On the elastic analysis of a circular lined tunnel considering the delayed installation of the support

International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

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On the elastic analysis of a circular lined tunnel considering the delayed installation of the support C. Carranza-Torres a,n, B. Rysdahl a, M. Kasim b a b

Department of Civil Engineering, University of Minnesota, Duluth Campus, MN, USA Swenson College of Sciences and Engineering, University of Minnesota, Duluth Campus, MN, USA

a r t i c l e i n f o

abstract

Article history: Received 1 March 2012 Received in revised form 8 January 2013 Accepted 24 January 2013 Available online 16 March 2013

This paper revisits the problem of excavation of a circular lined tunnel in an infinite elastic medium under plane strain conditions, subjected to non-uniform stresses, solved by Einstein and Schwartz in their article ‘Simplified Analysis for Tunnel Supports’ published in 1979 (ASCE J Geotech Eng Div, 106–7), and in related publications by Schwartz and Einstein. In contrast with the solution by Einstein and Schwartz from 1979 which considers that the support is installed at the very same time the tunnel is excavated, and also in contrast with related publications by Schwartz and Einstein which account for the delayed effect of support installation by multiplying the values of loads and displacements on the support by a reduction factor (in the same expressions from their 1979 article), this paper formulates the problem as two separate problems to account for the two-stage excavation process. In a first stage, the initial tractions on the periphery of the tunnel to be excavated are decreased (or ‘relaxed’) by a factor f R , to account for the presence of the tunnel front, as normally done in analysis of two dimensional sections of lined tunnels using commercial finite element software. For this first stage, the solution of the field quantities (stresses and displacements) in the ground are provided. In a second stage the support is installed, and the previously imposed tractions on the boundary of tunnel and support are removed, resulting in loading and deformation of the support as it interacts with the ground. For this second stage, the field quantities in the ground and the resulting values of load and displacements on the support are also provided. This paper shows that by decreasing the tractions on the tunnel periphery in a ratio f R of the initial ground stresses before the installation of the support (i.e., with 0 r f R o 1), the resulting final loads and displacements on the support are reduced in the same ratio f R , with respect to the corresponding loads and displacements computed by Einstein and Schwartz’s solution (i.e., when considering the case f R ¼ 1). This observation confirms the agreement of the two-stage procedure followed in this study and the procedure followed in publications by Schwartz and Einstein of multiplying the values of load and displacements on the support (for the case f R ¼ 1) by a correction factor to account for the delayed installation of the support. The paper also shows that when the stresses and displacements in the ground for the first stage are subtracted from the corresponding ground stresses and displacements for the second stage (after the support is installed), the resulting values of stresses and displacements are reduced in exactly the same ratio f R . All governing equations, boundary conditions and steps needed to arrive to a dimensionless form of the two-stage analytical solution of Einstein and Schwartz problem are presented, including the solutions for stresses and displacements in the ground which were not included in publications by Einstein and Schwartz. Also, particular forms of the two-stage solution are presented, including cases of an infinitely soft support, an infinitely rigid support, and uniform initial far-field stresses, which are shown to be equivalent to the classical expressions known as Kirsch and Lame´ solutions, respectively. Although the outstanding solution by Einstein and Schwartz published in 1979 is basically correct, Einstein and Schwartz dropped a term in the solution for bending moment because its effect is small in most of the situations (particularly when the thickness of the support is small compared with the radius of the tunnel). Since the two-stage solution for bending moment presented in this paper is equivalent to the Einstein and Schwartz’s solution only when the missing term in their solution is included, this paper presents a revised form of the expressions by Einstein and Schwartz published in 1979 which include

Keywords: Tunnel support Elastic analysis Convergence–confinement method Ground reaction curve Support characteristic curve Michell’s potential Kirsch’s solution Lame´’s solution Einstein and Schwartz’s solution

n

Corresponding author. Tel.: þ1 218 726 6460. E-mail address: [email protected] (C. Carranza-Torres).

1365-1609/$ - see front matter Published by Elsevier Ltd. http://dx.doi.org/10.1016/j.ijrmms.2013.01.010

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the missing term, and it examines what the effect of retaining or dropping the term is. Finally a comparison of results obtained with the two-stage analytical solution and the finite difference code FLAC3D is presented, together with the outline of a computer spreadsheet that implements the twostage analytical solution of this fundamental problem of tunnel ground–support interaction analysis. Published by Elsevier Ltd.

1. Introduction Closed-form solutions for tunnel excavation problems are important for understanding the mechanics of generation of stress and deformation that occur due to excavation of tunnels in rock and soil. Among others, closed-form solutions allow the examination of the fundamental relationships that exist between the different variables and parameters involved in the problem to be made, for example, relationships between quantities of stress and displacements in the ground, loading of the support and mechanical parameters of ground and support—see, for example, [1]. Indeed, analytical solutions of models of excavation of circular tunnels in elastic homogeneous materials subject to uniform initial stresses (i.e., the solution known as Lame´’s solution, after [2]), and non-uniform initial stresses (i.e., the solution known as Kirsch’s solution, after [3]) are discussed in textbooks of rock mechanics and excavation analysis due to their importance as basic teaching/learning tools for introducing the subject of the mechanics of underground excavations—see, for example, [4–8]. Some elastic models of excavation of supported tunnels in initially loaded elastic ground subject to non-uniform far-field stresses have been published in the past, with the Einstein and Schwartz’s solution [9,10] perhaps being the first of this type of solutions published in a major geomechanics journal. Prior to this outstanding journal publication, solutions of the complex problem of ground–support interaction, which at the time constituted doctoral thesis work, had been published mainly as technical reports [11,12]. Also, important extensions and further discussion of application of the Einstein and Schwartz’s solution [9,10] have been presented by [13,14]. The importance of Einstein and Schwartz’s solution has been realized with the advent of easy access of computing power needed to implement the complex equations conforming the solution, and publications that take as a basis the pioneering work done by these authors can be found in the form of books and journal publications— see, for example, [15–22], among others. Fig. 1 represents the elastic ground–support interaction problem solved by Einstein and Schwartz (the figure is adapted from their original article, [9]). Prior to excavation of the tunnel, the elastic ground is subjected to the non-uniform in situ (or far-field) stress state represented in the figure. In the vertical direction, the far-field stresses are equal to P; in the horizontal direction, the far-field stresses are equal to KP (i.e., K is the ratio of horizontal and vertical stresses). In the problem represented in Fig. 1, both P and KP are assumed not to vary with depth, so that forces associated with gravity are neglected (i.e., the tunnel problem considered by Einstein and Schwartz is that of a deep tunnel). In reference to Fig. 1, a tunnel of radius R is then excavated and, at the same time, an annular elastic support of thickness ts is installed. As a result of excavation, ground and support deform together, resulting in loading and deformation of the support (i.e., the generation of thrusts, bending moments, shear forces and normal and tangential displacements on the support). In reference to Fig. 1, plane strain conditions are assumed for both ground and support—i.e., the tunnel is considered to be infinitely long and a section of tunnel of unit length in the direction of the tunnel axis is analyzed. Also, ground and support are assumed to be linearly elastic and homogeneous materials, with Young Modulus and Poisson’s ratio E and n for the ground, respectively, and Es and ns for the support, respectively. Einstein

and Schwartz considered two situations for the interface between ground and support. A case of no slip, or continuity of radial and tangential stresses and displacements at the interface between ground and support, and a case of full slip, or continuity of radial displacements and normal stresses only, and zero tangential stresses at the interface—the two cases are treated in detail in Sections 6 and 7, and in Appendix A in this paper. Einstein and Schwartz [9] provided equations to compute the stresses and displacements resulting at the interface of ground and support, and the values of thrust and bending moment on the support, for both, the no slip and full slip cases mentioned above. Einstein and Schwartz’s solution, as presented in [9], does not account for the delayed effect of installation of the support, since the support is assumed to be installed at the very same time the tunnel is excavated. However, they [13,14], and particularly the underlying research report [12], discuss the implementation of the delayed effect of installing the support, through the application of a correction factor that multiplies the values of loads and displacements obtained for the support in the formulae presented in [9]. As noted in [13,14], inclusion of the delayed effect of tunnel support installation is an important aspect of the standard method for analyzing tunnel support sections in two dimensions under plane strain conditions, using the convergence–confinement method—see, for example, [23–26]. Fig. 2, adapted again from [9], represents the basis of the convergence–confinement method, for the case of axialsymmetry (i.e., the case in which K ¼1 in Fig. 1). The radial pressure ps on the support, represented by the ordinate of point Q in Fig. 2, is obtained as the intersection of the ground reaction curve (or GRC) and the support characteristic curve (or SCC). These curves represent the relationships between radial load and deformation of ground and support, respectively, as indicated by the two sketches on the right side of Fig. 2 (a comprehensive treatment of the method of construction of these curves can be found in [27]). Note that in Fig. 2, the SCC starts at point C, with an abscissa f(D), which is a function of the distance D between the tunnel front and the first section of support installed behind the tunnel front (i.e., the distance D is

Fig. 1. The tunnel ground–support interaction problem considered by Einstein and Schwartz—adapted from [9].

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

Fig. 2. The basis of the convergence–confinement method of tunnel support design for a circular tunnel in axi-symmetrical conditions of loading. Construction of ground reaction curve (or GRC) and support characteristic curve (or SCC)—adapted from [9].

Fig. 3. (a) Advancing tunnel and distribution of radial displacements, ur, as a function of the distance from the tunnel front, D, in the axi-symmetrical problem of tunnel excavation represented by the GRC in Fig. 2. (b) Relationship between the scaled radial displacements and scaled distance from the front. The longitudinal displacement profile (or LDP)—adapted from [27].

the unsupported length of the tunnel behind the front, as represented in the Fig. 3a). The different curves and squares in Fig. 3b, which represent the relationships between scaled displacement ur =uM and the distance D as obtained from r computational models and monitoring of tunnel convergence, suggest that at the tunnel front itself, the ground undergoes a displacement of approximately 30% the final displacement that the tunnel would have undergone if it was left unsupported. This means that if the support is installed at the front itself (i.e., the distance D in Fig. 3a is equal to zero), according to the

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convergence–confinement method the abscissa f(D) in Fig. 2 should be taken as approximately 30% of the maximum tunnel radial displacement for the unsupported tunnel (i.e., in Fig. 2, the length of the segment OC should be approximately 30% the length of the segment OB). Thus, if the material is assumed to be elastic, a radial displacement of approximately 30% the final radial displacement will be achieved when the internal pressure of the tunnel (i.e., the traction acting on the periphery of the tunnel to be excavated) is taken to be equal to the initial value (corresponding to the in situ stress) multiplied by a reduction factor equal to 100%  30% ¼ 70% prior to installing the support. This is indeed the method to be used in this paper to account for the delayed effect of the installation of the support and therefore to account for the beneficial effect of the presence of the tunnel front at the time the support is installed. It should be noted that the curves in Fig. 3b will normally depend on the way the tunnel is excavated and on the ground conditions in regard to values of initial stresses and mechanical properties of the ground; for example, the curve indicated as ‘Elastic Approximation’ in Fig. 3b, which is presented in [26], is obtained with a finite element axi-symmetric model of a circular tunnel, excavated in full section in homogeneous isotropic elastic material, subject to uniform initial stresses and Poisson’s ratio 0.25. In summary, it should be noted that Fig. 3b has been included as a reference only, to indicate that at the tunnel front itself, the ground has undergone some degree of deformation already. Also, it is important to emphasize that by assuming simultaneous excavation of the ground and installation of the support, which means considering that the distance f(D) in the diagram of Fig. 2 is null (actually, if the distance ur =uM r ¼ 0 in Fig. 3b then the distance D in Fig. 3a would be negative), the resulting support pressure ps acting on the support will be overestimated. This overestimation of the support load is the fundamental reason why consideration of the delay of installation of the support is critically important, as recognized already and accounted for by [13,14]. In contrast with the approach used in [13,12] which consists in multiplying the values of loads and displacements on the support obtained with the expressions in [9] by a reduction factor to account for the delayed installation of the support, this paper formulates the problem as two separate problems to account for the two-stage excavation process: in a first stage the tractions on the periphery of the tunnel to be excavated are reduced (or ‘relaxed’) in a specified ratio of the initial values of tractions associated with the initial ground stresses, and in a second stage the support is installed and the tractions on the periphery of the tunnel are removed, resulting in loading and deformation of the support as it interacts with the ground. This approach is chosen since it replicates the natural way in which lined tunnels are analyzed nowadays using commercial finite element or finite difference software to obtain the resulting loads and displacements on the ground and support—see, for example, [28,29]. Due to the academical importance of the Einstein and Schwartz’s problem, particularly for teaching undergraduate and graduate students about basic tunnel support design in the framework of the rigorous theory of elasticity, this paper presents a detailed formulation of the two-stage Einstein and Schwartz’s problem, including governing equations and boundary conditions needed to obtain the solution, as well as an application example and spreadsheet implementation of the two-stage solution.

2. Problem statement The problem considered here is similar to that in Fig. 1—also, plane strain conditions are assumed for both ground and support.

