failure mode. The modified Terzaghi theory, which was expanded to include complex strata and the corresponding recurrence formulas used to calculate vertical earth pressure, was obtained through theoretical derivation. The surrounding rock in the fractured zone of the tunnel side wall was analyzed as an isolated body using the limit equilibrium method to obtain the explicit calculation expressions of the horizontal earth pressure of a shallow buried tunnel. Case analysis indicates that this method is feasible. The effects of the lateral pressure coefficient of overlying strata and the nonlinear coefficient on the surrounding rock pressure with linear and nonlinear failure criteria are further studied, and results show that the surrounding rock pressure notably decreases with an increase in the lateral pressure coefficient of the overlying strata. Surrounding rock pressure slowly decreases with an increase in the nonlinear coefficient, and the effect of the nonlinear coefficient on horizontal earth pressure is significant. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The load, that is, surrounding rock pressure, is the chief condition that should be determined in the tunnel lining structure design process because surrounding rock pressure is directly related to the parameter design of the supporting structure. Theories on the calculation of the tunnel surrounding rock pressure include the pillar theory, Terzaghi theory, and the main methods recommended in the railway and road tunnel design code of China [1–3]. These methods are mostly dependent on experience or on a half experience stage because of the complexity of the existing environment and the variability of tunnel ground medium. In nearly half a century, much research on this problem was conducted scientifically and technically worldwide with the wide use of tunnel engineering. Main research methods include numerical simulations, site tests, model indoor tests, and limit theories. However, the methods referenced above are limited by environment conditions and technical means, and they possess certain disadvantages. Numerical simulations can simulate the furthest boundary condition and ground environment. However, the load being loaded onto the lining structure cannot be obtained directly. Therefore, forming a simple and convenient design method is difficult. The high cost and low reliability of sensors also limit the use of the site test method in some important projects. Moreover, the discreteness does not provide satisfactory ⁎ Corresponding author at: School of Civil Engineering, Central South University, 22 Shaoshan South Road, Changsha, 410075, China. Tel.: +86 13787232438. E-mail addresses: [email protected] (M. Lei), [email protected] (L. Peng), [email protected] (C. Shi). 0926-5805/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.autcon.2013.08.001
results because of the small data sample and the serious disturbance caused by external factors such as construction. The indoor experiment may be capable of obtaining results with a relatively good change law at a certain degree. However, the difference between experimental conditions and actual projects limits the wide use of this research method. The ultimate analysis method is favored by scientific researchers because of its distinct concept and mature theory. Atkinson et al. have conducted research on the stability of shallow buried tunnel using the upper or lower bound ultimate analysis method [4–6]. Jiang et al. assumed the failure modes and established the velocity field used to improve the ultimate analysis method in tunnel stability analysis based on existing research achievements, directly obtaining the upper bound solution to the surrounding rock pressure of the shallow buried tunnel [7–9]. The achievements referenced above promote the use of the ultimate analysis method in tunnel engineering and have undoubtedly developed basic tunnel theory. However, a general survey of these methods shows that they cannot directly obtain the horizontal load acting on the tunnel lining structure. However, these methods still rely on the assumption that the ratio of horizontal and vertical surrounding rock pressure is a constant. The vertical surrounding rock pressure is first obtained using the ultimate analysis method. The horizontal surrounding rock pressure is a multiple of the constant, which is the direct lateral pressure coefficient. The lateral pressure coefficient, however, is arbitrary and is calculated by either Poisson's ratio or is selected by experience without sufficient theoretical basis. The lateral pressure coefficient is also based on the soil linear failure criteria. Several research achievements [10–13] and engineering practices indicate that most types
192
M. Lei et al. / Automation in Construction 37 (2014) 191–195
p0
0
of soil follow the nonlinear failure criteria. Thus, the linear failure criteria are a special case. In the 1980s, Santarelli et al. found that the linear Mohr–Coulomb strength criteria overestimate the major principal stress in rock–soil yield [12,13]. In this thesis, the surrounding rock pressure of a shallow buried tunnel with a uniform formation and complex strata under linear and nonlinear failure criteria was discussed based on Terzaghi failure mode. In particular, the discussion on the horizontal surrounding rock pressure is expected to supplement and perfect the methods proposed in existing research achievements.
D z
H
C σv
dτ
dz σh
dτ σh
γ Ldz
σv+d σ v
Z
q A
B
2. Basic theory
h
2.1. Terzaghi failure mode Terzaghi theory used to calculate the surrounding rock is based on the limit equilibrium method. The basic concept of Terzaghi theory is that the stratum is seen as a loose body; when the tunnel is excavated, the soil body at the top of the tunnel moves down because of gravity. Two shear planes appear between the sides of the tunnel and the ground surface, and stress transfer occurs as soil particles rip one another. The soil body above the tunnel then hinders the downward movement of the soil, such that the minimum supporting pressure is greater or less than the primary stress of the soil. As a result, the vertical surrounding rock pressure acting on the lining is induced by stress transfer. The failure mode is shown in Fig. 1. 2.2. Linear and nonlinear failure criteria With reference to document [14], the nonlinear failure criteria of soil can be expressed as 1=m
τ ¼ c0 ð1 þ σ n =σ t Þ
ð1Þ
where τ and σn are the shear stress and normal stress of the rupture plane, respectively; c0 is the primary cohesion of soil; σt is the axial tensile stress of soil; and m is the nonlinear coefficient. c0, σt, and m can be obtained through the experiment. If m = 1 according to Eq. (1), the equation will transfer to the linear Mohr–Coulomb yield criteria. The tangential equation of nonlinear failure criteria: τ ¼ σ n tanϕt þ ct
ð2Þ
where ϕt and ct are the rake ratio and intercept of the tangent, respectively. When Eqs. (1) and (2) are simultaneously applied, then ct ¼
m−1 mσ t tanϕt 1=ð1−mÞ c0 þ σ t tanϕt : m c0
ð3Þ
3. Calculation of the Vertical surrounding rock pressure of a shallow buried tunnel 3.1. Uniform formation The vertical equilibrium of a micro thin layer dz at z depth, as shown in Fig. 1, is analyzed.
