Gravitational stress field around a tunnel in soft ground

B. HOYAUX AND B. LADANYI Department of Mining Engineering, Ecole Polytechnique de MontrPaI, MorztrPal, QuPbec

Received March 17, 1969

The finite element method has been used for determining the stress distribution and the displacements due to gravity around an unlined tunnel driven through a semi-infinite medium, characterized by three idealized material behaviors reflecting approximately a short term behavior of natural undisturbed insensitive and sensitive clays. The knowledge of stress and displacement fields around an unlined tunnel can be used for evaluating the need for supports according to the acceptability of expected deformations.

La mtthode des iliments finis a Btt appliqute B la dttermination de la distribution des contraintes et des dtplacements dus B la gravitt autour d'un tunnel non revgtu situt dans un milieu semi-infini, caracterist par trois comportements idtalists correspondant approximative- ment au comportement B court terme d'argiles sensibles et non sensibles non remanites. La connaissance des champs de contraintes et de diplacements autour d'un tunnel non revhu peut servir B tvaluer le besoin d'un revstement en fonction des diformations prtvues.

Introduction The determination of stress and displace-

ment field around a tunnel immediately after excavation is an important problem both in civil and in mining engineering. The knowledge is necessary in order to be able to predict the amount of overbreak as well as the short and long term stability of the surrounding mass and to determine the supports that may be required.

While some idea of the actual stress field can be obtained if a number of simplifying assumptions are made either of the material behavior or of the initial stress field (Tirno- shenko and Goodier 1951 ; Savin 1961 ; Caquot and KCrisel 1956; Ladanyi 1966; Blake l966), very few attempts have been made up to now to solve the problem rigorously, particularly by taking into account the effect of gravity forces.

One such solution, valid for a linear elastic material behavior and a gravitational medium, has been obtained by Mindlin (1939). How- ever, to the knowledge of the authors, there is no theoretical information available to date (1969) in literature on the stress field that would result from driving a tunnel through a gravitational medium composed of an elastic- plastic material simulating the short term be- havior of a saturated clay.

The solution shown in this paper was ob- tained by the finite element method and was for a horizontal tunnel of circular cross-section driven through a clay medium at a finite depth. It is assumed, moreover, that the clay medium

is originally under at-rest stress condition and that the clay is normally consolidated. The calculation was made for two idealized clay behaviors in undrained shear: a typically in- sensitive clay, represented by a linear elastic - perfectly plastic behavior (Fig. l a ) , and a typically sensitive clay, represented by a linear elastic - perfectly plastic behavior with a step- wise decrease of strength at the peak (Fig. 16) .

Characteristics of the Clay Assumed in the Calculation

The idealized stress-strain relationships of the normally consolidated clay assumed in the calculation are shown in Fig. l a and 6, in which the variation of the principal stress dif- ference (al - Q) is plotted against the maxi- mum engineering shear strain, defined by the principal strain difference (c1 - Q ) . A justifica- tion of the selected types of clay behavior and the amount of loss of strength with shear strain after the peak has been given elsewhere (Lad- anyi 1967; Ladanyi et al. 1968).

The slope of the elastic part of the curve is equal to 2G, where G is the shear modulus of elasticity, equal to E/[2(1 -l- V)], E and V denot- ing Young's modulus and Poisson's ratio of the material, respectively. In the solution, it is assumed that Poisson's ratio of the clay is constant and corresponds approximately to a no-volume-change case (short term behavior) . According to the theory of elasticity, for an incompressible material, Poisson's ratio should

Canadian Geotechnical Journal, 7, 54 (1970)

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HOYAUX AND LADANYI: GR

STRESS STRESS 1

(a) STRAIN €1-63 ( b ) STRAIN 6-63

FIG. 1. Types of stress-strain curves.

DEPTH

FIG. 2. Variation of undrained shear strength with depth.

be equal to 0.50. However, this limiting value cannot be used directly in any analytical or numerical method in which stresses are deter- mined from displacements, but has to be re- placed by a slightly smaller value of, say, 0.48 (Christian 1968).

