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Support pressure for circular tunnels in two layered undrained clay
Journal Pre-proof Support pressure for circular tunnels in two layered undrained clay Bibhash Kumar, Jagdish Prasad Sahoo
PII:
S1674-7755(19)30739-5
DOI:
https://doi.org/10.1016/j.jrmge.2019.04.007
Reference:
JRMGE 617
To appear in:
Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 29 October 2018 Revised Date:
3 April 2019
Accepted Date: 25 April 2019
Please cite this article as: Kumar B, Sahoo JP, Support pressure for circular tunnels in two layered undrained clay, Journal of Rock Mechanics and Geotechnical Engineering, https://doi.org/10.1016/ j.jrmge.2019.04.007. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. All rights reserved.
Support pressure for circular tunnels in two layered undrained clay Bibhash Kumara, Jagdish Prasad Sahoob,* a
Department of Civil Engineering, National Institute of Technology, Uttarakhand, 246174, India
b
Department of Civil Engineering, Indian Institute of Technology, Roorkee, 247667, India
ARTICLEINFO
ABSTRACT
Article history: Received 29 October 2018 Received in revised form 3 April 2019 Accepted 25 April 2019 Available online
To estimate the required support pressure for stability of circular tunnels in two layered clay under undrained condition, numerical solutions are developed by performing finite element lower bound limit analysis in conjunction with secondorder cone programming. The support system is assumed to offer uniform internal compressive pressure on its periphery. From the literature, it is known that the stability of tunnels depends on the overburden pressure acting over it, which is a function of undrained cohesion and unit weight of soil, and cover of soil. When a tunnel is constructed in layered undrained clay, the stability depends on the undrained shear strength, unit weight, and thickness of one layer relative to the other layer. In the present study, the solutions are presented in a form of dimensionless charts which can be used for design of tunnel support systems for different combinations of ratios of unit weight and undrained shear strength of upper layer to those of lower layer, thickness of both layers, and total soil cover depth. © 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. T h is is a n o p e n a c c e s s a r t ic l e u n d e r t he C C BY - N C - N D l ic e n s e ( h t t p : / / c re a t i v e c o m mo n s . o r g/ licenses/by-nc-nd/4.0/).
Keywords: Finite elements Layered clay Limit analysis Stability Tunnels
for square and circular tunnels advanced in undrained cohesive soil taking into 1. Introduction Tunnels and underground openings are constructed for better transport facilities, underground pipeline, canal and hydropower projects, etc. The stability of tunnels depends on the resistance of support system to prevent failure of soil mass. Thus, the support pressure required must be known prior to its design to maintain tunnel stability. In order to evaluate tunnel stability in homogeneous ground conditions, a number of investigations have been performed (Atkinson and Potts, 1977; Mair, 1979; Davis et al., 1980; Assadi and Sloan, 1991; Wu and Lee, 2003; Lee et al., 2006; Osman et al., 2006; Yang and Yang, 2010; Sahoo and Kumar, 2014a; Yang et al., 2015; Zhang et al., 2017). Atkinson and Potts (1977) have conducted centrifuge model tests on assessing the stability of circular tunnels driven in sand supported by means of either compressed air or bentonite slurry. By conducting centrifuge model tests, Mair (1979) examined the response of tunnel driven in soft clay supported by compressed air pressure from inside of tunnel. Davis et al. (1980) computed the support pressure required for preventing collapse of underground openings/tunnels formed in soft clay under undrained condition based on the concept of limit analysis. Wu and Lee (2003) carried out centrifuge model tests to monitor ground movements and the associated collapse mechanism for both single and group of parallel tunnels in clay under undrained condition. Lee et al. (2006) conducted a series of centrifuge model tests and performed numerical analysis of these test results for evaluating the surface settlement trough, generation of excess pore water pressure and arching effect developed during the excavation process of single and parallel tunnels through soft clay. Osman et al. (2006) analyzed the stability of tunnel by examining the deformation pattern around unlined tunnel excavated in undrained clay using upper bound limit analysis. In order to determine the magnitude of required support pressure for tunnels located in various types of soils, solutions have been developed on the basis of finite element limit analysis with implementation of different mathematical programmings (Yang and Yang, 2010; Sahoo and Kumar, 2012, 2014a, 2018; Yang et al., 2015; Zhang et al., 2017). Limited studies are available in the literature to obtain the support pressure of tunnels located in the non-homogeneous soil (Sloan and Assadi, 1991; Hagiwara et al., 1999; Grant and Taylor, 2000; Nunes and Meguid,
account linear variation of undrained strength with depth. For unlined circular tunnels in undrained clayey soil, Sahoo and Kumar (2013) performed stability analysis by employing finite element upper bound limit analysis considering a linear variation of undrained cohesion with depth. In layered soil, the ground movements induced by tunneling process depend on the stiffness and type of overlying soil mass. The effect of type and stiffness of overlying soil mass on the ground movements during tunneling was investigated by Hagiwara et al. (1999) with the help of centrifuge model tests for tunnels constructed in layered soil. By conducting a few centrifuge model tests, Grant and Taylor (2000) studied the stability of tunnels located in the purely cohesive soil underlain by granular soil. Nunes and Meguid (2009) carried out plane strain elastoplastic finite element analyses and laboratory tests to investigate the variation in bending stresses developed in lining of tunnel support in purely cohesive soil overlain by coarse-grained sand layer. Using lower bound finite element limit analysis in conjunction with second-order cone programming, Sahoo and Kumar (2019b) determined required support pressure to maintain the stability of tunnel advanced in undrained clay layer overlain by sand layer. Nevertheless, few researches seem to be available for predicting the required compressive pressure that should be offered by a support system of tunnel advanced in two layered clayey soil to maintain the tunnel stability. In the present study, we aimed to produce solutions for estimating the compressive pressure required to support the soil surrounding the tunnel advanced in an undrained clay layer overlain by another undrained clay layer. The required support pressure is assumed to be acting normally and uniformly along the periphery of the tunnel. For the computations, finite element lower bound limit analysis and second-order cone programming were employed. The finite element limit analysis has been extensively used for solving various stability problems in geomechanics (Sahoo and Kumar, 2014b; Keawsawasvong and Ukritchon, 2017 a,b,c,d; Khuntia and Sahoo, 2017). The influences of thickness of lower clay layer above tunnel crown, thickness of upper clay layer, strengths of both the layers in terms of undrained shear strength, and unit weights of both layers have been studied. The failure patterns, i.e. the proximity of stress states at failure, have also been presented for a few cases. 2. Problem definition and chosen domain
2009; Wilson et al., 2011; Sahoo and Kumar, 2013, 2019a). By performing finite element limit analysis and rigid block upper bound method, Sloan and
A circular tunnel having diameter D is excavated in undrained clay (φu1= 0°)
Assadi (1991) and Wilson et al. (2011) computed the required support pressure
overlain by another layer of undrained clay (φu2= 0°) at a cover depth of H below the horizontal ground surface, as illustrated in Fig. 1. H1 is the thickness
*Corresponding author. Fax: +91-1332-285568; E-mail address: [email protected]
of lower soil layer above the tunnel crown, and H2 is the thickness of upper layer. To perform plane strain analysis, it is assumed that tunnel length is very
large as compared to its diameter. Keeping the undrained shear
domain. The location of the boundaries JK and IJ with Lh = 3D–30D and Lv =
strength/undrained cohesion (cu1) and unit weight (γ1) of the lower layer the
3D–20D has been found to be sufficient depending upon the values of H/D by
same, the analysis was performed for the following cases: (i) undrained shear
performing sensitivity analysis as discussed in Section 5.
strength (cu1) and unit weight (γ1) of lower layer smaller than that of upper layer (cu2 > cu1 and γ2 > γ1), and (ii) undrained shear strength and unit weight of lower layer greater than that of upper layer (cu2 < cu1 and γ2 < γ1). The parameters c, γ and φ refer to the cohesion, unit weight and internal friction angle of soil mass, respectively, whereas the subscript u indicates the undrained condition, and 1 and 2 refer to the lower and upper layers, respectively. The undrained strengths of both the clay layers are modeled by employing Tresca yield criterion and an associated flow rule. In order to support soil mass surrounding the tunnel periphery for different cases as mentioned above, it is intended in the present study to calculate the required support pressure (σi) in the form of lining and associated anchorage system. The support pressure is expressed in the form of a dimensionless factor defined as σi/cu1. Taking into account the symmetric nature of loading and geometry of the problem, only half of the domain as illustrated in Fig. 2a was used to perform the present analysis. The horizontal boundary (IJ) and vertical boundary (JK)
Fig. 1. Definition of problem.
