Geotechnical Aspects of Underground Construction in Soft Ground – Yoo, Park, Kim & Ban (Eds)© 2014 Korean Geotechnical Society, Seoul, Korea, ISBN 978-1-138-02700-8

Tunnel modelling: Stress release and constitutive aspects

T.G.S. DiasGhent University, Ghent, Belgium

A. BezuijenGhent University, Ghent, BelgiumDeltares, Delft, The Netherlands

ABSTRACT: Tunnel construction in soft ground has evolved significantly over the last 20 years, especially onthe matter of settlement control. This was achieved by guiding the TBM operation to control the main factors thatinduced soil displacements, like the face pressure and the soil-lining void closure. However, the design methodsand numerical modeling procedures where not adapted to these new conditions, sometimes applying boundaryconditions, constitutive parameters or state variables with no physical meaning to match field measurements.This paper presents an analysis of the basic principles of plane strain numerical modeling of tunnel construction.The literature review is followed by an analysis of the stress release factor and the effects of different constitutivemodels to represent the soil. The tunneling convergence and settlement trough as well as the stress paths on soilelements at the crown and at springline will be presented.

1 INTRODUCTION

Tunnel construction in soft ground has evolved signifi-cantly over the last 20 years, especially on the matter ofsettlement control. The routine volume loss of mech-anized tunnels decreased from 6% to less than 1%.This was achieved by guiding the TBM operation tocontrol the main factors that induced soil displace-ments, like the face pressure and the soil-lining voidclosure. This approach was effective but tended to ana-lyze these factors individually with no account for theirpossible influence on the global mechanism of tunnelconstruction.

The same pattern emerged from tunnel design meth-ods that are particularly focused on the surface set-tlements. Again, significant contributions came fromthis; the settlement trough has been thoroughly ana-lyzed since its effect on buildings was addressed byBurland & Wroth (1974). Its description and quantifi-cation was of utmost importance for the developmentof underground construction in urban centers withoutjeopardizing the stability of surface structures.

Alongside of these trends, the use of the finite ele-ment method for the analysis of geotechnical problemshas increased notably. However, the vastness of choicesin modelling procedures, which include the consti-tutive model, mesh, parameters and boundary condi-tions, tend to produce results that are user-dependentand often limited in the aspects they can reproducefrom real cases or physical models. Ergo a great dealof tunnel modelling aimed solely on predicting the set-tlement trough, and did so without accounting for theeffect of the new construction techniques. To achieve

a better prediction, boundary conditions, constitutiveparameters or state variables were sometimes adaptedwithout a clear physical meaning. This normally hin-dered the model’s reliability and capacity to cope withthe different tunneling or soil conditions.

Considering these aspects, this paper presents ananalysis of the basic principles of plane strain numer-ical modeling of tunnel construction. The literaturereview presents how the basic stress release conceptevolved to account for TBM tunnel construction. Thatis followed by two groups of analyzes: a simple linearelastic model to remark on the effects of the soil com-pressibility and initial stress state and a comparativeanalysis on the effects of different constitutive mod-els to represent the soil. The tunneling convergenceand settlement trough as well as the stress paths onsoil elements at the crown and at springline will bepresented.

2 LITERATURE REVIEW

The underground opening of a tunnel can be evaluatedmechanically as a simple process. For an unlined tun-nel the stress state on the imaginary boundary surfaceof the excavation is taken from the initial in-situ stressto a condition of zero normal and shear stress. Thispath can be done in increments that are normally calledstress release factors (λ) and represent a percentage ofthe full path. For the case of a lined tunnel it is normallyassumed that part of the stress release will developwith a free boundary (λ), as in the unlined tunnels.After the lining is installed the remaining stress (1-λ)

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will be released and reach equilibrium with the lining.The soil-lining interaction and the lining rigidity willdictate the equilibrium state of the tunnel.

A series of analyzes on modeling procedures forTBM tunnels was initiated by Rowe et al. (1983),that also remarks the importance of modeling thesoil’s cross-anisotropic deformability to achieve bet-ter displacement predictions. Knowing the gap thatexists between the excavated boundary and the lin-ing, the partial stress release factor will be the one thatinduces a tunnel converge that closes that gap. Fromthen on soil-lining interaction develops. On subse-quent studies this gap parameter was revised to accountfor the quality of workmanship, face protrusion andother tunneling aspects. Bottom line is that a dis-placement criterion controlled the partial stress releasefactor.