σ v L þ γLdz ¼ ðσ v þ dσ v ÞL þ 2dτ
θ
θ
ð4Þ
When the micro thin layer generates downward displacement, upward friction dτ is generated at two sides because of the existence
B L Fig. 1. Terzaghi failure mode.
of initial horizontal stress. If the surrounding rock material satisfies the nonlinear Mohr–Coulomb yield criteria, then dτ ¼ ðσ h tanϕt þ ct Þdz
ð5Þ
where σh, calculated by σh = λσv, is the horizontal stress at depth z; and λ is the lateral pressure coefficient of the overlying stratum, which is calculated by λ = 1.0 to 1.5 as suggested by the Terzaghi experimental results. The lateral pressure coefficient in this study is notably different from the normal concept of the horizontal supporting force and vertical supporting force ratio. In Terzaghi theory, the lateral pressure coefficient of the overlying stratum is the stress ratio between the horizontal and vertical stresses above the tunnel roof, whereas the lateral pressure coefficient used to calculate the horizontal surrounding rock pressure of the tunnel is computed according to Rankine active earth pressure theory. Substituting Eq. (5) into Eq. (4) and rewriting the equation result in dσ v 2λ tanϕt 2c þ σ v ¼ γ− t dz L L
ð6Þ
where L = B + 2h tan θ; B is the excavation breadth of the tunnel; and θ is the rupture angle of the surrounding rock, which is usually calculated by θ = π/4 − ϕt/2. Eq. (6) determines the vertical stress at any depth σ v ðzÞ ¼
2λ tanϕ γL−2ct − L tz 1 þ Ae 2λ tanϕt
ð7Þ
where A is an undetermined parameter determined according to the boundary condition, which is evidently σv(z)|z = 0 = p0; and p0 is ground load, such as the overload of the pavement when the urban metro tunnel passes through the road, such that A¼
2p0 λ tanϕt −1: γL−2ct
ð8Þ
As a result,
σ v ðzÞ ¼
2λ tanϕ 2λ tanϕ γL−2ct − L tz − L tz 1−e : þ p0 e 2λ tanϕt
ð9Þ
M. Lei et al. / Automation in Construction 37 (2014) 191–195
0
As regards the usual mountain tunnel, p0 = 0, thereby σ v ðzÞ ¼
193
2λ tanϕ γL−2ct − L tz 1−e : 2λ tanϕt
p0
ð10Þ
pi-1
z The vertical pressure acting on the top of tunnel AB is obtained using Eq. (10)
dτ
2Hλ tanϕ γL−2ct − L t q ¼ σ v ðzÞz¼H ¼ 1−e 2λ tanϕt
σh
ð11Þ
Z
where H is the buried depth of the tunnel. This equation is also the computational formula of vertical surrounding rock pressure of the tunnel, as solved using Terzaghi theory.
m m m m m X X X X X γ jh j= h j ; ctp ¼ ctj h j = h j ; ϕtp ¼ ϕtj h j = hj
j¼1
dz
γ Ldz
dτ σh Hi
σv+d σ v
Fig. 3. Force analysis of micro-body in the ith strata.
Terzaghi theory is processed based on uniform formulation. However, a stratum is composed of multilayer rock and soil in practical engineering. Therefore, the expansion of Terzaghi theory in uniform formation to include complex strata is necessary. The failure mode shown in Fig. 2 is established according to Terzaghi basic theory. The overlying stratum is divided into a total of n layers, with Hi being the thickness of each soil layer, γi being volume weight, ϕti being the internal friction angle, cti being cohesion, and i = 1, ⋯, n. The strata through which the tunnel barrel passed is divided into a total of m layers, with hj being the thickness of each soil layer, γj being volume weight, ϕtj being the internal friction angle, ctj being cohesion, and j = 1, ⋯, m. Given that the research object is a shallow buried tunnel, the surrounding rock strengths of the strata through which the tunnel barrel passed are not far from one another. Therefore, the weighted mean method used to deal with the mechanical parameters of the strata through which the tunnel barrel passed is adopted to simplify the calculation m X
σv
Z
3.2. Complex strata
γp ¼
z
j¼1
j¼1
j¼1
j¼1
ð12Þ
j¼1
where γp, ctp, ϕtp are the weighted density, cohesion, and internal friction of the strata through which the tunnel barrel passed, respectively. The corresponding weighted rupture angle θp = π/4 − ϕtp/2, according to the related flow rule [15].
p0 C
0 D
1
H1
z n layers in all
i
H
Hi
Z q
n
Hn
A
B
θ
p
θp
m layers in all
h
B L Fig. 2. Modified Terzaghi failure mode for complex strata.
