In this paper, first, the method of a complex variable taking time dependency and anisotropy of the ground into consideration is newly proposed to estimate surface displacements caused by shallow tunnel driving in soft ground subject to gravity. Secondly, theoretically calculated values are shown to have good agreements with field data.


Die oberflachlichen Verschiebungen verursacht durch den Graben des Tunnels in einem anisotropen viskoelastischen Grund werden durch das Berechnungsverfahren der komplexen Veranderlichen gelöst. Dann werden die Feldmessungen mit den theoretischen Resultaten verglichen.


Cel article analyse les tassements superficiels induits par le creusement d'un tunnel à faible profondeur dans un sol mou soumis à son poids propre par une methode faisant appel au domaine complexe et incluant les effets de fluage et d'anisotropie du sol. Finalement, plusieurs mesures Sur le terrain sont comparees aux predictions theoriques.


The excavation of a shallow tunnel in soft ground brings about surface displacements of the ground. Therefore, it may be required, in advance, to estimate the magnitude of surface displacements and their effects on neighboring structures. As surface displacements are affected not only by such geometrical conditions as the depth of burial for a tunnel and inclination of ground surface but also by several geological properties, little information has been available up to now about the accurate estimate of the magnitude of the displacements. It is of importance to take into account the fact with respect to deformation phenomena that the surface displacements over a tunnel in soft ground generally increase with time and are affected, in no small quantities, by directions of sedimentation, stratification and joints which bring out the anisotropic properties of the ground as a whole. In this paper, problems of surface displacements resulting from shallow tunnel driving in soft ground subject to gravity are analyzed by the method of a complex variable, considering time dependency and anisotropy of the ground. Subsequently, the results of several field measurements with respect to displacements are compared with the theoretical ones.

1. Fundamental relationships for an anisotropicelasticity

The authors refer the ground to Cartesian coordinate system (X1,X2,X3) shown in Fig.l.

2. Determination of stress functions

The tunnel excavation can be expressed by releasing the initial stresses σijnj which exist on the virtual tunnel boundary B1(Fig.1) before the excavation, where nj are the components of the inward unit vector normal to B1. The initial stresses can be considered as functions of parameter θ having period 2π, and they can be expanded in Fourier series in cos (mθ) and sin(mθ). The expressions of the infinite series on the right-side of the above equations are obtained as a result of the integration of the,initial stresses with respect to arc-length along the contour B1. Then the coefficients bjk (m) become known complex quantities. On the other hand, 1bjo must be determined by the condition that the displacements caused by the tunnel excavation must be equal to zero at infinity.

3. Initial stresses of the ground due to gravity

Initial stresses in the anisotropic elastic ground due to gravity can be obtained by considering such conditions:

  1. All of the stresses σij are independent of the X1-axis.

  2. σ22=0, 012=0, 02,=0 on B2.

  3. Strain E11 is caused by σ11 only, which is the normal stress component obtained from equilibrium equations with respect to gravity.

  4. ε13=0 in the ground.

  5. ε33=0 in the ground.

4. Displacements caused by tunnel excavation in anisotropic viscoelastic ground

These equations independent of time correspond to those of anisotropic elastic ground. It has been shown that the way to solve boundary value problems in the theory of viscoelasticity is to apply the Laplace transform, with respect to time, to the time dependent field equations and the boundary conditions. By so doing, the solution to the original problem is reduced to the transform inverse of a solution to the transformed problem.

1. Convergent characteristics of analytical functions

Fig.2 shows convergent characteristics of the analytical functions. It may be recognized that the convergent characteristics of analytical functions are fairly good and that if the first two terms of the infinite series of the functions are used, engineering accuracy will be satisfied. In the following analysis the first four terms of the infinite series are used.

2. Analytical results

The horizontal and vertical displacements (Uh and Uv respectively) of ground surface are summarized in Figs.3 and 4. From Fig.3 we can see that the value of anisotropic parameter k=C22/22/C11/11 considerably affects the displacements. When the value of k becomes smaller than 1.0 (=isotropic), Uh increases greatly; on the other hand, if the value of k becomes greater than 1.0, Uv decreases remarkably. The directions of sedimentation, stratification and joints which bring out the anisotropic properties of the ground can, therefore, be thought to have very important effects on the displacements.

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