were kept at a sufficient large distance from the centroid of tunnel by satisfying two conditions: (i) the plastic zone must be within the chosen
3. Analysis
domain; and (ii) the solutions will not change with further increase in the size of
The analysis was performed following the finite element formulation based
the
on lower bound theorem of limit analysis developed by Sloan (1988) and the second-order conic optimization technique of Makrodimopoulos and Martin (2006). In the lower bound approach, the magnitude of the load at failure is determined in a statically admissible stress field, that is, the stress field satisfies the equilibrium and stress boundary conditions without violating yield criterion. For computing the magnitude of limit pressure required to maintain stability of the tunnel, the objective function is maximized subjected to a set of equality and inequality constraints imposed on the unknown nodal stresses. The equality constraints are produced in order to satisfy equilibrium of elements, statically admissible stress discontinuities, and stress boundary conditions; whereas, for the satisfaction of yield criterion, the inequality constraints are generated. The chosen domain as illustrated in Fig. 2a is discretized into a number of three-noded triangular elements. None of the nodes are linked with more than a single element, and consequently, every interface between the adjacent elements becomes always the line of the stress discontinuity. It should be noted that each node remains unique to a particular e
(a)
l
(b)
e
m
e
n
t
,
a
n
d
(c)
Fig. 2. (a) Chosen problem domain and associated stress boundary conditions; (b) Sign convention for stresses; and (c) A typical finite element mesh for H/D = 6 with H1/D = 3 and H2/D = 3. ‘+ve’ shows that stresses are positive in these directions.
consequently, two or more nodes often share the same coordinates. There are
are generated in MATLAB (MathWorks, 2015) by writing programming
three basic unknowns at each node, namely, horizontal normal stress (σxx),
codes.
vertical normal stress (σyy), and shear stress (τxy). A typical finite element mesh
For modeling the stress field under plane strain condition, a linear variation
for H/D = 6 with H1/D = 3 and H2/D = 3 is shown in Fig. 2c. All the meshes
of stresses is chosen within each element and linear shape functions are
employed. The stresses within each triangular element must satisfy the
σ n ,1 = σ x x ,1 cos 2 θ1 + σ y y ,1 sin 2 θ1 − τ x y ,1 sin (2θ1)
(4a)
equilibrium conditions which are defined as follows: ∂σ x x ∂τ x y + =0 ∂x ∂y
σ n ,2 = σ x x ,2 cos θ 2 + σ y y ,2 sin θ 2 − τ x y ,2 sin(2θ2 )
(4b)
∂τ x y ∂x
+
∂σ y y ∂y
=γ
2
2
(1a) (1b)
where γ is the body force per unit volume of soil mass (unit weight of soil mass) acting vertically in the downward direction. In Eqs. (1a) and (1b), normal tensile stresses are taken as positive and the positive direction of shear stresses and coordinate axes are shown in Fig. 2b. Since each node is unique to an element, the unit weight of upper layer (γ2) and lower layer (γ1) are assigned at the nodes of elements of upper and lower layers, respectively, along the interface of two layers for satisfying the equilibrium condition given in Eq. (1b). In order to obtain lower bound solutions close to the true solution, statically admissible stress discontinuities are included in the lower bound analysis (Drucker, 1953; Lysmer, 1970; Sloan, 1988), and to obtain statically admissible stress discontinuity, the shear and normal stresses are required to be continuous along the common sides of any two adjacent elements. No constraints on the stresses were imposed along the boundaries IJ and JK. Along boundary LK, the normal and shear stresses are made equal to zero while along the surfaces LM and NI, only shear stress is zero. Along the interfaces of the adjoining clayey soil and the tunnel lining, it has been specified that the developed shear stress is less than or equal to the undrained cohesion of the adjoining soil, that is, | τ x y | ≤ c u 1 . In order to obtain strictly lower bound solutions, the state of stresses should satisfy yield criterion everywhere within the problem domain. Under plane strain condition, the Tresca yield criterion for undrained clay may be expressed as
Fig. 3. Various parameters associated with a very small length of segment of the tunnel periphery. Nodes 1 and 2 lie on the tunnel periphery and node 3 lies within the soil mass. The difference between chord length and arc length between nodes 1 and 2 is negligible as they are very close to each other.
(σ x x − σ y y ) 2 + 4τ x2y ≤ 2c u
(2)
The inequality constraint (Eq. (2)) can be expressed as a set of second-order cone (Makrodimopoulos and Martin, 2006) by introducing a vector
z i = {z 1i z 2i z 3i }T at each node, satisfying the equation of second-order cone, i.e. z 1i ≥ z 22i + z 32i . Eq. (2) is represented by second-order cone constraints as i Asoc σi
i Asoc
i + z i = b soc
(i = 1, 2, L, N )
σ 0 0 0 z 1i 2c u x x ,i i = −1 1 0 , σ i = σ yy ,i , z i = z 2 i , b soc = 0 0 0 −2 z 3 i 0 τ x y ,i
(3a)
It is assumed that the internal support pressure (σi) acts uniformly on the tunnel periphery; hence, the following condition needs to be satisfied: σ i = σ n ,1 = σ n ,2 = L = σ n ,N t
(5)
where Nt is the total number of nodes along the tunnel periphery. The lining used as support system is generally elastic in nature and considered as thin-walled tube under plane strain condition (Wood, 1975; Croll, 2001; Tamura and Hayashi, 2005; Wang and Koizumi, 2010; Wang et al., 2015). Though the required support pressure to make the surrounding soil
(3b)
mass in stable condition is not uniform along the tunnel periphery, to calculate the stiffness and thickness of tunnel lining based on the theory of thin-walled
where σxx,i, σyy,i and τxy,i imply the horizontal normal stress, vertical normal
cylinders, the pressure distribution is assumed to be uniform, which can be
stress and shear stress, respectively, at the i-th node; and N refers to the total
calculated as follows: σ 1 = σ iD / (4t )
(6)
σ 2 = σ iD / (2t )
(7)
number of nodes in the domain. For modeling the support system, no exclusive elements were used. Following Sahoo and Kumar (2019b), the sides of triangular elements along the periphery of tunnel are treated as the segments of lining surface since for a given segment, the difference between chord length and arc length is negligible when the size of triangular elements is very small. The constraints developed for computing the uniform compressive support pressure to be exerted by the support system was already given in Sahoo and Kumar (2019b); however, for the sake of completeness, this part is again described herein. Fig. 3 shows a typical segment of the tunnel periphery, where σxx,1 and σxx,2 refer to the horizontal normal tractions in x-direction at nodes 1 and 2, respectively;
σyy,1 and σyy,2 refer to the vertical normal tractions in y-direction at nodes 1 and 2, respectively; and τxy,1 and τxy,2 are the shear tractions on the vertical/horizontal planes at nodes 1 and 2, respectively. The angles θ1 and θ2 are the inclinations of radial lines passing through nodes 1 and 2, respectively, with the horizontal plane. The normal stresses σn,1 and σn,2 acting at nodes 1 and 2 on the tunnel periphery are expressed as
ε1 = ε2 =
σ1 νσ 2 E
−
E
σ 2 νσ1 E
−
E
(8) (9)
where σ1 and σ2 are the longitudinal and hoop stresses in the lining, respectively; ε1 and ε2 are the longitudinal and hoop strains in the lining, respectively; and t, E andν are the thickness, Young’s modulus and Poisson’s ratio of the lining, respectively. For plane strain condition, ε1= 0, thus we have σ iD σ 1 = νσ 2 , ε 2 = 2tE (1 − ν 2 )
(10)
Assuming that tc is the change allowed in the thickness of lining of the cylinder, then we can obtain 2π(D / 2 + t c ) − π D 2t c = ε2 = πD D
(11)
t =
σ iD 2
(12)
4t c E (1 − ν 2 )
σi
4t E = t c (1 − ν 2 )D 2
Lining stiffness per unit tunnel length is expressed as π Dσ i 4tE k = = tc (1 − ν 2 )D
4. Results and discussion
(14)
1991; Osman et al., 2006; Wilson et al., 2011; Sahoo and Kumar, 2014a), it is
From the previous studies on homogeneous clayey soil (Assadi and Sloan,
the constraints imposed on the nodal stresses. For performing second-order cone programming in a statically admissible stress field, the objective function along with the constraints may be expressed as follows:
noted that for a given diameter of the tunnel, the magnitude of required support pressure depends on the unit weight and cohesion of soil, and cover depth of the tunnel. In the present analysis, numerical solutions were obtained for a tunnel where soil mass lying over the tunnel is of a two-layered medium, that is, tunnel located in clay under undrained condition (φu1 = 0°) with an overlay
Objective function: Maximize {σ i }
(15)
Constraints: Ax = B
(16)
x = {σ
(17)
of relatively stiff or soft clayey soil under undrained condition (φu2 = 0°), where the unit weight and undrained cohesion of upper layer are greater or
z }T
0 0 0 I 0
B = {Bequil Bbound
Keawsawasvong, 2018, 2019).
