A closed-form elastic solution for stresses and displacements around tunnels

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Abstract

A closed-form plane strain solution is presented for stresses and displacements around tunnels based on the complex potential functions and conformal mapping representation. The tunnel is assumed to be driven in a homogeneous, isotropic, linear elastic and pre-stressed geomaterial. Further, the tunnel is considered to be deep enough such that the stress distribution before the excavation is homogeneous. Needless to say that tunnels of semi-circular or “D” cross-section, double-arch cross-section, or tunnels with arched roof and parabolic floor, have a great number of applications in soil/rock underground engineering practice. For the specific type of semi-circular tunnel the distribution of stresses and displacements around the tunnel periphery predicted by the analytical model are compared with those of the FLAC2D numerical model, as well as, with Kirsch's “circular” solution. Finally, a methodology is proposed for the estimation of conformal mapping coefficients for a given cross-sectional shape of the tunnel.

Introduction

Underground openings in soils and rocks are excavated for a variety of purposes and in a wide range of sizes, ranging from boreholes through tunnels, drifts, cross-cuts and shafts to large excavations such as caverns, etc. A feature common to all these openings is that the release of pre-existing stress upon excavation of the opening will cause the soil or rock to deform elastically at the very least. However, if the stresses around the opening are not high enough then the rock will not deform in an inelastic manner. This is possible for shallow openings in relative competent geomaterials where high tectonic stresses are absent. An understanding of the manner in which the soil or rock around a tunnel deforms elastically due to changes in stress is quite important for underground engineering problems. In fact, the accurate prediction of the in situ stress field and deformability moduli through back-analysis of tunnel convergence measurements and of the ‘Ground Reaction Curve’ is essential to the proper design of support elements for tunnels [1], [2].

The availability of many accurate and easy to use finite element, finite difference, or boundary element computer codes makes easy the stress-deformation analysis of underground excavations. However, Carranza-Torres and Fairhurst note explicitly in their paper [3]: “…Although the complex geometries of many geotechnical design problems dictate the use of numerical modeling to provide more realistic results than those from classical analytical solutions, the insight into the general nature of the solution (influence of the variables involved etc.) that can be gained from the classical solution is an important attribute that should not be overlooked. Some degree of simplification is always needed in formulating a design analysis and it is essential that the design engineer be able to assess the general correctness of a numerical analysis wherever possible. The closed-form results provide a valuable means of making this assessment…”.

One of the simplifying assumptions always made by various investigators during studying—usually in a preliminary design stage—analytically stresses and displacements around a tunnel is that it has a circular cross-section [1], [2], [4]. This is due to the fact that the celebrated Kirsch's [5] analytical solution of the circular-cylindrical opening in linear elastic medium is available in the literature and it is rather simple for calculations [6]. On the other hand, Gercêk [7], [8] was the first investigator who presented a closed-form solution for the stresses around tunnels with arched roofs and with either flat or parabolic floor having an axis of symmetry and excavated in elastic media subjected to an arbitrarily oriented in situ far-field biaxial stress state. Gercêk has used the method of conformal mapping and Kolosov–Muskhelishvili complex potentials [9] along with the “modified method of undetermined coefficients” of Chernykh [10]. However, Gercêk did not consider (a) the incremental release of stresses due to excavation of the tunnel, (b) the solution for the displacements,1 (c) the influence of support pressure on tunnel walls on the stresses and displacements, and (d) the methodology to derive the constant coefficients of the series representation of the conformal mapping for prescribed tunnel cross-sections.

In order to add the above essential elements for appropriate tunnel and support design, we present here the closed-form solution for the elastic stresses and displacements around tunnels with rounded corners in pre-stressed soil/rock masses. This solution is derived by virtue of Muskhelishvili's [9] complex potential representation, the conformal mapping technique and the properties of Cauchy integrals.2 The proposed closed-form solution for the stresses and displacements that is presented here is original, although Gercêk following a different methodology has derived the solution for a different boundary value problem, that is appropriate only for stress and not for deformation analysis of underground excavations. The results of the analytical solution pertaining to the stresses and displacements around the tunnel with “D” cross-sectional shape are compared with the predictions of the FLAC2D numerical code [11], [13] for two far-field stress states. It is shown that the numerical model predictions compare very well with the analytical solution apart from the corner and invert regions. Further, in Appendix A, we present a simple methodology for the derivation of the coefficients of the series representation of the complex conformal mapping function that corresponds to a given tunnel cross-section shape.

Section snippets

The closed-form full-field elastic solution for the tunnel

In this section, a plane strain elastic model is considered for the influence of the excavation of an underground opening on a homogeneous stress-deformation state described by in situ principal stresses σx and σy referred to a Cartesian coordinate system Oxy. That is, it is assumed that the tunnel-axis is aligned with the direction of the third out-of-plane principal stress σz. The direction of σx forms an angle α with Ox-axis. The cross-section of the tunnel has a vertical axis of

Verification of the proposed closed-form solution with known solutions

The solution of the complex potentials φ(ζ) and ψ(ζ) that has been found above, is compared here with existing solutions for the elliptical opening subjected to internal pressure and to far-field uniaxial stress, and with the square opening subjected to uniaxial stress σ along Ox-axis.

Stress–deformation analysis of the semi-circular tunnel

In order to demonstrate the potential applications of the proposed solution in soil/rock engineering, a number of examples have been worked out and they are illustrated below. Namely, the comparability of analytical model results concerning the distribution of stresses and displacements around the semi-circular tunnel with those predicted by FLAC2D numerical code is demonstrated. It may be argued that a boundary element code would be more suitable for the comparison of boundary stresses and

Conclusions

An exact solution has been presented for stresses and displacements around tunnels with rounded corners. It has been shown that the complex potential formulation together with the conformal mapping representation can be used successfully for the solution of plane elasticity problems for any tunnel cross-sectional shape with an axis of symmetry with prescribed surface tractions. The solution method has been compared with the FLAC2D numerical model for the particular case of the semi-circular

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