It is assumed, further, that shear failure of the clay is governed by Tresca's failure crite- rion: *(c1 - as) = S,,, where S,, is the peak value of the undrained shear strength of the undisturbed clay. As expected for a normally consolidated clay, the value of S,, is assumed to vary linearly with depth (Fig. 2 ) . Since in the case of a sensitive clay, the average post-peak strength S' , was taken equal to a half of the undisturbed strength S,,, it implies that it would also vary linearly with the depth (Fig. 2 ) .

It is assumed, moreover, that the clay cannot sustain any tensile stresses.

Method of Calculation

To determine numerically, in each point of an assumed homogeneous, isotropic, and grav- itational medium, the values of stresses and displacements caused by the tunnel excavation, for a plane strain case, the finite element method has been used (Wilson 1963, 1965; Zienkiewicz and Cheung 1967). The method consists in decomposing a continuous medium into a finite number of elements (N) connected with one another at the nodal points ( M ) . The application of the principle of virtual work to

.AVlTATIONAL STRESS FlELD 55

each deformed element gives for the whole medium a system of 2M equilibrium equations:

where { U ) is the column matrix of nodal dis- placements, { F ) the column matrix of nodal forces (in the present case, the gravitational forces, in terms of the unit weight y ) and [ K ] the total stiffness matrix.

Because of the symmetry of the problem with respect to the vertical axis passing through the center of the tunnel, only the right half of the medium will be considered (Fig. 3 ) .

The following boundary conditions are pre- scribed: (a) On AB, CD, and EF, only vertical displacements allowed; (b ) on DE, only hori- zontal displacements allowed; and (c) BC and the rest of the medium are free.

The calculation has been carried out for tunnels located at four different relative depths, defined by the ratio H/R, where H is the depth at which the tunnel center is located and R is the radius of the tunnel.

FIG. 3. Considered portion of the medium with the basic finite element mesh and with imposed boundary conditions.

FIG. 4. Iterative method of pseudo-elastic modulus.

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CANADIAN GEOTECHNICAL JOURNAL. VOL. 7 1970

2.0!"/4H

LINEAR ELASTIC

LINEAR ELASTIC PERFECTLY PLASTIC

FIG. 5. Distribution of stresses along (a) vertical and (b) horizontal sections for two different behaviors of the medium: H/R = 1.5.

LINEAR ELASTIC

LINEAR ELASTIC PERFECTLY PLASTIC ( INSENSITIVE' CLAY)

------ LINEAR ELASTIC PERFECTLY PLASTIC WITH A LOWER RESIDUAL STRENGTH (SENSITIVE CLAY 1

FIG. 6. Distribution of stresses along (a) vertical and (b) horizontal sections for three different behaviors of the medium: H/R = 2.

If in the solution of the system [l], a con- used. The principle of this method is shown in stant value of Young's modulus E for the Fig. 4 and may be expressed analytically as whole medium is assumed, the analysis yields (Girijavallabhan 1967, pp. 65-83; Havner the elastic solution of the problem. In order to 1967) : obtain an elastic-plastic solution, the iterative method of pseudo-elastic modulus has been [2] { F ) = [Kli-I {U),

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HOYAUX AND LADANYI: GRAVITATIONAL STRESS FIELD

LINEAR ELASTIC

---- LINEAR ELASTIC PERFECTLY PLASTIC ( INSENSITIVE CLAY)

------- LINEAR ELASTIC PERFECTLY PLASTIC WITH A LOWER RESIDUAL STRENGTH (SENSITIVE C L A Y )

FIG. 7. Distribution of stresses along (a) vertical and (b) horizontal sections for three different behaviors of the medium: H / R = 5.

where {uIi is the column matrix of nodal dis- placements for the it"teration and [K]i-l is an estimate of the total staness matrix based on the previous solution, i - 1; this estimate de- pends only on the estimation of the stress- strain relationship for each element. In a plastic element, the pseudo-elastic moduli for the it" iteration are defined as:

The number of iterations depends on the speed of the convergence to the stable solution, which is considered to be reached when:

in all the elements of the plastic zone. Gen- erally, in the case of an insensitive clay, the

number of necessary iterations is about 10, while in the case of a sensitive clay, it is about 15 to 20.