The three-dimensional aspect of a TBM tunnel wasattempted to be simulated by combining two plane-strain models, for the transversal and longitudinalsections of a tunnel (Finno & Clough 1985). Cohe-sive soils were represented by the modified cam claymodel and cohesionless soils by a nonlinear pseudo-elastic law. The tunnel face in the longitudinal modelwas displaced until the horizontal forces were equalthe measured jacking forces of the TBM. The ratiobetween the vertical and the out-of-plane horizontaldisplacement was applied in the transversal modeluntil the horizontal displacements matched the incli-nometer measurements by the side of the tunnel.From this state the first stress release was applieduntil the tunnel convergence closed the gap to thelining. The subsequent steps were soil-lining inter-action and consolidation of the excess pore waterpressure (PWP).

Abu-Farsakh & Voyiadjis (1999) tried to model thesame TBM tunnel, with the same soil models, butrelying less on measured parameters. The longitudinalmodel advanced until the specified face pressure wasachieved. The outward displacement of the transver-sal model had an elliptical profile with a major/minoraxis ratio of five and was applied until the incrementof PWP on the tunnel springline was the same as theone calculated on the longitudinal section. After thecorrect displacement was determined a new tunnel sec-tion was modelled with a smaller diameter so that afterthe displacements are imposed the boundary geome-try matches the excavated diameter. The stress releasefactors were applied along the tunnel boundary by anelliptical profile, this time with a major/minor axisratio of 1.50.Again the final phases were the soil-lininginteraction and consolidation of excess PWP.

Other studies only analyzed the tunnel transversalsection, but accounted for the pressure increase due togrout injection on the tail-void. Bernat et al. (1999)modelled the TBM excavations of the Lyons-Vaisemetro by calibrating the partial stress release factorsof an unlined tunnel with the measured tunnel crowndisplacements. The soil was represented by the CJSmodel that account for kinematic strain-hardening. Asingle partial stress release was compared to a cycle

of stress reduction, increase (λ<0) and reduction toaccount for the grout injection at the tail void.

The tunnel excavation can also be modeled directlyby imposing displacements on the excavation bound-ary. However, there might be a case where the resultantboundary stresses are not representative of an exca-vation and the procedure only mimics the measureddisplacements. Dias et al. (2000) present a 2D modelwith a fixed tunnel invert and a ring plate elementon the excavation boundary. The soil was modeled asa drained Mohr-Coulomb material. As the excavationadvances the plate rigidity changed fromTBM to groutproperties. The passage of the cutter-head was simu-lated by reducing the soil-plate interface strength toinduce sliding and by reducing the diameter of the platering. The TBM passage, grout injection and grout con-solidation were simulated by a sequence of decrease,increase and decrease in the ring diameter.

Ding et al. (2004) analyzed the different TBMphases by combining stress release factors with specialinterface elements and a beam-joint discontinuous lin-ing. A simple model and empirical relations were pre-sented to calibrate the normal and tangential interfacestiffness by the properties of the fresh and consoli-dated grout. Two distributions of grout pressure arealso tested against measurements of an Osaka subwayline. Just recently, Konda et al. (2013) modeled the dif-ferent phases of the TBM by a set of normal forces onthe tunnel boundary together with a full stress release(λ=1). That allowed the internal tunnel pressures tobe determined with no relation to the in-situ stress.The soil was assigned the tij constitutive model on anundrained analysis.

Among these stress-based approaches for plane-strain modeling of TBM tunnels, a progression canbe traced from basic stress release factors to represen-tations of boundary pressure gradients related to thedifferent TBM elements. As mentioned, when usingstress release factors, the boundary stresses of theexcavation are always connected to the gradients ofthe initial stress state. The gradient adjustment canbe done through an asymmetric distribution of λ, asin Abu-Farsakh’s model, by a combination of partialstress release factors and internal pressures as in Ding’sor by a total stress release and internal pressures as inKonda’s model. How to assess the real pressure gradi-ent is still not a straightforward procedure, but it hasbeen analyzed at the excavation face (Bezuijen et al.2006a), along the TBM (Bezuijen 2009) and on thelining (Bezuijen et al. 2006). For the practice of tun-nel design this is an important understanding in orderto create numerical models with consistent physicalsignificance.

Regarding the constitutive model for the soil, thereis also no agreement in geotechnical engineering ingeneral. Compromise is always necessary over theaspects to be modeled and the availability of soil teststo assess the required parameter. Even so, Shirlaw(2000) presents how when tunneling through the samegeological strata, using the same tunneling methodand crew, the settlements can change over 2.5 times.

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Table 1. Constitutive parameters.

LE MC MC.D HS HS.s

E (MPa) 45 45 45 – –ν 0.2 0.2 0.2 0.2 0.2ϕ (◦) – 37 37 37 37ψ (◦) – – 7 7 7E50

ref (MPa) – – – 45 45Eoed

ref (MPa) – – – 45 45Eur

ref (MPa) – – – 135 135m – – – 0.466 0.466G0

ref (MPa) – – – – 111γ0.7 – – – – 1.25 10−4

ForTBM tunnels several aspects cannot be determinedaccurately before the construction. Nevertheless thesoil model is a necessary input, and how their charac-teristics will affect the results of a tunnel analysis issomething that will be analyzed hereafter.

