Statistics of imprecise data arising in rock mass characterization are tackled through random set theory and bounds to the reliability of a tunnel lining are then calculated.


A travers la theorie Random Set on analyse les statistiques des donnees imprecis qui caracterisenr la masse de la roche, et ensuite on calcule les limites à la resistance des revêtements des tunnels.


Ungenaue statistische Angaben bei der Analyse der Gesteine und ihrer Masse können mit der Theorie Random Set ueberprueft und wieder berechnet werden. Es ist weiterhin möglich, die eventuelle Unzuverlassigkeit einer Tunnelverkleidung festzustellen.


On the occasion of the 35th Rankine Lecture, R.E. Goodman pointed out [1]: "Charged with responsibility for design, an engineer hopes to have available tools appropriate to the applicable materials and conditions. When the materials are natural rock, the only thing known with certainty is that this material will never be known with certainty ". It is therefore advisable, in a design situation, to take account of every kind of uncertainty experienced in data collection, in order not to overestimate the safety of the projected work in or on rock. Although Probability Theory has been extensively and successfully used to tackle this problem quantitatively, mainly three drawbacks have recently been recognized about it [2,3]:

  • probability theory can only deal with statistics of precise data (i.e. each field observation has to give only one real number, no imprecision being permitted);

  • the amount of data necessary to define a reliable probability density function is so large that it is rarely available to the designer, in fact subjective assumptions about probability density functions are often made (Sakurai and Shimizu referred to this problem in [4]: "Compared with materials such as steel and concrete, the determination of a probability density function for the mechanical constants of rock masses is extremely difficult. In other words, there is no reliable way to determine the input data for the probabilistic approach. This means that the probabilistic approach may be less applicable to practical engineering problems. ");

  • the calculated probability of failure is instead quite sensitive to input data (variations of orders of magnitude in front of variations of some percent).

Conversely, Random Set Theory (RST) [5,6,7,8] and its companion Evidence Theory or Dempster- Shafer Theory (DST) [9, I 0, 11,12] provide us with an appropriate mathematical model of uncertainty when the information about the parameters is not complete to define probability density functions or when the result of each field observation is not point-valued but set valued. Moreover, RST has proved to be a general framework for computation under uncertainty because it incorporates as particular cases [8]: Interval Analysis [13], Convex Models [2,3], Fuzzy Measures [14] and Probabilistic Measures when the information becomes more and more precise. After introducing some basic notions about RST, this paper shows how the various kinds of uncertainty affecting a rock mass classification survey can be dealt with in a typical design situation and how this information can be used to predict the response of the rock mass surrounding an underground excavation, using the convergence confinement method. Particularly, it is possible to determine, in a very efficient way from a computational point of view, an upper and a lower bound to the probability that the pressure exerted on the lining by the rock mass exceeds the maximum pressure sustainable by the lining itself. Reliability of a tunnel lining was instead dealt with through probability theory by Oreste and Peila in [15]. A numerical example illustrates in detail every step of the proposed approach.

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