Results The particular values assumed in the calcula-

tion were: H / R = 1.5, 2.0, 5.0, and 18.0; the finite element grid (Fig. 3) is taken as: for H / R = 1.5, 2.0, and 5.0, C D / R = 11.0 and D E / R = 11.0; for H / R = 18.0, C D / R = 6.0 and D E / R = 11.0; 7 = 2.0 t/m3; s,,/yH = 0.66; sl,,/yH = 0.33; E = 2000 t/m2; V = 0.48; N = 161 a n d M = 189.

The results are shown in Figs. 5 to 11. Figures 5 to 8 represent the variation on hori- zontal and vertical sections of the gravity stresses, cr, (radial stress), and a, (tangential stress) divided by YH to become dimensionless, for the four considered depths and for the three types of behavior (linear elastic, linear elastic - perfectly plastic and linear elastic - perfectly plastic with a step-wise decrease of strength

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CANADIAN GEOTECHNICAL JOURNAL. VOL. 7. 1970

LINEAR ELASTIC ---- LINEAR ELASTIC PERFECTLY PLASTIC ( I N S E N S I T I V E C L A Y )

W----- LINEAR ELASTIC PERFECTLY PLASTIC WITH A LOWER RESIDUAL STRENGTH ( S E N S I T I V E C L A Y )

FIG. 8. Distribution of stresses along (a) vertical and (b) horizontal sections for three different behaviors of the medium: H/R = 18.

at peak). Figure 9 shows the extent of the esting to note that the conditions assumed on plastic zones, Fig. 10 the variation of the the lateral surfaces, which correspond to the tangential stresses u J y H around the tunnel, at-rest condition, lead to a non zero-value of and Fig. 11 the deformation of the originally the horizontal stress ut/yH along the vertical cylindrical contour. section (cases H / R = 1.5, 2.0, and 5.0) at the

In the case of an elastic behavior, it is inter- free surface.

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HOYAUX AND LADANYI: GRAVITATIONAL STRESS FIELD

( a ) H/R = 1.7 (b) H/R = 2.0 ( C ) H / R = ~ . o ( d ) H/R=~S.O

INSENSITIVE CLAY

SENSITIVE CLAY

FIG. 9. Extent of plastic zones for two different plastic behaviors of the medium.

(a ) H/R= (b) H/R = 2.0 ( C ) H/R=5.0 ( d ) H / R = I ~ . O

ELASTIC 1.0

--p- ELASTIC-PLASTIC ( INSENSITIVE CLAY)

--- STRESS SCALE ELASTIC-PLASTIC ( S E N S I T I V E CLAY)

FIG. 10. Variation of tangential stress oJyH in the tunnel wall.

In the case of an elastic - perfectly plastic behavior of the medium, the extent and the shape of plastic zones do not vary much with the value of H/R, and the tangential stress around the tunnel corresponds practically to the hydrostatic weightless state.

In the case of an elastic - perfectly plastic behavior with a drop of strength at the peak, it is found that the extent of plastic zones is considerably increased with respect to the former.

It is interesting to note that for both elastic plastic cases there exists a limiting depth at which the solution still converges. The depth is a function of assumed stress strain and

strength properties of the clay and corresponds approximately to the depth at which the plastic zone attains the free surface above the tunnel. Trial calculations have given the following critical depths for the two considered cases:

(H/R) ,,it = 1.7 (elastic - perfectly plastic)

(H/R) ,,it = 2.0 (elastic - perfectly plastic with lower residual strength).