The acting force loading first layer soil onto the second layer is evidently obtained based on Terzaghi theory with uniform formation and Eq. (9).
p1 ¼
2λ tanϕ 2λ tanϕ γ 1 L−2ct1 − 1 L t1 H1 − 1 L t1 H 1 1−e þ p0 e 2λ1 tanϕt1
ð13Þ
The vertical acting force loading i–1th layer soil onto the ith layer is considered overload, which can be obtained through a derivation of the uniform formation method when the ith layer soil is assumed to be a micro thin layer, as shown in Fig. 3.
pi ¼
2λ tanϕ 2λ tanϕ γi L−2cti − i L ti H i − i L ti Hi 1−e þ pi−1 e 2λi tanϕti
ð14Þ
Thus, the vertical surrounding rock pressure pn acting on the tunnel vault can be easily obtained. q ¼ pn ¼
2λ tanϕ 2λ tanϕ γn L−2ctn − n L tn Hn − n L tn Hn 1−e þ pn−1 e 2λn tanϕtn
ð15Þ
Eq. (15) is the calculation expression of the vertical surrounding rock pressure of a shallow buried tunnel under the complex strata condition. 4. Calculation of the horizontal surrounding rock pressure of a shallow buried tunnel The triangle slider at tunnel side was taken into force analysis, assuming that the resultant supporting resistance supplied by the supporting structure that maintains the stability of tunnel surrounding rock is F. The acting direction between the resultant supporting resistance and vertical line of the triangle is undoubtedly θ + ϕt according to the related flow rule, as shown in Fig. 4. Based on the static equilibrium condition,
qh tanθ þ G−N sinθ−T cosθ− F cosðθ þ ϕt Þ : F sinðθ þ ϕt Þ þ T sinθ−N cosθ ¼ 0
ð16Þ
hj The rupture plane satisfies the nonlinear Mohr–Coulomb yield criteria with T ¼ N tanϕt þ ct l where l is the length of the rupture plane, l = h/cos θ.
ð17Þ
194
M. Lei et al. / Automation in Construction 37 (2014) 191–195
q
F h
θ
+
t
G
T
θ
l N
Fig. 4. Force analysis of side wall fractured zone.
When Eqs. (16) and (17) are simultaneously applied and solved, we derive ðqh tanθ þ GÞð cosθ− tanϕt sinθÞ−ct l cosϕt þ tanϕt sinϕt
F¼
ð18Þ
where G is the self-weight of the triangle rupture body, which is calculated as G = γh2 tan θ/2. Thus, the horizontal supporting resistance to maintain the stability of surrounding rock needs the supporting structure. Eq. (19) solves for the horizontal supporting resistance and clearly shows that the supporting resistance is the horizontal load of the structure design, where θ, which is calculated using θ = π/4 − ϕt/2, is the rupture angle of the surrounding rock if the strata through which the tunnel barrel passed are uniform formations. However, if the strata through which the tunnel barrel passed are complex strata, then θ = π/4 − ϕtp/2, and ϕtp is computed using Eq. (12). F x ¼ F sinðθ þ ϕt Þ ¼
ðqh tanθ þ GÞð cosθ− tanϕt sinθÞ−ct l cosϕt þ tanϕt sinϕt sinðθ þ ϕt Þ
ð19Þ
If the horizontal load distributes as the line moves along the vertical direction, then the horizontal surrounding rock pressure in the depth range of the tunnel is e¼
ðqh tanθ þ GÞð cosθ− tanϕt sinθÞ−ct l sinðθ þ ϕt Þ : cosϕt þ tanϕt sinϕt h
ð20Þ
5. Case study
[8,9] is analyzed and compared with the current calculation method as an example. The buried depth of tunnel H = 20 m, tunnel span B = 10 m, height h = 10 m, volume weight of surrounding rock γ = 20 kN/m3, internal friction angle ϕ0 = 18°, cohesion c0 = 10 kPa, and the lateral pressure coefficient of overlying strata λ = 1.0 to 1.6. The computed results based on Eqs. (11) and (20) and the corresponding results in reference document [8,9] under the linear failure criteria, that is, m = 1.0, are listed in Table 1 and shown in Fig. 5. The calculation method for the vertical surrounding rock pressure is similar to Terzaghi theory. Therefore, the comparative analysis is only performed on the horizontal load. The results of the analysis are as follows. The change laws of the horizontal resistance maintaining the stability of the tunnel side wall differ among the referenced calculation methods when the lateral pressure coefficient of the overlying strata is increased. The results calculated using the method derived in this study are consistent with the results of Terzaghi theory: horizontal resistance decreases when the lateral pressure coefficient increases. The results in document [8,9] are similar. However, horizontal resistance increases when the lateral pressure coefficient increases. When the lateral pressure coefficient is rather small, the results calculated using the method in the current study are closer to the results of document [9], and when the lateral pressure coefficient is rather big, the results calculated using the method in the current study are closer to document [8]. The results of the analysis by synthesis, the change law, and the values calculated using the method in the current study are closest to Terzaghi theory and fall in between other methods. Therefore, the method used in this study is feasible. 6. Effect of the nonlinear failure criterion To discuss further the effect of the nonlinear failure criteria on surrounding rock pressure, a calculation using the nonlinear pressure coefficient m = 1.1 to 1.4 is considered and conducted based on reference document [8]. The axial tensile strength of the surrounding rock σt = 30 kPa, whereas the other calculation parameters are similar to the ones referenced in Section 5. The results are shown in Figs. 6 and 7. The analysis of these results is as follows. Under the nonlinear failure criteria, surrounding rock pressure significantly decreases when the lateral pressure coefficient increases, which is consistent with the result under the linear failure criteria. Therefore, the selection of the lateral pressure coefficient of overlying strata in calculating the surrounding rock pressure is significant under both linear and nonlinear failure criteria. The surrounding rock pressure decreases along with the nonlinear coefficient when the lateral pressure coefficient is fixed. However, the reduction amplitude is minimal. The effect of the nonlinear coefficient on horizontal surrounding rock is slightly stronger.