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The main differences are in the notation and sign convention used for the quantities involved. These will be explained in detail in following sections, as the solutions for the various problems comprising the derivation of the proposed two-stage solution are presented. In Fig. 1, the field quantities (stresses, strains and displacements in the ground) depend on the radial distance r and the angle y—measured from the right side springline of the tunnel and assumed positive in the counterclockwise direction (this sign convention for the angle y will be retained in the solution presented here). The load and deformation quantities for the support (thrusts, bending moments, shear forces and radial and tangential displacements) depend only on the angle y. Also, according to the theory of elastic shells the neutral axis of the section of support, which is assumed to be located at the midheight of the tunnel section, is assumed to coincide with the wall of the tunnel. In the solution presented here, the tunnel radius is called a; the solution considers then the angle y and the scaled radial distance r as independent variables. The latter is defined as follows:

r¼

r a

ð1Þ

The same dimensionless coefficients Cn and Fn (the compressibility and flexibility coefficients, respectively) introduced by [9] are considered in this paper, i.e., Cn ¼

a E 1n2s As Es 1n2

ð2Þ

a3 E 1n2s Is Es 1n2

ð3Þ

and Fn ¼

In Eqs. (2) and (3), E and Es are Young’s moduli of the ground and support, respectively, and n and ns are Poisson’s ratios for the ground and support, respectively. Also, in Eqs. (2) and (3), As and Is represent the area and moment of inertia, respectively, of a section of support of thickness ts, and unit length in the tunnel axis direction (see lower-right sketch in Fig. 1); these are computed as follows: As ¼ t s  1

ð4Þ

and t3 Is ¼ s  1 12

ð5Þ

To account for the presence of the tunnel front before installation of the support, and according to the convergence–confinement method (see, for example, [23–26]), the solution presented in this paper considers the two stages of excavation represented in Fig. 4. In the first stage (Fig. 4a), the tunnel is excavated and at the f f wall of the tunnel, normal stresses sr R and shear stresses sryR are applied. These stresses are obtained by multiplying the radial stress sor , and the shear stress sory (where sor and sory are the stresses existing in the medium before excavation of the tunnel), by a scalar relaxation factor f R , i.e.

sfr R ¼ sor  f R

ð6Þ

and

sfryR ¼ sory  f R

ð7Þ

where

sor ¼

1 þK 1K P P cos 2y 2 2

ð8Þ

Fig. 4. The tunnel ground–support interaction problem considered in this study. (a) First stage of excavation: relaxation of the initial ground stresses on the tunnel wall prior to installation of the support. (b) Second stage of excavation: installation of the support and removal of the ‘relaxed’ initial stresses on the tunnel wall.

and

sory ¼

1K P sin 2y 2

ð9Þ

The origin of the expressions for sor and sory given by Eqs. (8) and (9), respectively, is discussed in Section 5 and Appendix A. In the second stage (Fig. 4b), the support is installed and the f f stresses sr R and sryR applied in the first stage are removed. As a result, the tunnel and the support deform together, inducing thrusts, bending moments, shear stresses and radial and tangential displacements on the support. The relaxation factor f R in Eqs. (6) and (7), which will be assumed to lie in the range of 0–1, controls the amount of relaxation of the ground stresses prior to installation of the support. If f R ¼ 1, then there is no relaxation of the ground stresses in the first stage of excavation (Fig. 4a) and the loads and displacements for the support in the second stage (Fig. 4b) are the same as those obtained by [9]—i.e., the Einstein and Schwartz’s solution is a particular case of

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

the two-stage solution presented here when f R ¼ 1. If f R ¼ 0, then total relaxation of the ground stresses occurs in the first stage of excavation (Fig. 4a) and neither loads nor displacements occur on the support in the second stage of excavation (Fig. 4b). Note that in terms of the convergence–confinement method, this latter situation corresponds to the case of installing the support too far behind the tunnel front, so that the support does not take any load once the tunnel front has advanced far from the tunnel section. For a support that is installed at an arbitrary distance D behind the tunnel front (see Fig. 3a), the factor f R can be computed from the ground reaction curve (GRC) in Fig. 2 and the longitudinal displacement profile (LDP) in Fig. 3b—see, for example, [27]. For the particular case in which the ground is assumed to be elastic and the assumption of axi-symmetry of the convergence–confinement method is satisfied (i.e., considering K¼ 1 in Fig. 1), installing a support at the distance D from the front will imply considering a factor f R equal to 1ur =uM r , where ur is the radial displacement of the tunnel section without support at the distance D, and uM r is the maximum radial displacement far from the tunnel front—see Fig. 3a. Note that the use of the expression f R ¼ 1ur =uM r is justified by noting that when the material is elastic, the relationship between p and u in the GRC represented in Fig. 2 is linear; also, that if u¼0 then p ¼ so , where so is the initial stress in the ground; and that if u ¼ uM r then p¼0—see Fig. 2. Therefore, for the cases in which the ground is assumed elastic and K¼1, if the support is installed just at the tunnel front (the distance D is equal to zero in Fig. 3a) then the ratio ur =uM r is approximately equal to 0.3 (see Fig. 3b) and f R is approximately equal to 0.7—the example cases to be considered in this paper (see, for example, Figs. 9 and 14) will consider a relaxation factor f R ¼ 0:7 implying that the support is installed just at the tunnel front. Publications [30,31] present a discussion on the amount of ground stress relaxation to consider in the analysis of two-dimensional supported tunnel sections according to the convergence–confinement method, in the general cases for which the ground is assumed to be elasto-plastic and the value of the horizontal-to-vertical stress ratio K is different from one. In this paper, the solution for the first stage of excavation represented in Fig. 4a will be referred to as Solution A and this solution is discussed in Sections 4 and 5 (and Appendix A). The solution for the second stage of excavation, which involves the installation of the support (see Fig. 4b), will be divided into the two cases considered by [9]; these are the no slip and full slip conditions at the ground–support interface. The solutions for these two cases will be referred to as Solution B and Solution C, respectively, and these are presented in Sections 6 and 7 (and Appendix A).

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3. Scaling of variables In the formulation of Solutions A, B and C presented in this paper, load and deformation quantities are considered dimensionless by normalizing these quantities with respect to properly selected variables. The scaled quantities will be denoted by the use of the tilde symbol (  ) on top of the variable. For example, the far-field vertical stress P (see Fig. 1) is used to scale the components of normal and shear stresses, sr , sy and sry , respectively, in the ground, i.e.

s~ r ¼

sr P

,

s~ y ¼

sy P

s~ ry ¼

,

sry

ð10Þ

P

Similarly, the radial and tangential stresses acting on the support, the quantities pr and py , respectively, are scaled as follows: p~ r ¼

pr , P

p~ y ¼

py P

ð11Þ

The far-field vertical stress P, the Young Modulus of the ground E, and Poisson’s ratio of the ground n are used to scale the components of normal and shear strains; thus, the quantities er , ey and ery are scaled as follows:

e~ r ¼ er

E 1 , P 1þn

e~ y ¼ ey

E 1 , P 1þn

e~ ry ¼ ery

E 1 P 1þn

ð12Þ

The radial and tangential displacements are scaled similarly as the strains, but also including the radius of the tunnel a. Then the quantities ur and uy are scaled as follows: u~ r ¼

ur E 1 , a P 1þn

u~ y ¼

uy E 1 a P 1þn

ð13Þ

Similarly, the radial and tangential displacements of the support, the quantities usr and usy , are scaled as u~ sr ¼

usr E 1 , a P 1þn

u~ sy ¼

usy E 1 a P 1þn

ð14Þ

Finally, the loads generated on the support, i.e., the thrust force N, the shear force Q and the bending moment M, are scaled with respect to the far-field stress P and the tunnel radius a as follows: N , N~ ¼ Pa

Q Q~ ¼ , Pa

~ ¼ M M Pa2

ð15Þ

The following sections present the solution for the three problems (Solutions A, B and C) in which the two-stage proposed solution can be decomposed. These solutions are written in terms of the scaled variables introduced above.

Fig. 5. Solution A: stresses and displacements in the ground for the first stage of excavation prior to installation of the support (see Fig. 4a).

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4. Solution A: tunnel excavated in elastic ground with stress boundary conditions and no structural support Solution A is the solution for the first stage of excavation discussed in Section 2—see Fig. 4a. Fig. 5 represents the problem of excavation of the circular tunnel in the elastic material subject to normal stress sor and shear stress sory at the far-boundary (i.e., at r-1), and normal f f stress sr R and shear stress sryR at the tunnel wall (i.e., at r ¼a)—see Eqs. (6)–(9), respectively. The figure indicates the sign convention adopted for the field quantities. For example, for a point P located at an arbitrary distance r from the center of the tunnel, the stress components sr , sy and sry are negative as indicated (i.e., normal stresses are negative in compression). The framed sketch in Fig. 5 indicates the sign convention assumed for the radial and tangential displacements, ur and uy , respectively, which are also negative, as indicated. For the problem represented in Fig. 5, the solution of field quantities for the ground expressed in terms of the dimensionless quantities introduced in Section 3 is as follows (the superscript ‘[A]’ in the expressions that follow, refer to Solution A): The scaled radial stress is ½A ½A s~ ½A r ¼ 2B01 þ B03

1

4B½A 22

r2

1

½A cos 2y2B½A 23 cos 2y6B24

r2

1

r4

cos 2y ð16Þ

the scaled hoop stress is 1

½A ½A s~ ½A y ¼ 2B01 B03

½A þ 2B½A 23 cos 2y þ6B24

r2

1

r4

cos 2y

ð17Þ

s

¼

B½A 04

1

1

r

r2

2B½A 22 2

sin

2y þ 2B½A 23

sin

2y6B½A 24

1

r4

sin 2y

ð18Þ

¼ 2B½A 01 ð12

B½A 03

nÞr

1

r

þ4B½A 22 ð1

½A 2B½A 23 r cos 2y þ 2B24

nÞ

1

r3

1

r

cos 2y

1

r

cos 2y

2B½A 22 ð12nÞ

1

r

½A sin 2y þ2B½A 23 r sin 2y þ 2B24

1

r3

ð21Þ

1 B½A 03 ¼  ð1 þKÞð1f R Þ 2

ð22Þ

B½A 04 ¼ 0

ð23Þ

1 B½A 22 ¼  ð1KÞð1f R Þ 2

ð24Þ

B½A 23

1 ¼ ð1KÞ 4

B½A 24 ¼

1 ð1KÞð1f R Þ 4

o

and u~ y are In Eqs. (27) and (28), the scaled displacements the displacements associated with the application of the far-field stresses P and KP, in Fig. 4, before the tunnel is excavated. The stresses u~ or and u~ oy are computed as follows: 1 þK 1K ð12nÞr r cos 2y 2 2

ð29Þ

1K r sin 2y 2

ð30Þ

and

5. Particular cases of solution A sin 2y

½A ½A ½A ½A In Eqs. (16)–(20), the coefficients B½A 01 , B03 , B04 , B22 , B23 and B½A are 24

1 ð1 þKÞ 4

ð28Þ u~ or

ð19Þ

ð20Þ

B½A 01 ¼

~ ½A ~ o u~ ½A y IND ¼ u y u y

The origin of Eqs. (29) and (30) is discussed in the following section.

and the scaled tangential displacement is ½A u~ ½A y ¼ B04

ð27Þ

and the scaled tangential component of the induced displacement is computed as

u~ oy ¼

the scaled radial displacement is u~ ½A r

~ ½A ~ o u~ ½A r IND ¼ u r u r

u~ or ¼

the scaled shear stress is ~ ½A ry

It should be noted that the solution given by Eqs. (16)–(26) assume that the ground is initially unloaded. Therefore, the displacements given by Eqs. (19) and (20) are due to both, application of the far-field stresses (i.e., initial loading of the ground at the far-boundary) and excavation of the tunnel. For tunnel excavation problems, the displacements corresponding to the application of the far-field stresses are typically irrelevant (i.e., these have occurred throughout the geological history of the site). Instead, the displacements that are relevant (e.g., for engineering purposes) are those associated with excavation of the tunnel. These displacements are referred to as induced displacements, and, for the problem represented in Fig. 5, they can be computed by subtracting the displacements corresponding to the application of the far-field stresses only from the total displacements associated with both, application of the far-field stresses and excavation of the tunnel. Therefore, considering Eqs. (19) and (20), the scaled radial component of the induced displacement is computed as follows:

The first particular case for Solution A presented in Section 4 is obtained by making f R ¼ 1 in the coefficients given by Eqs. (21)–(26), and replacing these coefficients in Eqs. (16)–(20). Assuming f R ¼ 1 implies that the tunnel is excavated but that the normal and shear stresses acting on the wall of the tunnel are identical to the normal and shear stresses that were acting before the tunnel was excavated (see Eqs. (6) and (7)). The situation, which is represented in Fig. 6, is as if no tunnel was excavated at all, and as if only the far-field stresses were acting on the ground, producing the initial stresses and displacements of the ground. For this particular case, the solution, which will be denoted with the superscript ‘o’, is as follows. The scaled radial stress is

s~ or ¼ ð25Þ

ð26Þ

A detailed demonstration of the equations above (including the origin of the numbering of the coefficients ‘B’) is provided in Appendix A (see Section A.5, in particular).