For H/R ratio lower than the critical value, the h i t e element method does not converge and cannot be considered to give a solution to

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60 CANADIAN GEOTECHNICAL JOURNAL. VOL. 7, 1970

DEFORMED L

( a ) H/R = 1.7 ( b l H/R = 2.0 ( C ) H/R= 5.0 ( d ) H/R= 18.0

- ELASTIC ---- ELASTIC -PLASTIC (INSENSITIVE CLAY ) - - - - - - ELASTIC-PLASTIC ( SENSITIVE CLAY

FIG. 11. Original and deformed contours of the tunnel.

the problem. An idea about the clay behavior in that case can be obtained by the usual methods of the theory of plasticity (Mandel 1959).

Discussion The solutions obtained give numerical

values for the stress and displacement fields and the extent of plastic zones when a tunnel is excavated in a gravitational medium, taking into account three different types of the be- havior of the medium. The analysis does not show, however, how an eventual rupture of the medium overlying the tunnel would occur. This information should obviously be obtained from kinematic considerations of sliding along rup- ture lines. As, in the case of a deep tunnel, the line rupture is much less likely to occur than for a shallow one, the calculated stress field for the former corresponds to a real one around a stable tunnel. On the contrary, in the case of a shallow tunnel, especially for the plastic be- havior with a drop in post-peak strength, the calculated stress field corresponds only to a pre-rupture state, the rupture being defined by kinematic considerations, as mentioned above.

Conclusions The finite element method has been used for

determining the stress distribution and the dis- placements due to gravity around an unlined tunnel driven through a semi-infinite medium, characterized by three idealized material be- haviors reflecting approximately a short-term behavior of natural undisturbed insensitive and sensitive clays. The solution obtained gives complete information on the stress and dis- placement fields around an unlined tunnel.

In the case of a deep tunnel ( (H/R) 3 (H/R)c,it), the solution is stable and furnishes the real stress and displacement fields. In the case of a shallow tunnel ((H/R) < (H/R),,,,), the solution is divergent and leads to a con- tinuous plastic flow of the mass located above the tunnel. It is concluded that the finite ele- ment method can give a correct solution up to a critical depth of the tunnel. For tunnels at a depth shallower than critical, an answer can be obtained by the usual method of the theory of plasticity.

The knowledge of the stress and displace- ment fields around an unlined tunnel can be used for evaluating the need for supports ac- cording to the acceptability of expected defor- mations. There is no dficulty, however, in applying the same method to the case of a lined tunnel, as shown by Felippa (1966, p. 187).

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HOYAUX AND LADANYI: GRAVITATIONAL STRESS FIELD 61

Acknowledgments This work was supported by National Re-

search Council of Canada grant No. A-1801. the University of Montreal Computer Center provided computer time for this st6dy.

BLAKE, W. 1966. Application of the finite element method of analysis in solving boundary value problems in rock mechanics. Intern. J. Rock Mech. Min. Sci., 3, pp. 169-180.

CAQUOT, A. and K~RISEL, J. 1956. Trait6 de mica- nique des soh. Gauthier-Villars, Paris, pp. 475- 482.

CHRISTIAN, J. T. 1968. Undrained stress distribution by numerical methods. Proc. Amer. Soc. Civil Engrs., 94, No. SM6, pp. 1333-1346.

FELIPPA, C. A. 1966. Refined finite element analysis of linear and non linear two-dimensional struc- tures. Unpublished Ph.D. thesis, Univ. California, Berkeley, Calif.

GIRIJAVALLABHAN, C. V. 1967. Application of the finite element method to problems in soil and rock mechanics. Unpublished Ph.D. thesis, Univ. Texas, Austin, Texas.

HAVNER, K. S. 1967. An evaluation of iterative solu- tions in thermo-plasticity. Engineering Mechanics Division. Specialty Conference, Raleigh, North Carolina, pp. 91-94.

LADANYI, B. 1966. Short-term behaviour of clay around a circular tunnel. Internal report S.8, Laval Univ. Quibec, P.Q.

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MANDEL, J. 1959. Les calculs en matikre de pressions des terrains. La Revue de 1'Industrie MinBrale, 41, pp. 78-92.

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WILSON, E. L. 1963. Finite element analysis of two- dimensional structures. Unpublished Ph.D. thesis, Univ. California, Berkeley, Calif.

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