3 METHODOLOGY

A set of drained plane-strain finite element calcu-lations were performed to analyze the patterns ofboundary stress and the effects of the constitutive mod-els on the response on an unlined tunnel modeled byincrements of stress release. The stress path, tunnelconvergence and surface settlements were assessed onthe models.The calculations were performed on Plaxis2D v2012.01 software.

The first analysis considered a circular tunnel (8 min diameter; 30 m deep at springline) and a dry linearelastic soil (Young’s modulus E=50 MPa; Poisson’sratio ν = 0.30; Volumetric weight γ = 20 kN/m3). Theeffects of variations of the coefficient of earth pres-sure at rest (k0) and ν were assessed. A second groupof models considered the same layout and k0 = 0.50.The soil was analyzed with four constitutive models:LE – Linear Elastic; MC – Linear elastic perfectly-plastic with a Mohr-Coulomb failure criterion and anon-associated flow rule; HS – Hardening Soil andHS.s – Hardening Soil with small-strain stiffness. Theparameters were obtained from the empirical correla-tions from Brinkgreve et al. (2010) for sand at a relativedensity of 0.75, but with the same volumetric weightof the first analysis (Tab. 1). For the formulations ofthe models, the reader is referred to Brinkgreve et al.(2013).

There is a recurrent discussion on whether soildilatancy should be considered when employing theMohr-Coulomb model. The formulation implied thaton drained analyzes, shear strains can develop indef-initely without reaching the critical state. To evaluatethis aspect for tunnel modeling the Mohr-Coulombmodel was employed considering dilatancy (MC.D)and with zero dilatancy (MC). A triaxial test was sim-ulated for each constitutive model by the Soil Test

Figure 1. Triaxial test simulations with various constitutivemodels (see also text).

software in the Plaxis Program and the results are onFigure 1.

The hyperbolic stress-strain relation of the Harden-ing Soil models (Hs and Hs.s) can be seen on the uppercurves. All models, with the clear exception of theLE, converge to the Mohr-Coulomb failure criterion,corresponding to a deviatoric stress of 300 kPa. Thedilatant response of the MC.D model can be seen incontrast with the MC. Both the Hardening soil modelstogether with the MC.D present unrestrained dilatancyas no critical state is reached and no dilatancy cut-offwas assigned.

4 RESULTS AND DISCUSSION

The results will be divided by the linear elastic analysisand the comparison of different constitutive models.As discussed, with the stress release factors the bound-ary stresses remain related to the original stress state,and not the constitutive model. That also implies thatwhen that state is not isotropic (k0 �= 1), both nor-mal and shear stresses will be acting on the boundary(Fig. 2).

4.1 Linear elastic analyses

Considering k0 equals 1, 0.5 and 2, the incrementsof isotropic (p) and deviatoric (q) stress invariantson the tunnel crown and springline were evaluated(Fig. 3). For the isotropic state (k0 = 1) the incrementis predominantly of deviatoric stress. For the otherstates the response is distinguishable on whether thenormal stress is the initial major (σ1) or minor (σ3)principal stress. The initial normal stress is the minorprincipal stress on the tunnel crown for k0 = 2 and onthe tunnel springline for k0 = 0.50. In those cases thetunnel excavation, a reduction in the normal stress,represents a decrease in σ3 and an increase in the hoopstress, which is σ1. Therefore a path of increase ofdeviatoric and mean stress is the result. On the oppo-site conditions, the decrease of σ1 and the increase inσ3 result in negative increment of isotropic stress.

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Figure 2. Boundary stresses. The type of line representsdifferent stress release factors and the color represents thedifferent coefficients of earth pressure at rest.

Figure 3. Increments of p and q for different k0 values.

The difference on the initial stress state also affectsthe volume loss measured by the tunnel convergenceand by the settlement trough (Fig. 4). The increasein the horizontal stress increases the tunnel conver-gence without significantly affecting the settlementtrough. It is important to notice that the notion ofequivalency between the tunnel convergence and thesettlement trough can only be applied in undrainedconditions, when the soil is practically incompressible(ν = 0.5). However it is common in practice to assumethis equivalency even for compressible soils.

For a linear elastic analysis, being all the otherparameters fixed, a relation can be traced between theratio of volume losses measured on the surface andby the tunnel convergence and the Poisson’s ratio as itcan be seen on Figure 5. Leca & New (2007) reportedthat the so called deformation dampening can also bea consequence of the presence of a stiffer or dilatingsoil over the tunnel.