1 þ K 1K  cos 2y 2 2

ð31Þ

the scaled hoop stress is

s~ oy ¼

1 þ K 1K þ cos 2y 2 2

ð32Þ

the scaled shear stress is

s~ ory ¼

1K sin 2y 2

ð33Þ

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63

Fig. 6. Particular case of Solution A for a relaxation factor f R ¼ 1. Stresses and displacements due to application of the far-field stresses only (i.e., situation before excavation of the tunnel).

and

the scaled radial displacement is u~ or ¼

1þK 1K ð12nÞr r cos 2y 2 2

ð34Þ

and the scaled tangential displacement is 1K u~ y ¼ r sin 2y 2 o

ð35Þ

Note that Eqs. (34) and (35) are the same as Eqs. (29) and (30) used to define the induced components of displacements for Solution A in Section 4—see Eqs. (27) and (28). These equations are also the same ones listed in [9] and are referred to there as the initial displacements prior to excavation. The second particular case for Solution A is obtained by making f R ¼ 0 in Eqs. (21)–(26), and replacing these in Eqs. (16)– (20). Assuming f R ¼ 0 implies considering zero internal pressure on the wall of the tunnel in Fig. 5. This gives the following expressions. The scaled radial stress is 1 þ K 1K 1þK 1  cos 2y s~ r ¼ 2 2 2 r2 1 3 1 þ 2ð1KÞ 2 cos 2y ð1KÞ 4 cos 2y 2 r r

1 þK 1K 1þK 1 3 1 þ cos 2y þ þ ð1KÞ 4 cos 2y 2 2 2 r2 2 r

ð36Þ

1K 1 3 1 sin 2y þ ð1KÞ 2 sin 2y ð1KÞ 4 sin 2y 2 2 r r

1 þK 1K ð12nÞr r cos 2y 2 2 1 þK 1 1K 1 1 þ þ cos 2y2ð1KÞð1nÞ cos 2y 2 r 2 r3 r

1K 1K 1 1 r sin 2y þ sin 2y þ ð1KÞð12nÞ sin 2y 2 2 r3 r

1 þ K 1 1K 1 1 þ cos 2y2ð1KÞð1nÞ cos 2y 2 r 2 r3 r

ð43Þ

the scaled hoop stress is 1

r2

ð44Þ

the scaled shear stress is

u~ IND ¼ ð1p~ i Þ r

ð45Þ

1

r

ð46Þ

and the scaled induced tangential displacement is ð38Þ

¼0 u~ IND y

ð47Þ

The set of Eqs. (43)–(47) are Lame´’s solution for non-zero internal pressure acting on the wall of the tunnel—see, for example, [4,32]. ð39Þ

ð40Þ

The scaled radial and tangential components of induced displacements can be computed using the same Eqs. (27) and (28), respectively, which lead to u~ IND ¼ r

1

r2

the scaled induced radial displacement is

and the scaled tangential displacement is u~ y ¼

s~ r ¼ 1ð1p~ i Þ

ð37Þ

the scaled radial displacement is u~ r ¼

ð42Þ

The set of Eqs. (36)–(42) are Kirsch’s solution for zero stress on the wall of the tunnel—see, for example, [4,32]. The third particular case for Solution A is obtained by making K¼ 1 in Eqs. (21)–(26), and again replacing these in Eqs. (16)–(20). In this case (K ¼1) the shear stresses and tangential displacements (Eqs. (18) and (20), respectively) become zero, implying that the problem is axi-symmetric—this being a basic assumption of the convergence–confinement method discussed in Section 1. Therefore, considering K¼1 and making f R ¼ p~ i in Eqs. (16)–(26), the following particular solution is obtained. The scaled radial stress is

s~ ry ¼ 0

the scaled shear stress is

s~ ry ¼

1K 1 1 sin 2y þ ð1KÞð12nÞ sin 2y 2 r3 r

s~ y ¼ 1 þ ð1p~ i Þ

the scaled tangential stress is

s~ y ¼

u~ IND ¼ y

ð41Þ

6. Solutions B and C: supported tunnel with no slip (Solution B) and full slip (Solution C) conditions at the ground–support interface Solutions B and C are the solutions for the second stage of excavation (Fig. 4b) and for the no slip and full slip conditions at the ground–support interface, respectively. Fig. 7 represents the quantities characterizing the response of the support in Solutions B and C, which are the thrust N, bending moment M, shear force Q, radial displacement usr, and tangential displacement usy . The figure indicates the sign convention assumed

64

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

for these quantities (i.e., the positive signs preceding the quantities imply that these quantities are positive as indicated). The stresses pr and py , indicated in Fig. 7, represent the loads transmitted from the ground to the support, which according to the theory of shells are assumed to act at the mid-height of the support section (also, as discussed in Section 2, the tunnel wall is assumed to coincide with the mid-height of the support section). Fig. 8 represents the ground–support interaction problem, where the indicated quantities for the ground have a similar interpretation as in Figs. 5 and 6 (i.e., the negative signs preceding the stresses and displacements mean that these are negative as represented). On the left side of Fig. 8, the stress and displacement boundary conditions at the ground–support interface that distinguish Solutions B and C are summarized (these conditions are discussed in detail in Appendix A, and Sections A.7 and A.8, in particular). For Solution B (no slip case), the induced radial and tangential displacements of the ground at the ground–support interface (note that in Fig. 8, the subscript ‘i’ refers to the interface) are equal to the

corresponding displacements for the support. In this case there is full continuity of radial and tangential displacements (and therefore of radial and shear stresses) across the ground–support interface, and the ground–support interface behaves as a ‘glued’ interface. For Solution C (full slip case), only the induced radial displacements of the ground at the ground–support interface are equal to the radial displacements for the support, while the shear components of stresses along this interface are considered to be zero—i.e., there is continuity of stresses and displacements only in the radial direction, and the ground–support interface behaves as a frictionless interface. For the problem represented in Fig. 8 (see also Fig. 7), the solution of quantities for the ground and for the support, scaled according to the rules presented in Section 3, is as follows (the superscript ‘[B,C]’ in the equations below means that the quantities apply to both, Solutions B and C, while the superscript ‘[B]’ means they apply to Solution B only, and the superscript ‘[C]’ means they apply to Solution C only). For the ground, the scaled radial stress is ½B,C s~ ½B,C ¼ 2B½B,C r 01 þ B03

1

r2

4B½B,C 22

½B,C 2B½B,C 23 cos 2y6B24

1

r2

1

r4

cos 2y

cos 2y

ð48Þ

the scaled hoop stress is ½B,C s~ ½B,C ¼ 2B½B,C 01 B03 y

1

r2

½B,C þ2B½B,C 23 cos 2y þ6B24

1

r4

cos 2y

ð49Þ

the scaled shear stress is

s~ ½B,C ¼ B½B,C 04 ry

1

r2

2B½B,C 22

1

r2

½B,C sin 2y þ 2B½B,C 23 sin 2y6B24

1

r4

sin 2y ð50Þ

the scaled radial displacement is ½B,C u~ ½B,C ¼ 2B½B,C r 01 ð12nÞrB03

Fig. 7. Sign convention for load and displacement quantities of the support.

1

r

þ 4B½B,C 22 ð1nÞ

½B,C 2B½B,C 23 r cos 2y þ 2B24

1

r3

cos 2y

1

r

cos 2y ð51Þ

Fig. 8. Solutions B and C: stresses and displacements in the ground and support for the second stage of excavation after installation of the support (see Fig. 4b)—Solutions B and C correspond to the no slip and full slip conditions at the ground–support interface, respectively.

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

and the scaled tangential displacement is u~ ½B,C ¼ B½B,C 04 y

1

r

2B½B,C 22 ð12nÞ

þ 2B½B,C 24

1

1

r

B½B 22

sin 2y þ 2B½B,C 23 r sin 2y

r

ð52Þ

In view of Eqs. (27) and (28), the scaled radial component of the induced displacement is ~ ½B,C u~ or u~ ½B,C r IND ¼ u r

½B,C

¼ u~ y

o

u~ y

u~ or

½B,C ½B,C ½B,C ½B,C 2B½B,C 01 þ B03 2ð2B22 þ B23 þ3B24 Þ

¼

n n B½B 22N ¼ ½24 þ 9C ð1nÞ þ F ð1nÞð34nÞf R

ð66Þ

n n B½B 22D ¼ 2½24 þ 9C ð1nÞ þ F ð1nÞð12nÞ

þ 4½6 þ3C n ð1nÞ þF n ð1nÞ þ 2C n F n ð1nÞ2

ð67Þ

B½B 23 ¼

1 ð1KÞ 4

ð68Þ

B½B 24 ¼

" # B½B 1 ð1KÞ 1 24N 4 B½B 24D

ð69Þ

ð54Þ o

and u~ y are given by the same Eqs. (29) and (30) or where Eqs. (34) and (35), respectively. For the support, the scaled radial stress is p~ ½B,C r

ð65Þ

ð53Þ

and the scaled tangential component of the induced displacement is u~ ½B,C y IND

" # B½B 1 22N ¼  ð1KÞ 1 ½B 2 B22D

where

sin 2y

3

65

cos 2y

ð55Þ

where n n B½B 24N ¼ ½12 þ 3C ð1nÞ þ F ð1nÞð34nÞf R

ð70Þ

the scaled tangential stress is ½B,C ½B,C ½B,C p~ ½B,C ¼ B½B,C 04 2ðB22 B23 þ 3B24 Þ sin 2y y

ð56Þ

n n B½B 24D ¼ ½24 þ9C ð1nÞ þF ð1nÞð12nÞ

þ 2½6 þ3C n ð1nÞ þF n ð1nÞ þ C n F n ð1nÞ2

the scaled radial displacement is 2 n ½B,C ½B,C ½B,C 2 ¼ F n ð1nÞB½B,C u~ s½B,C r 04 y 3 F ð1nÞðB22 þ B23 þ B24 Þ cos y D½B,C cos yD½B,C ðy cos ysin yÞ þ D½B,C 1 2 3

ð57Þ

1 n ½B,C ½B,C 2 n u~ s½B,C ¼ C n ð1nÞð2B½B,C 01 þB03 Þy ðC þ F Þð1nÞB04 y y 2 1 ½B,C ½B,C þ ðC n þ F n Þð1nÞðB½B,C 22 þ B23 þB24 Þy 3 1  ð3C n F n Þð1nÞB½B,C 22 sin 2y 6 1 þ ð3C n þ F n Þð1nÞB½B,C 23 sin 2y 6 1 ½B,C  ð9C n F n Þð1nÞB½B,C sin y 24 sin 2y þ D1 6 n n C þF ½B,C þD½B,C ð2 cos y þ y sin yÞ D3 y 2 Fn

N~

ð72Þ

D½B 2 ¼0

ð73Þ

D½B 3 ¼

D½B 3N ½B D½B 3D1 D3D2

Fnf R

ð74Þ

n n2 2 D½B 3N ¼ 5C ð1nÞð34nÞð1 þ KÞ þC ð1nÞ ð67nÞð1 þ KÞ

þ F n ð1nÞð34nÞð1KÞ þ C n F n ð1nÞ2 ð35nÞð1KÞ þ C n2 F n ð1nÞ3 ð12nÞð1KÞ þ2C n ð1nÞð34nÞ þ C n 2 ð1nÞ2 ð34nÞ þ C n F n ð1nÞ2 ð32nÞ þ C n2 F n ð1nÞ3 ð58Þ

the scaled thrust is ½B,C ½B,C ½B,C ¼ 2B½B,C 01 þB03 þ 2ðB23 B24 Þ

D½B 1 ¼0

and

the scaled tangential displacement is

½B,C

ð71Þ

2 D½B,C sin y cos 2y þ n F ð1nÞ 2 ð59Þ

n n n n D½B 3D1 ¼ C þF þ C F ð1nÞ

  ~ ½B,C ¼ B½B,C yðB½B,C þ B½B,C þ B½B,C Þ 1 cos 2y M 04 22 23 24 3 1 ½B,C ½B,C ð2D2 sin y þ D3 Þ þ n F ð1nÞ

þ 2½6 þ3C n ð1nÞ þF n ð1nÞ þ C n F n ð1nÞ2 For the full slip case (Solution C), the coefficients ½C B23 ,

B½C 01 ¼ ð60Þ

ð76Þ

n n D½B 3D2 ¼ ½24 þ 9C ð1nÞ þ F ð1nÞð12nÞ

B½C 22 ,

the scaled bending moment is

ð75Þ

B½C 24 ,

D½C 1 ,

D½C 2

and

D½C 3

ð77Þ B½C 01 ,

½C B03 ,

½C B04 ,

in Eqs. (62)–(76) are

1 ð1 þ KÞ 4

ð78Þ

  1 C n þF n fR B½C n n n 03 ¼  ð1 þ KÞ 1 n 2 C þF þ C F ð1nÞ

ð79Þ

B½C 04 ¼ 0

ð80Þ

  1 9ð34nÞ ð1KÞ 2 f B½C ¼  22 4 F n ð1nÞ þ 3ð56nÞ R

ð81Þ

and the scaled shear force is ½B,C ½B,C ½B,C ½B,C ¼ B½B,C Q~ 04 2ðB22 þ B23 þ B24 Þ sin 2y þ

2 D½B,C cos y F n ð1nÞ 2 ð61Þ

For the no slip case (Solution B), the coefficients ½B ½B ½B ½B ½B B22 , B½B 23 , B24 , D1 , D2 and D3 in Eqs. (48)–(61) are ½B ¼ B01

1 ð1þ KÞ 4

B½B 01 ,

B½B 03 ,

B½B 04 ,

ð62Þ

B½C 23 ¼

1 ð1KÞ 4

ð82Þ

B½C 24 ¼

  1 3ð34nÞ ð1KÞ 1 n fR 4 F ð1nÞ þ3ð56nÞ

ð83Þ

  1 Cn þ Fn ½B B03 ¼  ð1 þKÞ 1 n fR n n n 2 C þ F þ C F ð1nÞ

ð63Þ

D½C 1 ¼0

ð84Þ

½B B04 ¼0

ð64Þ

D½C 2 ¼0

ð85Þ

66

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

and D½C 3N D½C 3D

D½C 3 ¼

Fnf R

ð86Þ

where n n D½C 3N ¼ 2C ð1nÞð67nÞð1þ KÞ þF ð1nÞð34nÞð1KÞ

þ 2C n F n ð1nÞ2 ð12nÞð1KÞ þ 2C n ð1nÞ½34n þ F n ð1nÞ

ð87Þ

to recover the expected linear relationship with the factor f R . Therefore, operating with Eqs. (48)–(54), and with Eqs. (16)–(28), and after some algebra manipulation, the following expressions are obtained (as done earlier on, the values of f R considered are indicated within parentheses in the equations below). For the radial, hoop and shear stresses in the ground ~ ½A ~ ½B,C s~ ½B,C ðf R Þs~ ½A s~ ½B,C ðf R Þs~ ½A r r ðf R Þ y ðf R Þ ¼ s r y ðf R Þs ry ðf R Þ ¼ f ¼ y½B,C R ½B,C ½A ½A ½B,C s~ r ð1Þs~ r ð1Þ s~ y ð1Þs~ y ð1Þ s~ ry ð1Þs~ ½A ð1Þ ry ð93Þ

n n n n n D½C 3D ¼ 2½C þ F þ C F ð1nÞ½3ð56nÞ þ F ð1nÞ

ð88Þ

A detailed demonstration of the equations above (including the origin of the numbering of the coefficients ‘B’ and ‘D’) is provided in Appendix A (see Sections A.6–A.8, in particular). With the exception of the equations for bending moment M (this is explained further in Section 9 and Appendix B), the solution for the support quantities given by the equations above are identical to those presented by [9] when the coefficient f R is considered to be equal to one—i.e., when no relaxation of the ground stresses is assumed prior to installation of the support. When the coefficient f R is assumed to be less than one, the solution for the support quantities given by Eqs. (55)–(61) are reduced in exactly the same ratio f R , with respect to the solution for the support quantities for the case f R ¼ 1. This observation, that is the expected according to the principle of superposition of linear elasticity, can be verified by computing the ratios of Eqs. (55)–(61) with respect to the corresponding equations when f R ¼ 1. Indeed, after some algebra manipulation, the following ratios are obtained (the values of f R considered are indicated within parentheses in the equations below). For the radial stress on the support ðf R Þ p~ ½B,C r ¼ fR p~ ½B,C ð1Þ r

