NATM-tunnelling in softening stiff clays and weak rocks

S.C. Möller & P. A. Vermeer Institute of Geotechnical Engineering, University of Stuttgart, Germany

T. Marcher ILF Consulting Engineers ZT GmbH, Innsbruck, Austria

ABSTRACT: This numerical study concerns the mechanical behaviour of tunnels in non-structured stiff clays and weak rocks. A constitutive model will be presented, which accounts for isotropic hardening as well as for softening. Attention will be focused on cohesion softening rather than friction softening. The hardening-softening model is formulated by adopting a non-local continuum, as softening behaviour yields ill-posed problems when formulated within classical continuum mechanics. For computing ground response curves, numerical analyses are carried out for a very shallow and a medium deep NATM tunnel. It is shown that such situations differ from very deep tunnels in the sense that material softening can produce progressive failure, as demonstrated by a trough-like ground response curve.

1 INTRODUCTION

Stiff clays and weak rocks tend to show a peak strength and a much lower residual strength. The transition from the peak to the residual strength is referred to as softening and it usual occurs in com-bination with the localization of deformations in shear bands, i.e. thin zones of intensively shearing material. Softening shear bands imply a reduction of shear stresses both inside and outside the band; oth-erwise there would be no equilibrium. In adjacent regions outside the band the shear stress reduction causes a quasi-elastic unloading so that one observes more or less rigid block movements. The softening related effect of progressive failure is well-known for clay slopes, e.g. by the papers of Skempton (1964), Chen et al. (1992) and Potts et al. (1997). In tunnelling, softening may result in a con-cave ground-response curve as considered e.g. by Pacher (1964) and more recently by Vavrovsky (1994). In stiff clays softening occurs both for the friction angle and the cohesion, but friction softening is not as dangerous as cohesion softening. This relates to the fact that bonds between particles conferring ef-fective cohesion are destroyed after small deforma-tions, finally resulting in zero cohesion. In contrast, friction angles tend to drop much more slowly down to a residual value often well above zero. A gradual loss of stability due to ductile friction softening is thus less severe and more readily observed within

the framework of a monitoring programme, whereas more brittle cohesion softening may lead to a more sudden loss of stability. It is thus logic to concen-trate first of all on cohesion softening, at least when considering effects in tunnelling. The constitutive model will be formulated within the framework of rate-independent plasticity with iso-tropic hardening and softening. A particular idea in which friction hardening is combined with softening in the cohesion is elaborate. This idea seems promis-ing, as it yields brittle behaviour at low stress levels and increasing ductility with higher stress levels. Moreover in this paper attention will be focussed on isotropic ground without a macro structure due to stratification, schistarity or jointing. The hardening-softening model is formulated by adopting a non-local (De Borst et al., 1993) contin-uum, as softening behaviour yields ill-posed prob-lems when formulated within classical continuum mechanics. A relatively simple method is obtained by using a single non-local parameter, namely the softening parameter. It will be shown that this leads to mesh-independent finite-element analyses. Results of numerical analyses will be shown for NATM-tunnelling. Here the so-called ground re-sponse curve, also referred to as Fenner-Pacher curve, is computed for two different tunnels. The hypothesis by Pacher (1964) concerning a trough-shaped ground response curve in softening ground is to be investigated.

For triaxial conditions with σ1=σ2, Equation 4 re-duces to Equation 3b, but surfaces of constant mobi-lized φm-values are smooth as indicated in Figure 1. According to Equation 2, φm increases as a function of γp and approaches the failure value φa. The as-ymptotic surface for φm= φa is a failure surface as proposed by Matsuoka & Nakai (1982).

