Carranza-Torres, C., T. Reich, and D. Saftner (2013). Stability of shallowcircular tunnels in soils using analytical and numerical models. In Proceed-ings of the 61st Minnesota Annual Geotechnical Engineering Conference.University of Minnesota, St. Paul Campus. February 22, 2013.

Stability of shallow circular tunnels in soils using analyticaland numerical models

C. Carranza-Torres, T. Reich & D. SaftnerDepartment of Civil Engineering, University of Minnesota, Duluth Campus, Minnesota, USA

ABSTRACT: This paper revisits a classical problem of geotechnical engineering involving thestability of shallow circular tunnels excavated in frictional cohesive materials. The problem is ofpractical interest since, among others, it allows establishing conditions of stability for the frontof tunnels in soils excavated manually or using mechanized methods. A historical background ofcomputational methods developed to establish the stability conditions of shallow cavities in soilsis presented first. In particular, analytical models based on lower and upper bound theories ofplasticity are discussed. Thereafter a classical lower bound model due to Caquot is analyzed andextended to account for the presence of a surface surcharge and water in the soil being excavated.This model is proposed as a means of getting a first estimate of the stability conditions of shallowtunnels under various hydraulic conditions, using a closed-form solution. The concept of factor ofsafety, traditionally used in the assessment of stability of slopes in frictional cohesive materials,is also included in the model. Results obtained with the extended Caquots model are shown tobe in accordance with those obtained with more sophisticated finite element and finite differencemethods. A computer spreadsheet including the implementation of Caquots extended solution isalso provided in the paper.

1 INTRODUCTION

The stability of shallow circular cavities excavated in soils is an important topic of practical sig-nificance in geotechnical engineering. The problem is relevant for design of buried pipes, sewerand transportation tunnels. In the case of tunnels excavated using mechanized methods, such asthe Earth Pressure Balance (or EPB) method and the Slurry Shield method, the problem is that ofdetermining the minimum pressure to be applied at the front of the tunnel to avoid formation of acave at the front (see, for example, Guglielmetti, Grasso, Mahtab & Xu 2008).

Impressive collapses of shallow tunnels excavated in soils with manual or mechanical methods,have occurred in the past due to inadequate support considered for the openings or other unforeseenconditions in the ground. For example, Figure 1a shows a bus swallowed into a sinkhole formedby the collapse of a tunnel front during construction of the Munich Metro in 1994 (ConstructionToday, 1994a, 1994b). Figure 1b shows another collapse that occurred due to failure of tunnelsupport below Hollywood Boulevard in Los Angeles, during construction of the metro in 1995(Civil Engineer International 1995; Oliver 1995). Figure 1c shows another collapse that occurredduring construction of the underground Mass Rapid System in Singapore in 2005 (Government ofSingapore 2005).

Figure 1. a) Collapse during construction of the Munich Metro (after Construction Today, 1994a). b) Col-lapse during construction of the LA Metro (after Civil Engineer International, 1995). c) Collapse duringconstruction of the Singapore underground Mass Rapid Transit (MRT) system (after Government of Singa-pore, 2005).

Economic yet safe and flexible designs of tunnel supports can be achieved with better under-standing of the relationships that govern the stability of shallow tunnels, especially the relationshipbetween support pressure, soil cover above the tunnel, size of the tunnel and the mechanical strengthof the ground hosting the tunnel. Such understanding could be gained by availability of simple ana-lytical relationships that allow examination of the dependence of variables involved in the problem.Perhaps more important, these analytical relationships can allow implementation of probabilisticanalyses for computing probability of failure (and therefore reliability of designs) for tunnel frontsand sections, as it is current practice for other geotechnical structures, such as natural and man-madeslopes (including embankments).

The following sections discuss analytical and numerical methods developed to analyze stability ofshallow tunnels and propose an extended form of an analytical solution known as Caquots solutionas a means of quickly establishing stability conditions for tunnel sections and tunnel fronts underdifferent conditions of loading and presence of water in the ground.

2 STABILITY MODELS FOR SHALLOW TUNNELS

In his book Theoretical Soil Mechanics, Terzaghi (1943) presented one of the first models forcomputing the stability of sections of shallow tunnels excavated in soils, based on the principle ofsoil arching. This model, which is shown in Figure 2a, is based on the condition that a mass of soilextending laterally and above the tunnel is in a state of limit equilibrium. A method for computingsupport requirements for tunnels in both soils and rock (and both, shallow and deep tunnels) waslater developed from the original Terzaghis model. This became to be known as the TerzaghisRock Load method (Proctor & White 1946) see Figure 2b. Terzaghis models, which will bediscussed in more detail in the next section of this paper, became popular methods for designingtunnel supports in the US, after their introduction.

Figure 2. a) Limit equilibrium model proposed by Terzaghi (after Terzaghi, 1943). b) Terzaghis Rock LoadMethod for designing tunnel support (after Proctor and T.L. White, 1946).

Other limit equilibrium models for shallow tunnels based on the original Terzaghis model (Figure2a) have also been proposed for assessing the stability of tunnel fronts i.e., the vertical plane thatremains unsupported as a tunnel is advanced and support is installed behind it. Figure 3a showssuch a limit equilibrium model for the front of an unsupported tunnel as proposed by Horn (1961).Figure 3b shows another model due to Tamez et al. (1997), as reported by Cornejo (1989) (see alsoTanzini 2001), which accounted for tunnel support behind the face. Also, Anagnostou & Kovari(1996) used as a basis the model by Horn (1961) to account for the negative effect of seepage forcesat the front, in particular reference to excavation using EPB and Slurry Shield excavation methods.

Besides the limit equilibrium methods mentioned above, there exists another group of analyticalmethods for assessing the stability of shallow tunnels. This group is based on the construction ofstatically and kinematically admissible solutions to take advantage of what are known the lowerand upper bound theorems of plasticity see, for example, Salenon (1977); Lancellotta (1993);Davis & Selvadurai (2002).

The lower bound theorem of plasticity states that a statically admissible solution (a solution thatdoes not violate the equilibrium conditions nor the yield condition of the material) is a lower boundand therefore gives a safeestimate of the loads required to stabilize an excavation. Thus, a staticallyadmissible solution provides a conservative estimate of the internal support pressure required forstability. On the other hand, the upper bound theorem of plasticity states that a kinematically

Figure 3. Limit equilibrium models for analyzing the stability of tunnel fronts for a) unsupported tunnel(after Horn, 1961); and b) supported tunnel (as reported in Cornejo, 1989).

admissible solution, a solution in which compatibility of displacements between regions (e.g.,blocks) of material during failure is not violated, is an upper bound and therefore gives an unsafeestimate of the loads required to stabilize an excavation. Thus, a kinematically admissible solutionprovides a non-conservative estimate of the internal support pressure required for stability. Becauseof the safe nature of the values of support pressure derived from statically admissible solutions,these types of solutions have been predominantly used in the estimation of stability of shallowtunnels.