(13)
The magnitude of limit support pressure at failure is optimized by satisfying
Aequil Abound A = Adis Asocp Aucs
2018; Sahoo and Khuntia, 2018; Ukritchon et al., 2018; Ukritchon and
smaller than that of lower layer. The present solutions have been obtained for two cases: (1)
In the first case, the results are obtained for a homogeneous clayey soil with γ1D/cu1 = γ2D/cu2 equal to 1, 3 and 5;
(18)
(2)
In the second case, keeping the unit weight and cohesion of the lower layer constant with γ1D/cu1 equal to 1, 3 and 5, the values of unit weight and cohesion of the upper layer are varied with:
Bdis
Bsocp
T
Bucs }
(19)
(i)
relatively higher cohesion (cu2 > cu1) for cu2/cu1 = 10, 2.5 and 1.43 (or cu1/cu2 = 0.1, 0.4 and 0.7) with relatively higher unit
where A is the global matrix of coefficients of all the constraints; Aequil, Abound,
weight (γ2 > γ1) for γ2/γ1 = 1.2, 1.4 and 1.6 (or γ1/γ2 = 0.83,
Adis, Asocp and Aucs are the coefficient matrices of constraints owing to element equilibrium, imposed boundary conditions, satisfying statically admissible stress discontinuities, a set of second-order cones and uniform compressive support pressure, respectively; I is the identity matrix; B is the global vector of
0.71 and 0.625), and (ii)
relatively lower cohesion (cu2 < cu1) for cu2/cu1 = 0.1, 0.4 and 0.7 with relatively lower unit weight (γ2 < γ1) for γ2/γ1 = 0.83, 0.71 and 0.625.
constants; Bequil, Bbound, Bdis, Bsocp and Bucs are the global vectors of constants on account of element equilibrium, imposed boundary conditions, satisfying
It is noted that the values of undrained cohesion and unit weight of soft to
statically admissible stress discontinuities, a set of second-order cones and
very stiff clay generally lie in the range of 14–22 kN/m3 and 25–100 kPa,
uniform compressive support pressure, respectively; σ is the global variable
respectively. The computations were performed by choosing the values of (i)
consisting of the nodal stresses, which can be defined as
σ = {σ x x ,1 σ y y ,1 τ x y ,1 σ x x ,2 σ y y ,2 τ x y ,2 ... σ x x ,N σ y y ,N z is the global auxiliary variable vectors defined as T
z T = {z 11 z 21 z 31 z 12 z 22 z 32 L z 1N z 2 N z 3 N }
τ x y ,N } (20)
H/D ranging from 1 to 10, (ii) γ1D/cu1 or γ2D/cu2 varying from 1 to 5, and (iii) cu2/cu1 between 0.1 and 10.
4.1. Variation of support pressure (21)
In the present study, the support pressure is normalized with respect to the
It is worth mentioning that for determining the resistance to be offered by
undrained cohesion of the lower layer (σi/cu1), which has been kept the same as
internal support pressure against active failure induced by overburden pressure
that used for obtaining results in the case of homogeneous clay. In two-layered
and surcharge (if any) on the ground surface, the objective function is
clay medium, the support pressure (σi) required for maintaining stability of
maximized. On the other hand, for computing the resistance to be provided by
tunnel depends on the thicknesses of upper layer (H2) and lower layer (H1)
the overburden pressure and surcharge against the collapse caused by internal
above the crown of tunnel, unit weight and cohesion of both layers. For
pressure which is termed as passive failure or blow out failure, the objective
various combinations of H1/D, cu2/cu1 and γ2/γ1, the variation of σi/cu1 with
function is minimized, that is, Minimize {σ i }. The pre-processing computer
H2/D is shown in Figs. 4–12. It may be noted that the movement of soil mass
codes were written in MATLAB (MathWorks, 2015) and optimized solutions
towards tunnel at failure due to the action of gravitational load is resisted by
were developed on the basis of cone optimization using MOSEK (MOSEK
the internal pressure offered by the support system. From the results presented
ApS, 2015), an optimization toolbox available for MATLAB. To obtain the
in Figs. 4–12, the positive value of σi implies that support system is required to
optimal lower bound solutions, the MOSEK optimizer is employed for solving
be provided to prevent the collapse of tunnel; whereas, negative values of σi
different stability problems in geotechnical engineering (Khuntia and Sahoo,
indicate that the tunnel remains stable without any support system.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 4. Variation of σi/cu1 with H2/D for H1/D = 1 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=1.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 5. Variation of σi/cu1 with H2/D for H1/D = 3 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1 = γ2D/cu2=1.
(a)
(c)
(b)
(d)
(e)
(f)
Fig. 6. Variation of σi/cu1 with H2/D for H1/D = 5 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1= 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1 = γ2D/cu2=1,
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 7. Variation of σi/cu1 with H2/D for H1/D = 1 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1= 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1= 0.4, and (f) cu2/cu1= 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=3.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 8. Variation of σi/cu1 with H2/D for H1/D = 3 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1= 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=3.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 9. Variation of σi/cu1 with H2/D for H1/D = 5 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1= 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=3.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 10. Variation of σi/cu1 with H2/D for H1/D = 1 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1= 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=5.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 11. Variation of σi/cu1 with H2/D for H1/D = 3 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1= 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=5.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 12. Variation of σi/cu1 with H2/D for H1/D = 5 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1= 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=5.
Keeping the shear strength of lower layer constant, corresponding to
offered by the undrained cohesion of soil mass. For smaller thickness of upper
particular values of H/D, H1/D, γ1D/cu1 and γ2/γ1, the magnitude of required
layer, the influence of driving force due to overburden pressure of upper soil
support pressure (i) decreases with increase in the undrained shear strength of
layer may be less compared to the resistance owing to the cohesion of upper
upper layer when the upper layer soil is relatively stiffer compared to the lower
layer, leading to nonlinear variation of σi/cu1. On the other hand, with increase
one, that is, unit weight and shear strength of lower layer are smaller than that
in the thickness of upper layer, the effect of driving force due to increase in
of upper layer (γ2>γ1 and cu2/cu1>1) as shown in parts (a–c) in Figs. 4–12; and
overburden pressure of soil becomes dominant versus that of increase in the
(ii) increases with reduction in the undrained shear strength of upper layer
resistance owing to the cohesion of soil mass, which causes the stiffer or linear
when the upper layer soil is relatively softer compared to the lower one, that is,
variation of σi/cu1.
the unit weight and shear strength of lower layer are greater than those of
For H2/D = 5, the variations of σi/cu1 with (i) cu2/cu1 for different values of
upper layer (γ2<γ1 and cu2/cu1< 1) as shown in parts (d–f) in Figs. 4–12. It can
γ2/γ1 with H1/D = 5 and γ1D/cu1 = 3, (ii) γ2/γ1 for different values of cu2/cu1 with
be seen from Figs. 4–12 that the variation of σi/cu1 is found to increase
H1/D = 5 and γ1D/cu1 = 3, (iii) γ1D/cu1 for different values of γ2/γ1 and cu2/cu1
nonlinearly with increase in H2/D up to a certain limit, after which it increases
with H1/D = 5, and (iv) H1/D for different values of γ2/γ1 and cu2/cu1 with
linearly. It is known that the failure is mainly due to the weight of soil mass
γ1D/cu1 = 3 are shown in Figs. 13–16. It is observed that there is increase in the
(overburden pressure) lying over the tunnel and the resistance to failure is
value of support pressure with (i) reduction in the undrained cohesion and
increase in the unit weight of upper layer with respect to the lower layer for
expressed in dimensionless term, namely, t/s, where t = (σ x x − σ y y ) 2 + 4τ x2y
given values of H1/D, H2/D and γ1D/cu1, (ii) increase in γ1D/cu1 for a given
and s = 2 c u , where cu becomes equal to cu1 and cu2 for the elements in the
combination of H1/D, H2/D, γ2/γ1 and cu2/cu1, and (iii) increase in H1/D for a
lower and upper layers, respectively. A point within the chosen domain will be
given combination of H2/D, γ1D/cu1, γ2/γ1 and cu2/cu1.
in a state of shear failure when the ratio t/s tends to unity. The failure patterns
From Figs. 4–12, it shows that compared to tunnel driven in homogeneous
are generated in a manner such that the color of an element becomes darker
clay, the magnitude of σi/cu1 required for the case of tunnel driven in two-
when approaching the shear failure (t/s = 1). The failure patterns have been
layered clay medium (where the properties of lower layer remain the same as
generated for H/D = 6 (H1/D = 3 and H2/D = 3) and γ1D/cu1= 3, as shown in
those of homogeneous clay) has been found to be dependent on the combined
Fig. 17 in the case of layered clay for different values of cu2/cu1 and γ2/γ1
influence of H1/D, H2/D, γ1D/cu1, cu2/cu1 and γ2/γ1, i.e.
keeping the cohesion of lower layer constant.
(1)
Greater only when cu2/cu1 = 1.43 for any values of H/D and γ1D/cu1, and dependent on the combined influence of H1/D, H2/D, γ1D/cu1, cu2/cu1 and γ2/γ1 for cu2/cu1 equal to 2.5 and 10 for lower values of
γ1D/cu1 = 1, as provided in Figs. 4–6. (2)
Smaller only when cu2/cu1 = 0.7 for any values of H/D and γ1D/cu1, and dependent on the combined influence of H1/D, H2/D, γ1D/cu1, cu2/cu1 and γ2/γ1 for cu2/cu1 equal to 0.4 and 0.1 for lower values of
γ1D/cu1 = 1, as illustrated in Figs. 4–6. (3)
Always greater when cu2/cu1 > 1 and always lower when cu2/cu1 < 1 for higher values of γ1D/cu1 = 3 and 5, which can be noted from Figs. 7–12.