2 CONSTITUTIVE MODEL WITH COHESION SOFTENING

2.1 Plastic shear strain in primary loading One of the ideas for the formulation of the model is the hyperbolic relationship between the shear strain γ and the deviatoric stress q=σ1′-σ3′ in a standard drained triaxial test. For primary loading, we adopt the hyperbolic equation 2.2 Yield function for friction hardening

)/(5031 qqq apppp −⋅=−≡ γεεγ (1)

A proper elastoplastic model is obtained by convert-ing Equation 4 into the yield function

where γ50p is the plastic shear strain for q=qa/2 and

qa is the asymptotical failure stress. The above equa-tion may also be written as m

ma

aa

IIIf

ϕϕ

2

2

3

21

sin1sin9

−−

−⋅

= (7)

)sin/(sinsin mamp ϕϕϕαγ −⋅= (2)

where φa is the asymptotic friction angle and

)sin1(50 ap ϕγα −⋅= (3a) p

For plastic yielding with f=0, this equation reduces again to Equation 4. Stress-hardening is included as φm relates to γp. Indeed, Equation 2 may be inverted to obtain

)/(sinsin pam γαγϕϕ +⋅= (8a)

)2/(sin 31 aqm +′+′= σσϕ , aca ϕcot⋅′= . (3b) with γp=∫γ& p dt and

213

232

221 )()()(

21 ppppppp εεεεεεγ &&&&&&& −+−+−⋅= (8b) On using the mobilized friction angle φm instead of

the ratio q/qa as done in Equation 1, one obtains a shear strain formula that can be extended to general three dimensional states of stress and strain. Indeed, the formula implies Mohr-Coulomb type shear strain contours in principal stress space. As test data sug-gest smooth cones rather than irregular hexagonal cones, it is appropriate to define

Equations 7 and 8 define an isotropic yield function with the plastic shear strain as a hardening parame-ter. Important input parameters are used in the ex-pression

aaa

aaa

m IIIIII

213

2132 9sin⋅−⋅−⋅

≡ϕ (4) 50

mp

ap

a apap

o

−

++′

⋅⋅−=⋅−=1

050)sin1()sin1( γϕγϕα (9)

where p0 is a fixed reference stress of 100 kPa, p′ is the effective mean stress and γ50

p as well as m and φa are material constants. For m=1 successive yield loci in p-q-plane are straight lines, but a curvature is ob-tained for m<1. In addition there is a curvature due to the definition of the cohesion c´.

where aaaaI 3211 σσσ ++=aaaaI 2212 σσσ −⋅−=

, , , (5)

aaaaI 3213 σσσ ⋅⋅=aaa133 σσ ⋅−σ⋅

and

aiai +′= σσ , ϕ′⋅′= cotca (6) 2.3 Extension to cohesion softening

The hardening function can easily be extended to in-clude both friction softening (Marcher & Vermeer, 2001) and cohesion softening. To this end both φa and c′ have to be related to a distortion measure γ*. In the present paper, however, φa is taken constant and only c′ is linked to γ* by the relation

*γ⋅−=′ cpeak hcc (10)

where hc is a softening modulus. From a view point of consistency it would be logical to take the (local) plastic shear strain γp, but this (local) softening does not allow for stable numerical procedures and mesh-independent results. Hence, the model has to be regularised by introducing either a polar continuum, second-order gradients of strain or non-local strains.

Figure 1. Generalized definition of mobilized friction using stress invariants.

In coarse-grained materials, particle rotations may occur and one might adopt a polar continuum. In this paper, however, the focus is on fine-grained materi-als and we adopt the non-local plastic shear strain

dt∫= ** γγ & , dVrwV

p

w

γγ && ⋅⋅= ∫ )(1* (11)

with 2

exp1)(

−⋅

⋅=

lr

lrw

π, V (12) ∫= dVww

Figure 2. Typical curve of a drained triaxial compression test.

denoted as νur. The cohesion dependent stress a and the input parameter m have already been used to de-fine the yield function f. According to Equation 9 the yield function also involves a plastic shear strain as an input parameter. As engineers have more affinity with stiffness moduli than with magnitudes of strain, we consider stiffness moduli as more suitable input parameters. For this reason, we consider the secant modulus E50, as indicated in Figure 2. This modulus relates to the elastic and the plastic strain in a stan-dard drained triaxial test. From the yield function for the plastic strains and the formulation of elastic strains, it can be deduced that

where r is the distance to the material point consid-ered and l is an internal length. Marcher (2003) found a strong coupling between the internal length l and the thickness ts of shear bands in finite element analysis, namely such that ts≈2l. Hence the thickness of numerical shear bands is fixed by the choice of the internal length.