Caquot (Caquot 1934; Caquot & Kerisel 1949) presented one of the first statically admissiblesolutions for analyzing the stability of shallow tunnels. Caquots model, which is represented inFigure 4a, relies on integration of the force equilibrium equations in a circular zone surroundinga circular tunnel, using the condition that the soil reaches the yield state on the crown of thetunnel, along the segment AB represented in Figure 4a. dEscatha & Mandel (1974) and Mandelet al. (1974), presented statically admissible solutions for both cylindrical and spherical cavitiesfor various configurations of internal pressure by integrating the hyperbolic equilibrium equationsthat satisfy the material yield condition, along the associated characteristic lines see Figure 4b.Both Caquots model and dEscatha & Mandel models represented in Figure 4 allow determinationof a lower bound (i.e., safe) value of support pressure required for equilibrium of the cavity.

Atkinson & Potts (1977), Davis et al. (1980) and Muhlhaus (1985) presented statically and kine-matically admissible solutions for shallow circular cavities under different loading configurations(surcharge at the ground surface and internal support pressure). As shown in Figure 5, Davis et al.(1980) and Muhlhaus (1985) were perhaps the first to propose that a statically admissible solutionfor a spherical opening, could be used to analyze the stability of the front of a lined tunnel (Figure5a) or the unsupported span in the vicinity of the tunnel front (Figure 5b), respectively.

Leca & Dormieux (1990) and Chambon & Cort (1994) presented statically and kinematicallyadmissible solutions for the tunnel front, in particular, with regard to the pressure required at thefront when excavating tunnels using mechanized methods (EPB and Slurry Shield methods). Forthe interested reader, Guglielmetti et al. (2008) and ITA/AITES (2007) provide a comprehensivereview of different existing analytical methods for computing stability of tunnel fronts, includingother methods that, for space reasons, are not mentioned in this paper.

Figure 4. Statically admissible solutions proposed by a) Caquot (1934) and b) dEscatha & Mandel (1974).

Figure 5. Statically admissible solutions for spherical cavities by a) Davis et al. (1980) and b) Muhlhaus(1985) used to analyze stability of the tunnel front and the unsupported region behind the tunnel front,respectively.

3 NUMERICAL MODELS OF TUNNEL STABILITY

Although lower and upper bound solutions for shallow tunnel problems could be directly com-puted using finite element methods and other non-linear programming techniques (see for example,Lyamin & Sloan 2002a and 2002b), the common way for analyzing stability of shallow tunnels is byrepeated application of non-linear finite element or finite difference computations using commercialsoftware, to obtain, for example, the minimum support pressure required for equilibrium see, forexample, Peila (1994), Vermeer et al. (2002), Fairhurst & Carranza-Torres (2002).

Figure 6 (after Fairhurst & Carranza-Torres 2002) shows application of the software FLAC3D(Itasca, Inc. 2012) for determining the minimum support pressure required to maintain stability of asection of a shallow cylindrical tunnel (under plane strain conditions) excavated in Mohr-Coulombmaterial. For the given geometrical and mechanical characteristics of the tunnel (see Figure 6a),the model is solved repeatedly for decreasing values of support pressure, starting with the valueassociated with in-situ stresses prior to excavation. As the internal pressure falls below a criticalvalue (see triangle pointing upwards in the diagram of Figure 6b), the tunnel collapses (i.e., the codeis not able to find a state of equilibrium for the model anymore) and the critical pressure requiredfor equilibrium is interpreted to be just above this critical value. Figure 7a through 7d, show theextent of the plastic region (i.e., the region defined by elements in the model that have reachedthe yield state) for the decreasing values of internal pressure represented in the diagram of Figure6b. The figures show that stability is lost when the plastic region has reached the surface and hasdeveloped well enough above the crown of the tunnel to produce a kinematic mechanism by whicha mass of soil above the crown slides into the tunnel (note the similarity of the development ofplastic regions in Figure 7 and Figure 4).

Figure 6. Computation of minimum internal pressure required for stability of a shallow tunnel using FLAC3Dafter Fairhurst and Carranza-Torres (2002).

Figure 8 summarizes the results obtained with FLAC3D together with results obtained with thesolutions by Terzaghi, Caquot and dEscatha & Mandel described in the previous section (see Figures2 and 4, respectively). In Figure 8, the horizontal axis represents the scaled depth of the tunnel (his the depth of the tunnel axis and a is the radius of the tunnel) and the vertical axis represents thescaled critical internal pressure below which the tunnel collapses (ps is the internal pressure and is the unit weight of the soil). The figure shows that Terzaghis solution overestimates significantlythe support pressure required to maintain equilibrium i.e., Terzaghis solution leads to too safeor too uneconomical support designs. In contrast, the values of support pressure obtained with

Caquot and dEscatha & Mandel solutions are in better agreement with the FLAC3D results thesenumerical results can be considered to be the closest to the actual solution.

The results in Figure 8 suggests that design of shallow tunnels based on limit equilibrium models,e.g., Terzaghis models in Figure 2, and their 3D extensions in Figure 3, would lead to significantoverestimation of required support. A statically admissible solution like Caquots solution shall beexpected to give better (more economical) estimation of required support for the shallow tunnel andthus can be considered to be a reasonable starting point for assessing stability of a shallow tunnel.In the following sections, the authors describe the application of an extended form of Caquotssolution to the analysis of stability of two-dimensional cylindrical sections of shallow tunnels, andthe application of the three-dimensional spherical version of solution to the analysis of stability oftunnel fronts.

Figure 7. Sequence of internal pressure reduction in the FLAC3D model of Figure 6 after Fairhurst andCarranza-Torres (2002).

4 COMPUTATION OF FACTOR OF SAFETY USING CAQUOTS MODELThe introduction of a factor of safety (or similar dimensionless value) for characterizing the stabilityconditions for shallow underground openings excavated in soils can be a useful in preliminary design

Figure 8. Comparison of required support pressure for shallow tunnels according to Terzaghis solution,Caquot and dEscatha & Mandel solutions and the numerical FLAC3D solution after Fairhurst and Carranza-Torres (2002).

of shallow tunnels, particularly at the stage in which support requirements have not been fully definede.g., in cases in which the design engineer is interested to quickly assess whether heavy, light orno support at all would be required for a planned opening.