To test and verify the reliability of the calculation method of the horizontal surrounding rock pressure of the shallow buried tunnel derived as previously mentioned, the project in reference document
The paper method
Terzaghi method
Method of [8]
1.0 1.1 1.2 1.3 1.4 1.5 1.6
141.5 138.8 136.1 133.6 131.2 128.8 126.5
163.6 – 156.2 – 149.4 – 142.9
115.0 122.8 129.9 136.6 142.9 148.9 –
160
e /kPa
Table 1 Results of horizontal earth pressure comparison [kPa]. λ
200
Method of [9] Mode A
Mode B
145.9 – 159.3 – 169.3 – 177.2
145.8 – 160.9 – 175.7 – 190.2
120 the paper method Terzaghi method method of [8] mode A mode B
80 40 0 1.0
1.1
1.2
1.3 λ
1.4
1.5
1.6
Fig. 5. Change of horizontal earth pressure to lateral pressure coefficient of overlying strata.
M. Lei et al. / Automation in Construction 37 (2014) 191–195
b) horizontal earth pressure m=1.1 m=1.2 m=1.3 m=1.4
300
q /kPa
290 280 270
m=1.1 m=1.2 m=1.3 m=1.4
140
e /kPa
a) vertical earth pressure
195
260
135 130 125
250 1.1
1.2
1.3
1.4
1.5
120 1.1
1.6
1.2
1.3
λ
1.4
1.5
1.6
λ
Fig. 6. Change of earth pressure to lateral pressure coefficient using nonlinear failure criterion.
a) vertical earth pressure
b) horizontal earth pressure
=1.1 =1.5
=1.2 =1.6
=1.3
=1.0 =1.4
300
145
290
140
e /kPa
q /kPa
=1.0 =1.4
280 270
=1.2 =1.6
=1.3
1.3
1.4
135 130 125
260 250
=1.1 =1.5
1.1
1.2
1.3
1.4
120 1.1
m
1.2
m
Fig. 7. Change of earth pressure to nonlinear coefficient using nonlinear failure criterion.
7. Conclusions
References
A method to calculate the surrounding rock pressure of a shallow buried tunnel under linear and nonlinear failure criteria is presented based on Terzaghi failure mode. Terzaghi theory, which was limited to uniform formations, is expanded to include complex strata. A limit equilibrium analysis of the surrounding rock bodies in a tunnel side rupture zone is conducted, and the computational expressions of the horizontal surrounding rock pressure are obtained by theoretical derivation. The case study indicates that the method presented in this paper is feasible and perfects the insufficiency of the exit theory, which calculates horizontal surrounding rock pressure by setting the vertical surrounding rock pressure as a multiple of a direct constant. The value of the lateral pressure coefficient is significant in obtaining the surrounding rock pressure of the tunnel under linear and nonlinear failure criteria. When the value of the lateral coefficient is greater, the surrounding rock pressure becomes smaller. The surrounding rock pressure gradually decreases when the nonlinear coefficient increases under the nonlinear failure criterion, and the effect on horizontal load is slightly stronger.
[1] Ministry of Railways of the People's Republic of China, Code for Design on Tunnel Of Railway (TB 10003-2005), China Railway Publishing House, Beijing, 2005. [2] Professional Standards Compilation Group of People's Republic of China, Code for Design of Road Tunnel (JTG D70-2004), China Communications Press, Beijing, 2004. [3] Limin Peng, Xiaobin Liu, Tunnel Engineering, Central South University Press, Changsha, 2009. [4] J.H. Atkinson, D.M. Potts, Stability of a shallow circular tunnel in cohesion-less soil, Geotechnique 27 (1977) 203–215. [5] E.H. Davis, M.J. Gunn, R.J. Mair, H.N. Seneviratne, The stability of shallow tunnels and underground openings in cohesive material, Geotechnique 30 (1980) 397–416. [6] E. Leca, L. Dormieux, Upper and lower bound solutions for the face stability of shallow circular tunnels in frictional material, Geotechnique 40 (1990) 581–606. [7] Jiang Gongliang, Limit analysis of the stability of shallow tunnels in soft ground, China Civ. Eng. J. 31 (1998) 65–72. [8] Yang Xiaoli, Wang Zuowei, Limit analysis of earth pressure on shallow tunnel using nonlinear failure criterion, J. Cent. South Univ.: Sci. Technol. Ed. 41 (2010) 299–302. [9] Yang Feng, Yang Junsheng, Limit analysis method for determination of earth pressure on shallow tunnel, Eng. Mech. 25 (2008) 179–184. [10] P.V. Lade, Elasto-plastic stress–strain theory for cohesionless soil with curved yield surface, Int. J. Solids Struct. 13 (1977) 1019–1035. [11] E. Hoek, E.T. Brown, Empirical strength criterion for rock masses, ASCE J. Geotech. Eng. Div. 106 (1980) 1013–1035. [12] J.G. Agar, N.R. Morgenstern, J.D. Scott, Shear strength and stress–strain behavior of Athabasca oil sand at elevated temperatures and pressure, Can. Geotech. J. 24 (1987) 1–10. [13] F.J. Santarelli, Theoretical and Experimental Investigation of the Stability of the Axisymmetric Wellhole, University of London, London, 1987. [14] X.J. Zhang, W.F. Chen, Stability analysis of slopes with general nonlinear failure criterion, Int. J. Numer. Anal. Methods Geomech. 11 (1987) 33–50. [15] Lei Mingfeng, Peng Limin, Shi Chenghua, Research on construction spatial effects in large-long-deep foundation pit, Rock Soil Mech. 31 (2010) 1579–1584(+1596).
