Theapplicationof a strong-discontinuity FEM to the analysis of strainlocalization induced by underground openings

C. Callari,Universita di Roma “Tor Vergata”, Dipartimento di Ingegneria CivileNumer. Models in Geomech. - NUMOG VIII, Pande & Pietruszczak (eds),c© Balkema 2002, pp.163-170, ISBN 905809359 X

ABSTRACT: In this work, the excavation of shallow tunnels is simulated by a strong-discontinuity finite ele-ment method. The enhanced FEM shows to be able to capture the onset of localization and its propagation fromthe tunnel up to the ground surface, leading to a solution independent of the considered mesh size and align-ment. These results are contrasted with the response obtained by a standard formulation of the finite elementmethod. In this comparison, the attention is focused on the distribution of ground surface settlements.

1 INTRODUCTIONAs an effect of the excavation of shallow tunnels, thelocalization of strains along narrow bands has beenobserved in a number of laboratory and field studies(Hansmire & Cording 1985, Ribacchi 1993, Mair &Taylor 1997, Sterpi 1999, Leca et al. 2000). The prop-agation of these “shear bands” from the cavity up tothe ground surface can lead to a collapse mechanismof the tunnel roof or of the excavation face. As a con-sequence, an effective numerical method for the simu-lation of localization has important applications in thestability analysis of underground openings and in theprediction of surface settlements induced by shallowtunnels.

It is well known that conventional numerical meth-ods show difficulties in capturing localization phe-nomena. Furthermore, localized solutions obtained bythese standard formulations are strongly dependent onthe adopted discretization. As observed by many au-thors, these deficiencies could explain why standardfinite-element methods often fail in reproducing thedistribution of ground surface displacements inducedby shallow tunnels (Eisenstein 1986, Ribacchi 1993,Gioda & Locatelli 1999, Schuller & Schweiger 1999,Leca et al. 2000). In particular, the calculated surfacesettlement troughs are flatter and wider than the ob-served ones, leading to the unsafe underestimation ofdistortion effects on existing buildings.

Among the different approaches to localization, theso-called “strong-discontinuity” method considers so-lutions involving a discontinuous displacement field.The resulting enhanced finite-element formulation isable to capture localization and avoids the patho-logical mesh-size dependence observed in classicalimplementations of continuum elasto-plastic models

with strain softening (Simo et al. 1993, Armero &Garikipati 1995, 1996, Oliver 1996, Borja & Regueiro2001, Callari & Lupoi 2001). An extension of thestrong-discontinuity approach to the coupled case ofsaturated poro-plastic media is presented in (Armero& Callari 1999, Callari & Armero 2001).

In this paper, the excavation of shallow tunnelsin a dry soil is simulated by means of the uncou-pled strong-discontinuity FEM reported in (Callari &Lupoi 2001). Main scope of the present work is tocontrast the obtained displacement fields with the re-sults of a standard formulation.

In the following, we briefly recall the analysis ofstrong discontinuities in elasto-plastic solids (Sect. 2)and the resulting enhanced finite-element formulation(Sect. 3). The numerical simulations are discussed inSect. 4.

2 STRONG DISCONTINUITIES IN ELASTO-PLASTIC MEDIA

In this Section, we report the main results of the anal-ysis of strong discontinuities in uncoupled associatedmodels.

Figure 1. Kinematics of strong discontinuities in plane strainconditions

Figure 2. Displacement discontinuous interpolation functionΨΓe

2.1 KinematicsIn the solidΩ, we denote byu the “large-scale” dis-placement field, satisfying standard regularity condi-tions. In a local neighborhoodΩx of the pointx ∈ Ω,we consider the following “small-scale” displacementfield (Armero 1999), exhibiting a discontinuity acrossa surfaceΓx with unit normaln in x (Fig. 1):

uµ = u + ζ ΨΓx with ΨΓx = HΓx −NΓx (1)

whereHΓx is the Heaviside step function acrossΓx

andNΓx is a general smooth function. Note that thelocal field ζ corresponds to the displacement jump:[[uµ]] = ζ.

The strain field corresponding to (1) is given in thedistributional sense by:

εµ := ∇suµ = ε︸︷︷︸

regular distribution

+ (ζ⊗ n)s δΓx︸ ︷︷ ︸singulardistribution

(2)

with ε := ε(u)+ΨΓx ∇sζ− (ζ⊗∇NΓx)s andδΓx the

Dirac delta distribution acrossΓx.