Indeed, the use of factor of safety evaluations for shallow tunnels can be traced back to the workby Broms & Bennemark (1967) and Peck (1969) who proposed an overload factor N, defined asthe ratio of surface surcharge plus overburden pressure minus support pressure over the undrainedshear strength of soils. Based on observations, these authors suggested values for the overload factorN for which shallow excavations would be unstable, potentially unstable or definitely unstablesee Broms & Bennemark (1967) and Peck (1969). It is worth mentioning that in their definition,these authors disregarded the size of the opening, so their conclusions would have applied to specifictunnel sizes.

The implementation of a factor of safety definition into any of the statically admissible solutionsdiscussed in Section 2 is a relatively straight forward procedure. Below, the implementation isillustrated for the case of Caquots model represented in Figure 4a.

The problem to be analyzed is shown in Figure 9. It involves excavation of a cylindrical tunnelor a spherical cavity of radius, a, located at a depth, h, below the surface. The material has a unitweight, , and a shear strength defined by Mohr-Coulomb parameters c and the cohesion andthe internal friction angle, respectively. The distribution of vertical stresses before excavation islithostatic (i.e., v = qs + h) and the ratio of horizontal to vertical stresses is assumed to beunitary (i.e., h = v). A structural support pressure, ps , (e.g., provided by a liner) is applied at thecrown of the tunnel, while a uniform surcharge, qs , is acting on the ground surface this surchargerepresents, for example, the load transmitted by the foundation of existing buildings to the ground.

For the geometry and loading conditions represented in Figure 9, Caquots solution defines thevalue of internal pressure, ps , at the crown of the opening required to maintain the stability ofthe excavation i.e., ps is the minimum or critical pressure below which the tunnel collapses.Appendix A presents a demonstration of the fundamental relationship in Caquots solution to beused here (a complete derivation of Caquots solution can be found in Caquot 1934 and Caquot &

Figure 9. Extended Caquots model considered in this study as applied to the analysis of stability of cylindricaland spherical cavities.

Kerisel 1949).According to Caquots model, the following relationship relates the internal pressure, ps , and the

remaining parameters introduced above (see equation A-14),

ps

h=

(qs

h+ 2 c

h

N

N 1

) (h

a

)k(N1)(1)

1k(N 1

) 1[(

h

a

)k(N1)(h

a

)1] 2 c

h

N

N 1where the parameter k dictates the type of excavation being considered i.e., k = 1 is for acylindrical excavation and k = 2 is for a spherical excavation, and the parameter N is the passivereaction coefficient defined as follows (see, for example, Terzaghi, Peck, & Mesri 1996)

N = 1 + sin 1 sin = tan2(

4+

2

)(2)

It should be noted that equation (1) is valid only when the given Mohr-Coulomb parameters forthe soil lead to a state of limit equilibrium for the tunnel the situation for which the excavationis about to collapse. In general, the strength of the material, as defined by the cohesion, c, and theinternal friction angle, , will be larger than the strength associated with the critical equilibriumstate of the tunnel (i.e., the parameters involved in equation 1).

Next the factor of safety, FS, for the shallow tunnel is defined to be the ratio of actual Mohr-Coulomb parameters and critical Mohr-Coulomb parameters that lead to failure of the tunnel (seeFigure 10), i.e.,

FS = cccr

= tan tan cr

(3)

The definition of factor of safety given by equation (3) (see also Figure 10) is the very samedefinition of factor of safety implemented in the strength reduction technique for computation offactor of safety for slopes in Mohr-Coulomb materials with non-linear finite element and finitedifference codes (see, for example, Zienkiewicz et al. 1975; Donald & Giam 1988; Dawson et al.1999; Hammah et al. 2007 and 2008). In the last few years, this technique has become a standardmethod for computing stability conditions of surface excavations (typically slopes) and can be found

Figure 10. Scheme of shear strength reduction used to the compute factor of safety (FS) according to Caquotsextended model in the case of Mohr-Coulomb material.

implemented in the most popular commercial numerical codes for analysis of excavations forexample, FLAC (Itasca, Inc. 2011); Phase2 (Rocscience, Inc. 2011); Plaxis (Plaxis, bv 2012).

With the definition of factor of safety given by equation (3), Caquots fundamental relationship(equation 1) can now be written as follows

ps

h=

(qs

h+ 2 c

h

N

N 1

) (h

a

)k(NFS 1)(4)

1k(NFS 1

) 1(h

a

)k(NFS 1)(h

a

)1 2 c h

N

N 1

where NFS is

NFS =1 + sin (tan1 tan FS )1 sin (tan1 tan FS ) (5)

Equation (4), which is valid for any given values of Mohr-Coulomb parameters c and , allowscomputation of a factor of safety for the case of tunnels in frictional cohesive materials.

When the material is frictionless (i.e., = 0 degrees and therefore N = 1), a series ofsingularities appear in equation (4), which can be overcome by application of LHospital rule.Indeed, for frictionless materials, equation (4) becomes,

ps

h= 1 + qs

h(h

a

)1 2cFS

hk ln

(h

a

)(6)

wherecFS = cFS (7)

and therefore, solving for FS in equations (6) and (7), the factor of safety for the shallow tunnel can

be explicitly written as follows

FS = 2c h

k ln(ha

)1 + qs

h ps

h (h

a

)1 (8)As seen from the equations above, computation of the factor of safety for the general case of

frictional cohesive material requires solving the non-linear equation (4) by means of some numericaltechnique. Appendix B in this paper presents a computer spreadsheet and associated programmingcode required to compute the factor of safety, FS, from the transcendental equation (4).

5 CONSIDERATION OF WATER PORE-PRESSURE IN CAQUOTS MODELWhen excavating shallow tunnels in soils, and as it is typically the case with other geotechnicalstructures like slopes and foundations, water in the ground and water on the face of the excavationitself can be expected to have an influence in the stability of the opening.

Although the undrained condition for the ground as it applies to saturated clays can be accountedfor readily with equations (6) through (8) in Caquots extended model, for the general case ofpermeable soils, as a first approximation to solving the problem, the effect of water can be accountedfor by using Terzaghis effective stress principle. This implies decomposing total stresses in theground into effective stresses and water pressure, and computing the strength of the material interms of effective stresses only (see, for example, Terzaghi et al. 1996). A comprehensive analysisof this type for the case of deep tunnels in permeable porous media and various hydraulic conditionsfor the tunnel itself (i.e., whether water pressure exists inside the tunnel or not) has been presentedin Carranza-Torres & Zhao (2007). In this section, Caquots model is further extended to considerwater pressure in the ground according to Terzaghis principle and various hydraulic conditions.