2.4 Completion of the model In this section the model will be completed by add-ing a plastic potential function g. Adopting a non-associated model the plastic potential differs from the yield function, but it is formulated in analogy to the yield function of Equation 7 by defining

m

50 apapE= E

++

0

050

´ (16)

m

mb

bb

IIIg

ψψ

2

2

3

21

sin1sin9

−−

−⋅

= (13) where E500 is obtained for p′=p0=100kPa. The se-

cant modulus E50 is closely related to the shear strain γ50 at least for incompressible soils. Indeed, consid-ering a triaxial test with ε2=ε3, it yields where ψm is a mobilized angle of dilatancy and the

invariants Ib are obtained by using σb=σ′+b. The added stress b is computed such that we have g=0. Hence b depends both on the state of stress and the mobilized dilatancy angle. The latter is computed from Rowe’s so-called stress-dilatancy equation

3131 εεεεγ ∆−∆=−= (17)

cvm

cvmm ϕϕ

ϕϕψsinsin1

sinsinsin⋅−

−= (14)

where ∆ε1 and ∆ε3 are measured in the shearing phase of loading after isotropic compression. On in-troducing the equation ∆εvol=∆ε1+2∆ε3, it follows that

50

5050150150 5.15.13

21

Eq

vol ⋅=∆⋅≈∆−∆⋅= εεεγ (18) The dilatancy angle is thus positive as soon as ψm exceeds a constant-volume angle φcv. Considering dense materials contraction is excluded by taking ψm=0 for φm<φcv. Together with the yield function the plastic potential determines the rates of plastic strain on the basis of the corresponding flow rule and the consistency condition.

Especially in case of hard soils and soft rocks volu-metric strains tend to be small and can be disre-garded in the above expression. The subscript 50 im-plies that the deviatoric stress is chosen such that q50=qa/2. The plastic shear strain at q=qa/2 is ob-tained by subtracting elastic strains, i.e. Elastic strain rates are formulated using Hooke’s

law in hypoelastic form. The stress dependent Young’s modulus is chosen to be

m

urur apapE= E

++

0

0 ´ (15)

5050

50505015.1 q

EE ur

urep

+−=−=

νγγγ (19)

The plastic shear strain constant γ50p0, as used in the

yield function f, is obtained for a stress state with p′=p0.

where Eur0 is the tangent modulus for a reference

mean stress of p′=p0=100kPa. The indices ur refer to unloading-reloading. Similarly Poisson’s ratio is

For the analyses presented in this paper, tensile stresses are excluded by using a tension cut-off.

3 ON THE GROUND RESPONSE CURVE

In NATM-tunnelling the ground response curve , as indicated in Figure 3, is used to illustrate the ground pressure on the lining as a function of deformations. A steep ground response curve with a low minimum indicates a stiff and strong ground which needs little support of a lining. Such a ground is able to carry the overburden load by arching around the tunnel. Vice versa, a relatively flat ground response curve with a high minimum corresponds to a relatively soft ground that needs significant support from a lining.

Figure 3. Geometry and typical pressure-displacement curve (ground response curve).

elements inside the tunnel. This does not disturb the equilibrium as equivalent pressures are applied on the inside of the entire tunnel. The minimum amount of pressure needed to support the tunnel is then de-termined by a stepwise reduction of the supporting pressure.

Attention will be focussed on relatively shallow tunnels and the constitutive model will be used in fi-nite element analyses to compute ground response curves. Most recently this was done for deep tunnels by Bliem & Fellin (2001) to find non-concave curves. In contrast, we will consider shallow tunnels to find trough-like ground response curves.

4 VERY SHALLOW UNLINED TUNNEL All subsequent analyses of the ground response curve have been obtained by using an earlier version of the hardening-softening model. In this early ver-sion we used the Mohr-Coulomb yield function f=(σ1′-σ3′)/(σ1′+σ3′+2a)-sinφm. For f=0 this yields a hexagonal yield surface in principal stress space, rather than the smooth one after Matsuoka & Nakai (1982).