Acknowledgment Financial support from the National Basic Research Program of China (973 Program: 2011CB013802) is gratefully acknowledged.
Contents lists available at ScienceDirect
Automation in Construction journal homepage: www.elsevier.com/locate/autcon
Calculation of the surrounding rock pressure on a shallow buried tunnel using linear and nonlinear failure criteria Mingfeng Lei, Limin Peng, Chenghua Shi ⁎ School of Civil Engineering, Central South University, Changsha, Hunan 410075, China
a r t i c l e
i n f o
Article history: Accepted 8 August 2013 Available online 11 September 2013 Keywords: Shallow buried tunnel Surrounding rock pressure Terzaghi theory Linear and nonlinear Complex strata Limit equilibrium method
a b s t r a c t A method to calculate the surrounding rock pressure on a shallow buried tunnel was established using linear and nonlinear failure criteria based on Terzaghi failure mode. The modified Terzaghi theory, which was expanded to include complex strata and the corresponding recurrence formulas used to calculate vertical earth pressure, was obtained through theoretical derivation. The surrounding rock in the fractured zone of the tunnel side wall was analyzed as an isolated body using the limit equilibrium method to obtain the explicit calculation expressions of the horizontal earth pressure of a shallow buried tunnel. Case analysis indicates that this method is feasible. The effects of the lateral pressure coefficient of overlying strata and the nonlinear coefficient on the surrounding rock pressure with linear and nonlinear failure criteria are further studied, and results show that the surrounding rock pressure notably decreases with an increase in the lateral pressure coefficient of the overlying strata. Surrounding rock pressure slowly decreases with an increase in the nonlinear coefficient, and the effect of the nonlinear coefficient on horizontal earth pressure is significant. © 2013 Elsevier B.V. All rights reserved.
1. Introduction The load, that is, surrounding rock pressure, is the chief condition that should be determined in the tunnel lining structure design process because surrounding rock pressure is directly related to the parameter design of the supporting structure. Theories on the calculation of the tunnel surrounding rock pressure include the pillar theory, Terzaghi theory, and the main methods recommended in the railway and road tunnel design code of China [1–3]. These methods are mostly dependent on experience or on a half experience stage because of the complexity of the existing environment and the variability of tunnel ground medium. In nearly half a century, much research on this problem was conducted scientifically and technically worldwide with the wide use of tunnel engineering. Main research methods include numerical simulations, site tests, model indoor tests, and limit theories. However, the methods referenced above are limited by environment conditions and technical means, and they possess certain disadvantages. Numerical simulations can simulate the furthest boundary condition and ground environment. However, the load being loaded onto the lining structure cannot be obtained directly. Therefore, forming a simple and convenient design method is difficult. The high cost and low reliability of sensors also limit the use of the site test method in some important projects. Moreover, the discreteness does not provide satisfactory ⁎ Corresponding author at: School of Civil Engineering, Central South University, 22 Shaoshan South Road, Changsha, 410075, China. Tel.: +86 13787232438. E-mail addresses: [email protected] (M. Lei), [email protected] (L. Peng), [email protected] (C. Shi). 0926-5805/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.autcon.2013.08.001
results because of the small data sample and the serious disturbance caused by external factors such as construction. The indoor experiment may be capable of obtaining results with a relatively good change law at a certain degree. However, the difference between experimental conditions and actual projects limits the wide use of this research method. The ultimate analysis method is favored by scientific researchers because of its distinct concept and mature theory. Atkinson et al. have conducted research on the stability of shallow buried tunnel using the upper or lower bound ultimate analysis method [4–6]. Jiang et al. assumed the failure modes and established the velocity field used to improve the ultimate analysis method in tunnel stability analysis based on existing research achievements, directly obtaining the upper bound solution to the surrounding rock pressure of the shallow buried tunnel [7–9]. The achievements referenced above promote the use of the ultimate analysis method in tunnel engineering and have undoubtedly developed basic tunnel theory. However, a general survey of these methods shows that they cannot directly obtain the horizontal load acting on the tunnel lining structure. However, these methods still rely on the assumption that the ratio of horizontal and vertical surrounding rock pressure is a constant. The vertical surrounding rock pressure is first obtained using the ultimate analysis method. The horizontal surrounding rock pressure is a multiple of the constant, which is the direct lateral pressure coefficient. The lateral pressure coefficient, however, is arbitrary and is calculated by either Poisson's ratio or is selected by experience without sufficient theoretical basis. The lateral pressure coefficient is also based on the soil linear failure criteria. Several research achievements [10–13] and engineering practices indicate that most types
192
M. Lei et al. / Automation in Construction 37 (2014) 191–195
p0
0
of soil follow the nonlinear failure criteria. Thus, the linear failure criteria are a special case. In the 1980s, Santarelli et al. found that the linear Mohr–Coulomb strength criteria overestimate the major principal stress in rock–soil yield [12,13]. In this thesis, the surrounding rock pressure of a shallow buried tunnel with a uniform formation and complex strata under linear and nonlinear failure criteria was discussed based on Terzaghi failure mode. In particular, the discussion on the horizontal surrounding rock pressure is expected to supplement and perfect the methods proposed in existing research achievements.