The five different cases considered here are listed in Table 1 and represented in Figures 11through 13. The solution of these different cases can be obtained by applying a similar procedureas the one in Appendix A, this time decomposing the total (stress) problem into effective and waterpressure components and applying the appropriate stress boundary conditions (which may or maynot include water pressure depending on the case considered) at the crown of the tunnel and on theground surface.

Table 1. The five hydraulic conditions considered for Caquots extended model.Case A. Dry ground see Figure 11.Case B1. Wet ground. Water level above ground level (WL>GL). Dry tunnel see Figure 12a.Case B2. Same as above, but with flooded tunnel see Figure 12b.Case C1. Wet ground. Phreatic level below ground level (PL

follows.

For dry conditions (Case A, in Table 1 and Figure 11) the terms are h

c

= dh

c(10)

qs

h

= qs

dh(11)

andps

h

= ps

dh(12)

For wet conditions and water level above the ground level (Cases B1 and B2, in Table 1 andFigures 12a and 12b, respectively), the terms are

h

c

= dh

c

(s

d w

d

)(13)

andqs

h

= qs

dh

dh

c

h

c

(14)The remaining pressure terms are, for Case B1,

ps

h

= ps

dh

dh

c

h

c

w

d

(1 a

h+hwh

) dh

c

h

c

(15)and for Case B2,

ps

h

= ps

dh

dh

c

h

c

(16)Finally, for wet conditions and water level below the ground level (Cases C1 and C2 in Table 1

and Figures 13a and 13b, respectively), the terms are

h

c

= dhc

hwh

1 a

h

+ dhc

1 ah

hwh

1 a

h

(s

d w

d

)(17)

andqs

h

= qs

dh

dh

c

h

c

(18)The remaining pressure terms are, for Case C1,

ps

h

= ps

dh

dh

c

h

c

w

d

(1 a

hhwh

) dh

c

h

c

(19)and for Case C2,

ps

h

= ps

dh

dh

c

h

c

(20)

It is important to note that equations (17) through (20) are derived taking an average value of unitweight for the material above the crown of the tunnel a weighted average of dry unit weight dand saturated unit weight s , weighted with respect to the heights hw and ha hw, respectively(see Figure 13). Therefore equations (17) through (20), are valid only when hw < ha (see Figure13). For cases in which the phreatic surface is below the crown of the tunnel (hw > h a in Figure13), the excavation should be analyzed as if the ground is dry i.e., considering it to be Case A(Figure 11).

Figure 11. Extended Caquots model for dry ground conditions.

Figure 12. Extended Caquots model for the case of wet ground, water level above ground level, andconditions of a) dry tunnel and b) flooded tunnel.

Figure 13. Extended Caquots model for the case of wet ground, phreatic level below ground level, andconditions of a) dry tunnel and b) flooded tunnel.

6 COMPARISON OF RESULTS FROM CAQUOTS EXTENDED MODEL AND NUMERICALMODELS

As mentioned in Section 4, the definition of factor of safety introduced in equation (3) is the samedefinition of factor of safety considered in the implementation of the strength reduction techniquefor Mohr-Coulomb materials available in commercial finite difference and finite element codes suchas FLAC (Itasca, Inc. 2011), FLAC3D (Itasca, Inc. 2012) and Phase2 (Rocscience, Inc. 2011). Inview of this, factors of safety obtained with the extended Caquots solution presented in previoussections can be compared on a one-to-one basis with the factors of safety that are obtained using thestrength reduction technique in these commercial packages again, understanding that Caquotsextended solution, being a statically admissible solution, is a lower bound to the true solution, andtherefore factors of safety obtained with this solution will be on the conservative side.

In the following two subsections, examples of stability analysis for cylindrical sections of tunneland the front of the tunnel, regarded as a spherical cavity, are presented.

6.1 Stability analysis of long cylindrical tunnelThe problem is represented in Figure 14. The material properties, geometry and loading con-

ditions of the shallow tunnel are indicated in the figure. The problem is solved for the five casesdiscussed in Section 5 (see Table 1 and Figures 11 through 13), applying the equations listed in thatsection.

Using the computer spreadsheet described inAppendix B (that implements the extended Caquotssolution discussed Section 5) the results listed in the second column of Table 2, and referred to asCaquots extended results, have been obtained. Note that the resulting values of factor of safetyin Table 2 vary depending on whether water pore pressure is considered for the ground and, inparticular, on whether the water is filling the tunnel or not. Comparing, for example, the factor ofsafety for the cases of dry ground (Case A), FS = 1.71, with the factor of safety for the cases ofwet ground and flooded tunnel (Cases B2 and C2), FS = 1.96 and FS = 1.76, respectively, the

Figure 14. Example of cylindrical tunnel section analyzed with Caquots extended model and with FLAC.

Figure 15. FLAC grid used to solve the problem in Figure 14.

effect of water is seen to increase the factor of safety i.e., the water inside the tunnel has the effectof providing a mechanical support pressure to the excavation above the given value of structuralsupport pressure ps . Also, the factor of safety for cases of wet ground and dry tunnel (CasesB1 and C1), FS = 0.97 and FS = 1.61, respectively, result to be smaller than the factor of safetyfor dry ground (FS = 1.71). This is because the lack of pore pressure inside the tunnel implies areduction of support pressure, from the existing value ps , equal to the value of pore pressure in theground at the level of the tunnel crown. Therefore, for the cases B1 and C1, the larger the value ofpore pressure in the ground surrounding the tunnel, the lower the factor of safety obtained for thetunnel.

It is important to realize that for cases B1 and C1 in this example, in which water is removed fromthe excavation, a minimum support pressure, pmins , is required to avoid tensile failure of the tunnel

wall. The value of pmins , provided, for example, by installation of a temporary or a permanent liner,should be equal at least to the value of pore pressure of the surrounding ground (see, for example,Carranza-Torres & Zhao 2007).

The problem in Figure 14 has also been solved using the strength reduction technique implementedin FLAC. For Cases B and C, a total stress analysis with fixed pore pressure specified at the zonesin the model has been performed. The mesh used to analyze the problem is represented in Figure15. To further illustrate the implementation of the strength reduction technique in FLAC, Figure16 shows views of the extent of the plastic region (plots on the left) and contours of magnitude ofdisplacement (plots on the right) for the last stable state (Figure 16a) and first unstable state (Figure16b) for the model presented in Figures 14 and 15, in the case of dry soil (i.e., the Case A).

Values of factor of safety obtained with FLAC for all five cases considered are summarized inthe third column of Table 2 and referred to as Numerical. Comparing the values of factor ofsafety obtained with Caquots solution and FLAC, the former values are observed to be smaller (i.e.,more conservative) than the latter ones. This is in agreement with the results discussed in Figure 8and the fact that a statically admissible solution like Caquots solution underestimates the stabilityconditions for the tunnel.Table 2. Factor of safety results for the problem in Figure 14.