2.2 Strong discontinuities in elasto-plastic mediaWe consider the rate-independent associated elasto-plastic model defined by the following relations:

σ = C(εµ − εpµ) (3)

εpµ = γ ∂σf H−1q := −αµ = −γ ∂qf

with C the 4th order elasticity tensor,εpµ the plastic

strain tensor,f(σ, q) the yield function andH theisotropic hardening modulus. In (3), the strain-likeand the stress-like conjugated hardening variables aredenoted byαµ and q, respectively. The usual load-ing/unloading and consistency conditions are addedto these equations.

The appearance of strong discontinuities (1) ischaracterized by the localization conditionQepm = 0(Simo et al. 1993), where:

Qep = n ·[C− 1

∂σf : C∂σfC∂σf ⊗C∂σf

]· n(4)

is theelastic perfectly plastic acoustic tensor andm isthe direction of the displacement jump rate (i.e.ζ =‖ζ‖m with ‖m‖ = 1).

(a) (b)Figure3. Circular tunnel (a) and tunnel crown (b) excavations.Configuration of the problems with boundary conditions and“excavation forces”f := (1− λ)f0

Table 1. Soil parameters considered in the numerical simulations

soil unit weight γ 20 kN/m3

Young modulus E 20000 kPaPoisson coefficient ν 0.35pressure coefficient β 0.56 (ϕ ∼= 24)initial yield stress σy 63.41 kPa (c ∼= 30 kPa)

Themagnitudeof the rate of the displacement jumpis given by the localized relation:

‖ζ‖ =∂σf : σ

H Ξwith Ξ = (∂qf)2C∂σf : (m⊗ n)

∂σf : C∂σf(5)

whereH is the localizedsofteningmodulus. In fact,the softening law (3)2 must be understood in thedistributional sense (H−1q := −α = −γ∂qf δΓx ⇒H−1 = H−1 δΓx).

2.3 Strong discontinuities in a dilatant modelEmploying the results obtained in (Callari & Lupoi2001), we particularize the equations reported in theprevious Section to an associated dilatant model de-fined by the following Drucker-Prager yield function:

f (σ, αµ) = ‖s‖+ β1

3σ : 1+

√2

3q(αµ) (6)

wheres= dev(σ) is the deviatoric part of stress tensorand the pressure coefficientβ is a material parameter.

Denoting byt the unit tangent toΓx, the localizeddilatancy is defined under the assumption of planestrain as the ratio:Φ := ζn/|ζt|, whereζn := ζ · n andζt := ζ · t are the normal and the tangential compo-nents of displacement jump rate, respectively (Fig. 1).For the considered model, the expressions for the lo-calized dilatancy and for the angleθ betweenn andthe major (in-plane) principal stress directione1 are,respectively (Fig. 1):

Φ =

√2 β

2√

1− 2β2/3θ =

1

2arctan

[sign(ζt)

Φ

](7)

Time = 7.83E-01Time = 7.99E-01

-1.87E-01

-1.36E-01

-8.45E-02

-3.33E-02

1.78E-02

6.90E-02

-2.38E-01

1.20E-01

DISPLACEMENT 2

Current ViewMin = -2.38E-01X = 2.64E+00Y = 2.83E+01

Max = 1.20E-01X = 0.00E+00Y = 2.04E+01

Time = 7.99E-01Time = 7.99E-01

(a)

Time = 7.83E-01Time = 7.99E-01

-1.91E-01

-1.39E-01

-8.69E-02

-3.50E-02

1.68E-02

6.86E-02

-2.42E-01

1.20E-01

DISPLACEMENT 2

Current ViewMin = -2.42E-01X = 2.63E+00Y = 2.83E+01

Max = 1.20E-01X = 0.00E+00Y = 2.04E+01

Time = 7.99E-01Time = 7.99E-01

(b)Figure 4. Circular tunnel excavation. Enhanced strong-discontinuity FEM: deformed meshes of 378 elements (a) and672 elements (b) with vertical displacement distributions (λ=0.799)

Denoting byσ1, σ2 the in-plane principal stresses(σ1 ≥ σ2) and bys3 the out-of-plane principal devi-atoric stress, the localization condition (4) is satisfiedfor:

‖s‖r

=

√2

1− β2/6with r =

σ1 − σ2

2(8)

or, equivalently, for s3 + β/3 = 0. The post-localization relation (5) between the stress and thedisplacement jump rate reads in this case:

τΓx sign(ζt) + Φ σΓx =H

3− 2β2|ζt| (9)

whereτΓx := tΓx · t and σΓx := tΓx · n are the tan-gential and the normal components of traction vectortΓx = σn|Γx on the discontinuityΓx.Remark. The continuumdilatancy is commonly de-fined as the ratio:(εp : 1)/‖dev(εp)‖. It is well

(a) (b)Figure 5. Circular tunnel excavation. Enhanced strong-discontinuity FEM: localized elements in the meshes of 378 el-ements (a,λ = 0.847) and 672 elements (b,λ = 0.853)

known that in plane strain conditions, the followingexpression of the Drucker-Prager pressure coefficientβ = [(6 sin2 ϕ)/(3 + sin2 ϕ)]1/2 in terms of the fric-tion angleϕ, leads to the same continuum dilatancyprediction of Mohr-Coulomb model. The substitutionof this expression in equations (7) gives:Φ = tanϕandθ = ±(π/4− ϕ/2), as expected. In particular, asa consequence ofΦ = tanϕ, the Mohr-Coulomb formof the post-localization law (9) is apparent.

(a)

(b)Figure6. Circular tunnel. Strong-discontinuity FEM: a) verticaldisplacements at tunnel axis vs. unloading factorλ; b) verticaldisplacements at ground surface (λ = 0.831)

Time = 8.47E-01Time = 8.63E-01

-2.77E-01

-2.11E-01

-1.45E-01

-7.93E-02

-1.36E-02

5.21E-02

-3.42E-01

1.18E-01

DISPLACEMENT 2

Current ViewMin = -3.42E-01X = 1.79E+00Y = 2.84E+01

Max = 1.18E-01X = 0.00E+00Y = 2.54E+01

Time = 8.63E-01Time = 8.63E-01

(a)

Time = 8.47E-01Time = 8.63E-01

-2.86E-01

-2.19E-01

-1.51E-01

-8.36E-02

-1.60E-02

5.16E-02

-3.54E-01

1.19E-01

DISPLACEMENT 2

Current ViewMin = -3.54E-01X = 1.89E+00Y = 2.84E+01

Max = 1.19E-01X = 0.00E+00Y = 2.54E+01

Time = 8.63E-01Time = 8.63E-01

(b)Figure 7. Tunnel crown excavation. Enhanced strong-discontinuity FEM: deformed meshes of 268 elements (a)and 642 elements (b) with vertical displacement distributions(λ = 0.863)

3 ENHANCED FINITE ELEMENT FORMULA-TION

In this section, we report the main equations of theenhanced finite element formulation developed for di-latant plasticity in (Callari & Lupoi 2001, Callari &Armero 2001) on the basis of the approach presentedin (Armero & Garikipati 1995, Armero 1999).

The multi-scale framework considered in previoussections is implemented in the finite element methodby identifying the local neighborhoodΩx with the ele-mentΩe,loc ⊂ Ω where localization has been detected.In particular,ΨΓx in (1) is approximated by theunreg-ularized discontinuous function:ΨΓe(x) = HΓ(x) −N (i)(x) whereN (i)(x) = 1 − (x(i) − x) · n(i)/h(i) isthe linear shape function associated to node(i) sus-taining the discontinuity lineΓe (Fig. 2). The corre-sponding assumed enhanced strain field is expressedin terms of the element nodal displacementsde and of

(a) (b)

Figure 8. Tunnel crown excavation. Enhanced strong-discontinuity FEM: localized elements in the meshes of 268elements (a,λ = 0.872) and 642 elements (b,λ = 0.874)

the displacement jump componentsze = [ζne ; ζte ]T :

εe = Bede

︸︷︷︸conforming

−Ge ze + Pe ze δΓ

︸ ︷︷ ︸enhanced

(10)

for thestandard strain matrixBe and for the enhancedoperators:Ge = [(n ⊗ n(i))s/h(i) ; (t ⊗ n(i))s/h(i)],Pe = [n⊗ n ; (t ⊗ n)s].