From Fig. 17a, for a given value of γ2/γ1 in the case of two-layered clayey soil, it can be noted that the yielding of soil mass depends upon the magnitude of undrained cohesion of both layers relative to each other, that is, the zone of yielding is larger as expected in the lower layer where the cohesion is relatively lower compared to that of upper layer (cu2/cu1 = 10). By comparing Fig. 17a and b, it is observed that the size of failure zone tends to reduce in the lower layer and increase in the upper layer with the reduction in the values of cohesion of upper layer. Further, though the cohesion of upper layer is less, the zone of yielding is found to be smaller in the upper layer for cu2/cu1 = 0.7 as shown in Fig. 17c, when compared with that of cu2/cu1 = 1.43 as shown in Fig. 17b. It needs to be mentioned that the zone of failure in the upper layer also depends on the ratio of unit weight of both layers, that is, the unit weight of
Further, for cu2/cu1>1 in the case of layered soil, though the upper layer offers higher resistance than the homogeneous soil where the cohesion is the same as that of lower layer (cu1), the magnitude of σi/cu1 corresponding to all the values of cu2/cu1 > 1 for any combinations of H1/D, H2/D and γ1D/cu1 is not always lower than that of homogeneous soil. This is due to the fact that the
upper layer is smaller for the case shown in Fig. 17c than that presented in Fig. 17b. 4.3. Remarks For H1/D = 3, H2/D = 3, γ1D/cu1= 3, cu2/cu1 = 10 and γ2/γ1= 1.2, 1.4 and 1.6, the variation of σi/cu1 with Lh/D is presented in Fig. 18. It could be noted that
driving force causing failure is also higher for the tunnel driven in layered soil
the magnitude of σi/cu1 increases continuously with increase in domain size up
than that in homogeneous soil as γ2 > γ1. Similarly, when the resistance of
to a certain limit (Lh/D is approximately equal to 12), beyond which with
upper layer is lower compared to that of homogeneous soil (cu2 < cu1), the
further increase in the domain size, the values of σi/cu1 remains constant.
magnitude of σi/cu1 corresponding to all the values of cu2/cu1 < 1 for any
Further, from Fig. 17a, the developed plastic zone is observed to be well
combinations of H1/D, H2/D and γ1D/cu1 is not always greater than that of
contained within the chosen domain with Lh and Lv equal to 25D and 15D,
homogeneous soil because the driving force causing failure is also lower for
respectively. Thus, the chosen domain with Lh = 25D and Lv = 15D for
the tunnel driven in layered soil in comparison to the tunnel driven in
determining the magnitude of σi/cu1 when H1/D = 3, H2/D = 3, γ1D/cu1= 3,
homogenous soil as γ2<γ1.
cu2/cu1 = 10 and γ2/γ1= 1.6 is proved to be sufficient. Similarly, for different combinations of H1/D, H2/D, γ1D/cu, cu2/cu1 and γ2/γ1, the domain size was
4.2. Failure patterns The proximity of the stress state to shear failure, that is, the failure pattern,
decided in the present analysis.
has been examined from the stress state obtained at the element centroid and
(a)
(b)
Fig. 13. Variation of σi/cu1 with cu2/cu1 for (a) cu2/cu1>1 and γ2/γ1> 1 and (b) cu2/cu1<1 and γ2/γ1 < 1, for H1/D = 5, H2/D = 5 and γ1D/cu1 = 3.
(a)
(b)
Fig. 14. Variation of σi/cu1 with γ2/γ1 for (a) cu2/cu1> 1 and γ2/γ1> 1, and (b) cu2/cu1 < 1 and γ2/γ1< 1, for H1/D = 5, H2/D = 5 and γ1D/cu1= 3.
(a)
(b)
Fig. 15. Variation of σi/cu1 with γ1D/cu1 for (a) cu2/cu1 = 10 and γ2/γ1>1 and (b) cu2/cu1= 0.1 and γ2/γ1<1, for H1/D = 5 and H2/D = 5.
(a)
(b)
Fig. 16. Variation of σi/cu1 with H1/D for (a) cu2/cu1= 10 and γ2/γ1>1, and (b) cu2/cu1 = 0.1 and γ2/γ1< 1, for H2/D = 5 and γ1D/cu1 = 3.
t/s
x/D
t/s
y/D
y/D
y/D
t/s
x/D
x/D
(a) (b) (c) Fig. 17. Proximity of shear failure for layered clay with H1/D = 3, H2/D = 3 and γ1D/cu1= 3: (a) cu2/cu1= 10 and γ2/γ1=1.6; (b) cu2/cu1=1.43 and γ2/γ1=1.6; and (c) cu2/cu1=0.7 and γ2/γ1=0.625.
properties of both layers corresponding to undrained condition. The required support pressure is presented in terms of normalized normal compressive stress, defined as σi/cu1, and the variations of σi/cu1 for various combinations of H1/D, H2/D, γ1D/cu1, cu2/cu1 and γ2/γ1 have been established in the form of dimensionless charts, which can be used for the purpose of design of tunnel support system in a two-layered undrained clayey soil. The following conclusions can be drawn from the present study: (1)
For a particular combination of H1/D, H2/D, γ1D/cu1 and γ2/γ1, the magnitude of σi/cu1 is found to be continuously reducing with increase in the undrained cohesion of the upper layer when tunnel is located in the lower layer of relatively smaller stiffness than that of the upper layer (cu2/cu1 > 1 and γ2 > γ1), and continuously increasing with reduction in the undrained cohesion of the upper layer when tunnel is driven in the lower layer of relatively greater stiffness than
Fig. 18. Variation of σi/cu1 with Lh/D keeping Lv/D = 15 for H1/D = 3, H2/D = 3, γ1D/cu1= 3, cu2/cu1 = 10 and different values of γ2/γ1.
that of the upper layer (cu2/cu1 < 1 and γ2 < γ1). (2)
With increase in the unit weight of upper layer with respect to the lower layer for given values of H1/D, H2/D and γ1D/cu1, the
5. Comparison of present solutions with those available in the literature
magnitude of required support pressure increases. Since no solutions seem to be available in the literature for determining the
(3)
The increase in support pressure for a particular value of tunnel cover depth is nonlinear for lower values of H2/D and becomes linear
required support pressure to maintain the tunnel stability in two-layered clayey soil, for the purpose of validation of present analysis, the variation of σi/cu with
for higher values of H2/D. The rate of nonlinearity is found to be
H/D for different values of γD/cu obtained from the present analysis is
higher when the undrained cohesion of upper layer is relatively
compared with the numerical solutions of Wilson et al. (2011) and centrifuge
greater than that of lower layer. Furthermore, the support pressure
experimental results of Mair (1979) for a tunnel located in homogeneous clay
also increases with increase in the thickness of lower layer when the thickness of upper layer is constant.
under undrained condition. The comparisons are provided in Fig. 19a and b. In these figures, cu and γ are the undrained shear strength and unit weight of
(4)
With increase in the values of γ1D/cu1, the magnitude of required
homogenous undrained clay (cu = cu1= cu2, and γ = γ1 = γ2), respectively.
support pressure also increases when the thicknesses of both layers,
Wilson et al. (2011) analyzed the stability of tunnel using finite element lower
ratio of unit weight of upper to lower layer, ratio of cohesion of upper layer to lower layer remain constant.
bound limit analysis in conjunction with nonlinear programming. The solutions obtained from these two analyses are found to be almost merging
(5)
When the support pressure of a tunnel driven in layered soil with cu2
with each other. The present results also match closely with the experimental
< cu1 and γ2 < γ1 is compared with that of a tunnel driven in
results of Mair (1979).
homogeneous soil having the same properties as that of lower layer
6. Conclusions
required is not always lower for a tunnel driven in a layered soil,
(cu1 and γ1), it has been noted that the magnitude of support pressure
The support pressure (σi) required for the stability of a circular tunnel formed in clay layer overlain by relatively stiffer and softer clay layer has been computed by using lower bound finite element limit analysis in combination
although the cohesion of upper layer is higher than that of homogeneous soil. This is because the driving force inducing failure is also higher for the tunnel driven in layered soil than that in homogeneous soil as γ2 > γ1.
with the second-order conic programming. The analysis was performed considering
the
(a)
(b)
Fig. 19. Comparison of present solutions with (a) numerical solutions of Wilson et al. (2011) and (b) experimental results of Mair (1979) for tunnel located in homogenous clayey soil.
Conflict of interest The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial
support for this work that could have influenced its outcome.
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Dr. Jagdish Prasad Sahoo is presently working as Assistant Professor at Indian Institute of Technology Roorkee, India. He has obtained his MSc and PhD degrees from Indian Institute of Science Bangalore, India. He has been working in the areas of stability of various problems in Geotechnical Engineering for last 10 years and published more than 35 papers in different international journals. Presently he is the editorial board member of International Journal of Geotechnical Engineering. He is the life member of Indian Geotechnical Society (IGS), Indian Roads Congress (IRC), Indian Society for Construction Materials and Structures (ISCMS), and Indian Society of Earthquake Technology (ISET).