Case Caquots extended NumericalType (Spreadsheet) (FLAC)A- Dry ground 1.71 2.01B1- WL > GL; dry tunnel 0.97 1.03B2- WL > GL; flooded tunnel 1.96 2.25C1- PL < GL; dry tunnel 1.61 1.74C2- PL < GL; flooded tunnel 1.76 2.02

6.2 Stability analysis of tunnel frontFollowing the analysis by Davis et al. (1980) and Muhlhaus (1985) see Figures 5a and Figures

5b, respectively the extended Caquots solution for spherical cavities is used here to analyze thestability of the unsupported region near the tunnel front. Figure 17a shows a schematic representationof the unsupported span behind the tunnel front regarded as a spherical cavity. If the unsupportedspan length is denoted as L, and the tunnel has a radius, R, then the radius of the enclosing sphericalsurface, a, can be computed according to the equation a = L2/4 + R2. Figure 17b shows twoside views of tunnel front regions enclosed by spherical surfaces, computed with the equation above,for ratios L/2R equal to 0.5 and 1.0, respectively.

As an application example, the problem represented in Figure 18 is solved first using the extendedCaquots solution and the finite difference commercial code FLAC3D. The geometry of the problemand properties of the material (including loading conditions) are indicated in Figure 18. To simplifythe analysis, the problem is solved for the dry situation only (Case A in Figure 11). Using thecomputer spreadsheet presented in Appendix B, the factor of safety according to Caquots extendedmodel results to be FS = 1.0 (see first row in Table 3). The same problem is solved next usingthe strength reduction technique implemented in FLAC3D. Figure 19 represents the mesh used inthe numerical model. According to FLAC3D, the factor of safety results to be 1.1 (see second rowin Table 3). This factor of safety, as expected, is larger than the one obtained with the Caquotsextended solution. To illustrate the type of failure obtained with the strength reduction techniqueimplemented in FLAC3D, Figure 20 shows views of the extent of the plastic region (plots on theleft) and contours of magnitude of displacement (plots on the right) for the last stable state (Figure20a) and first unstable state (Figure 20b) obtained for the model represented in Figures 18 and 19.

The problem in Figure 18 has also been solved using a limit equilibrium model for the tunnelfront, with the equations provided in Cornejo (1989) and Tamez et al. (1997) (see Figure 3). The

Figure 16. Extent of plastic region (left) and contours of magnitude of displacement (right) for the problem inFigures 14 and 15 in the case dryground, as solved with the strength reduction technique implementation inFLAC. Plots represented are for a) last equilibrium state; and b) first unstable state obtained with the strengthreduction technique.

resulting factor of safety, FS = 0.75 (see last row in Table 3), is significantly lower than the factorof safety obtained earlier on with Caquots extended model and with FLAC3D. This is an expectedresult since limit equilibrium solutions (those derived from Terzaghis model) have been shownalready to lead to significantly conservative values of support pressure required for equilibrium, ascompared with statically admissible solutions and numerical models (see Figure 8).Table 3. Factor of safety results for the problem in Figure 18.

Type of solution Factor of safetyCaquots extended solution (spreadsheet) 1.00Numerical, FLAC3D 1.10Limit equilibrium solution (model in Figure 3b 0.75

Figure 17. a) Schematic representation of the unsupported span near the tunnel front regarded as a sphericalcavity. b) Side view of tunnel front region enclosed by spherical surface for different ratios of unsupportedlength and tunnel radius.

Figure 18. Example of unsupported span of a tunnel front regarded as a spherical cavity, analyzed withCaquots extended model and with FLAC3D for dry ground conditions.

Figure 19. FLAC3Dgrid used to solve the problem in Figure 18.

Figure 20. Extent of plastic region (left) and contours of magnitude of displacement (right) for the problemin Figures 18 and 19, as solved with the strength reduction technique implementation in FLAC3D. Plotsrepresented are for a) last equilibrium state; and b) first unstable state obtained with this technique.

7 FINAL COMMENTS

From a mechanical point of view, instability of shallow tunnels and instability of slopes in soilsshare many similarities. In both cases, gravity or external applied loads (e.g., a surcharge on thecrest of the slope or a surcharge on the ground surface) tend to produce mechanisms by which amass of soil detaches and slides along shear surfaces when collapse takes place. In both cases,additional resisting forces are typically incorporated in the design (e.g., as reinforcement in theform of soil nails, buttress fills or ponded water, in the case of slopes; and typically reinforcementand support in the case of shallow tunnels) so as to increase the stability condition for the slope ortunnel. This is especially important when the strength of the soil is not high enough to guaranteestability of the slope or tunnel by itself. In view of the similarities between the two problems, itseems strange that the concept of a factor of safety to characterize stability of slopes in soils hasgained so much popularity among geotechnical engineers, while the very same concept apparentlyhas been ignored when trying to assess stability conditions for shallow tunnels (including stabilityof tunnel fronts).

With the popularity of finite element and finite difference software in the practical design andanalysis of geotechnical engineering in the last years and, in particular, with the availability of thestrength reduction technique for computing factor of safety for basically any problem that can bemodelled with the software (not necessarily slopes), there exists now the opportunity to investigatefurther the potential application of a factor of safety concept for the case of shallow openings insoils. One of the main objectives of this paper has been, indeed, to be a starting point to revivediscussions on the subject.

One of the advantages the authors see in using the concept of factor of safety for the case ofshallow tunnels in combination with simple models, like the extended Caquots solution consideredin this paper, is in being able to easily incorporate uncertainty and variability into the analysis ofstability of shallow tunnels by performing probability analyses of failure. By using simple models(e.g., closed-form solutions), instead of computing-intensive and time-consuming reduction strengthmodels implemented in commercial codes, it should be feasible to apply probability techniques likeMonte Carlo simulations as part of design. These techniques are already standard procedures forconducting slope stability evaluations, especially, when used in combination with relatively fast (tocompute) limit equilibrium models for slopes see, for example, Abramson et al. (2002).

The work presented in this paper is by no means complete, and many further developments arepossible. Some of these are mentioned below.

Although the Caquots solution used in this paper has been said to be an admissible staticallysolution, the solution disregards the stress state beyond the circular zone of integration, so strictlyspeaking, it does not satisfy equilibrium throughout the medium. This would certainly have influ-ences on predicted values of factor of safety and how far (or close) these values would be fromthe ones that could be computed with the strength reduction technique implemented in commercialsoftware. To address this problem, further analysis is needed, especially to quantify the differencesbetween factor of safeties obtained with the extended Caquots solution and the actualones (which,in principle, could be assumed to be given by the strength reduction models). Such analyses shouldalso account for the influence of other variables that are expected to influence the resulting valuesof factor of safety, like distribution of support pressure inside the tunnel, initial state of stress in theground, and others.