The following finite-element residual equation isobtained from the weak formulation of the continuum

(a)

(b)Figure9. Tunnel crown excavation. Strong-discontinuity FEM:a) vertical displacements at tunnel axis vs. unloading factorλ; b)vertical displacements at ground surface (λ = 0.863)

Time = 7.83E-01Time = 7.99E-01

-1.30E-01

-8.86E-02

-4.70E-02

-5.36E-03

3.63E-02

7.79E-02

-1.72E-01

1.20E-01

DISPLACEMENT 2

Current ViewMin = -1.72E-01X = 0.00E+00Y = 2.95E+01

Max = 1.20E-01X = 0.00E+00Y = 2.04E+01

Time = 7.99E-01Time = 7.99E-01

Figure10.Circular tunnel excavation. Standard FEM: deformedmesh of 672 elements with vertical displacement distributions(λ = 0.799).

equilibrium:

r d = fext −nelem

Ae=1

∫

Ωe

BeT σ dΩ = 0 (11)

for the nodal external force vectorfext. The symbolAdenotes the assembly of all thenelem element contri-butions. The stressesσ are obtained through the re-lation (3)1 in terms of the regular partεe := Bede −Ge ze of the enhanced strain field (10).

The weak formulation of the equilibrium on the dis-continuity leads to the following finite-element resid-ual equation:

renh = − 1

Ae

D∫

Ωe,loc

PTe σ dΩ + sΓ = 0 (12)

for the areaAe of the localized elementΩe,loc andthe projection operatorD = [sign(ζte)Φ ; 1 ]. In (12),the notationsΓ := τΓ + sign(ζte)ΦσΓ is introduced,where sΓ = sΓ(ζte) is obtained by integrating the

Figure11.Circular tunnel: vertical displacements at tunnel axisvs. unloading factorλ. Comparison between standard FE formu-lation and enhanced FEM with strong discontinuities

given localized softening law (9) on the discontinu-ity Γe, that is:

˙sΓ(ζte) =H

3− 2β2ζte (13)

In the generic time step[tn, tn+1], nonlinear equa-tions (11) and (12) are solved through a Newton-Raphson iterative procedure. Return-mapping algo-rithms are employed in the integration of the post-localization stress-displacement relation (13) and ofthe bulk elasto-plastic model.

Introducing the notation∆(·) = (·)(k+1)n+1 − (·)(k)

n+1,with (k) the iteration number, the linearization ofequation (11) can be written:

r (k)d −

nelem

Ae=1

(K e ∆de −LGe ∆ζte) = 0 (14)

for the enhanced operatorLGe = LGeDT and the stan-

dard element stiffness matrixK e, where:

LGe =

∫

Ωe,loc

BeTC Ge dΩ K e =

∫

Ωe

BeTC Be dΩ (15)

Linearization of the local equation (12) leads to:

r(k)enh = LT

Pe∆de −He ∆ζte where: (16)

(a)

(b)Figure 12. Circular tunnel. Vertical displacements (a) and an-gular distortion (b) at ground surface obtained by enhancedFEM with strong discontinuities and by standard FE formula-tion (λ = 0.831)

Time = 8.47E-01Time = 8.63E-01

-1.94E-01

-1.42E-01

-8.95E-02

-3.74E-02

1.46E-02

6.66E-02

-2.46E-01

1.19E-01

DISPLACEMENT 2

Current ViewMin = -2.46E-01X = 2.87E+00Y = 2.81E+01

Max = 1.19E-01X = 0.00E+00Y = 2.54E+01

Time = 8.63E-01Time = 8.63E-01

Figure13. Tunnel crown excavation. Standard FEM: deformedmesh of 642 elements with vertical displacement distributions(λ = 0.863).

LPe =1

Ae

∫

Ωe,loc

BeTC Pe dΩ (17)

He =1

Ae

∫

Ωe,loc

PTe C Ge dΩ +

dsΓ

dζte

(18)

with Pe = PeDT andGe = GeDT . Thesecond term ofthe enhanced tangent operator (18) is obtained from(13). The combination of the linearized equations (14)and (16) leads to the local elimination of the enhancedstrain parameter increments∆ζte through a static con-densation at the element level.