D z
H
C σv
dτ
dz σh
dτ σh
γ Ldz
σv+d σ v
Z
q A
B
2. Basic theory
h
2.1. Terzaghi failure mode Terzaghi theory used to calculate the surrounding rock is based on the limit equilibrium method. The basic concept of Terzaghi theory is that the stratum is seen as a loose body; when the tunnel is excavated, the soil body at the top of the tunnel moves down because of gravity. Two shear planes appear between the sides of the tunnel and the ground surface, and stress transfer occurs as soil particles rip one another. The soil body above the tunnel then hinders the downward movement of the soil, such that the minimum supporting pressure is greater or less than the primary stress of the soil. As a result, the vertical surrounding rock pressure acting on the lining is induced by stress transfer. The failure mode is shown in Fig. 1. 2.2. Linear and nonlinear failure criteria With reference to document [14], the nonlinear failure criteria of soil can be expressed as 1=m
τ ¼ c0 ð1 þ σ n =σ t Þ
ð1Þ
where τ and σn are the shear stress and normal stress of the rupture plane, respectively; c0 is the primary cohesion of soil; σt is the axial tensile stress of soil; and m is the nonlinear coefficient. c0, σt, and m can be obtained through the experiment. If m = 1 according to Eq. (1), the equation will transfer to the linear Mohr–Coulomb yield criteria. The tangential equation of nonlinear failure criteria: τ ¼ σ n tanϕt þ ct
ð2Þ
where ϕt and ct are the rake ratio and intercept of the tangent, respectively. When Eqs. (1) and (2) are simultaneously applied, then ct ¼
m−1 mσ t tanϕt 1=ð1−mÞ c0 þ σ t tanϕt : m c0
ð3Þ
3. Calculation of the Vertical surrounding rock pressure of a shallow buried tunnel 3.1. Uniform formation The vertical equilibrium of a micro thin layer dz at z depth, as shown in Fig. 1, is analyzed.
σ v L þ γLdz ¼ ðσ v þ dσ v ÞL þ 2dτ
θ
θ
ð4Þ
When the micro thin layer generates downward displacement, upward friction dτ is generated at two sides because of the existence
B L Fig. 1. Terzaghi failure mode.
of initial horizontal stress. If the surrounding rock material satisfies the nonlinear Mohr–Coulomb yield criteria, then dτ ¼ ðσ h tanϕt þ ct Þdz
ð5Þ
where σh, calculated by σh = λσv, is the horizontal stress at depth z; and λ is the lateral pressure coefficient of the overlying stratum, which is calculated by λ = 1.0 to 1.5 as suggested by the Terzaghi experimental results. The lateral pressure coefficient in this study is notably different from the normal concept of the horizontal supporting force and vertical supporting force ratio. In Terzaghi theory, the lateral pressure coefficient of the overlying stratum is the stress ratio between the horizontal and vertical stresses above the tunnel roof, whereas the lateral pressure coefficient used to calculate the horizontal surrounding rock pressure of the tunnel is computed according to Rankine active earth pressure theory. Substituting Eq. (5) into Eq. (4) and rewriting the equation result in dσ v 2λ tanϕt 2c þ σ v ¼ γ− t dz L L
ð6Þ
where L = B + 2h tan θ; B is the excavation breadth of the tunnel; and θ is the rupture angle of the surrounding rock, which is usually calculated by θ = π/4 − ϕt/2. Eq. (6) determines the vertical stress at any depth σ v ðzÞ ¼
2λ tanϕ γL−2ct − L tz 1 þ Ae 2λ tanϕt
ð7Þ
where A is an undetermined parameter determined according to the boundary condition, which is evidently σv(z)|z = 0 = p0; and p0 is ground load, such as the overload of the pavement when the urban metro tunnel passes through the road, such that A¼
2p0 λ tanϕt −1: γL−2ct
ð8Þ
As a result,
σ v ðzÞ ¼
2λ tanϕ 2λ tanϕ γL−2ct − L tz − L tz 1−e : þ p0 e 2λ tanϕt
ð9Þ
M. Lei et al. / Automation in Construction 37 (2014) 191–195
0
As regards the usual mountain tunnel, p0 = 0, thereby σ v ðzÞ ¼
193
2λ tanϕ γL−2ct − L tz 1−e : 2λ tanϕt
p0
ð10Þ
pi-1
z The vertical pressure acting on the top of tunnel AB is obtained using Eq. (10)
dτ
2Hλ tanϕ γL−2ct − L t q ¼ σ v ðzÞz¼H ¼ 1−e 2λ tanϕt
σh
ð11Þ
Z
where H is the buried depth of the tunnel. This equation is also the computational formula of vertical surrounding rock pressure of the tunnel, as solved using Terzaghi theory.
m m m m m X X X X X γ jh j= h j ; ctp ¼ ctj h j = h j ; ϕtp ¼ ϕtj h j = hj
j¼1
dz
γ Ldz
dτ σh Hi
σv+d σ v
Fig. 3. Force analysis of micro-body in the ith strata.