In this section a tunnel with a cover of H = 8 m is considered. Firstly a response curve was computed for a non-cohesive ground to obtain the dashed up-per curve in Figure 4 with a failure pressure of pf = 0.4γD. Secondly a non-softening cohesive ground with c′ = 40 kPa was considered to obtain the lower curve in Figure 4 with a slightly negative failure pressure indicating a stable situation. Analy-ses involving cohesion softening should obviously render ground response curves in between the upper curve for c′ = 0 kPa and the lower curve for c′ = 40 kPa.

We consider an unlined circular tunnel with a di-ameter of 8 m and concentrate on a plane strain situation. As symmetrical tunnels are considered, calculations are based on only half a circular tunnel. The ground is represented by 6-noded triangular elements. The boundary conditions of the finite ele-ment mesh are as follows: The ground surface is free to displace, the side surfaces have roller boundaries and the base is fixed. Initial stresses are assumed to be geostatic according to σ´h = K0 σ´v, where σ´h is the horizontal effective stress and σ´v is the vertical one. The coefficient of lateral earth pressure at rest is taken as K0 = 1.

In order to model softening in narrow shear bands sufficiently accurate, we used a very fine mesh around the tunnel, as indicated in Figure 5. In fact, such a fine mesh is needed when applying a non-local model in combination with a small internal length (see Section 2). First of all a softening analy-sis was carried out for a stiff clay with a softening modulus hc = 600 kPa. This yields the ground

The considered ground has a unit weight of γ = 20 kN/m³, a friction angle of ϕ′a = 30° and an initial effective cohesion of cpeak = 40 kPa and dila-tancy angel ψ=0. The stiffness moduli are taken as Eur

0=90 MPa and E500=30 MPa. The Poisson’s ratio

is νur=0.2 and m=1.

The softening modulus hc can be obtained on the basis of high-quality triaxial tests with a relatively homogeneous post-peak sample deformations. At Stuttgart University such tests were carried out on a particular stiff clay, named Beaucaire Marl (Marcher, 2002), to find hc = 600 kPa. As different clays will give different values of hc, we used 600 kPa as a reference value in the context of a sen-sitivity analysis, by varying hc=300; 600; 1200 kPa. Figure 4. Computed ground response curves for a shallow tun-

nel with H/D = 1. The first stage of the calculations is to remove the

response curve in the middle of Figure 4 with a marked peak in point A. Well before peak this curve deviates already from the non-softening lower bound and it meets the non-cohesive upper bound finally in point B. The peak point A yields a peak pressure of pf = 0.23γD, being about half way in between the failure pressures for non-softening materials with c′ = 0 kPa and c′ = 40 kPa respectively.

5 DEEPER UNLINED TUNNEL IN SOFTENING GROUND

In this section a deeper tunnel with a cover of H = 32 m is considered and all other parameters conform to the shallow tunnel of the previous sec-tion. The initial supporting pressure is given by p0 = γ (H + 0.5D) and this pressure is stepwise re-duced to failure. As in the previous section, upper and lower bounds to the ground response curve are obtained for non-softening material with c′ = 0 and c′ = 40 kPa respectively. Instead of showing the full curves starting at p = 4.5γD, Figure 6 focuses on the lower part from p = 2γD down to failure. As in the previous section the upper curve reaches a failure pressure of pf = 0.4γD and the lower curve reaches a slightly negative failure pressure.

As different clays will have different softening moduli, we have varied the softening modulus around the above value of hc = 600 kPa. Resulting ground response curves for hc = 300 kPa and hc = 1200 kPa are indicated by the curves next to the middle one in Figure 4. A very slight decrease of the minimum pressure is observed for hc = 300 kPa and a noticeable increase for hc = 1200 kPa.

Figure 5 shows a close up around the tunnel with softening zones at and beyond peak, i.e. for point A, B and C of the middle curve in Figure 4. The lightest zones indicate regions where cohesion has softened down to about 10 kPa. States B and C in Figure 5 show post-peak softening zones with a shear band starting at the tunnel side and gradually growing to-wards the surface. In the middle of these shear bands the soil has fully softened. Similar results have been obtained by Schuller & Schweiger (2002) using a multilaminate model that includes softening behav-iour.

