Page 192 - Numerical Analysis and Modelling in Geomechanics

BACK ANALYSIS OF GEOTECHNICAL PROBLEMS 173
errors on the computed mechanical parameters. Among them, two will be
mentioned here.
A first approach, is based on the so-called Monte Carlo, or simulation,
technique. Following this method, the influence of the measurement error, and of
the number of input data, is evaluated through a series of numerical tests [9].
Each of them consists of a set of back analyses based on suitable generated input
measurements (e.g. displacements).
The input data are obtained by adding to the “exact” measurements a
disturbance that represents the experimental “errors”. Independent generators of
random numbers, with chosen probability distributions and zero mean value, are
used to work out these errors. The number of generators coincides with that of
the input measurements. Their probability distributions depend on the
characteristics of the measuring devices and of the measured quantities.
The “exact” displacements either can be evaluated on the basis of actual field
measurements or could be simulated through a preliminary stress analysis of the
problem at hand, in which reference values of the material parameters are
introduced [9].
This procedure permits establishing a probabilistic correlation between the
resolution of the measuring device, the number of measurements and the
accuracy of the computed parameters characterising the soil/rock mass.
The simulation technique offers the advantage of an extremely simple
implementation, but requires a computational effort rapidly increasing with the
number of free variables of the numerical model and with the number of unknown
parameters.
This drawback could be limited by making recourse to probabilistic
approaches [10–13]. Among them, the so-called Bayesian approach will
discussed here, which was adopted in [14] and [15] for a rock mechanics problem
and for the back analysis of the field measurements performed during the
construction of a railroad embankment.
A typical feature of the Bayes approach is that “a priori” information on the
unknown parameters can be introduced in the back analysis, together with the
data deriving from in situ measurements. In most cases, the a priori information
consists of an estimation of the unknown parameters based on the engineer’s
judgement or on available general information. This leads to a numerical
calibration procedure that combines the knowledge deriving from previous,
similar problems with the results of the in situ investigation. For the sake of
briefness, only the bases of the approach are outlined here.
Consider the experimental measurements, collected in vector u*, and the
corresponding errors, seen as random variables, grouped in vector ≥ u. Let us
assume that the expected average value of the error vector, expressed by the
“expectation” operator E , vanishes,
x
(6.12)