Terzaghi theory is processed based on uniform formulation. However, a stratum is composed of multilayer rock and soil in practical engineering. Therefore, the expansion of Terzaghi theory in uniform formation to include complex strata is necessary. The failure mode shown in Fig. 2 is established according to Terzaghi basic theory. The overlying stratum is divided into a total of n layers, with Hi being the thickness of each soil layer, γi being volume weight, ϕti being the internal friction angle, cti being cohesion, and i = 1, ⋯, n. The strata through which the tunnel barrel passed is divided into a total of m layers, with hj being the thickness of each soil layer, γj being volume weight, ϕtj being the internal friction angle, ctj being cohesion, and j = 1, ⋯, m. Given that the research object is a shallow buried tunnel, the surrounding rock strengths of the strata through which the tunnel barrel passed are not far from one another. Therefore, the weighted mean method used to deal with the mechanical parameters of the strata through which the tunnel barrel passed is adopted to simplify the calculation m X
σv
Z
3.2. Complex strata
γp ¼
z
j¼1
j¼1
j¼1
j¼1
ð12Þ
j¼1
where γp, ctp, ϕtp are the weighted density, cohesion, and internal friction of the strata through which the tunnel barrel passed, respectively. The corresponding weighted rupture angle θp = π/4 − ϕtp/2, according to the related flow rule [15].
p0 C
0 D
1
H1
z n layers in all
i
H
Hi
Z q
n
Hn
A
B
θ
p
θp
m layers in all
h
B L Fig. 2. Modified Terzaghi failure mode for complex strata.
The acting force loading first layer soil onto the second layer is evidently obtained based on Terzaghi theory with uniform formation and Eq. (9).
p1 ¼
2λ tanϕ 2λ tanϕ γ 1 L−2ct1 − 1 L t1 H1 − 1 L t1 H 1 1−e þ p0 e 2λ1 tanϕt1
ð13Þ
The vertical acting force loading i–1th layer soil onto the ith layer is considered overload, which can be obtained through a derivation of the uniform formation method when the ith layer soil is assumed to be a micro thin layer, as shown in Fig. 3.
pi ¼
2λ tanϕ 2λ tanϕ γi L−2cti − i L ti H i − i L ti Hi 1−e þ pi−1 e 2λi tanϕti
ð14Þ
Thus, the vertical surrounding rock pressure pn acting on the tunnel vault can be easily obtained. q ¼ pn ¼
2λ tanϕ 2λ tanϕ γn L−2ctn − n L tn Hn − n L tn Hn 1−e þ pn−1 e 2λn tanϕtn
ð15Þ
Eq. (15) is the calculation expression of the vertical surrounding rock pressure of a shallow buried tunnel under the complex strata condition. 4. Calculation of the horizontal surrounding rock pressure of a shallow buried tunnel The triangle slider at tunnel side was taken into force analysis, assuming that the resultant supporting resistance supplied by the supporting structure that maintains the stability of tunnel surrounding rock is F. The acting direction between the resultant supporting resistance and vertical line of the triangle is undoubtedly θ + ϕt according to the related flow rule, as shown in Fig. 4. Based on the static equilibrium condition,
qh tanθ þ G−N sinθ−T cosθ− F cosðθ þ ϕt Þ : F sinðθ þ ϕt Þ þ T sinθ−N cosθ ¼ 0
ð16Þ
hj The rupture plane satisfies the nonlinear Mohr–Coulomb yield criteria with T ¼ N tanϕt þ ct l where l is the length of the rupture plane, l = h/cos θ.
ð17Þ
194
M. Lei et al. / Automation in Construction 37 (2014) 191–195
q
F h
θ
+
t
G
T
θ
l N
Fig. 4. Force analysis of side wall fractured zone.