The scaling law used in deriving the extended Caquots solution in Appendix A could be used forobtaining convenient dimensionless representations of factor of safety values for shallow tunnels.Proper scaling should allow to obtain similar stability charts for shallow tunnels as those for slopesin the best (i.e., most compact) form possible, at least the authors are aware of, as presented inHoek & Bray (1981) and Wyllie & Mah (2004).

The spreadsheet presented in Appendix B could be easily modified to implement a Monte Carlosimulation scheme, and so to compute probability of failure and a statistical reliability quantification

for the design.Finally, the constitutive model for the soil considered in this paper for the tunnel problems was

the simple Mohr-Coulomb failure criterion. Most commercial software implementing the strengthreduction technique allows application of other constitutive models. A popular one is the Hoek-Brown failure criterion (Hoek & Brown 1980; Hoek et al. 2002), which is widely used nowadaysin design of excavations in rock masses. The analysis presented in this paper (with the additionalimprovements discussed above) could be extended for the case of weak rocks that satisfy the Hoek-Brown failure criterion.

ACKNOWLEDGEMENTS

Many of the developments presented in this paper have been carried out by the first author whileworking with Geodata S.p.A. in Turin, Italy, during a two-month working visit in 2004 (see Carranza-Torres 2004). These initial developments are serving as a basis for research work conducted by thesecond author of this paper for his MSc graduate work, which is being supervised by the first andthird authors. The first author would like to thank Geodatas president, Dr. Piergiorgio Grasso, andhis associates, Dr. Giordano Russo and Dr. Shulin Xu, for the support and hospitality providedwhile working with their group in 2004.

REFERENCESAbramson, L. W., Thomas, S. L., Sharma, S. & Boyce, G. M. 2002. Slope Stability and Stabilization Methods

(Second ed.). John Wiley & Sons.Anagnostou, G. & Kovari, K. 1993. Significant parameters in elastoplastic analysis of underground openings.

ASCE J. Geotech. Eng. Div. 119(3), 401419.Anagnostou, G. & Kovari, K. 1996. Face stability conditions with earth-pressure-balanced shields. Tunnelling

and Underground Space Technology 11(2), 165173.Atkinson, J. H. & Potts, D. M. 1977. Stability of a shallow circular tunnel in cohesionless soil. Geotech-

nique 27(2), 203215.Broms, B. B. & Bennemark, H. 1967. Stability of clay at vertical openings. ASCE Journal of Soil Mechanics

and Foundation Engineering Division 93, 7194.Caquot, A. 1934. quilibre des massifs a frottement interne. Paris: Gauthier-Villars.Caquot, A. & Kerisel, J. 1949. Traite de Mecanique des Sols. 2e dition de lOuvrage: quilibre des massifs a

frottement interne. Stabilit des terres pulvrulentes ou cohrentes, de A. Caquot. Paris: Gauthier-Villars.Carranza-Torres, C. 2003. Dimensionless graphical representation of the elasto-plastic solution of a circular

tunnel in a Mohr-Coulomb material. Rock Mechanics and Rock Engineering 36(3), 237253.Carranza-Torres, C. 2004. Technical Report to Geodata S.p.A.: Computation of factor of safety for shallow

tunnels using Caquots lower bound solution. Itasca Consulting Group. Minneapolis.Carranza-Torres, C. & Zhao, J. 2007. Analytical and numerical study of the effect of water pressure on the

mechanical response of cylindrical lined tunnels in elastic and elasto-plastic porous media. InternationalJournal of Rock Mechanics and Mining Sciences 46, 531547.

Chambon, P. & Cort, J. F. 1994. Shallow Tunnels in Cohesionless Soils: Stability of Tunnel Face. Journalof Geotechnical Engineering 120(7), 11481165.

Chapra, S. C. 2010. Introduction to VBA for Excel (Second ed.). Pearson. Prentice Hall.Civil Engineer International 1995. Tunnel lining removal prompts LA Metro cave in. Institution of Civil

Engineers, July Issue, 10.Construction Today 1994a. Police probe repeat Munich tunnel breach. Construction Today, October Issue,

45.Construction Today 1994b. Unstable ground triggers Munich tunnel collapse. Construction Today, October

Issue, 5.

Cornejo, L. 1989. Instability at the face: its repercussions for tunnelling technology. Tunnels and Tunnelling,6974.

Davis, E. H., Gunn, M. J., Mair, R. J. & Seneviratne, H. N. 1980. The stability of shallow tunnels andunderground openings in cohesive material. Geotechnique 30(4), 397416.

Davis, R. O. & Selvadurai, A. P. S. 2002. Plasticity and geomechanics. Cambridge University Press.Dawson, E. M., Roth, W. H. & Drescher, A. 1999. Slope stability analysis by strength reduction. Geotech-

nique 49(6), 835840.dEscatha,Y. & Mandel, J. 1974. Stabilite dune galerie peu profunde en terrain meuble. Industrie Minerale 6,

19.Donald, I. B. & Giam, S. K. 1988. Application of the nodal displacement method to slope stability analysis.

In Proceedings of the 5th Australia-New Zealand Conference on Geomechanics. Sydney. Australia, pp.456460.

Fairhurst, C. & Carranza-Torres, C. 2002. Closing the Circle. In J. Labuz & J. Bentler (Eds.), Proceedingsof the 50 th Annual. Geotechnical Engineering Conference. February 22, 2002. University of Minnesota.(Available for downloading at Fairhurst Files, www.itascacg.com; and at www.cctrockengineering.com).

Government of Singapore 2005. Report of the Committee of Inquiry into the Incident at the MRT Circle LineWorksite that led to the Collapse of Nicoll Highway on 20 April 2004. Government of Singapore, LandTransport Authority.

Guglielmetti, V., Grasso, P., Mahtab, A. & Xu, S. 2008. Mechanized Tunnelling in Urban Areas: DesignMethodology and Construction Control. London: Tylor & Francis.

Hammah, R. E., Yacoub, T. E., Corkum, B. & Curran, J. H. 2008. The practical modelling of discontinuousrock masses with finite element analysis. In Proceedings of the 42nd US Rock Mechanics Symposium(USRMS). San Francisco, California.

Hammah, R. E., Yacoub, T. E., Corkum, B., Wibowo, F. & Curran, J. H. 2007. Analysis of blocky rockslopes with finite element shear strength reduction analysis. In Proceedings of the 1st Canada-US RockMechanics Symposium. Vancouver. Canada, pp. 329334.