PII:
S1674-7755(19)30739-5
DOI:
https://doi.org/10.1016/j.jrmge.2019.04.007
Reference:
JRMGE 617
To appear in:
Journal of Rock Mechanics and Geotechnical Engineering
Received Date: 29 October 2018 Revised Date:
3 April 2019
Accepted Date: 25 April 2019
Please cite this article as: Kumar B, Sahoo JP, Support pressure for circular tunnels in two layered undrained clay, Journal of Rock Mechanics and Geotechnical Engineering, https://doi.org/10.1016/ j.jrmge.2019.04.007. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. All rights reserved.
Support pressure for circular tunnels in two layered undrained clay Bibhash Kumara, Jagdish Prasad Sahoob,* a
Department of Civil Engineering, National Institute of Technology, Uttarakhand, 246174, India
b
Department of Civil Engineering, Indian Institute of Technology, Roorkee, 247667, India
ARTICLEINFO
ABSTRACT
Article history: Received 29 October 2018 Received in revised form 3 April 2019 Accepted 25 April 2019 Available online
To estimate the required support pressure for stability of circular tunnels in two layered clay under undrained condition, numerical solutions are developed by performing finite element lower bound limit analysis in conjunction with secondorder cone programming. The support system is assumed to offer uniform internal compressive pressure on its periphery. From the literature, it is known that the stability of tunnels depends on the overburden pressure acting over it, which is a function of undrained cohesion and unit weight of soil, and cover of soil. When a tunnel is constructed in layered undrained clay, the stability depends on the undrained shear strength, unit weight, and thickness of one layer relative to the other layer. In the present study, the solutions are presented in a form of dimensionless charts which can be used for design of tunnel support systems for different combinations of ratios of unit weight and undrained shear strength of upper layer to those of lower layer, thickness of both layers, and total soil cover depth. © 2019 Institute of Rock and Soil Mechanics, Chinese Academy of Sciences. Production and hosting by Elsevier B.V. T h is is a n o p e n a c c e s s a r t ic l e u n d e r t he C C BY - N C - N D l ic e n s e ( h t t p : / / c re a t i v e c o m mo n s . o r g/ licenses/by-nc-nd/4.0/).
Keywords: Finite elements Layered clay Limit analysis Stability Tunnels
for square and circular tunnels advanced in undrained cohesive soil taking into 1. Introduction Tunnels and underground openings are constructed for better transport facilities, underground pipeline, canal and hydropower projects, etc. The stability of tunnels depends on the resistance of support system to prevent failure of soil mass. Thus, the support pressure required must be known prior to its design to maintain tunnel stability. In order to evaluate tunnel stability in homogeneous ground conditions, a number of investigations have been performed (Atkinson and Potts, 1977; Mair, 1979; Davis et al., 1980; Assadi and Sloan, 1991; Wu and Lee, 2003; Lee et al., 2006; Osman et al., 2006; Yang and Yang, 2010; Sahoo and Kumar, 2014a; Yang et al., 2015; Zhang et al., 2017). Atkinson and Potts (1977) have conducted centrifuge model tests on assessing the stability of circular tunnels driven in sand supported by means of either compressed air or bentonite slurry. By conducting centrifuge model tests, Mair (1979) examined the response of tunnel driven in soft clay supported by compressed air pressure from inside of tunnel. Davis et al. (1980) computed the support pressure required for preventing collapse of underground openings/tunnels formed in soft clay under undrained condition based on the concept of limit analysis. Wu and Lee (2003) carried out centrifuge model tests to monitor ground movements and the associated collapse mechanism for both single and group of parallel tunnels in clay under undrained condition. Lee et al. (2006) conducted a series of centrifuge model tests and performed numerical analysis of these test results for evaluating the surface settlement trough, generation of excess pore water pressure and arching effect developed during the excavation process of single and parallel tunnels through soft clay. Osman et al. (2006) analyzed the stability of tunnel by examining the deformation pattern around unlined tunnel excavated in undrained clay using upper bound limit analysis. In order to determine the magnitude of required support pressure for tunnels located in various types of soils, solutions have been developed on the basis of finite element limit analysis with implementation of different mathematical programmings (Yang and Yang, 2010; Sahoo and Kumar, 2012, 2014a, 2018; Yang et al., 2015; Zhang et al., 2017). Limited studies are available in the literature to obtain the support pressure of tunnels located in the non-homogeneous soil (Sloan and Assadi, 1991; Hagiwara et al., 1999; Grant and Taylor, 2000; Nunes and Meguid,
account linear variation of undrained strength with depth. For unlined circular tunnels in undrained clayey soil, Sahoo and Kumar (2013) performed stability analysis by employing finite element upper bound limit analysis considering a linear variation of undrained cohesion with depth. In layered soil, the ground movements induced by tunneling process depend on the stiffness and type of overlying soil mass. The effect of type and stiffness of overlying soil mass on the ground movements during tunneling was investigated by Hagiwara et al. (1999) with the help of centrifuge model tests for tunnels constructed in layered soil. By conducting a few centrifuge model tests, Grant and Taylor (2000) studied the stability of tunnels located in the purely cohesive soil underlain by granular soil. Nunes and Meguid (2009) carried out plane strain elastoplastic finite element analyses and laboratory tests to investigate the variation in bending stresses developed in lining of tunnel support in purely cohesive soil overlain by coarse-grained sand layer. Using lower bound finite element limit analysis in conjunction with second-order cone programming, Sahoo and Kumar (2019b) determined required support pressure to maintain the stability of tunnel advanced in undrained clay layer overlain by sand layer. Nevertheless, few researches seem to be available for predicting the required compressive pressure that should be offered by a support system of tunnel advanced in two layered clayey soil to maintain the tunnel stability. In the present study, we aimed to produce solutions for estimating the compressive pressure required to support the soil surrounding the tunnel advanced in an undrained clay layer overlain by another undrained clay layer. The required support pressure is assumed to be acting normally and uniformly along the periphery of the tunnel. For the computations, finite element lower bound limit analysis and second-order cone programming were employed. The finite element limit analysis has been extensively used for solving various stability problems in geomechanics (Sahoo and Kumar, 2014b; Keawsawasvong and Ukritchon, 2017 a,b,c,d; Khuntia and Sahoo, 2017). The influences of thickness of lower clay layer above tunnel crown, thickness of upper clay layer, strengths of both the layers in terms of undrained shear strength, and unit weights of both layers have been studied. The failure patterns, i.e. the proximity of stress states at failure, have also been presented for a few cases. 2. Problem definition and chosen domain
2009; Wilson et al., 2011; Sahoo and Kumar, 2013, 2019a). By performing finite element limit analysis and rigid block upper bound method, Sloan and
A circular tunnel having diameter D is excavated in undrained clay (φu1= 0°)
Assadi (1991) and Wilson et al. (2011) computed the required support pressure
overlain by another layer of undrained clay (φu2= 0°) at a cover depth of H below the horizontal ground surface, as illustrated in Fig. 1. H1 is the thickness
*Corresponding author. Fax: +91-1332-285568; E-mail address: [email protected]
of lower soil layer above the tunnel crown, and H2 is the thickness of upper layer. To perform plane strain analysis, it is assumed that tunnel length is very
large as compared to its diameter. Keeping the undrained shear
domain. The location of the boundaries JK and IJ with Lh = 3D–30D and Lv =
strength/undrained cohesion (cu1) and unit weight (γ1) of the lower layer the
3D–20D has been found to be sufficient depending upon the values of H/D by
same, the analysis was performed for the following cases: (i) undrained shear
performing sensitivity analysis as discussed in Section 5.
strength (cu1) and unit weight (γ1) of lower layer smaller than that of upper layer (cu2 > cu1 and γ2 > γ1), and (ii) undrained shear strength and unit weight of lower layer greater than that of upper layer (cu2 < cu1 and γ2 < γ1). The parameters c, γ and φ refer to the cohesion, unit weight and internal friction angle of soil mass, respectively, whereas the subscript u indicates the undrained condition, and 1 and 2 refer to the lower and upper layers, respectively. The undrained strengths of both the clay layers are modeled by employing Tresca yield criterion and an associated flow rule. In order to support soil mass surrounding the tunnel periphery for different cases as mentioned above, it is intended in the present study to calculate the required support pressure (σi) in the form of lining and associated anchorage system. The support pressure is expressed in the form of a dimensionless factor defined as σi/cu1. Taking into account the symmetric nature of loading and geometry of the problem, only half of the domain as illustrated in Fig. 2a was used to perform the present analysis. The horizontal boundary (IJ) and vertical boundary (JK)
Fig. 1. Definition of problem.
were kept at a sufficient large distance from the centroid of tunnel by satisfying two conditions: (i) the plastic zone must be within the chosen
3. Analysis
domain; and (ii) the solutions will not change with further increase in the size of
The analysis was performed following the finite element formulation based
the
on lower bound theorem of limit analysis developed by Sloan (1988) and the second-order conic optimization technique of Makrodimopoulos and Martin (2006). In the lower bound approach, the magnitude of the load at failure is determined in a statically admissible stress field, that is, the stress field satisfies the equilibrium and stress boundary conditions without violating yield criterion. For computing the magnitude of limit pressure required to maintain stability of the tunnel, the objective function is maximized subjected to a set of equality and inequality constraints imposed on the unknown nodal stresses. The equality constraints are produced in order to satisfy equilibrium of elements, statically admissible stress discontinuities, and stress boundary conditions; whereas, for the satisfaction of yield criterion, the inequality constraints are generated. The chosen domain as illustrated in Fig. 2a is discretized into a number of three-noded triangular elements. None of the nodes are linked with more than a single element, and consequently, every interface between the adjacent elements becomes always the line of the stress discontinuity. It should be noted that each node remains unique to a particular e
(a)
l
(b)
e
m
e
n
t
,
a
n
d
(c)
Fig. 2. (a) Chosen problem domain and associated stress boundary conditions; (b) Sign convention for stresses; and (c) A typical finite element mesh for H/D = 6 with H1/D = 3 and H2/D = 3. ‘+ve’ shows that stresses are positive in these directions.