When Eqs. (16) and (17) are simultaneously applied and solved, we derive ðqh tanθ þ GÞð cosθ− tanϕt sinθÞ−ct l cosϕt þ tanϕt sinϕt
F¼
ð18Þ
where G is the self-weight of the triangle rupture body, which is calculated as G = γh2 tan θ/2. Thus, the horizontal supporting resistance to maintain the stability of surrounding rock needs the supporting structure. Eq. (19) solves for the horizontal supporting resistance and clearly shows that the supporting resistance is the horizontal load of the structure design, where θ, which is calculated using θ = π/4 − ϕt/2, is the rupture angle of the surrounding rock if the strata through which the tunnel barrel passed are uniform formations. However, if the strata through which the tunnel barrel passed are complex strata, then θ = π/4 − ϕtp/2, and ϕtp is computed using Eq. (12). F x ¼ F sinðθ þ ϕt Þ ¼
ðqh tanθ þ GÞð cosθ− tanϕt sinθÞ−ct l cosϕt þ tanϕt sinϕt sinðθ þ ϕt Þ
ð19Þ
If the horizontal load distributes as the line moves along the vertical direction, then the horizontal surrounding rock pressure in the depth range of the tunnel is e¼
ðqh tanθ þ GÞð cosθ− tanϕt sinθÞ−ct l sinðθ þ ϕt Þ : cosϕt þ tanϕt sinϕt h
ð20Þ
5. Case study
[8,9] is analyzed and compared with the current calculation method as an example. The buried depth of tunnel H = 20 m, tunnel span B = 10 m, height h = 10 m, volume weight of surrounding rock γ = 20 kN/m3, internal friction angle ϕ0 = 18°, cohesion c0 = 10 kPa, and the lateral pressure coefficient of overlying strata λ = 1.0 to 1.6. The computed results based on Eqs. (11) and (20) and the corresponding results in reference document [8,9] under the linear failure criteria, that is, m = 1.0, are listed in Table 1 and shown in Fig. 5. The calculation method for the vertical surrounding rock pressure is similar to Terzaghi theory. Therefore, the comparative analysis is only performed on the horizontal load. The results of the analysis are as follows. The change laws of the horizontal resistance maintaining the stability of the tunnel side wall differ among the referenced calculation methods when the lateral pressure coefficient of the overlying strata is increased. The results calculated using the method derived in this study are consistent with the results of Terzaghi theory: horizontal resistance decreases when the lateral pressure coefficient increases. The results in document [8,9] are similar. However, horizontal resistance increases when the lateral pressure coefficient increases. When the lateral pressure coefficient is rather small, the results calculated using the method in the current study are closer to the results of document [9], and when the lateral pressure coefficient is rather big, the results calculated using the method in the current study are closer to document [8]. The results of the analysis by synthesis, the change law, and the values calculated using the method in the current study are closest to Terzaghi theory and fall in between other methods. Therefore, the method used in this study is feasible. 6. Effect of the nonlinear failure criterion To discuss further the effect of the nonlinear failure criteria on surrounding rock pressure, a calculation using the nonlinear pressure coefficient m = 1.1 to 1.4 is considered and conducted based on reference document [8]. The axial tensile strength of the surrounding rock σt = 30 kPa, whereas the other calculation parameters are similar to the ones referenced in Section 5. The results are shown in Figs. 6 and 7. The analysis of these results is as follows. Under the nonlinear failure criteria, surrounding rock pressure significantly decreases when the lateral pressure coefficient increases, which is consistent with the result under the linear failure criteria. Therefore, the selection of the lateral pressure coefficient of overlying strata in calculating the surrounding rock pressure is significant under both linear and nonlinear failure criteria. The surrounding rock pressure decreases along with the nonlinear coefficient when the lateral pressure coefficient is fixed. However, the reduction amplitude is minimal. The effect of the nonlinear coefficient on horizontal surrounding rock is slightly stronger.
To test and verify the reliability of the calculation method of the horizontal surrounding rock pressure of the shallow buried tunnel derived as previously mentioned, the project in reference document
The paper method
Terzaghi method
Method of [8]
1.0 1.1 1.2 1.3 1.4 1.5 1.6
141.5 138.8 136.1 133.6 131.2 128.8 126.5
163.6 – 156.2 – 149.4 – 142.9
115.0 122.8 129.9 136.6 142.9 148.9 –
160
e /kPa
Table 1 Results of horizontal earth pressure comparison [kPa]. λ
200
Method of [9] Mode A
Mode B
145.9 – 159.3 – 169.3 – 177.2
145.8 – 160.9 – 175.7 – 190.2
120 the paper method Terzaghi method method of [8] mode A mode B
80 40 0 1.0
1.1
1.2
1.3 λ
1.4
1.5
1.6
Fig. 5. Change of horizontal earth pressure to lateral pressure coefficient of overlying strata.
M. Lei et al. / Automation in Construction 37 (2014) 191–195
b) horizontal earth pressure m=1.1 m=1.2 m=1.3 m=1.4
300
q /kPa
290 280 270
m=1.1 m=1.2 m=1.3 m=1.4
140
e /kPa
a) vertical earth pressure
195
260
135 130 125
250 1.1
1.2
1.3
1.4
1.5
120 1.1
1.6
1.2
1.3
λ
1.4
1.5
1.6
λ
Fig. 6. Change of earth pressure to lateral pressure coefficient using nonlinear failure criterion.
a) vertical earth pressure
b) horizontal earth pressure
=1.1 =1.5
=1.2 =1.6
=1.3
=1.0 =1.4
300
145
290
140
e /kPa
q /kPa
=1.0 =1.4
280 270
=1.2 =1.6
=1.3
1.3
1.4
135 130 125
260 250
=1.1 =1.5
1.1
1.2
1.3
1.4
120 1.1
m
1.2
m
Fig. 7. Change of earth pressure to nonlinear coefficient using nonlinear failure criterion.
7. Conclusions
References
A method to calculate the surrounding rock pressure of a shallow buried tunnel under linear and nonlinear failure criteria is presented based on Terzaghi failure mode. Terzaghi theory, which was limited to uniform formations, is expanded to include complex strata. A limit equilibrium analysis of the surrounding rock bodies in a tunnel side rupture zone is conducted, and the computational expressions of the horizontal surrounding rock pressure are obtained by theoretical derivation. The case study indicates that the method presented in this paper is feasible and perfects the insufficiency of the exit theory, which calculates horizontal surrounding rock pressure by setting the vertical surrounding rock pressure as a multiple of a direct constant. The value of the lateral pressure coefficient is significant in obtaining the surrounding rock pressure of the tunnel under linear and nonlinear failure criteria. When the value of the lateral coefficient is greater, the surrounding rock pressure becomes smaller. The surrounding rock pressure gradually decreases when the nonlinear coefficient increases under the nonlinear failure criterion, and the effect on horizontal load is slightly stronger.
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Acknowledgment Financial support from the National Basic Research Program of China (973 Program: 2011CB013802) is gratefully acknowledged.