ð89Þ

For the tangential stress on the support (and for the Solution B only, since for Solution C the tangential stress is null and therefore the ratio is indeterminate) p~ ½B y ðf R Þ p~ ½B y ð1Þ

¼ fR

ð90Þ

For the radial and tangential displacements of the support u~ s½B,C ðf Þ u~ s½B,C ðf R Þ r ¼ ys½B,C R ¼ f R s½B,C u~ y ð1Þ u~ r ð1Þ

ð91Þ

Similarly, for the trust, bending moment and shear force of the support ½B,C ½B,C ~ ½B,C ðf Þ N~ M ðf R Þ ðf Þ Q~ R ¼ ¼ ½B,C R ¼ f R ½B,C ½B,C ~ N~ M ð1Þ ð1Þ ð1Þ Q~

ð92Þ

Eqs. (89)–(92) confirm the equivalence of the two-stage procedure adopted in this paper to account for the delay of installation of the support with the procedure adopted by [13,14], which based on the principle of superposition of linear elasticity consists in multiplying the support quantities in the equations in [9] by a reduction factor. Although [9] and related publications by the same authors did not present the solution for the ground quantities, it is worth noting that similar relationships as indicated by Eqs. (89)–(92) hold for stresses and displacements in the ground—i.e., for Eqs. (48)–(54). In this case, due to the fact that the total displacements in the ground are obtained by application of two different solutions (Solution A and Solutions B or C), and therefore the problem of obtaining displacements on the ground is a non-linear problem, the quantities corresponding to Solution A must be subtracted from the corresponding quantities for Solutions B or C

Similarly, for the radial and tangential induced displacements in the ground ~ ½A u~ ½B,C r IND ðf R Þu r IND ðf R Þ ~ ½A u~ ½B,C r IND ð1Þu r IND ð1Þ

¼

~ ½A u~ ½B,C y IND ðf R Þu y IND ðf R Þ ~ ½A u~ ½B,C y IND ð1Þu y IND ð1Þ

¼ fR

ð94Þ

~ ½A Note that in Eq. (94), the quantities u~ ½A r IND ð1Þ and u y IND ð1Þ are zero, and have been included for clarity only—i.e., for one-to-one comparison with the denominators in Eq. (93). The need to subtract the displacements for the ground that occur before the support is installed can be understood considering the ground reaction curve and support characteristic curve construction in Fig. 2. The displacement on the ground, u, is measured on the abscissa of the diagram from point O, while the displacement on the support, us, is measured from point C. Considering an arbitrary degree of relaxation before installation of the support, given by the abscissa f(D) in Fig. 2, and considering that the ground reaction curve is linear from point A through point B (as it will be the case for an elastic formulation as in this paper), due to similarity of triangles, the sides of the resulting triangle CQU will decrease linearly with the relaxation parameter, as the abscissa f(D) increases from zero to the current value indicated in Fig. 2. This implies that the values of ps and us obtained for the support will be linearly dependent on the relaxation factor—as indicated by Eqs. (89) and (91), respectively. Also, since the loading on the support (thrust, bending moment, and shear forces) is linearly dependent on the values of external loading acting on the support, these quantities will also be linearly dependent on the relaxation factor, as indicated by Eq. (92). For the ground, and for the same reasons of similarity of the resulting triangle CQU, the displacement, u, indicated in Fig. 2 will only be linearly dependent on the relaxation factor, provided the initial displacement of the ground, f(D), is subtracted from the total displacement, u, first—as indicated by Eq. (94). Also for the ground, considering that displacements are linearly related to stresses, the stresses will also be linearly dependent on the relaxation factor, provided that the initial stresses of the ground, i.e., those associated with the relaxation stage, are subtracted from the final ones as well—this is indicated by Eq. (93). The diagrams in Fig. 9, which are similar to the ones presented in [9] to summarize results for the support, represent the ~ scaled bending moment M, ~ relationship of the scaled thrust N, and scaled support radial displacement u~ sr , for the tunnel springline (i.e., y ¼ 01) and the ratio of Young’s moduli for ground and support (E=Es ) for the no slip and full slip conditions at the ground– support interface. These relationships, which were obtained by application of Eqs. (57)–(61), were computed assuming values of f R equal to 1 and 0.7 and values of K ¼0.5, t s =a ¼ 0:1 and n ¼ ns ¼ 0:25 (as discussed in Section 2, a value f R ¼ 0:7 implies that the support is installed at the tunnel front itself, while a value f R ¼ 1 implies the practical impossible situation of installing the support ahead of the tunnel front). In agreement with Eqs. (91) and (92), the diagrams in Fig. 9, ~ M ~ and u~ r for f ¼ 0:7 are exactly indicate that the values of N, R equal to 70% the corresponding values for f R ¼ 1—this is indicated by the pairs of points B and B0 , and C and C 0 , corresponding to

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

Fig. 9. Distribution of (a) scaled thrust, (b) scaled bending moment, and (c) scaled radial displacement of the support, as a function of the ratio E=Es , for the no slip and full slip conditions at the ground–support interface, and cases of no ground stress relaxation f R ¼ 1, and ground stress relaxation f R ¼ 0:7.

Solutions B and C, respectively, and for an arbitrary constant value of abscissa E=Es ¼ 4  103 . For example, the ordinate of the point B0 is 70% the ordinate of point B, and the ordinate of the point C 0 is 70% the ordinate of point C. Also, as observed by Einstein and Schwartz, the values of scaled thrust at the springline of the tunnel are larger for the no slip case (Solution B) than for the full slip case (Solution C)—see Fig. 9a. The values of scaled bending

67

Fig. 10. Distribution of increments of (a) scaled radial stress, (b) scaled hoop stress, and (c) scaled induced radial displacement, as a function of the ratio r, for the no slip and full slip conditions at the ground–support interface, and cases of no ground stress relaxation f R ¼ 1, and ground stress relaxation f R ¼ 0:7.

moment are smaller for the no slip case (Solution B) than for the full slip case (Solution C)—see Fig. 9b. The latter observation is also valid for the scaled radial displacement of the support (when analyzed in absolute value), for cases in which the ratio E=Es is approximately smaller than 5  10  2—see Fig. 9c. The diagrams in Fig. 10 represent the relationship of the increment of scaled radial stress s~ ½B,C s~ ½A r r , increment of scaled

68

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

~ ½A hoop stress s~ ½B,C y s y , and increment of scaled induced radial ~ ½A displacement u~ ½B,C  r IND u r IND , for the tunnel springline (i.e., y ¼ 01) and the scaled radial distance, r, for the no slip and full slip conditions at the ground–support interface. These relationships, which were obtained by application of Eqs. (48)–(54), and Eqs. (16)–(28), were computed assuming values of f R equal to 1 and 0.7 and values of K ¼0.5, t s =a ¼ 0:1, n ¼ ns ¼ 0:25 and E=Es ¼ 4  103 . Again, in agreement with Eqs. (93) and (94), the diagrams ~ ½B,C s~ ½A in Fig. 10 indicate that the values of s~ ½B,C s~ ½A r r , sy y and ½B,C ½A u~ r IND u~ r IND for f R ¼ 0:7 are exactly equal to 70% the corresponding values for f R ¼ 1. This is indicated by the pairs of points B and B0 , and C and C 0 , corresponding to Solutions B and C, respectively, and for an arbitrary constant value of abscissa r ¼ 1:5.

~ ~s Disregarding the expressions for the quantities s~ ry , u~ IND y , py, uy and Q~ (Eqs. (97), (99), (101), (103), and (106), respectively) that for the case K ¼1 are zero, the rest of the Eqs. (95)–(106) can be expressed in terms of the scaled radial stress p~ r acting on the support (see Eq. (100)) as follows. For the ground, the scaled radial stress is

s~ r ¼ 1ð1p~ r Þ

s~ y ¼ 1 þ ð1p~ r Þ

ð96Þ

the scaled tangential stress is

s~ ry ¼ 0

ð97Þ

the scaled induced radial displacement is   Cn þ Fn 1 f ¼ 1 u~ IND R r r C n þ F n þC n F n ð1nÞ

ð98Þ

and the scaled induced tangential displacement is ¼0 u~ IND y

ð99Þ

For the support, the scaled radial stress is p~ r ¼

C n þF n fR C þF þ C n F n ð1nÞ n

n

ð100Þ

the scaled tangential stress is p~ y ¼ 0

C n F n ð1nÞ fR C þ F n þ C n F n ð1nÞ n

ð102Þ

ð103Þ

n

C þF fR C n þ F n þ C n F n ð1nÞ

ð104Þ

and the scaled shear force is Q~ ¼ 0

N~ ¼ p~ r

ð111Þ

and the scaled bending moment is ~ ¼ M

Cn p~ r C þ Fn

ð112Þ

n

Note that Eqs. (107)–(109) are identical to Lame´’s solution (Eqs. (43), (44), and (46), respectively) obtained as a particular case of Solution A, when the internal pressure acting on the wall of the tunnel is equal to the scaled support pressure p~ r . Eqs. (107)–(112) will be used in Section 8 to verify that Solutions B and C for the particular case K ¼1, give the exact same results for the support loads, as with previously published solutions derived from construction of the ground reaction curve and support characteristic curves in the convergence–confinement method. The second particular case for Solutions B and C, presented in Section 6, is obtained by considering an infinitely rigid support (i.e., t s -1 or Es -1 in Fig. 8), and the condition K ¼1. In view of Eqs. (2) and (3) considering an infinitely rigid support implies taking values C n ¼ 0 and F n ¼ 0 in Eqs. (48)–(88)— the process actually involves computing the limits of the expressions C n -0 and F n -0 by applying the L’Hospital rule, since simply stating C n ¼ 0 and F n ¼ 0 in Eqs. (48)–(88), would give indeterminate expressions. In that case, both Solutions B and C give the same expressions, which are as follows. For the ground, the scaled radial stress is 1

r2

ð113Þ

the scaled hoop stress is 1

r2

ð114Þ

and the scaled induced radial displacement is 1

r

ð115Þ

For the support, the scaled radial stress is p~ r ¼ f R

ð105Þ

ð110Þ

the scaled thrust is

u~ IND ¼ ð1f R Þ r

n

C fR C n þF n þC n F n ð1nÞ

ð109Þ

C n F n ð1nÞ p~ r Cn þ Fn

n

the scaled bending moment is ~ ¼ M

u~ sr ¼

s~ y ¼ 1 þ ð1f R Þ

the scaled thrust is N~ ¼

1

r

s~ r ¼ 1ð1f R Þ

the scaled tangential displacement is u~ sy ¼ 0

ð108Þ

and the scaled induced radial displacement is

ð101Þ

the scaled radial displacement is u~ sr ¼

1

r2

For the support, the scaled radial displacement is

The first particular case for Solutions B and C presented in Section 6 is obtained by making K ¼1 in the coefficients given by Eqs. (62)–(88), and replacing these coefficients in Eqs. (48)–(61). As discussed in Section 1, the assumption K ¼1 implies that the problem is axi-symmetric, and for this case both Solutions B and C give the same results. These are as follows. For the ground, the scaled radial stress is   C n þF n 1 fR 2 s~ r ¼ 1 1 n n n n ð95Þ r C þF þ C F ð1nÞ the scaled hoop stress is   Cn þ Fn 1 fR 2 s~ y ¼ 1 þ 1 n n n n r C þF þ C F ð1nÞ

ð107Þ

the scaled hoop stress is

u~ IND ¼ ð1p~ r Þ r 7. Particular cases of Solutions B and C

1

r2

ð116Þ

the scaled radial displacement is u~ sr ¼ 0

ð117Þ

the scaled thrust is ð106Þ

N~ ¼ f R

ð118Þ

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

the scaled bending moment is ~ ¼0 M

ð119Þ

and the scaled shear force is Q~ ¼ 0

ð120Þ IND

Note that for K ¼1 the expressions for s~ ry , u~ y , p~ y and u~ sy are all equal to zero (these expressions have not been listed above). Eqs. (113)–(120) give the intuitively expected results. If the support is infinitely rigid, after installation of the support and f f removal of the relaxed stresses sr R (note that sryR ¼ 0 when K¼ 1), the particular forms of Solutions B and C predict no deformation for the support (Eq. (117)) and predict a scaled radial pressure for the support equal to f R (Eq. (116)) which, according to Eq. (6), is the relaxed scaled radial stress applied in the first stage of excavation (Fig. 4a). Also, as expected for the ground, the particular form of Solutions B and C given by Eqs. (113)–(115), are identical to Lame´’s solution Eqs. (43)–(47) when the scaled pressure at the tunnel wall is equal to f R . The third particular case for Solutions B and C is obtained by considering an infinitely soft support (i.e., t s -0 or Es -0 in Fig. 8, and therefore C n -1 and F n -1 in Eqs. (48)–(88). For this case, again, both Solutions B and C give the same expressions, which are as follows. For the ground, the scaled radial stress is

s~ r ¼ 1

1

r

2

ð121Þ

the scaled hoop stress is

s~ y ¼ 1 þ

1

r

2

Indeed, for the second stage of excavation (Fig. 4b), the scaled (induced) radial displacement of the ground at the wall of the tunnel will be u~ IND ¼ 1 (this is obtained by making r ¼ 1 in Eq. (123)) minus r the corresponding displacement in the first stage of excavation (Fig. 4a), which is u~ IND ¼ 1f R (this is obtained by making r ¼ 1 r and p~ i ¼ f R in Eq. (46)); therefore, the scaled radial displacement of the support (and ground) for the second stage of excavation will be f R , which is exactly what Eq. (125) states.