The strong discontinuity is propagated through themesh, without the spatial discretization knowing a pri-ori its location. The adopted procedure employs thelocalization condition (8), the discontinuity direction(7)2 and geometric arguments based on the connectiv-ity graph of the finite element mesh.

4 NUMERICAL SIMULATIONSIn the following, the excavation of shallow under-ground openings in a dry soil is simulated by meansof the strong-discontinuity FEM recalled in the pre-

Figure 14. Tunnel crown excavation: vertical displacements attunnel axis vs. unloading factorλ. Comparison between standardFE formulation and enhanced FEM with strong discontinuities

vious Section. In particular, we consider two differentplane-strain schemes: the full-face boring of a circulartunnel (Fig. 3a) and a tunnel crown excavation (Fig.3b).

According to a common simplified procedure, theexcavation is simulated by reducing the “excavationforces”f := (1− λ)f0 applied to the tunnel boundary,whereλ ∈ [0; 1] is an unloading coefficient andf0 arethe surface actions equilibrating the pre-excavationstate of stress (Panet & Guellec 1974).

We recall that in such astress-controlledproblem,it is impossible to simulate the system response afterthe attainment of the collapse limit value of the un-loading factorλ. We refer to (Callari & Armero 2001,Callari & Lupoi 2001) for an analysis of some repre-sentative strain-controlled problems.

After localization, a linear softening law is consid-ered with the following settings for the localized soft-ening modulus:H=−1 kPa/m (circular tunnel) andH=−100 kPa/m (crown excavation). The other soilparameters are reported in Table 1. The initial coeffi-cient of horizontal stress isk0 = 0.54.

Both the analyses are performed with two differ-ent unstructured discretizations in quadratic triangles.The solutions obtained with the two meshes are prac-tically coincident, as shown by Figures 4, 6 for the

(a)

(b)Figure15. Tunnel crown excavation. Vertical displacements (a)and angular distortion (b) at ground surface obtained by en-hanced FEM with strong discontinuities and by standard FE for-mulation (λ = 0.863)

circular tunnel,and by Figures 7, 9 for the crown ex-cavation.

In spite of the coarseness of the adopted discretiza-tions, the deformed meshes demonstrate the ability ofthe enhanced FEM in capturing the onset of local-ization and its propagation from the tunnel up to theground surface (Figures 4 and 7; displacement am-plification factor= 3). The localized elements for thetwo considered tunneling schemes are pointed out inFigures 5 and 8.

These results are contrasted with the response ob-tained by standard finite elements with continuumstrain-softening and no enhanced fields. The de-formed meshes show that the standard formulationhas difficulties in capturing the onset and the propa-gation of localization (Figures 10 and 13). Such diffi-culties are particularly apparent for the circular tunnelcase, where an unrealistic deformation of the cavityboundary is obtained (Fig. 10).

In standard FEM computations, the continuumsoftening modulaH = −100 kPa (circular tunnel)and H = −50 kPa (crown excavation) are set toobtain approximately the same collapse limit valuesof λ given by the strong-discontinuity method (Fig-ures 11 and 14). With respect to the enhanced FEMwith strong discontinuities, these figures show that thestandard formulation leads to a stiffer response.

Compared to the strong-discontinuity method re-sults, the surface settlement troughs predicted by thestandard FEM are flatter above the tunnel and wideras the distance from the cavity grows (Figures 12aand 15a). As a consequence, the maximum valuesof surface angular distortion given by the strong-discontinuity method are significantly greater than theones obtained with the conventional formulation (Fig-ures 12b and 15b).

5 CONCLUDING REMARKSAccording to the results presented in this paper, thestandard FEM inability to simulate localization canbe one of the main reasons of unsatisfying predic-tions of tunnelling induced displacements. In particu-lar, without considering constitutive modelling issues,more realistic surface displacement distributions areobtained with a strong-discontinuity FEM.

The application of the coupled method presented in(Callari & Armero 2001) to the case of tunneling in asaturated soil is currently under study.

Aknowledgements.The finite element method considered inthis work has been implemented in the general code FEAP, cour-tesy of Professor R.L. Taylor at UC Berkeley. The author isgrateful for the invaluable guidance and continuous support ofprofessors R. Jappelli and F. Maceri. The research was partiallysupported by the “Progetto giovani ricercatori” 2001 of the Uni-versity of Rome “Tor Vergata”.

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