Hoek, E. & Bray, J. 1981. Rock Slope Engineering. Spon Press.Hoek, E. & Brown, E. T. 1980. Underground Excavations in Rock. London: The Institute of Mining and

Metallurgy.Hoek, E., Carranza-Torres, C. & Corkum, B. 2002. Hoek-Brown failure criterion 2002 edition. In Pro-

ceedings of NARMS-TAC 2002, Mining Innovation and Technology. Toronto 10 July 2002, pp. 267273.University of Toronto. (Available for downloading at Hoeks Corner, www.rocscience.com).

Horn, M. 1961. Horizontaler Erddruck auf senkrechte Abschlussflchen von Tunneln (Horizontal earth pres-sure on vertical tunnel fronts). Landeskonferenz der ungarischen Tiefbauindustrie (National Conferenceof Hungarian Civil Engineering Industry). Translation into German by STUVA, Dsseldorf.

ITA/AITES 2007. Report 2006 on Settlements induced by tunnelling in soft ground. Presented by workinggroup Research of the International Tunelling and Underground Space Association. Tunnelling andUnderground Space Technology 22, 119149.

Itasca, Inc. 2011. FLAC (Fast Lagrangian Analysis of Continua) Version 7.0. Itasca Consulting Group, Inc.Minneapolis. Minnesota.

Itasca, Inc. 2012. FLAC3D (Fast Lagrangian Analysis of Continua in 3 Dimensions) Version 5.0. ItascaConsulting Group, Inc. Minneapolis. Minnesota.

Lancellotta, R. 1993. Geotecnica. Seconda edizione. Zanichelli Editore S.p.A., Bologna.Leca, E. & Dormieux, L. 1990. Upper and lower bound solutions for the face stability of shallow circular

tunnels in frictional material. Geotechnique 40(4), 581606.Lyamin, A.V. & Sloan, S. W. 2002a. Lower bound limit analysis using non-linear programming. Int. J. Numer.

Anal. Methods Geomech. 55, 573611.Lyamin, A. V. & Sloan, S. W. 2002b. Upper bound limit analysis using linear finite elements and non-linear

programming. Int. J. Numer. Anal. Methods Geomech. 26, 181216.Mandel, J., Habib, P., dEscatha, Y., Halphen, B., Luong, M. & Zarka, J. 1974. tude thorique et expri-

mentale de la stabilit des cavits souterraines. Symposium franco-polonais Problmes de Rhologie etde Mcanique des Sols. Nice. France.

Muhlhaus, H. B. 1985. Lower bound solutions for circular tunnels in two and three dimensions. Rock Me-chanics and Rock Engineering 18, 3752.

Oliver, A. 1995. Los Angeles metro collapse investigation focuses on tunnel lining replacement. GroundEngineering 28(6), 45.

Peck, R. B. 1969. Deep excavations and tunnelling in soft ground. In Proceedings of the 7th InternationalConference on Soil Mechanics and Foundation Engineering. Mexico City, August 1969, pp. 225281.

Peila, D. 1994. A theoretical study of reinforcement influence on the stability of a tunnel face. Geotechnical& Geological Engineering 12(3), 145168.

Plaxis, bv 2012. Plaxis 2D (Finite Element Software for Geotechnical Professionals). Plaxis bv. Delft. TheNetherlands.

Proctor, R. V. & White, T. L. 1946. Rock Tunnelling with Steel Supports. Commerical Shearing, Inc., Ohio.Rocscience, Inc. 2011. Phase2 (Finite Element Analysis for Excavations and Slopes) Version 8.0. Rocscience.

Toronto.Salenon, J. 1977. Applications of the theory of plasticity in soil mechanics. Series in geotechnical engineering.

Wiley. New York.Tamez, E., Rangel, J. L. & Holgun, E. 1997. Diseno Geotecnico de Tuneles. Editorial TGC, Mexico.Tanzini, M. 2001. Gallerie. Aspetti Geotecnici nella Progettazione e Costruzione. Dario Flaccovio Editore.

Palermo. Italia.Terzaghi, K. 1943. Theoretical Soil Mechanics. New York: John Wiley & Sons, Inc.Terzaghi, K., Peck, R. & Mesri, G. 1996. Soil Mechanics in Engineering Practice. New York: John Wiley &

Sons, Inc.Timoshenko, S. P. & Goodier, J. N. 1970. Theory of Elasticity (Third ed.). New York: Mc. Graw Hill.Vermeer, P. A., Ruse, N. & Marcher, T. 2002. Tunnel heading stability in drained ground. Felsbau 20(6),

818.Wyllie, D. C. & Mah, C. W. 2004. Rock Slope Engineering. Civil and Mining. Taylor & Francis.Zienkiewicz, O. C., Humpheson, C. & Lewis, R. W. 1975. Associated and non-associated visco-plasticity

and plasticity in soil mechanics. Geotechnique 25(4), 671689.

APPENDIX A. DEMONSTRATION OF CAQUOTS STABILITY EQUATION

Caquots solution (Caquot 1934; Caquot & Kerisel 1949) is a classical solution for determining thestress field around a circular tunnel located below a flat surface. This appendix presents a demon-stration of equation (1) in the main text, which is one of the fundamental expressions conformingCaquots solution. The analysis that follows refer to the same problem presented in Figure 9.

In reference to Figure 9, and to simplify the formulation, the radial distance, r , is first scaled withrespect to the tunnel radius, a. This defines the dimensionless ratio, , i.e.,

= ra

(A-1)

Note that according to equation (A-1), the position of the ground surface, r = h in Figure 9, isdetermined by the variable , i.e.,

= ha

(A-2)The material surrounding the tunnel is assumed to obey the Mohr-Coulomb failure criterion, so

that the relationship between major and minor principal stresses at failure, the quantities 1 and 3,respectively, is

1 = 3N + 2c

N (A-3)

In equation (A-3), c is the cohesion and N is the passive reaction coefficient of the material (see,for example, Terzaghi et al. 1996), which is computed from the internal friction angle, , as follows

N = 1 + sin 1 sin = tan2(

4+

2

)(A-4)

Also, to simplify the formulation, a particular form of scaling that applies to Mohr-Coulombshear failure will be used. The scaling consists in adding the term c tan , or equivalently, the term2c

N/(N 1) to the normal stresses (see, for example, Anagnostou & Kovari 1993; Carranza-Torres 2003). Therefore, when scaled according to the rule mentioned above, the radial and hoopstresses, r and , respectively (see Figure 9), become

Sr = r + 2c

N

N 1 (A-5)

S = + 2c

N

N 1 (A-6)

Note that in equations (A-5) and (A-6), and also in the equations that follow, capital letters are usedto denote scaled stresses.

