The paper presents an analysis of stresses and ground response curves around a circular tunnel, introducing a seepage and using the Hoek-Brown failure criterion, a perfect elasto-brittle-plastic rock behaviour and a treatment of plastic strains according to an associated flow rule.
L'article present une analyse des contraintes et des curves caracteristiques autour d'une cavite circulaire, en tenant compte d'une venue d'eau et en utilisant le critere de rupture de Hoek-Brown, un comportement elastoplastique de la roche et en supossant que les deformations plastiques derivent d'un potential plastique.
Die Mitteilung prasentiert eine Analise der Spannungen und Dehnungskurven um einen Kreisförmigen Tunnel herum mit Durchströmungen, indem man das Bruckkriterium von Hoek-Brown benutzt hat ein Sprödbruch des Felsen und man angenommen hat, daβ die Spannungsfunktion von einem plastischen Potential abhangt.
The problem of calculating the "Ground Response Curves" or "Characteristic lines" is related with the problem of obtaining the stresses and strains around the tunnel. Due to its complexity, there are several analytical solutions in the case of circular tunnel, hydrostatic stress-field and plane strain, depending on the failure criterion, the treatment of the plastic strains and the ground behaviour model. Being a tridimensional problem, studies on the face are very scarce: Lombardi (1970) assumes it to be a "nucleus" in equilibrium before the excavation; Amberg and Lombardi(1974) present a pseudo-tridimensional method to evaluate its influence; the solution proposed by Egger(1980) of evaluating the face as a semi-spherical cavity is quite simple. With seepage the problem becomes much more complicated. Jimenez-Salas (1981) evaluated the "plastic zone radius" using the concept of "influence radius'. Moreover, Adachi and Tamura (1978) have developed for the Seikan tunnel an elastoplastic model with a coulombian material to study the effects of water and the effectiveness of a ring of drains or/and grounting ground around a circular tunnel, admitting Terzaghi's and Darcy' laws. This paper shows the results of an analysis made in order to evaluate stresses and strains in the tunnel cross section and in the face (Izquierdo,1984).
A simple way of introducing seepage and heterogeneities consists of idealizing the real problem (figure 1) by a system of concentric rings which can have different properties. The simplest model -but allowing to study the influence of many parameters- consists of two rings: the inner one is plastic and the outer elastic (figure 2).
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The tunnel is circular with radius R.
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The problem has cylindrical (spherical) symmetry for the tunnel cross section (face).
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The geometric heigth is not considered and the stress-field is hydrostatic (σ o).
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The radius of the outer surface of the model is called "influence radius" and it is there where the pore pressure (ρo) is applied.
Each ring can be admitted as a continuous medium. The material is isotropic, homogeneous, without viscosity or weight, with permeability (k) constant and it is assumed to be linear-elastic and characterised by constant Young's modulus (E) and constant Poisson ratio (υ). When strength reachs its peak it falls down to the residual value (figure 3). Herein the analysis is made for a brittle-elastoplastic behaviour, being the perfect elastoplastic behaviour a particular case of this one: it is sufficient to substitute in the "brittle" equations the original ground parameters (with out subindex "r") (do b 1 =b" in figure 3).
With this condition, the derivate of Y with respect to r is zero if r=R and Δ*=l, and it is negative for greater values. As Y has been defined as positive, it can be deduced that, as in the plane strain case, there is a - limit value for the seepage so that for values greater than this is not possible to find a solution for the effective stresses. However, in the spherical case such value depends on the boundary conditions imposed at r=R.
The analysis presented evaluates stresses and ground response curves in the tunnel cross section and in the face. We have derived some close-form solutions and results considering seepage and using the Hoek- Brown criterion (1980), a treatment of plastic volumetric strains and an elastic-brittle-residual-plastic stress-strain model with dilatation.