8. Convergence–confinement analysis using the particular forms of Solutions B and C for the case K ¼1 The publication [33] analyzed the mechanical response of a circular closed liner of radius a and thickness ts using the same governing equations for the support considered in this paper (i.e., corresponding to the thick shell formulation and plane strain conditions, as described in Appendix A, and Section A.3, in particular) for various external loading configurations acting on the support. For the particular case in which the external load is radial only (i.e., py ¼ 0 in Fig. 7), the following expressions were obtained (the notation and sign convention of the quantities below, which are transcribed from [33], are the same ones indicated in Fig. 7). The radial displacement is usr ¼

¼

1

r

rCF pr a2 1þ rCF D

ð122Þ

usy ¼ 0

ð130Þ

where D¼ ð123Þ

Es t s 1n2s

ð124Þ

the scaled radial displacement is u~ sr ¼ f R

ð127Þ

N ¼ pr a

M¼

1 p a2 1 þ rCF r

ð132Þ

ð133Þ

ð128Þ

~ y and u~ sy are all equal Note that the expressions for s~ ry , u~ IND y , p to zero when K ¼1 (these equations have not been listed above). Eqs. (121)–(128) give, again, the intuitively expected results. If the support is infinitely soft, after installation of the support f and removal of the radial stress sr R applied in the fist stage of excavation (see Fig. 4a), the particular form of Solutions B and C predicts that the support will react with a null radial stress (Eq. (124)) and, therefore, that the stresses and induced displacements in the ground will be governed by the same Lame´’s solution equations, derived in Section 5, when the scaled internal pressure p~ i is equal to zero (see Eqs. (121)–(123); and Eqs. (43), (44), and (46), respectively). Also, if the support takes no radial stress (Eq. (124)), then no thrust, no bending moment, and no shear force can be expected to occur on the support—see Eqs. (126)–(128). Finally, an infinitely soft support will show the same radial displacement as the induced ground displacements at the wall of the tunnel.

ð134Þ

and the shear force is Q ¼0

and the scaled shear force is Q~ ¼ 0

12 ðt s =aÞ2

the bending moment is ð126Þ

the scaled bending moment is ~ ¼0 M

rCF ¼

Also, the thrust is ð125Þ

the scaled thrust is N~ ¼ 0

ð131Þ

and

For the support, the scaled radial stress is p~ r ¼ 0

ð129Þ

and the tangential displacement is

and the scaled radial displacement is u~ IND r

69

ð135Þ

The relationship between the coefficient rCF and the scaled thickness of the liner t s =a (Eq. (132)) is represented in Fig. 11. The diagram also indicates the corresponding values of the coefficient rCF obtained for three different cases of scaled support thickness, namely the cases of t s =a ¼ 0:025, 0.10 and 0.25, respectively. For the case of the support loaded with radial stress only, and again, according to the theory of thick shells, Eq. (134) indicates that there exists a bending moment on the support that depends on the coefficient rCF represented in Fig. 11—and also on the radial stress acting on the support and on the radius of the support. Publication [33] discussed that this bending moment is generated due to compatibility of displacements in the curved section of a thick support according to the following mechanism: as fibers on the intrados of the liner have move more than fibers on the extrados of the liner when the section displaces radially, a linear variation of strains along the height of the section occurs, which in turn, produces bending moment. It should be emphasized that this mechanism of generation of bending moment for a

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Fig. 12. Convergence–confinement analysis of the ground–support interaction problem, using Solutions A, B and C, for the particular case K¼ 1.

Fig. 11. Variation of the compressibility/flexibility coefficient rCF , as a function of the scaled thickness of support t s =a—after [33].

Indeed, for the support, Eq. (137) indicates that the slope of the SCC (Fig. 2) must be equal to ðC n þ F n Þ=ðC n F n ð1nÞÞ; also, the abscissa f(D) in Fig. 2 must be equal to 1f R (see discussion in the last part of Section 7; note also that the abscissa f ðDÞ ¼ 1f R is readily obtained by considering p~ i ¼ f R and r ¼ 1 in Eq. (46)). Therefore, the equation describing the SCC must be p~ i ¼ ðu~ r 1þ f R Þ

radially loaded support occurs when the mechanical behavior of the support is described according to the theory of thick shells (as used in the original article of Einstein and Schwartz and as described in Appendix A, and Section A.3 in particular). When using the theory of thin shells, the bending moment (Eq. (134)) becomes zero—see [33]. The coefficient rCF , given by Eq. (132) (see Fig. 11), can also be written in terms of the coefficients Cn and Fn introduced in Eqs. (2) and (3). The relationship is

rCF ¼

Fn Cn

u~ sr ¼

C n F n ð1nÞ p~ r Cn þ Fn

p~ i ¼ 1u~ r

ð141Þ

Eqs. (140) and (141) are represented in Fig. 12. As explained in Section 1, the ordinate of the intersection point of the GRC and SCC, point Q in Fig. 12, defines the radial support pressure at the equilibrium point. Thus, solving for u~ r in Eqs. (140) and (141), and equating the corresponding results, the resulting value of p~ i , which will be called p~ r , is p~ r ¼

ð137Þ

ð140Þ

For the ground, Eq. (46) (with r ¼ 1) gives the relationship needed to construct the GRC (Fig. 2), i.e.

ð136Þ

Using Eq. (136) and the same scaling rules introduced in Section 3, the non-zero quantities in Eqs. (129)–(135) can be re-written in terms of the coefficients Cn and Fn as follows. The scaled radial displacement is

Cn þ Fn C F n ð1nÞ n

Cn þ Fn fR C þ F þC n F n ð1nÞ n

n

ð142Þ

Eq. (142) is identical to Eq. (100), obtained as the particular case of Solutions B and C for K ¼1 in Section 7. Also, considering that Eqs. (137)–(139) are exactly the same as Eqs. (110)–(112), respectively, then the intended demonstration is complete (i.e., the particular form of Solutions B and C for K ¼1 predict the same results as the convergence–confinement method).

the scaled thrust is N~ ¼ p~ r

ð138Þ

9. Revised form of the Einstein and Schwartz [9] solution for bending moment of the support

and the scaled bending moment is ~ ¼ M

Cn p~ r C þF n n

ð139Þ

It will be demonstrated now that the application of the convergence–confinement method using Eq. (137) for constructing the support characteristic curve (the curve SCC in Fig. 2) and the expressions of the classical Lame´’s solution for constructing the ground reaction curve (the curve GRC in Fig. 2) give exactly the same results as Solutions B and C for the particular case K ¼1, as discussed in Section 7.

Although the solution presented in [9] is basically correct, Einstein and Schwartz consciously dropped a term in their solution for bending moment because they judged its effect was small in most of the situations (particularly when the thickness of the support was small compared with the radius of the tunnel). In this regard it should be mentioned that the underlying research in report [12], which as mentioned in Section 1 was part of the doctoral work of Schwartz, arrived to a complete form of the solution for bending moment which, as also mentioned above, was later simplified by dropping a term. Also, in an almost

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concurrent research development of a similar solution presented in [12], in his doctoral work, Ranken [11] retained the term for bending moment that was dropped by Einstein and Schwartz in [9]. Since the solution for bending moment presented in Section 6 (see Eq. (60)) is equivalent to the Einstein and Schwartz’s solution [9] only when the missing term in their solution is included, this section presents a revised form of the solution by Einstein and Schwartz [9] which includes the missing term. This section also discusses the effect of retaining or dropping the term in question by considering supports of two different thicknesses. In essence, by dropping the term for bending moments, the solution by Einstein and Schwartz [9] does not fully satisfy the governing equation relating bending moment and radial and tangential displacements for the support (see Appendix B). Below the original equations presented by Einstein and Schwartz together with the revised equations for bending moment are summarized for the no slip and full slip conditions at the ground– support interface (the equations below refer to the notation and sign convention shown in Fig. 1). The compressibility coefficient, Cn, is Cn ¼

ð143Þ

and flexibility coefficient, F , is ð144Þ

1 1 sR ¼ Pð1 þ KÞð1ano Þ Pð1KÞð16an2 þ 4bn2 Þ cos 2y 2 2

ð145Þ

1 2

tRy ¼ Pð1KÞð1þ 6an2 2bn2 Þ sin 2y

ð146Þ

us E 1 1 n ¼ ð1 þKÞano þ ð1KÞ½4ð1nÞb2 2an2  cos 2y PRð1 þ nÞ 2 2

vs E n ¼ ð1KÞ½an2 þ ð12nÞb2  sin 2y PRð1 þ nÞ

ð148Þ

the thrust of the support is ð149Þ

and the bending moment of the support (as originally given by Einstein and Schwartz) is 1 n ð1KÞð12an2 þ 2b2 Þ cos 2y 4

ð150Þ n

In Eqs. (145)–(150), the coefficients ano , b, b2 and an2 (again, for the no slip case) are C n F n ð1nÞ C n þ F n þ C n F n ð1nÞ

n

ð6 þ F n ÞC n ð1nÞ þ 2F n n 3F n þ 3C n þ 2C n F n ð1nÞ

b2 ¼

C n ð1nÞ 2½C ð1nÞ þ 4n6b3bC n ð1nÞ n

n

an2 ¼ bb2

Cn C þF þ C n F n ð1nÞ n

n

ð156Þ

For the full slip case, the radial stress acting on the support is 1 2

1 2

sR ¼ Pð1 þKÞð1ano Þ Pð1KÞð36an2 Þ cos 2y

ð157Þ

the tangential stress acting on the support is

tRy ¼ 0

ð158Þ

the radial displacement of the support is us E 1 ¼ ð1þ KÞano ð1KÞ½ð56nÞan2 ð1nÞ cos 2y PRð1þ nÞ 2

vs E 1 ¼ ð1KÞ½ð56nÞan2 ð1nÞ sin 2y PRð1þ nÞ 2

T 1 1 ¼ ð1 þ KÞð1ano Þ þ ð1KÞð12an2 Þ cos 2y PR 2 2

ð159Þ

ð160Þ

ð151Þ

ð152Þ

ð153Þ

ð154Þ

ð161Þ

and the bending moment of the support (as originally given by Einstein and Schwartz) is ¼

1 ð1KÞð12an2 Þ cos 2y 2

ð162Þ

In Eqs. (157)–(162)), the coefficients ano and an2 (again, for the full slip case) are ano ¼

C n F n ð1nÞ C þF n þ C n F n ð1nÞ

ð163Þ

an2 ¼

ðF n þ 6Þð1nÞ 2F n ð1nÞ þ 6ð56nÞ

ð164Þ

n

The revised form of the solution for bending moment for the full slip case is M

T 1 1 ¼ ð1 þKÞð1ano Þ þ ð1KÞð1 þ 2an2 Þ cos 2y PR 2 2

b¼

an3 ¼

ð147Þ

the tangential displacement of the support is

ano ¼

ð155Þ

n

M

the radial displacement of the support is

¼

1 1þK n n ð1KÞð12an2 þ2b2 Þ cos 2y þ a3 4 2

¼

where b2 and an2 are given by Eqs. (153) and (154), respectively, and an3 is given as follows:

PR2

the tangential stress acting on the support is

M

M PR2

the thrust of the support is

R3 E 1n2s Is Es 1n2

For the no slip case, the radial stress acting on the support is

PR2

The expression for bending moment provided by Einstein and Schwartz (Eq. (150)) is missing a term (see Appendix B). The revised form of the solution for bending moment for the no slip case is

the tangential displacement of the support is

R E 1n2s As Es 1n2 n

Fn ¼

71

PR2

¼

1 1þK n ð1KÞð12an2 Þ cos 2y þ a3 2 2

ð165Þ

where the coefficient an2 is given by Eq. (164) and an3 is given by the same Eq. (156). The differences in results between the original solution for bending moment given by [9] and the revised forms given by Eqs. (155) and (165) (which are the same as the corresponding expressions given in Section 6 for the case f R ¼ 1) are presented in the diagrams of Fig. 13. Fig. 13a represents the variation of the scaled values of bending moment for both the no slip and full slip cases, as a function of the ratio E=Es used already in the diagrams in Fig. 9. The diagram in Fig. 13a corresponds to a thick support with a ratio t s =a ¼ 0:25—see point T3 in Fig. 11 (also, the diagram was constructed with parameters K ¼0.5, f R ¼ 1, n ¼ ns ¼ 0:25 and y ¼ 01). Fig. 13b presents a similar relationship as in Fig. 13a but with the pair of curves corresponding to ratios t s =a ¼ 0:10 and 0.025—see points T2 and T1 in Fig. 11. The diagrams in Fig. 13 suggest that the differences between the original and revised solution for bending moments are not significant for relatively thin supports (Fig. 13b), but that these

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Fig. 14. Example problem of circular lined tunnel, showing the mesh used to model the problem with the finite difference code FLAC3D [29].

Fig. 13. Variations in values of bending moment for the support, as computed with the original Einstein and Schwartz’s solution, and with Solutions B and C in this study, for the particular case f R ¼ 1. The results correspond to scaled thicknesses of support (a) t s =a ¼ 0:25 and (b) t s =a ¼ 0:10 and t s =a ¼ 0:025, respectively.

differences could become more important for relatively thick supports (Fig. 13a).