When the stresses are scaled according to equations (A-5) and (A-6), the Mohr-Coulomb failurecriterion (equation A-3) takes the following simple form

S1 = NS3 (A-7)Caquots solution is obtained by integrating two force equilibrium equations for the radial and

hoop directions in the circular area around the cavity in Figure 9, in particular, considering thatthe limit state of equilibrium is reached along the segment AB in Figure 9. This Appendix willnot present the demonstration of the full solution for the stress field obtained by the integrationmentioned above, but just the demonstration of equation (1) in the main text, which can be obtainedby stating equilibrium of forces along the segment AB in Figure 9. Indeed, considering that Figure 9represents either the case of a cylindrical cavity (for which the out-of-plane stress is an intermediateprincipal stress) or a spherical cavity (for which the out-of-plane stress is equal to the hoop stress)equilibrium of forces along the segment AB in terms of the scaled quantities introduced earlier oncan be written as (see, for example, Timoshenko & Goodier 1970)

dSrd

+ kSr S

+ a = 0 (A-8)

In equation (A-8), the parameter k takes the value 1 for the case of cylindrical cavities and the value2 for the case of spherical cavities.

When the material is at a state of plastic failure along the segmentAB in Figure 9 due to unloadingof the cavity (i.e., when the internal pressure in Figure 9 is reduced from the initial value definedby in-situ stress conditions), the radial stress will become a minimum principal stress and the hoopstress will become a major principal stress. Therefore, considering S1 = S and S3 = Sr in equation(A-7) and replacing this into equation (A-8), the equilibrium condition along the segment AB inFigure 9 is written as follows

dSrd

k(N 1

)Sr

+ a = 0 (A-9)

According to Figure 9, the unknown function, Sr , in equation (A-9) is required to satisfy the

following conditions

Sr = Qs for = (A-10)Sr = Ps for = 1 (A-11)

where Ps and Qs are scaled values of internal pressure and ground surcharge, respectively, i.e.,

Ps = ps + 2c

N

N 1 (A-12)

Qs = qs + 2c

N

N 1 (A-13)

Solving the ordinary differential equation (A-9) and applying the conditions (A-11) and (A-10),the following relationship between the scaled internal pressure, Ps , and the scaled surcharge, Qs ,is obtained

Ps

a= Qs

ak(N1) 1

k(N 1

) 1[ 1k(N1) 1

](A-14)

Equation (A-14) is one of the expressions conforming Caquots solution, which is transcribed asequation (1) in the main text. The reader is referred to Caquot (1934) and Caquot & Kerisel (1949)for a demonstration of the entire Caquots solution.

APPENDIX B. SPREADSHEET FOR THE COMPUTATION OF FACTOR OF SAFETYACCORDING TO CAQUOTS MODEL

This appendix describes an Excel R spreadsheet and associated VBA R code (see, for example,Chapra 2010) that implements the equations (9) through (20) in the main text and allows computationof the values of factor of safety for cylindrical and spherical openings, for the five cases (Cases A,B1/B2 and C1/C2) listed in Table 1 and represented in Figures 11 through 13. The layout of thespreadsheet is shown in Figure B-1. The yellow cells in the spreadsheet are input cells while thegreen cells are output cells. The input parameters to be entered in the spreadsheet are listed inTable B-1 (see yellow cells in Figure B-1).

The output variables (see green cells in Figure B-1) are as follows: i) the factor of safetycoefficient, FS, for the opening computed by numerical solution of the transcendental equation (9);and ii) a line of comments indicating whether the computation of factor of safety was successful ornot.

Note that the resulting value of factor of safety, FS, is computed after the user presses the buttonCompute results. The user can also clear all the input variables cells at once by pressing the buttonClear results. By pressing either of the two buttons mentioned above, Excel calls the VBA macrolisted in Figures B-2 through B-5.

Table B-1. Description of input parameters and required units for the spreadsheet in Figure B-1

- Analysis case [1 or 2]:This is the parameter k in the equations (9) through (20), in Section 5. As discussed in Section 4, k = 1is for cylindrical openings and k = 2 is for spherical openings.

- Hydraulic Conditions [1, 2, 3]:The value entered in this cell allows to distinguish between cases 1) A; 2) B1 or B2; and 3) C1 or C2 inTable 1 and Figures 11 through 13, and to apply the proper equations within the group of equations (9)through (20), in Section 5.

- Cylinder or sphere radius [m]:This is the variable a in the equations (9) through (20), in Section 5. The value entered here should beequal to the radius of the tunnel if k = 1, or the radius of the enclosing sphere if k = 2 see Figure17.

- Depth Tunnel Axis [m]:This is the variable h in the equations (9) through (20), in Section 5.

- Dry unit weight [kN/m3]:This is the variable d in the equations (9) through (20), in Section 5.

- Mohr-Coulomb effective friction angle [deg]:This is the variable , that allows computation of the coefficient N according to the equation (2), foruse in the equations (9) through (20), in Section 5.

- Mohr-Coulomb effective cohesion [kPa]:This is the variable c in the equations (9) through (20), in Section 5.

- Ground surface surcharge [kPa]:This is the variable qs in the equations (9) through (20), in Section 5.

- Tunnel support pressure [kPa]:This is the variableps in the equations (9) through (20), in Section 5. A representative value of (maximumpossible) support pressure at the crown of the tunnel for the installed liner should be specified.

- Water pressure case [1, 2]:The value entered in this cell allows to distinguish between cases 1) B1/C1 or 2) B2/C2, i.e., dry exca-vation or flooded excavation, respectively (see Table 1 and Figures 11 through 13), for the applicationof the correct set of equations, within the group of equations (9) through (20), in Section 5.

- Distance between water surface and ground surface [m]:This is the variable hw in the equations (9) through (20), in Section 5. According to these equations, thespreadsheet will convert the entered value to a positive value (the absolute value of the input value willbe considered).

- Saturated unit weight [kN/m3]:This is the variable s in the equations (9) through (20), in Section 5.

- Unit weight water [kN/m3]:This is the variable w in the equations (9) through (20), in Section 5.

Figure B-1. Excel spreadsheet for the computation of factor of safety using the extended Caquots solution.

Figure B-2. VBA program for Excel spreadsheet in Figure B-1.

Figure B-3. VBA program for Excel spreadsheet in Figure B-1. Continued from Figure B-2.

Figure B-4. VBA program for Excel spreadsheet in Figure B-1. Continued from Figure B-3.

Figure B-5. VBA program for Excel spreadsheet in Figure B-1. Continued from Figure B-4.