The computed ground response curve for the stiff clay with a softening modulus of hc = 600 kPa is found to be well in between the bound solutions. However, there is a distinct difference to the re-sponse curve of a shallow tunnel. Instead of follow-ing the lower bound for c′ = 40 kPa, the deep-tunnel response curve tends to remain closer to the upper bound for c′ = 0 kPa. Accordingly the computed peak of pf = 0.33γD at point A is only slightly below the upper bound of pf = 0.4γD. In fact, there is a dif-ference of only ∆ pf = 0.07γD with non-cohesive material. At the same time the deeper tunnel is sub-ject to much larger deformations than the shallow tunnel, as can be observed by comparing Figures 4 and 6. No doubt, the relatively large deformations in deep tunnelling induce a relatively large amount of cohesion softening.

Figure 7 shows the development of the softening zone around the tunnel. In the following we concen-trate on the fully softened zone. At failure (state A) one observes already a thin fully softened zone and post peak this zone increases rapidly. For state C, one observes the initiation of a shear band towards the surface. For state D this shear band has extended to the surface, but the fully softened part of the band has not yet reached the surface.

Figure 5. Development of softening zones for materials with hc = 600 kPa. Red indicates fully softened material with c´ = 0. First for failure state, then for intermediate state and finally for residual state.

Figure 6. Computed ground response curves for a deeper tun-nel with H/D = 4.

Figure 7. Development of softening zones for a deep tunnel.

6 CONCLUSIONS

On tunnelling: Attention has been focussed on tun-nels in softening ground. To study consequences of cohesion degradation, ground response curves have been computed both for a very shallow tunnel and a deeper one. The computed ground response curves appear to depend significantly on tunnel depth. For the very shallow tunnel, a trough-like Fenner-Pacher curve is computed with a marked minimum as fail-ure pressure. The deeper the tunnel, however, the smaller the softening behaviour on the structural level of the tunnel. The present study suggests that ground response curves for very deep tunnels will show no softening at all. This is conform to recent numerical studies by Bliem & Fellin (2001). More-over it confirms practical experiences by Vavrovsky (1994).

In practical tunnelling one has to consider the sta-bility of the tunnel heading. In closed-face shield tunnelling the heading is supported by a support pressure, but in (open-face) NATM-tunnelling there is no support pressure and the stability of the head-ing depends heavily on the cohesion as shown by Vermeer et al. (2003). Considering present computa-tional results for the ground-response curve, it would seem interesting to consider the three-dimensional tunnel-heading stability also on the basis of cohesion softening.

For shallow tunnels, it may also be of importance to use the present model in 3D settlement analyses as described by Vermeer et al. (2003).

On modelling: Stiff clays and clay stones do not only show cohesion softening but also friction sof-tening. The latter is especially important when the clay fraction dominates the silt fraction (Lupini et al., 1981) and it would seem that both phenomena can be well-described by means of the non-local sof-tening parameter γ*. The related softening moduli hc may for instance be measured in a direct-shear test or a ring-shear test. Unfortunately these softening moduli relate directly to the internal length l, which governs the thickness ts≈2l of shear bands. In reality clays have vanishing thin shear bands and a real thickness simulation requires vanishing small finite elements. In order to limit the computational effort, one has to simulate relatively thick shear bands by means of an artificially large internal length and an artificially high softening modulus. On increasing this modulus one approaches an abrupt type of softening which renders numerical proce-dures unstable again. As a consequence there is an upper bound for the shear band thickness and thus for the size of the elements. Therefore realistic simulations of yielding in stiff clays will require extremely fine meshes. No doubt

this finding applies to all regularisation methods, i.e. non-local, second-order gradient and the polar con-tinuum. For clay problems, it would seam appropri-ate to allow for strong (displacement) discontinuities rather than for finite element meshes with weak dis-continuities as considered in this paper.

7 ACKNOWLEGMENT

The authors are indebted to Dr. Paul Bonnier of the Plaxis software company for his work on the imple-mentation of the constitutive model.

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