consequently, two or more nodes often share the same coordinates. There are
are generated in MATLAB (MathWorks, 2015) by writing programming
three basic unknowns at each node, namely, horizontal normal stress (σxx),
codes.
vertical normal stress (σyy), and shear stress (τxy). A typical finite element mesh
For modeling the stress field under plane strain condition, a linear variation
for H/D = 6 with H1/D = 3 and H2/D = 3 is shown in Fig. 2c. All the meshes
of stresses is chosen within each element and linear shape functions are
employed. The stresses within each triangular element must satisfy the
σ n ,1 = σ x x ,1 cos 2 θ1 + σ y y ,1 sin 2 θ1 − τ x y ,1 sin (2θ1)
(4a)
equilibrium conditions which are defined as follows: ∂σ x x ∂τ x y + =0 ∂x ∂y
σ n ,2 = σ x x ,2 cos θ 2 + σ y y ,2 sin θ 2 − τ x y ,2 sin(2θ2 )
(4b)
∂τ x y ∂x
+
∂σ y y ∂y
=γ
2
2
(1a) (1b)
where γ is the body force per unit volume of soil mass (unit weight of soil mass) acting vertically in the downward direction. In Eqs. (1a) and (1b), normal tensile stresses are taken as positive and the positive direction of shear stresses and coordinate axes are shown in Fig. 2b. Since each node is unique to an element, the unit weight of upper layer (γ2) and lower layer (γ1) are assigned at the nodes of elements of upper and lower layers, respectively, along the interface of two layers for satisfying the equilibrium condition given in Eq. (1b). In order to obtain lower bound solutions close to the true solution, statically admissible stress discontinuities are included in the lower bound analysis (Drucker, 1953; Lysmer, 1970; Sloan, 1988), and to obtain statically admissible stress discontinuity, the shear and normal stresses are required to be continuous along the common sides of any two adjacent elements. No constraints on the stresses were imposed along the boundaries IJ and JK. Along boundary LK, the normal and shear stresses are made equal to zero while along the surfaces LM and NI, only shear stress is zero. Along the interfaces of the adjoining clayey soil and the tunnel lining, it has been specified that the developed shear stress is less than or equal to the undrained cohesion of the adjoining soil, that is, | τ x y | ≤ c u 1 . In order to obtain strictly lower bound solutions, the state of stresses should satisfy yield criterion everywhere within the problem domain. Under plane strain condition, the Tresca yield criterion for undrained clay may be expressed as
Fig. 3. Various parameters associated with a very small length of segment of the tunnel periphery. Nodes 1 and 2 lie on the tunnel periphery and node 3 lies within the soil mass. The difference between chord length and arc length between nodes 1 and 2 is negligible as they are very close to each other.
(σ x x − σ y y ) 2 + 4τ x2y ≤ 2c u
(2)
The inequality constraint (Eq. (2)) can be expressed as a set of second-order cone (Makrodimopoulos and Martin, 2006) by introducing a vector
z i = {z 1i z 2i z 3i }T at each node, satisfying the equation of second-order cone, i.e. z 1i ≥ z 22i + z 32i . Eq. (2) is represented by second-order cone constraints as i Asoc σi
i Asoc
i + z i = b soc
(i = 1, 2, L, N )
σ 0 0 0 z 1i 2c u x x ,i i = −1 1 0 , σ i = σ yy ,i , z i = z 2 i , b soc = 0 0 0 −2 z 3 i 0 τ x y ,i
(3a)
It is assumed that the internal support pressure (σi) acts uniformly on the tunnel periphery; hence, the following condition needs to be satisfied: σ i = σ n ,1 = σ n ,2 = L = σ n ,N t
(5)
where Nt is the total number of nodes along the tunnel periphery. The lining used as support system is generally elastic in nature and considered as thin-walled tube under plane strain condition (Wood, 1975; Croll, 2001; Tamura and Hayashi, 2005; Wang and Koizumi, 2010; Wang et al., 2015). Though the required support pressure to make the surrounding soil
(3b)
mass in stable condition is not uniform along the tunnel periphery, to calculate the stiffness and thickness of tunnel lining based on the theory of thin-walled
where σxx,i, σyy,i and τxy,i imply the horizontal normal stress, vertical normal
cylinders, the pressure distribution is assumed to be uniform, which can be
stress and shear stress, respectively, at the i-th node; and N refers to the total
calculated as follows: σ 1 = σ iD / (4t )
(6)
σ 2 = σ iD / (2t )
(7)
number of nodes in the domain. For modeling the support system, no exclusive elements were used. Following Sahoo and Kumar (2019b), the sides of triangular elements along the periphery of tunnel are treated as the segments of lining surface since for a given segment, the difference between chord length and arc length is negligible when the size of triangular elements is very small. The constraints developed for computing the uniform compressive support pressure to be exerted by the support system was already given in Sahoo and Kumar (2019b); however, for the sake of completeness, this part is again described herein. Fig. 3 shows a typical segment of the tunnel periphery, where σxx,1 and σxx,2 refer to the horizontal normal tractions in x-direction at nodes 1 and 2, respectively;
σyy,1 and σyy,2 refer to the vertical normal tractions in y-direction at nodes 1 and 2, respectively; and τxy,1 and τxy,2 are the shear tractions on the vertical/horizontal planes at nodes 1 and 2, respectively. The angles θ1 and θ2 are the inclinations of radial lines passing through nodes 1 and 2, respectively, with the horizontal plane. The normal stresses σn,1 and σn,2 acting at nodes 1 and 2 on the tunnel periphery are expressed as
ε1 = ε2 =
σ1 νσ 2 E
−
E
σ 2 νσ1 E
−
E
(8) (9)
where σ1 and σ2 are the longitudinal and hoop stresses in the lining, respectively; ε1 and ε2 are the longitudinal and hoop strains in the lining, respectively; and t, E andν are the thickness, Young’s modulus and Poisson’s ratio of the lining, respectively. For plane strain condition, ε1= 0, thus we have σ iD σ 1 = νσ 2 , ε 2 = 2tE (1 − ν 2 )
(10)
Assuming that tc is the change allowed in the thickness of lining of the cylinder, then we can obtain 2π(D / 2 + t c ) − π D 2t c = ε2 = πD D
(11)
t =
σ iD 2
(12)
4t c E (1 − ν 2 )
σi
4t E = t c (1 − ν 2 )D 2
Lining stiffness per unit tunnel length is expressed as π Dσ i 4tE k = = tc (1 − ν 2 )D
4. Results and discussion
(14)
1991; Osman et al., 2006; Wilson et al., 2011; Sahoo and Kumar, 2014a), it is
From the previous studies on homogeneous clayey soil (Assadi and Sloan,
the constraints imposed on the nodal stresses. For performing second-order cone programming in a statically admissible stress field, the objective function along with the constraints may be expressed as follows:
noted that for a given diameter of the tunnel, the magnitude of required support pressure depends on the unit weight and cohesion of soil, and cover depth of the tunnel. In the present analysis, numerical solutions were obtained for a tunnel where soil mass lying over the tunnel is of a two-layered medium, that is, tunnel located in clay under undrained condition (φu1 = 0°) with an overlay
Objective function: Maximize {σ i }
(15)
Constraints: Ax = B
(16)
x = {σ
(17)
of relatively stiff or soft clayey soil under undrained condition (φu2 = 0°), where the unit weight and undrained cohesion of upper layer are greater or
z }T
0 0 0 I 0
B = {Bequil Bbound
Keawsawasvong, 2018, 2019).