4.2 Constitutive models analyses

With the conditions described in the methodol-ogy, each analysis was conducted in stress release

Figure 4. Volume losses for different k0 values. Bars corre-spond to the left Y-axis, the red line with the right.

Figure 5. Volume losses as a function of the Poisson’s ratio.

increments of 10% the original state until the solu-tion did not reach convergence. The maximum λ forthe different models was: MC 0.7; MC.D 0.8; HS 0.9;HS.s 0.8. The linear elastic analysis does not accountfor a failure criterion. For the sake of comparison thesurface settlements are presented for λ = 0.7 (Fig. 6)while the volume losses (Fig. 7) and the stress paths(Fig. 8) are presented from λ = 0.1 to λ = 0.7.

The typical problem with the settlement trough of2D models is evident in the LE results: the troughis shallow and narrow. It can be observed that whenplastic deformations are taken into account the troughbecomes deeper. On the other hand, when hardeningis considered the trough becomes wider. The consid-eration of small strain stiffness reduces the maximumsettlements without affecting the trough extent. TheHS settlement troughs were reported to be signifi-cantly deeper than the MC trough for higher overcon-solidation ratio (OCR) values (Vermeer et al. 2003).The parameters of the typical Gaussian model for thesettlement trough are on Figure 6.

As discussed in the previous sections, drained ana-lyzes do not hold the equivalency of volume lossmeasured on the surface and on the tunnel boundary.There is a direct correlation between stress release and

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Figure 6. Surface settlements for different constitutivemodels.

Figure 7. Tunnel convergence and surface/tunnel volumeloss ratio for different constitutive models. All the lines thatare intercepted by the dotted arrow should be read on the leftaxis.

the tunnel volume loss. For the MC and MC.D mod-els the relation deviates from the linear elastic fromλ = 0.5 on, as the model is actually linear elastic untilplasticity is reached. On the other hand, the HardeningSoil models that present a hyperbolic stress-strain rela-tion deviate from the linear elastic model from earlystages and present always smaller volume losses at thetunnel level.

The ratios between surface and tunnel volume lossesare also on Figure 7. Again there is a clear distinc-tion between the LE, MC and MC.D models and theHardening Soil models (HS and HS.s). As presentedin Figure 5, for the LE model the ratio is constant andsmaller than 1 for all values of λ. The MC modelspresent a ratio increase from λ = 0.5 on, when bothvolume losses increase, but on a higher pace on thesurface. In contrast to that, both HS models presentratios above 2 for all values of λ, as the volume losson the surface was higher than on the tunnel.

From the previous section it is understood that thefinal stress boundary condition results in zero verticalstress on the tunnel crown and zero horizontal stress onthe tunnel springline, inducing deviatoric and positive

Figure 8. Stress paths for vertical x horizontal stress andisotropic x deviatoric stress for different constitutive models.

isotropic stress increment on the springline and a neg-ative isotropic stress increment on the tunnel crown ask0 = 0.5. The stress path on the springline is of typi-cal loading, leading to failure. For the MC and MC.Dmaterials the path is the same until the Mohr-Coulombfailure line is reached. From there on there is a decreasein the vertical stress that reduced the deviatoric stressalong the failure line. For the Hardening Soil modelsthe decrease in the horizontal stress is not followed bythe increase in the vertical stress, therefore there is anegative increment on isotropic stress that leads moredirectly to the failure line.

5 CONCLUSION

From the literature review it was possible to demon-strate how the concept of plane-strain modeling ofTBM tunnel is evolving. The gradient of boundarypressure due to a partial stress release factor holds arelation to the initial stress state and not to the TBMelements acting on that section. However, the basic lin-ear elastic analysis presents how the initial stress stateis important for the stress paths that are imposed inthe soil elements due to the excavation. The effects ofthe soil’s compressibility on the relation between thevolume losses measured on the surface and on the tun-nel boundary were also assessed by the linear elasticanalysis.

When different constitutive models were consid-ered, and a maximum strength was assigned to the soil,the tunnels were not stable over the whole path of stressrelease. The surface settlements and their relation tothe volume losses measured on the tunnel boundarywere significantly different among different models,especially between the models with linear and non-linear stress-strain relations.The recurrently discussedconsideration of the soil dilatancy when applying theMohr-Coulomb model did not results in significantdifferences for the results.

This clear understanding of the modelling con-ditions and implications is very important for the

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practice of tunnel design, so that the analyses canreach reliable predictions with consistent physicalsignificance.

ACKNOWLEDGEMENTS

The first author would like to acknowledge the finan-cial support of “Conselho Nacional de Desenvolvi-mento Científico e Tecnológico – CNPq, Brasil”.

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