10. Application example and comparison of analytical and numerical solutions An example of application of Solutions A, B and C presented in Sections 4 and 6, respectively, is discussed below. The implementation of Solutions A, B and C was done using the computer spreadsheet described in Appendix C—i.e., the spreadsheet was used to obtain the analytical results indicated in the diagrams in this section. Fig. 14 represents the problem considered and the mesh used in the code FLAC3D [29] to solve the problem. The FLAC3D model considered the same two-stage sequence adopted in the derivation of the analytical solution in this paper —i.e., in a first stage the initial tractions on the tunnel periphery were reduced and the equilibrium state was computed, and in a second stage the support was installed, the reduced tractions on the tunnel periphery were removed, and the equilibrium state was again computed. The problem involves a circular lined tunnel with the geometrical and mechanical properties for the ground and support indicated in Fig. 14—in particular, note that the indicated relaxation factor f R ¼ 0:7 implies that the support is

installed at the tunnel front itself (see Section 2). The FLAC3D mesh considers a ‘slice’ of the problem of unit thickness in the out-of-plane direction and in plane-strain conditions, with farboundaries located at a distance of 40 times the radius of the tunnel. Only half of the problem has been modelled—although due to symmetry of problem, a quarter of the problem could have been modelled as well. The mesh is a radial mesh (i.e., the inner boundary is circular and the outer boundary is rectangular) and has enough density of elements so as to provide accurate results (see [29]). The tunnel liner has been modelled using structural shell elements with a frictional interface between ground elements and support elements at the wall of the tunnel. By specifying an infinitely large value of cohesion and tensile strength for the frictional interface, the no slip case (i.e., Solution B) was modelled. By specifying zero value of cohesion and friction angle, and an infinitely large value of tensile strength for the frictional interface, the full slip case (i.e., Solution C) was modelled. It should be emphasized that the FLAC3D model shown in Fig. 14 corresponds to the second stage of excavation (after the support was installed). In a first stage (not represented in the figure) the initial tractions on the tunnel periphery were reduced and the model was brought to equilibrium. In the diagrams that follow, the ‘numerical’ results correspond to those obtained with FLAC3D while the ‘analytical’ results correspond to those obtained with the analytical solution presented in previous sections. Figs. 15 and 16 represent the results obtained before installation of the support and after relaxation of the ground stresses at the wall of the tunnel—i.e., the analytical results were obtained with Solution A discussed in Section 4. Fig. 15 represents the distribution of tangential and radial displacements for the ground at the tunnel wall as a function of the angle y, which due to symmetry of the problem, is taken to be in the range y ¼ 02901 only (note that according to the sign convention discussed in

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

Section 2, y ¼ 01 corresponds to the springline of the tunnel and y ¼ 901 corresponds to the crown of the tunnel). Fig. 16a represents the distribution of radial and tangential displacements for

Fig. 15. Distribution of radial and tangential displacements for the ground at the wall of the tunnel (i.e., at r=a ¼ 1), as obtained with the analytical and numerical solutions, for the first stage of excavation (see Fig. 4a).

Fig. 16. Distribution of (a) radial and tangential displacements and (b) radial, hoop and shear stresses for the ground, computed along scanlines inclined at angles y ¼ 451 and y ¼ 42:191, respectively, as obtained with the analytical and numerical solutions, for the first stage of excavation (see Fig. 4a).

73

the ground as a function of the radial distance along a radial scanline inclined 451 with respect to the springline of the tunnel. Fig. 16b represents the distribution of radial stress, hoop stress and shear stress on the ground as a function of the radial distance, along a radial scanline inclined 42.191 with respect to the springline of the tunnel. The reason for not using the same scanline as for results in Fig. 16a, is that in FLAC3D, the stresses are computed at the centroids of the elements representing the ground; then the centroid of the elements closest to the 451 diagonal in the FLAC3D mesh have an inclination of 42.191—see Fig. 14. The analytical and numerical results for the ground quantities before installation of the support, as represented by lines and squares in Figs. 15 and 16, are seen to agree well. Figs. 17–19 represent the analytical and numerical results after installation of the support and after removal of the stresses on the tunnel wall applied in the first excavation stage for the no slip condition at the ground–support interface (i.e., the analytical results were obtained with Solution B discussed in Section 6). The interpretation of these diagrams is equivalent to that of Figs. 15 and 16. For example, Fig. 17a represents the distribution thrust for the support as function of the angle y, while Fig. 17b represents the corresponding distribution of bending moment and shear force for the support as a function of y. Fig. 18

Fig. 17. Distribution of (a) thrust and (b) bending moment and shear force for the support, as obtained with the analytical and numerical solutions, for the second stage of excavation (see Fig. 4b) and the no slip condition at the ground–support interface.

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Fig. 18. Distribution of radial and tangential displacements for the support and for the ground (in the latter case for r=a ¼ 1) as obtained with the analytical and numerical solutions, for the second stage of excavation (see Fig. 4b) and the no slip condition at the ground–support interface.

Fig. 19. Distribution of (a) radial and tangential displacements and (b) radial, hoop and shear stresses for the ground, computed along scanlines inclined at angles y ¼ 451 and y ¼ 42:191, respectively, as obtained with the analytical and numerical solutions, for the second stage of excavation (see Fig. 4b) and the no slip condition at the ground–support interface.

represents the distribution of radial and tangential displacements for ground and support, as a function of the angle y (note that the displacements of the ground at the wall of the tunnel are different from the corresponding displacements of the support, because the displacements for the ground include the displacements associated with relaxation of the ground stresses before installation of the support). Fig. 19a represents the distribution of radial and tangential displacements for the ground as a function of radial distance (again for a radial scanline inclined 451 with respect to the springline of the tunnel), while Fig. 19b represents the distribution of radial stress, hoop stress and shear stress for the ground as a function of radial distance (again for a radial scanline inclined 42.191 with respect to the springline of the tunnel). The analytical and numerical results for the ground and support quantities for the no slip condition at the ground–support interface, as represented by lines and squares/circles in Figs. 17–19, are seen to agree well. Finally, Figs. 20–22 represent the analytical and numerical results after installation of the support (and after removal of the stresses acting on the tunnel wall applied in the first excavation stage) for the full slip condition at the ground–support interface (i.e., the analytical results were obtained with Solution C discussed

Fig. 20. Distribution of (a) thrust and (b) bending moment and shear force for the support, as obtained with the analytical and numerical solutions, for the second stage of excavation (see Fig. 4b) and the condition of full slip at the ground– support interface. The diagrams also indicate with discontinuous lines the corresponding distributions of quantities obtained with the analytical solution for the no slip condition at the ground–support interface.

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

75

in Section 6). The interpretation of the diagrams in Figs. 20–22 is similar to that of the diagrams in Figs. 17–19, respectively (for purposes of comparing Solutions B and C, the results for Solution B, obtained analytically, are also indicated in the diagrams with discontinuous lines). Again, the analytical and numerical results for ground and support quantities in the full slip condition at the ground–support interface, as represented by solid lines and squares/circles in Figs. 20–22, are seen to agree well.

11. Final comments

Fig. 21. Distribution of radial and tangential displacements for the support and for the ground (in the latter case for r=a ¼ 1) as obtained with the analytical and numerical solutions, for the second stage of excavation (see Fig. 4b) and the condition of full slip at the ground–support interface. The diagram also indicates with discontinuous lines the corresponding distributions of displacements obtained with the analytical solution for the no slip condition at the ground–support interface.

The solutions and their detailed demonstrations presented in this paper could serve as academical tools for teaching/learning the mechanism of ground–support interaction in tunnelling, particularly in the context of the convergence–confinement method of tunnel design and in the framework of the rigorous theory of elasticity. Under certain circumstances, the solutions and their numerical implementation through the computer spreadsheet discussed in this paper (see Appendix C) could be applied in the design of support systems for tunnels in engineering practice. For example, in tunnel excavation in soils using mechanized methods, like the Earth Pressure Balance (or EPB) method, the stress and displacement re-distribution of the ground occurring behind the shield may not be significant, so that in some cases the assumptions of elastic behavior for the soil could be acceptable. Under these circumstances, the two-stage solution for computing loading and deformation of the support presented in this paper could be useful in a first stage of design/verification of the circular closed liner (typically formed by pre-cast concrete segments), which is installed behind the shield. Certainly, the solution presented in this paper could be further improved or adapted by including other factors that are normally considered important in the design of support systems in mechanized excavations. For example, in the EPB method, the diameter of the cutting head is slightly larger than the diameter of the shield behind it. Therefore, a ground relaxation method prior to the installation of the support which would account for controlling the convergence of the tunnel wall, rather than controlling the initial stresses acting on the tunnel wall (as done in this paper), may be more appropriate—publications [34,35] presented a solution of this type for elastic shallow tunnels. Another improvement/adaptation of the solution presented in this paper could be in accounting for the mortar filling that is normally injected under pressure between the excavated soil and the concrete liner after the support installation in the EPB method.

Acknowledgments

Fig. 22. Distribution of (a) radial and tangential displacements and (b) radial, hoop and shear stresses for the ground, computed along scanlines inclined at angles y ¼ 451 and y ¼ 42:191, respectively, as obtained with the analytical and numerical solutions, for the second stage of excavation (see Fig. 4b) and the condition of full slip at the ground–support interface. The diagram also indicates with discontinuous lines the corresponding distributions of displacements and stresses obtained with the analytical solution for the no slip condition at the ground–support interface.

This research was supported in part by a University of Minnesota Undergraduate Research Opportunities Program (UROP) award, which is gratefully acknowledged. This research was also supported in part by a Geotechnical Engineering Fund award from the Department of Civil Engineering at University of Minnesota, Duluth Campus, which was contributed by Arup, Brisbane (Australia). This contribution is also gratefully acknowledged. The authors would also like to acknowledge the contribution of Prof. David Saftner, from the Department of Civil Engineering at University of Minnesota, Duluth Campus, who provided valuable comments for improving this work. Appendix A. Demonstration of solutions A, B and C This appendix presents the governing equations, boundary conditions, and procedures required to arrive to Solutions A, B

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and C presented in the main text. To facilitate reading, this appendix is divided into eight subsections describing various aspects of the derivation of the solutions. A.1. Initial stresses in the ground

textbooks on elasticity theory. The authors have referred mainly to [36–38]. The formulation presented here corresponds to plane strain conditions for the ground. For the case in which no body forces are considered in the ground (e.g., forces associated with gravity are disregarded), the equilibrium condition of forces for the radial direction, expressed in terms of scaled stresses, is as follows:

This subsection provides a demonstration of the equations for radial, hoop and tangential stresses existing in the ground prior to excavation of the tunnel. In particular, the initial radial and shear stresses are the stress boundary conditions for obtaining Solution A—i.e., the initial radial and shear stresses, after being multiplied by the relaxation factor f R , are the stresses considered to act at the wall of the tunnel prior to installation of the support (see Eqs. (6)–(9) and Fig. 4a). Referring to Fig. 6 and considering an orthogonal cartesian system of coordinates x and y (with the axis x being horizontal and positive when increasing towards the right), the initial stresses existing in the ground which are compatible with the horizontal far-field stress KP and the vertical far-field stress P (see Fig. 4), are

According to the theory of elasticity, for the case of an homogeneous isotropic elastic material with Young’s modulus E and Poisson’s ratio n, the relationships between scaled strains and scaled displacements, for plane strain conditions, are

sx ¼ KP, sy ¼ P, sxy ¼ 0

e~ r ¼ ð1nÞs~ r ns~ y

ðA:10Þ

e~ y ¼ ð1nÞs~ y ns~ r

ðA:11Þ

e~ ry ¼ s~ ry

ðA:12Þ

ðA:1Þ 0

Considering an orthogonal cartesian system of coordinates x and y0 which axes are rotated at an angle y with respect to the system of coordinates x and y (the angle y is assumed to have the same sign convention discussed in Section 2), the normal and shear stress components expressed in terms of the coordinates x0 and y0 are given by the following equations (see, for example, [36,37]): 2

2

@s~ r 1 @s~ ry s~ r s~ y þ þ ¼0 @r r @y r

Similarly, the equilibrium condition of forces for the tangential direction is 1 @s~ y @s~ ry 2s~ ry þ þ ¼0 r @y @r r

e~ r ¼

@u~ r @r

ðA:2Þ

sy0 ¼ sy cos2 y þ sx sin2 y2 sxy sin y cos y

ðA:3Þ

e~ y ¼

ðA:4Þ

e~ ry ¼

and

If the cylindrical reference system of coordinates r and y (see, for example, Fig. 6) is regarded as being equivalent to the system x0 and y0 in Eqs. (A.2)–(A.4), then sx0 ¼ sr , sy0 ¼ sy and sxy0 ¼ sry . Also, considering the trigonometry relationships sin y cos y ¼ 2 1 2 1 1 2 sin 2y, cos y ¼ 2 ð1 þ cos 2yÞ and sin y ¼ 2 ð1cos 2yÞ, replacing Eqs. (A.1) into Eqs. (A.2)–(A.4), and considering the scaling rule for stresses introduced in Section 3, the following expressions for the scaled components of the initial stresses in terms of the coordinates r and y are obtained

s~ r ¼

1 þ K 1K  cos 2y 2 2

ðA:5Þ

s~ y ¼

1þ K 1K þ cos 2y 2 2

ðA:6Þ

and

s~ ry ¼

1K sin 2y 2

ðA:7Þ

Eqs. (A.5)–(A.7) are the same as Eqs. (8) and (9) presented in Section 2. Also Eqs. (A.5)–(A.7) are the same as Eqs. (31)–(33) presented in Section 5. A.2. Governing equations for the ground This subsection lists the governing equations for the elastic ground expressed in terms of the cylindrical coordinates r and y—see Figs. 5, 6, and 8. The quantities involved (stresses, strains and displacements) have been scaled according to the scaling rules introduced in Section 3. The non-scaled form of the equations presented in this subsection can be found in most

ðA:9Þ

Finally, the relationships between scaled strains and scaled displacements are

sx0 ¼ sx cos y þ sy sin y þ2sxy sin y cos y

sxy0 ¼ sxy ðcos2 ysin2 yÞ þ ðsy sx Þ sin y cos y

ðA:8Þ

1 @u~ y

ðA:13Þ 1

u~ r

ðA:14Þ

  1 1 @u~ r @u~ y u~ y þ  2 r @y @r r

ðA:15Þ

r @y

þ

r

Eq. (A.15) is commonly referred to as the compatibility of deformation equation and expresses that no gap and/or no overlapping of material can occur as the continuum material deforms. A.3. Governing equations for the support This subsection lists the governing equations for the elastic support—see Fig. 7. The quantities involved (forces, bending moments, stresses and displacements for the support) have been scaled according to the scaling rules introduced in Section 3. The non-scaled form of the equations presented in this subsection can be found in most books on elastic structural shells. The authors have followed [39,40] mainly and have considered the formulation of thick shells in the framework of the theory of elastic shells (i.e., as Einstein and Schwartz did in [9]). In this regard, CarranzaTorres and Diederichs [33] discussed the differences in results obtained for circular tunnel supports when these are considered to obey the formulation of thick shells (as in this paper) and the formulation for thin shells. The formulation presented here corresponds to plane strain conditions for the support. Again, for the case in which no body forces are considered for the support (e.g., forces associated with gravity are disregarded), the equilibrium condition of forces for the radial direction is expressed in terms of scaled support quantities as follows: dQ~ ~ p~ ¼ 0 þ N r dy