(13)
The magnitude of limit support pressure at failure is optimized by satisfying
Aequil Abound A = Adis Asocp Aucs
2018; Sahoo and Khuntia, 2018; Ukritchon et al., 2018; Ukritchon and
smaller than that of lower layer. The present solutions have been obtained for two cases: (1)
In the first case, the results are obtained for a homogeneous clayey soil with γ1D/cu1 = γ2D/cu2 equal to 1, 3 and 5;
(18)
(2)
In the second case, keeping the unit weight and cohesion of the lower layer constant with γ1D/cu1 equal to 1, 3 and 5, the values of unit weight and cohesion of the upper layer are varied with:
Bdis
Bsocp
T
Bucs }
(19)
(i)
relatively higher cohesion (cu2 > cu1) for cu2/cu1 = 10, 2.5 and 1.43 (or cu1/cu2 = 0.1, 0.4 and 0.7) with relatively higher unit
where A is the global matrix of coefficients of all the constraints; Aequil, Abound,
weight (γ2 > γ1) for γ2/γ1 = 1.2, 1.4 and 1.6 (or γ1/γ2 = 0.83,
Adis, Asocp and Aucs are the coefficient matrices of constraints owing to element equilibrium, imposed boundary conditions, satisfying statically admissible stress discontinuities, a set of second-order cones and uniform compressive support pressure, respectively; I is the identity matrix; B is the global vector of
0.71 and 0.625), and (ii)
relatively lower cohesion (cu2 < cu1) for cu2/cu1 = 0.1, 0.4 and 0.7 with relatively lower unit weight (γ2 < γ1) for γ2/γ1 = 0.83, 0.71 and 0.625.
constants; Bequil, Bbound, Bdis, Bsocp and Bucs are the global vectors of constants on account of element equilibrium, imposed boundary conditions, satisfying
It is noted that the values of undrained cohesion and unit weight of soft to
statically admissible stress discontinuities, a set of second-order cones and
very stiff clay generally lie in the range of 14–22 kN/m3 and 25–100 kPa,
uniform compressive support pressure, respectively; σ is the global variable
respectively. The computations were performed by choosing the values of (i)
consisting of the nodal stresses, which can be defined as
σ = {σ x x ,1 σ y y ,1 τ x y ,1 σ x x ,2 σ y y ,2 τ x y ,2 ... σ x x ,N σ y y ,N z is the global auxiliary variable vectors defined as T
z T = {z 11 z 21 z 31 z 12 z 22 z 32 L z 1N z 2 N z 3 N }
τ x y ,N } (20)
H/D ranging from 1 to 10, (ii) γ1D/cu1 or γ2D/cu2 varying from 1 to 5, and (iii) cu2/cu1 between 0.1 and 10.
4.1. Variation of support pressure (21)
In the present study, the support pressure is normalized with respect to the
It is worth mentioning that for determining the resistance to be offered by
undrained cohesion of the lower layer (σi/cu1), which has been kept the same as
internal support pressure against active failure induced by overburden pressure
that used for obtaining results in the case of homogeneous clay. In two-layered
and surcharge (if any) on the ground surface, the objective function is
clay medium, the support pressure (σi) required for maintaining stability of
maximized. On the other hand, for computing the resistance to be provided by
tunnel depends on the thicknesses of upper layer (H2) and lower layer (H1)
the overburden pressure and surcharge against the collapse caused by internal
above the crown of tunnel, unit weight and cohesion of both layers. For
pressure which is termed as passive failure or blow out failure, the objective
various combinations of H1/D, cu2/cu1 and γ2/γ1, the variation of σi/cu1 with
function is minimized, that is, Minimize {σ i }. The pre-processing computer
H2/D is shown in Figs. 4–12. It may be noted that the movement of soil mass
codes were written in MATLAB (MathWorks, 2015) and optimized solutions
towards tunnel at failure due to the action of gravitational load is resisted by
were developed on the basis of cone optimization using MOSEK (MOSEK
the internal pressure offered by the support system. From the results presented
ApS, 2015), an optimization toolbox available for MATLAB. To obtain the
in Figs. 4–12, the positive value of σi implies that support system is required to
optimal lower bound solutions, the MOSEK optimizer is employed for solving
be provided to prevent the collapse of tunnel; whereas, negative values of σi
different stability problems in geotechnical engineering (Khuntia and Sahoo,
indicate that the tunnel remains stable without any support system.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 4. Variation of σi/cu1 with H2/D for H1/D = 1 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=1.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 5. Variation of σi/cu1 with H2/D for H1/D = 3 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1 = γ2D/cu2=1.
(a)
(c)
(b)
(d)
(e)
(f)
Fig. 6. Variation of σi/cu1 with H2/D for H1/D = 5 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1= 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1 = γ2D/cu2=1,
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 7. Variation of σi/cu1 with H2/D for H1/D = 1 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1= 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1= 0.4, and (f) cu2/cu1= 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=3.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 8. Variation of σi/cu1 with H2/D for H1/D = 3 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1= 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=3.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 9. Variation of σi/cu1 with H2/D for H1/D = 5 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1= 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=3.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 10. Variation of σi/cu1 with H2/D for H1/D = 1 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1= 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=5.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 11. Variation of σi/cu1 with H2/D for H1/D = 3 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1= 2.5, (c) cu2/cu1 = 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=5.
(a)
(b)
(c)
(d)
(e)
(f)
Fig. 12. Variation of σi/cu1 with H2/D for H1/D = 5 with different ratios of γ2/γ1: (a) cu2/cu1 = 10, (b) cu2/cu1 = 2.5, (c) cu2/cu1= 1.43, (d) cu2/cu1 = 0.7, (e) cu2/cu1 = 0.4, and (f) cu2/cu1 = 0.1, compared to homogeneous clay with γ1D/cu1= γ2D/cu2=5.
Keeping the shear strength of lower layer constant, corresponding to
offered by the undrained cohesion of soil mass. For smaller thickness of upper
particular values of H/D, H1/D, γ1D/cu1 and γ2/γ1, the magnitude of required
layer, the influence of driving force due to overburden pressure of upper soil
support pressure (i) decreases with increase in the undrained shear strength of
layer may be less compared to the resistance owing to the cohesion of upper
upper layer when the upper layer soil is relatively stiffer compared to the lower
layer, leading to nonlinear variation of σi/cu1. On the other hand, with increase
one, that is, unit weight and shear strength of lower layer are smaller than that
in the thickness of upper layer, the effect of driving force due to increase in
of upper layer (γ2>γ1 and cu2/cu1>1) as shown in parts (a–c) in Figs. 4–12; and
overburden pressure of soil becomes dominant versus that of increase in the
(ii) increases with reduction in the undrained shear strength of upper layer
resistance owing to the cohesion of soil mass, which causes the stiffer or linear
when the upper layer soil is relatively softer compared to the lower one, that is,
variation of σi/cu1.
the unit weight and shear strength of lower layer are greater than those of
For H2/D = 5, the variations of σi/cu1 with (i) cu2/cu1 for different values of
upper layer (γ2<γ1 and cu2/cu1< 1) as shown in parts (d–f) in Figs. 4–12. It can
γ2/γ1 with H1/D = 5 and γ1D/cu1 = 3, (ii) γ2/γ1 for different values of cu2/cu1 with
be seen from Figs. 4–12 that the variation of σi/cu1 is found to increase
H1/D = 5 and γ1D/cu1 = 3, (iii) γ1D/cu1 for different values of γ2/γ1 and cu2/cu1
nonlinearly with increase in H2/D up to a certain limit, after which it increases
with H1/D = 5, and (iv) H1/D for different values of γ2/γ1 and cu2/cu1 with
linearly. It is known that the failure is mainly due to the weight of soil mass
γ1D/cu1 = 3 are shown in Figs. 13–16. It is observed that there is increase in the
(overburden pressure) lying over the tunnel and the resistance to failure is
value of support pressure with (i) reduction in the undrained cohesion and
increase in the unit weight of upper layer with respect to the lower layer for
expressed in dimensionless term, namely, t/s, where t = (σ x x − σ y y ) 2 + 4τ x2y
given values of H1/D, H2/D and γ1D/cu1, (ii) increase in γ1D/cu1 for a given
and s = 2 c u , where cu becomes equal to cu1 and cu2 for the elements in the
combination of H1/D, H2/D, γ2/γ1 and cu2/cu1, and (iii) increase in H1/D for a
lower and upper layers, respectively. A point within the chosen domain will be
given combination of H2/D, γ1D/cu1, γ2/γ1 and cu2/cu1.
in a state of shear failure when the ratio t/s tends to unity. The failure patterns
From Figs. 4–12, it shows that compared to tunnel driven in homogeneous
are generated in a manner such that the color of an element becomes darker
clay, the magnitude of σi/cu1 required for the case of tunnel driven in two-
when approaching the shear failure (t/s = 1). The failure patterns have been
layered clay medium (where the properties of lower layer remain the same as
generated for H/D = 6 (H1/D = 3 and H2/D = 3) and γ1D/cu1= 3, as shown in
those of homogeneous clay) has been found to be dependent on the combined
Fig. 17 in the case of layered clay for different values of cu2/cu1 and γ2/γ1
influence of H1/D, H2/D, γ1D/cu1, cu2/cu1 and γ2/γ1, i.e.
keeping the cohesion of lower layer constant.
(1)
Greater only when cu2/cu1 = 1.43 for any values of H/D and γ1D/cu1, and dependent on the combined influence of H1/D, H2/D, γ1D/cu1, cu2/cu1 and γ2/γ1 for cu2/cu1 equal to 2.5 and 10 for lower values of
γ1D/cu1 = 1, as provided in Figs. 4–6. (2)
Smaller only when cu2/cu1 = 0.7 for any values of H/D and γ1D/cu1, and dependent on the combined influence of H1/D, H2/D, γ1D/cu1, cu2/cu1 and γ2/γ1 for cu2/cu1 equal to 0.4 and 0.1 for lower values of
γ1D/cu1 = 1, as illustrated in Figs. 4–6. (3)
Always greater when cu2/cu1 > 1 and always lower when cu2/cu1 < 1 for higher values of γ1D/cu1 = 3 and 5, which can be noted from Figs. 7–12.
