It has been shown recently (Gill and Ladanyi, 1983;, Ladanyi and Gill, 1984) that the conventional convergence-confinement method, based on isochronous characteristic lines, generally underestimates both the ground pressure on the lining and the tunnel wall convergence. By applying that method to a ground showing a Zener-type creep, and using an exact solution as a basis, the authors show in this paper that this underestimate decreases with time, tending to zero at long term, when the time tends to infinity. A simple numerical method for calculating isochronous characteristic lines, which takes into account the loading history, and requires only a pocket calculator, is finally proposed.


On a demontre recemment (Gill et Ladanyi, 1983; Ladanyi et Gill, 1984) que la methode convergence - confinement conventionnelle, basee sur les lignes caracteristiques isochrones, tend à sousestimer, tant la pression des terrains, que la convergence de la paroi du tunnel. En appliquant cette methode à un terrain caracterise par le fluage du type Zener, et en se basant sur une solution exacte du problème, les auteurs montrent que cette sousestimation diminue avec le temps, tendant vers zero lorsque le temps tend vers l'infini. On propose finalement une simple methode numerique, n'utilisant qu'une calculatrice de poche, permettant de determiner approximativement les lignes caracteristiques correctes, qui tiennent compte de l'histoire de chargement.


In zwei vorlaufigen Arbeiten (Gill und Ladanyi, 1983; Ladanyi und Gill, 1984) wurde gezeigt, dass die uebliche Gebirgskennlinienmethode, die isochrones Kennlinien benutzt, im Algemeinen zu einer Unterschatzung, sowohl des Gebirgsdruckes auf die Tunnelauskleidung, als auch zu einer Unterschatzung der Ausbruchrandverschiebungen fuehrt..In der vorliegenden Veröffentlichung wurde diese Methode auf ein viskoelastisches Gebirge angewandt, um zu beweisen, dass diese Unterschatzung mit der Zeit abnimmt und ganzlich verschwindet, wenn t unendlich wird. Es wird ebenfalls am Ende der Veröffentlichung ein einfaches numerisches Verfahren vorgeschlagen, das die Belastungsgeschichte beruecksichtigt und das auf einem einfachen Taschenrechner ausgefuehrt werden kann.


The design of tunnel linings has been extensively discussed in the recent literature, and many different design methods have been proposed for that purpose, as can be seen from the surveys made by Ladanyi(1980) and Brown et al.(1983). Among these methods, the characteristic- line ground support design approach, or the convergence-confinement method, has received a Considerable attention over the last two decades, even though this concept implies relatively simple conditions. The popularity of this method is mainly due to its clarity and versatility. Used primarily for the design of artificial ground supports, the technique has been recently applied also to the design of mine pillars (SHkkH, 1984). The method, introduced originally by Lauffer and Seeber (1961) and Pacher (1964), has since then been used and discussed by numerous authors. A review of that literature reveals that, when the concept is used for designing artificial tunnel supports, the load increase on the lining is usually considered to be due to one or both of the following two processes:

  • The wall convergence at a given cross-section of an underground opening, due to the advance of the driving face. This convergence can be controlled by the use of air pressure at the face(Panet,1976).

  • The long-term convergence of the opening, due either to the time-dependent deformation (creep) of the rock mass, or to its time-dependent deterioration (E.g., Ladanyi, 1974, 1980).

One of the first attempts to introduce the long - term strength concept in the determination of ground pressure on tunnel linings by the characteristic line method was made by Ladanyi (1974). In that approach only two limiting characteristic lines were considered: One for the short-term response (t → 0+), and another for the long-term response (t → ∞). For intermediate times, it was assumed that the surrounding rock mass undergoes a continuous deterioration, involving a decrease in its modulus of deformation and a gradual loss of strength. In a subsequent paper (Ladanyi, 1980), the author attempted to fill the time gap between the short - term and the long-term rock mass response, by assuming that the rock around the tunnel creeps according to a non-linear Maxwell (power law) model. However, since that solution did not take into account the effect of the loading history on the rock response, but only the rock deterioration with time, independent of the lining pressure, it falls clearly in the class of solutions based on the "aging theory of creep". It is noted that nearly all "conventional" characteristic line methods presently in use around the world, are based on the same "rock aging" assumption. The necessity of incorporating the loading history into the modelling of the tunnel wall displacement curves was recognized by Kaiser (1980), who applied it to the case in which the support loading process resulted from the advance of the driving face, in a tunnel driven through an imperfectly elastic rock mass.

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