ðA:16Þ

and the equilibrium condition of forces for the tangential

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

direction is dN~ ~ Q þ p~ y ¼ 0 dy

ðA:17Þ

In Eqs. (A.16) and (A.17), N~ is the scaled thrust, Q~ is the scaled shear force and p~ r and p~ y are the radial and tangential components of scaled stresses acting on the support, respectively (see Fig. 7). Considering the equilibrium condition of moment (of forces) for a section of support, the following relationship between scaled ~ and the scaled shear force is obtained: bending moment, M, ~ dM Q~ ¼ 0 dy

ðA:18Þ

According to the thick formulation of the theory of elastic shells, the relationship between the scaled thrust and the scaled radial and tangential displacements, u~ sr and u~ sy , respectively, is " #   2 1 du~ sy 1 d u~ sr u~ sr þ u~ sr þ N~ ¼ n þ n ðA:19Þ 2 dy F ð1nÞ C ð1nÞ dy and the relationship between the scaled bending moment and the scaled radial displacement is " # 2 1 d u~ sr s ~ ¼ ~ ðA:20Þ u þ M 2 F n ð1nÞ r dy In Eqs. (A.19) and (A.20), the coefficients Cn and Fn are the compressibility and flexibility coefficients, defined by Eqs. (2) and (3), respectively. Using Eq. (A.18) and considering the relationship (A.20), the following relationship between scaled shear force and scaled radial displacement is obtained: " # 3 1 du~ sr d u~ sr Q~ ¼ n þ ðA:21Þ 3 F ð1nÞ dy dy Replacing Eqs. (A.19) and (A.20) into Eq. (A.16), the equilibrium condition of forces in the radial direction can be written in terms of scaled radial and tangential displacements as follows: " #   2 4 1 d u~ sr d u~ sr 1 du~ y s ~ þ n ur þ 2 u~ sr þ þ p~ r ¼ 0 ðA:22Þ n 2 4 dy F ð1nÞ C ð1nÞ dy dy Similarly, replacing Eqs. (A.19) and (A.20) into Eq. (A.17), the equilibrium condition of forces in the tangential direction can be written as " # 2 1 du~ sr d u~ sy þ ðA:23Þ þ p~ y ¼ 0 2 C n ð1nÞ dy dy Solving for the second derivative of the scaled tangential displacement with respect to y in Eq. (A.23), and replacing this result into the corresponding term in Eq. (A.22) (after Eq. (A.22) has been differentiated once more with respect to y), the following differential equation that expresses the condition of equilibrium of forces for the support, for both radial and tangential directions, is obtained: " # 3 5 1 du~ sr d u~ sr d u~ sr dp~ þ2 þ ðA:24Þ  r p~ y ¼ 0 5 3 dy F n ð1nÞ dy dy dy Note that only the scaled radial displacement appears in the global equilibrium Eq. (A.24)—the scaled tangential displacement does not appear in this equation. So with appropriate boundary conditions (including also the definition of the scaled stresses acting on the support for both radial and tangential directions), the ordinary differential Eq. (A.24) can be solved to obtain the solution for the scaled radial displacement. Then integrating Eq. (A.23), the solution for the scaled tangential displacement

77

can also be obtained—and thereafter, the solutions for the scaled thrust, scaled bending moment and scaled shear force, can be obtained using Eqs. (A.19)–(A.21), respectively. A.4. Michell’s elastic potential This subsection describes the methodology for solving the governing equations for the ground (as listed in Section A.2) using Michell’s elastic potential approach. The methodology is described in detail in [36,38] and the main equations and steps required to obtain stress, strain and displacements fields are summarized below (again, the equations below have been scaled according to the rules introduced in Section 3). The general form of Michell’s potential in cylindrical coordinates is given by the following infinite series equation:

f~ ¼ B01 r2 þ B02 r2 ln r þB03 ln r þ B04 y þ ðB11 r3 þB12 r ln r þ B14 r1 Þ cos y þ B13 ry sin y þ ðC 11 r3 þ C 12 r ln r þ C 14 r1 Þ sin y þ C 13 ry cos y 1 X þ ðBn1 rn þ 2 þ Bn2 rn þ 2 þ Bn3 rn þ Bn4 rn Þ cos ny n¼2

þ

1 X

ðC n1 rn þ 2 þC n2 rn þ 2 þ C n3 rn þ C n4 rn Þ sin ny

ðA:25Þ

n¼2

The coefficients ‘B’ and ‘C’ in Michell’s potential above must be chosen so as to satisfy the boundary conditions of the problem, as it will be explained in the next subsections (it should be noted that the names ‘B’ and ‘C’ given to the coefficients in Eq. (A.25) have no relation at all with the names given to Solutions B and C described in this paper). According to Michell’s elastic potential approach, the scaled radial, hoop and shear stresses that satisfy the governing equations ((A.8)–(A.12) can be obtained from Eq. (A.25) as follows:

s~ r ¼

~ ~ 1 @f 1 @2 f þ 2 2 r @r r @r

ðA:26Þ

s~ y ¼

~ @2 f @r2

ðA:27Þ

s~ ry ¼

~ ~ 1 @2 f 1 @f  2 r @y r @r@y

ðA:28Þ

With the components of scaled stresses given by Eqs. (A.26)– (A.28), the corresponding components of scaled strains can be computed applying Eqs. (A.10)–(A.12), respectively. Thereafter, with the computed components of scaled strains, the scaled displacements can be obtained by integrating the strain–displacement relationships given by Eqs. (A.13) and (A.14) as follows. First, the scaled radial displacement is obtained from integration of Eq. (A.13) with respect to the variable r, which gives Z u~ r ¼ e~ r dr þ f 0 ðyÞ ðA:29Þ 0

In the equation above, f ðyÞ is a constant (actually, a function) of integration that depends on the variable y only. For convenience, 0 the function f ðyÞ is considered to be the derivative of a function f ðyÞ—i.e., the prime symbol represents derivative of the function with respect to the variable y. Second, the scaled tangential displacement is obtained from integration of Eq. (A.14), with respect to the variable y, which gives Z u~ y ¼ ðre~ y u~ r Þ dy þ gðrÞ ðA:30Þ

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C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

In the equation above, gðrÞ is another function of integration that depends on the variable r only. Replacing Eq. (A.29) into (A.30), the following expression for the scaled tangential displacement is obtained:  Z  Z u~ y ¼ re~ y  e~ r dr dyf ðyÞ þ gðrÞ ðA:31Þ Finally, the functions f ðyÞ and gðrÞ in Eqs. (A.29) and (A.30) are found from integration of two independent ordinary differential equations that are obtained by enforcing the compatibility Eq. (A.15). When solving the resulting differential equations, the integration constants are assumed to be zero, in view that the functions u~ r and u~ y should not admit rigid movement or rotation of the infinite elastic medium (see [36]). For the case of an infinite elastic medium with a circular tunnel considered in this study, a significant large number of the coefficients ‘B’ and ‘C’ in the generalized Michell’s potential (Eq. (A.25)) must be equal to zero. The coefficients that are equal to zero are those which are multiplying functions of the variables r and/or y that produce infinite stresses, strains and/or displacements when the variable r is taken to be infinite (this is described in detail in [36]). Therefore, for the Solutions A, B and C considered in this study, Michell’s potential takes the following simple form:

f~ ¼ B01 r2 þ B03 ln r þ B04 y þ B22 cos 2y þ B23 r2 cos 2y þ B24

1

r2

cos 2y

ðA:32Þ Replacing Eq. (A.32) into Eqs. (A.26)–(A.28), the scaled components of stresses are

s~ r ¼ 2B01 þ B03

s~ y ¼ 2B01 B03

1 2

r

1 2

r

4B22

1

cos 2y2B23 cos 2y6B24

2

r

þ2B23 cos 2y þ6B24

1 4

r

s~ ry ¼ B04

r2

2B22

1

cos 2y

4

r

cos 2y

r2

sin 2y þ 2B23 sin 2y6B24

1

r4

sin 2y

1

r

þ 4B22 ð1nÞ

2B23 r cos 2y þ 2B24

1

r3

1

r

and ¼ B04 u~ IND y

1

r

2B22 ð12nÞ

1

r

¼ 2B01 ð12nÞrB03

r

þ4B22 ð1nÞ

A.5. Approach used to obtain Solution A For Solution A, the six coefficients B01, B03, B04, B22, B23 and B24 in Eqs. (A.33)–(A.39) are obtained by applying the stress boundary conditions at infinity (i.e., at r-1) and at the wall of the tunnel (i.e., at r ¼ 1)—see Fig. 5. These boundary conditions are as follows. At r-1, s~ r ð1, yÞ ¼ s~ or (see Fig. 5). Considering Eqs. (A.33) and (A.5), this condition is written as follows: 2B01 2B23 cos 2y ¼

2B23 r cos 2y þ 2B24

1

r3

cos 2y

1 þ K 1K  cos 2y 2 2

ðA:40Þ

~ ory

Similarly, at r-1, s~ ry ð1, yÞ ¼ s (see Fig. 5). Again, considering Eqs. (A.35) and (A.7), this condition is written as 1K sin 2y 2

ðA:41Þ

ðA:35Þ

1þ K 1K f  f cos 2y 2 R 2 R ðA:42Þ

f Similarly, at r ¼ 1, s~ ry ð1, yÞ ¼ s~ ryR (see Fig. 5), and again, in view of Eq. (A.33) and the scaled form of Eqs. (7) and (9), the following expression is obtained:

B04 2ðB22 B23 þ 3B24 Þ sin 2y ¼

1K f sin 2y 2 R

ðA:43Þ

Equating the multipliers of the same functions of the variable

y in the left- and right-hand sides of Eqs. (A.40)–(A.43), the

2B01 

1 þK ¼0 2

ðA:44Þ

1

r

3

sin 2y

2B23 

1K ¼0 2

ðA:45Þ

2B01 þ B03 

1þ K f ¼0 2 R

2ð2B22 þ B23 þ 3B24 Þ þ

ðA:46Þ 1K f ¼0 2 R

B04 ¼ 0 2ðB22 B23 þ3B24 Þ

cos 2y

ðA:39Þ

Note that Eqs. (A.33)–(A.37) are the same as Eqs. (16)–(20) presented in Section 4—in Section 4 the superscript ‘[A]’ has been used to indicate that the quantities correspond to Solution A. Also, Eqs. (A.33)–(A.37) are the same as Eqs. (48)–(52) presented in Section 6—in Section 6 the superscript ‘[B,C]’ has been used to indicate that the quantities correspond to both Solutions B and C.

ðA:36Þ

r

1K r sin 2y 2

following system of six independent equations for the six unknowns (i.e., the coefficients B01, B03, B04, B22, B23 and B24) is obtained:

cos 2y

1

sin 2y

sin 2y þ2B23 r sin 2y

2B01 þ B03 2ð2B22 þB23 þ3B24 Þ cos 2y ¼

The displacements of the ground that are of significance for solving the problem of excavation of a tunnel are the induced components of displacements (see Section 4). Thus, in view of Eqs. (A.36) and (A.37), and using Eqs. (27) and (28), or alternatively, Eqs. (53) and (54), the scaled components of induced displacements are expressed as follows: 1

1

r3

1

r

ðA:34Þ

ðA:37Þ

u~ IND r

2B22 ð12nÞ

At r ¼ 1, s~ r ð1, yÞ ¼ s~ fr R (see Fig. 5); in view of Eqs. (A.33) and the scaled form of Eqs. (6) and (8), this condition results to be

cos 2y

sin 2y þ 2B23 r sin 2y þ 2B24

1

r

þ 2B24

and u~ y ¼ B04

ðA:38Þ

ðA:33Þ

Computing the scaled components of strains with Eqs. (A.10)– (A.12), and integrating the scaled strains as described earlier on, the scaled components of displacements in the radial and tangential directions are also obtained, i.e. u~ r ¼ 2B01 ð12nÞrB03

1 þK 1K ð12nÞr þ r cos 2y 2 2

2B23 sin 2y ¼

1

and 1



ðA:47Þ ðA:48Þ

1K f ¼0 2 R

ðA:49Þ

Solution of the system of equations above, allows the six coefficients B01, B03, B04, B22, B23 and B24 to be determined. The resulting coefficients are listed as Eqs. (21)–(26) in Section

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

2 D2 cos y F n ð1nÞ

79

4—again, in Section 4, the superscript ‘[A]’ has been used in the coefficients to indicate that they correspond to Solution A.

Q~ ¼ B04 2ðB22 þB23 þB24 Þ sin 2y þ

A.6. Approach used to obtain Solutions B and C

The computation of the nine coefficients (B01, B03, B04, B22, B23, B24, D1, D2 and D3) in Eqs. (A.38)–(A.56) is discussed separately for Solutions B and C in the following two subsections.