From Fig. 17a, for a given value of γ2/γ1 in the case of two-layered clayey soil, it can be noted that the yielding of soil mass depends upon the magnitude of undrained cohesion of both layers relative to each other, that is, the zone of yielding is larger as expected in the lower layer where the cohesion is relatively lower compared to that of upper layer (cu2/cu1 = 10). By comparing Fig. 17a and b, it is observed that the size of failure zone tends to reduce in the lower layer and increase in the upper layer with the reduction in the values of cohesion of upper layer. Further, though the cohesion of upper layer is less, the zone of yielding is found to be smaller in the upper layer for cu2/cu1 = 0.7 as shown in Fig. 17c, when compared with that of cu2/cu1 = 1.43 as shown in Fig. 17b. It needs to be mentioned that the zone of failure in the upper layer also depends on the ratio of unit weight of both layers, that is, the unit weight of
Further, for cu2/cu1>1 in the case of layered soil, though the upper layer offers higher resistance than the homogeneous soil where the cohesion is the same as that of lower layer (cu1), the magnitude of σi/cu1 corresponding to all the values of cu2/cu1 > 1 for any combinations of H1/D, H2/D and γ1D/cu1 is not always lower than that of homogeneous soil. This is due to the fact that the
upper layer is smaller for the case shown in Fig. 17c than that presented in Fig. 17b. 4.3. Remarks For H1/D = 3, H2/D = 3, γ1D/cu1= 3, cu2/cu1 = 10 and γ2/γ1= 1.2, 1.4 and 1.6, the variation of σi/cu1 with Lh/D is presented in Fig. 18. It could be noted that
driving force causing failure is also higher for the tunnel driven in layered soil
the magnitude of σi/cu1 increases continuously with increase in domain size up
than that in homogeneous soil as γ2 > γ1. Similarly, when the resistance of
to a certain limit (Lh/D is approximately equal to 12), beyond which with
upper layer is lower compared to that of homogeneous soil (cu2 < cu1), the
further increase in the domain size, the values of σi/cu1 remains constant.
magnitude of σi/cu1 corresponding to all the values of cu2/cu1 < 1 for any
Further, from Fig. 17a, the developed plastic zone is observed to be well
combinations of H1/D, H2/D and γ1D/cu1 is not always greater than that of
contained within the chosen domain with Lh and Lv equal to 25D and 15D,
homogeneous soil because the driving force causing failure is also lower for
respectively. Thus, the chosen domain with Lh = 25D and Lv = 15D for
the tunnel driven in layered soil in comparison to the tunnel driven in
determining the magnitude of σi/cu1 when H1/D = 3, H2/D = 3, γ1D/cu1= 3,
homogenous soil as γ2<γ1.
cu2/cu1 = 10 and γ2/γ1= 1.6 is proved to be sufficient. Similarly, for different combinations of H1/D, H2/D, γ1D/cu, cu2/cu1 and γ2/γ1, the domain size was
4.2. Failure patterns The proximity of the stress state to shear failure, that is, the failure pattern,
decided in the present analysis.
has been examined from the stress state obtained at the element centroid and
(a)
(b)
Fig. 13. Variation of σi/cu1 with cu2/cu1 for (a) cu2/cu1>1 and γ2/γ1> 1 and (b) cu2/cu1<1 and γ2/γ1 < 1, for H1/D = 5, H2/D = 5 and γ1D/cu1 = 3.
(a)
(b)
Fig. 14. Variation of σi/cu1 with γ2/γ1 for (a) cu2/cu1> 1 and γ2/γ1> 1, and (b) cu2/cu1 < 1 and γ2/γ1< 1, for H1/D = 5, H2/D = 5 and γ1D/cu1= 3.
(a)
(b)
Fig. 15. Variation of σi/cu1 with γ1D/cu1 for (a) cu2/cu1 = 10 and γ2/γ1>1 and (b) cu2/cu1= 0.1 and γ2/γ1<1, for H1/D = 5 and H2/D = 5.
(a)
(b)
Fig. 16. Variation of σi/cu1 with H1/D for (a) cu2/cu1= 10 and γ2/γ1>1, and (b) cu2/cu1 = 0.1 and γ2/γ1< 1, for H2/D = 5 and γ1D/cu1 = 3.
t/s
x/D
t/s
y/D
y/D
y/D
t/s
x/D
x/D
(a) (b) (c) Fig. 17. Proximity of shear failure for layered clay with H1/D = 3, H2/D = 3 and γ1D/cu1= 3: (a) cu2/cu1= 10 and γ2/γ1=1.6; (b) cu2/cu1=1.43 and γ2/γ1=1.6; and (c) cu2/cu1=0.7 and γ2/γ1=0.625.
properties of both layers corresponding to undrained condition. The required support pressure is presented in terms of normalized normal compressive stress, defined as σi/cu1, and the variations of σi/cu1 for various combinations of H1/D, H2/D, γ1D/cu1, cu2/cu1 and γ2/γ1 have been established in the form of dimensionless charts, which can be used for the purpose of design of tunnel support system in a two-layered undrained clayey soil. The following conclusions can be drawn from the present study: (1)
For a particular combination of H1/D, H2/D, γ1D/cu1 and γ2/γ1, the magnitude of σi/cu1 is found to be continuously reducing with increase in the undrained cohesion of the upper layer when tunnel is located in the lower layer of relatively smaller stiffness than that of the upper layer (cu2/cu1 > 1 and γ2 > γ1), and continuously increasing with reduction in the undrained cohesion of the upper layer when tunnel is driven in the lower layer of relatively greater stiffness than
Fig. 18. Variation of σi/cu1 with Lh/D keeping Lv/D = 15 for H1/D = 3, H2/D = 3, γ1D/cu1= 3, cu2/cu1 = 10 and different values of γ2/γ1.
that of the upper layer (cu2/cu1 < 1 and γ2 < γ1). (2)
With increase in the unit weight of upper layer with respect to the lower layer for given values of H1/D, H2/D and γ1D/cu1, the
5. Comparison of present solutions with those available in the literature
magnitude of required support pressure increases. Since no solutions seem to be available in the literature for determining the
(3)
The increase in support pressure for a particular value of tunnel cover depth is nonlinear for lower values of H2/D and becomes linear
required support pressure to maintain the tunnel stability in two-layered clayey soil, for the purpose of validation of present analysis, the variation of σi/cu with
for higher values of H2/D. The rate of nonlinearity is found to be
H/D for different values of γD/cu obtained from the present analysis is
higher when the undrained cohesion of upper layer is relatively
compared with the numerical solutions of Wilson et al. (2011) and centrifuge
greater than that of lower layer. Furthermore, the support pressure
experimental results of Mair (1979) for a tunnel located in homogeneous clay
also increases with increase in the thickness of lower layer when the thickness of upper layer is constant.
under undrained condition. The comparisons are provided in Fig. 19a and b. In these figures, cu and γ are the undrained shear strength and unit weight of
(4)
With increase in the values of γ1D/cu1, the magnitude of required
homogenous undrained clay (cu = cu1= cu2, and γ = γ1 = γ2), respectively.
support pressure also increases when the thicknesses of both layers,
Wilson et al. (2011) analyzed the stability of tunnel using finite element lower
ratio of unit weight of upper to lower layer, ratio of cohesion of upper layer to lower layer remain constant.
bound limit analysis in conjunction with nonlinear programming. The solutions obtained from these two analyses are found to be almost merging
(5)
When the support pressure of a tunnel driven in layered soil with cu2
with each other. The present results also match closely with the experimental
< cu1 and γ2 < γ1 is compared with that of a tunnel driven in
results of Mair (1979).
homogeneous soil having the same properties as that of lower layer
6. Conclusions
required is not always lower for a tunnel driven in a layered soil,
(cu1 and γ1), it has been noted that the magnitude of support pressure
The support pressure (σi) required for the stability of a circular tunnel formed in clay layer overlain by relatively stiffer and softer clay layer has been computed by using lower bound finite element limit analysis in combination
although the cohesion of upper layer is higher than that of homogeneous soil. This is because the driving force inducing failure is also higher for the tunnel driven in layered soil than that in homogeneous soil as γ2 > γ1.
with the second-order conic programming. The analysis was performed considering
the
(a)
(b)
Fig. 19. Comparison of present solutions with (a) numerical solutions of Wilson et al. (2011) and (b) experimental results of Mair (1979) for tunnel located in homogenous clayey soil.
Conflict of interest The authors wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial
support for this work that could have influenced its outcome.
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Dr. Jagdish Prasad Sahoo is presently working as Assistant Professor at Indian Institute of Technology Roorkee, India. He has obtained his MSc and PhD degrees from Indian Institute of Science Bangalore, India. He has been working in the areas of stability of various problems in Geotechnical Engineering for last 10 years and published more than 35 papers in different international journals. Presently he is the editorial board member of International Journal of Geotechnical Engineering. He is the life member of Indian Geotechnical Society (IGS), Indian Roads Congress (IRC), Indian Society for Construction Materials and Structures (ISCMS), and Indian Society of Earthquake Technology (ISET).