As discussed in Section 6, Solution B corresponds to the supported-tunnel case for which the no slip condition exists at the ground–support interface. Solution C, instead, corresponds to the supported-tunnel case for which the full slip condition exists at the ground–support interface (see Fig. 8). For Solutions B and C, the stresses and displacements in the ground are computed with the same Eqs. (A.33)–(A.39). To obtain the different coefficients ‘B’ in these equations, conditions of compatibility of displacements for ground and support at the ground–support interface must be stated. When doing so, the induced components of displacements for the ground given by Eqs. (A.38) and (A.39) must be used. For Solutions B and C, the scaled stresses acting on the support are obtained by computing the scaled radial and shear stresses for the ground (Eqs. (A.33) and (A.39) respectively evaluated at the wall of the tunnel (i.e., at r ¼ 1). Indeed, considering that p~ r ¼ s~ r ð1, yÞ and that p~ y ¼ s~ ry ð1, yÞ, the following expressions for the stresses acting on the support are obtained: p~ r ¼ 2B01 þB03 2ð2B22 þ B23 þ 3B24 Þ cos 2y

ðA:50Þ

and p~ y ¼ B04 2ðB22 B23 þ 3B24 Þ sin 2y

ðA:51Þ

The scaled radial displacement of the support is obtained from the integration of the differential Eq. (A.24), which together with the Eqs. (A.50) and (A.51) give: 2 u~ sr ¼ F n ð1nÞB04 y F n ð1nÞðB22 þ B23 þ B24 Þ cos2 y 3 D1 cos yD2 ðy cos ysin yÞ þ D3

ðA:52Þ

Replacing the first derivative of the scaled radial displacement with respect to the angular coordinate (as computed from Eq. (A.52)) into Eq. (A.23), and integrating the resulting equation, the scaled tangential displacement for the support is found to be 1 2 u~ sy ¼ C n ð1nÞð2B01 þB03 Þy ðC n þF n Þð1nÞB04 y 2 1 þ ðC n þ F n Þð1nÞðB22 þB23 þ B24 Þy 3 1  ð3C n F n Þð1nÞB22 sin 2y 6 1 þ ð3C n þF n Þð1nÞB23 sin 2y 6 1  ð9C n F n Þð1nÞB24 sin 2y 6 Cn þ Fn þ D1 sin y þD2 ð2 cos y þ y sin yÞ D3 y Fn

Note that the integration of Eqs. (A.23) and (A.24) has added three new unknown coefficients in Eqs. (A.52) and (A.53)—these are the coefficients D1, D2 and D3. Finally, the scaled thrust, scaled bending moment and scaled shear force of the support are computed based on the scaled radial and tangential displacements (Eqs. (A.52) and (A.53)), using the Eqs. (A.19)–(A.21), respectively. This gives N~ ¼ 2B01 þ B03 þ 2ðB23 B24 Þ cos 2y þ

2 D2 sin y F n ð1nÞ

ðA:54Þ

  1 ~ ¼ B04 yðB22 þ B23 þ B24 Þ 1 cos 2y þ M ð2D2 sin y þ D3 Þ 3 F n ð1nÞ ðA:55Þ and

A.7. Boundary conditions for Solution B For Solution B (and also for Solution C), the fist two boundary conditions correspond to the normal and shear stress boundary conditions at infinity. These two conditions are the same as those discussed in Section A.5 (see Eqs. (A.40) and (A.41)). These two boundary conditions allow the following equations involving the coefficients B01 and B23 to be obtained (see Eqs. (A.44) and (A.45)): 2B01 

1 þK ¼0 2

ðA:57Þ

2B23 

1K ¼0 2

ðA:58Þ

For Solution B, the other seven linearly independent equations needed to compute the seven remaining coefficients are obtained by stating compatibility of the scaled radial and tangential displacements of ground and support at the ground–support interface. These conditions are written as follows: ~s u~ iIND u~ i½A r r IND ¼ u r

ðA:59Þ

~s u~ iIND u~ i½A y y IND ¼ u y

ðA:60Þ

Note that in the equations above, the superscript ‘i’ means that the scaled radial and tangential displacements given by Eqs. (A.38) and (A.39), and Eqs. (27) and (28) respectively, have been evaluated at the tunnel wall, or ground–support interface (i.e., at r ¼ 1). Therefore, replacing Eqs. (A.38) and (A.39), and Eqs. (27) and (28) on the left side of Eqs. (A.59) and (A.60), and Eqs. (A.52) and (A.53) on the right side of Eqs. (A.59) and (A.60), the resulting Eqs. (A.59) and (A.60) become long polynomials with terms that are functions of the variable y only. The multipliers of the different terms in these polynomials have to be equated on both sides of the Eqs. (A.59) and (A.60) to obtain the values of the remaining seven linear independent equations. Indeed, equating the multipliers of the term cos y in Eq. (A.59), the following equation is obtained: D1 ¼ 0

ðA:53Þ

ðA:56Þ

ðA:61Þ

The equation above is also obtained when equating the multipliers of the term sin y in Eq. (A.60). Equating the multipliers of the term sin y in Eq. (A.59), the following equation is obtained: D2 ¼ 0

ðA:62Þ

The equation above is also obtained when equating the multipliers of the term y cos y in Eq. (A.59), the term y sin y in Eq. (A.60), and the term cos y in Eq. (A.60). Equating the multipliers of the term y in Eq. (A.59), the following equation is obtained: B04 ¼ 0

ðA:63Þ

The equation above is also obtained when equating the indepen2 dent terms in Eq. (A.60), and also the term y in Eq. (A.60). 2 Equating the multipliers of the term cos y in Eq. (A.59), the following equation is obtained: 2ð1nÞð12 þ F n ÞB22 2½6F n ð1nÞB23 þ 2½6 þ F n ð1nÞB24 þ 3ð1KÞ½4ð1nÞð34nÞf R  ¼ 0

ðA:64Þ

80

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

The equation above is also obtained when equating the multipliers of the term sin2 y in Eq. (A.59). Equating the independent terms in Eq. (A.59), the following equation is obtained: 12ð12nÞB01 6B03 þ 2F n ð1nÞðB22 þ B23 þ B24 Þ 6D3 3ð1 þ KÞ½2ð1nÞf R  ¼ 0

ðA:65Þ

Equating the multipliers of the term sin2 y in Eq. (A.60), the following equation is obtained: ½12ð12nÞð1nÞð3C n F n ÞB22 þ½12ð1nÞð3C n þ F n ÞB23 þ ½12 þ ð1nÞð9C n F n ÞB24 3ð1KÞ½4ð1nÞð34nÞf R  ¼ 0 ðA:66Þ Equating the multipliers of the term y in Eq. (A.60), the following equation is obtained, 6C n F n ð1nÞ2 B01 þ 3C n F n ð1nÞ2 B03 3ðC n þ F n Þð1nÞD3 þ F n ðC n þF n Þð1nÞ2 ðB22 þ B23 þ B24 Þ ¼ 0

ðA:67Þ

Solution of the system of nine linearly independent equations formed by Eqs. (A.57) and (A.58), and (A.61)–(A.67), allows the nine coefficients B01, B03, B04, B22, B23, B24, D1, D2 and D3 to be determined. The resulting coefficients are listed as Eqs. (62)–(77) in Section 6—again, in Section 6, the superscript ‘[B]’ has been used in the coefficients to indicate that they correspond to Solution B.

following equations:

s~ iry ¼ 0

ðA:68Þ

u~ sy ð0Þ ¼ 0

ðA:69Þ

and u~ sy ðp=2Þ ¼ 0

ðA:70Þ

Note that in Eq. (A.68), the superscript ‘i’ means that the shear stress given by Eq. (A.35) is evaluated at the tunnel wall (i.e., at r ¼ 1); since s~ ry ð1, yÞ ¼ p~ y , this condition could also have been written as p~ y ¼ 0 using Eq. (A.51) instead. Note also that Eqs. (A.69) and (A.70) arise from symmetry conditions of the problem. The same linear Eqs. (A.57) and (A.58), and (A.61)–(A.65) still apply to Solution C. The two new linearly independent equations are obtained as follows. Equating the multipliers of the term sin2 y in Eq. (A.68), the following equation is obtained: B22 B23 þ3B24 ¼ 0

ðA:71Þ

A.8. Boundary conditions for Solution C The boundary conditions to consider for Solution C are the same as for Solution B, except that Eq. (A.60) is replaced by the

Fig. C1. Layout of spreadsheet for the implementation of the two-stage solution of the Einstein and Schwartz’s problem. Implementation of input variables and intermediate variables.

Fig. C2. Layout of spreadsheet for the implementation of the two-stage solution of the Einstein and Schwartz’s problem. Implementation of Solution A.

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

Equating the independent term in Eq. (A.68) gives the same as Eq. (A.63), considered already when enforcing the condition given by Eq. (A.59). The condition stated by Eq. (A.69) gives the same as Eq. (A.62), considered already when enforcing the condition given by Eq. (A.59). Finally, the condition stated by Eq. (A.70) gives the following equation: 24pC n ð1nÞB01 þ 12pC n ð1nÞB03 3p2 ðC n þ F n Þð1nÞB04 þ4pðC n þF n Þð1nÞðB22 þB23 þB24 Þ þ 24D1 þ12pD2 12pð1 þ C n =F n ÞD3 ¼ 0

ðA:72Þ

Solution of the system of nine linearly independent equations formed by Eqs. (A.57) and (A.58), Eqs. (A.61)–(A.65) and Eqs. (A.71) and (A.72) allows the nine coefficients B01, B03, B04, B22, B23, B24, D1, D2 and D3 to be determined. The resulting coefficients are listed as Eqs. (78)–(88) in Section 6—in Section 6, the superscript ‘[C]’ has been used in the coefficients to indicate that they correspond to Solution C.

81

Appendix B. Governing equations of the Einstein and Schwartz [9] solution and the revised form of the solution for support bending moment This subsection provides further details about the effect of the missing term in the solution for bending moment presented by Einstein and Schwartz, as discussed in Section 9. Einstein and Schwartz [9] list the following equations ass governing equations for the support (the notation and sign convention in the equations below, follow the notation and sign convention indicated in Fig. 1). The condition of equilibrium of forces in the radial direction expressed in terms of displacements of the support is presented as ! 4 2 dvs DF d us d us R2 þ us þ þ 2 þ u sR ðB:1Þ ¼ s 2 4 2 dy D DC R C dy dy while the condition of equilibrium of forces in the tangential direction of the support is presented as 2

d vs dy

2

þ

dus R2 ¼ t dy DC R y

Fig. C3. Layout of spreadsheet for the implementation of the two-stage solution of the Einstein and Schwartz’s problem. Implementation of Solution B.

ðB:2Þ

82

C. Carranza-Torres et al. / International Journal of Rock Mechanics & Mining Sciences 61 (2013) 57–85

In Eqs. (B.1) and (B.2), the coefficients DC and DF are given by the following expressions: DC ¼

Es As 1n2s

ðB:3Þ

DF ¼

Es Is 1n2s

ðB:4Þ

Note that Eqs. (B.1) and (B.2) with the coefficients given by Eqs. (B.3) and (B.4), are the same equations listed as Eqs. (A.22) and (A.23) in Section A.3. Einstein and Schwartz [9] also present the following equations stating equilibrium of forces (and moment of forces) in terms of the loads on the support, i.e., the thrust T, the bending moment M and the shear force Q 2

RT þ

R

d M 2

dy

¼ R2 sR

dT dM  ¼ R2 tRy dy dy

Q¼

1 dM R dy

ðB:5Þ

ðB:6Þ

ðB:7Þ

Note that Eqs. (B.5)–(B.7) are the same equations listed as Eqs. (A.16)–(A.18), respectively, in Section A.3. Although not listed in the Einstein and Schwartz’s article, if the equilibrium conditions for forces in the support are given by Eqs. (B.1) and (B.2), and Eqs. (B.5) and (B.6), then the relationship between thrust and radial and tangential displacements for the support must be given by the following equation: " #   2 DC dvs DF d u s T¼ ðB:8Þ þus þ 3 þ u s R dy R dy2 while the relationship between bending moment and displacements must be given by " # 2 DF d u s M¼ 2 þ us ðB:9Þ R dy2 Again, Eqs. (B.8) and (B.9) are the same equations listed as Eqs. (A.19) and (A.20) in Section A.3. Although Eq. (B.8) is satisfied by the original expressions for thrust and radial and tangential displacements given by Einstein and Schwartz (Eqs. (147)–(149)) for the no slip case, and Eqs. (159)–(161) for the full slip case), Eq. (B.9) is not satisfied by the original expressions for bending moment given by Einstein and Schwartz (Eqs. (150) and (162)), for each of the two cases

Fig. C4. Layout of spreadsheet for the implementation of the two-stage solution of the Einstein and Schwartz’s problem. Implementation of Solution C.

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Table C1 Formulae to use in the named cells of the spreadsheet in Fig. C1.

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Table C3 Formulae to use in the named cells of the spreadsheet in Fig. C3.

Table C2 Formulae to use in the named cells of the spreadsheet in Fig. C2.

considered). In order for the bending moment to satisfy Eq. (B.9), the revised Eqs. (155) and (165) have to be used, for the no slip case and for the full slip case, respectively.

Appendix C. Spreadsheet for the implementation of Solutions A, B and C This subsection describes the numerical implementation of Solutions A, B and C presented in Sections 4 and 6, using a

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Table C4 Formulae to use in the named cells of the spreadsheet in Fig. C4.

implementation of Solution C, respectively. The variables in the spreadsheet in Figs. C1–C4 follow the sign convention for ground and support quantities discussed in Sections 4 and 6—see Figs. 5, 7, and 8. Also, to facilitate interpretation of input and output values in the spreadsheet, the units of the values are indicated within brackets after the variable name in Figs. C1–C4. In Figs. C.2–C.4 it can be seen that the solution is computed for 20 equally spaced points along a radial scanline of specified upper range rmax and specified inclination angle ysline (see upper right corner of the spreadsheet layout in Fig. C1); the solution is also computed for 15 equally spaced points along the wall of the tunnel, between the springline and the crown. In Figs. C1–C4, the text within parentheses located on the right side of the indicated numerical values (or below a column of values) refer to the names of the variables that have to be assigned to cells (or ranges of cells) through the variable declaration feature available in most computer spreadsheets—for example, in Excel [41], the variable declaration feature is accessed from the panel (or menu option) ‘Insert’, selecting the subpanel (or submenu option) ‘Names’. Tables C1–C4 list the formulas expressed in terms of the variable names assigned to cells (or ranges of cells) that have to be entered in cells (or ranges of cells) to the left (or above) the variable names listed within parentheses in Figs. C1–C4, respectively. References

computer spreadsheet—the implementation was done using the software Excel [41]. An electronic version of the spreadsheet may be requested by writing the contact author of this paper. Figs. C1–C4 show the layout of the spreadsheet implementing the solutions. For clarity of presentation, the spreadsheet has been divided into four different sectors; these are, the input variables and intermediate variables; the implementation of Solution A; the implementation of Solution B; and the

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