Page 188 - Numerical Analysis and Modelling in Geomechanics

BACK ANALYSIS OF GEOTECHNICAL PROBLEMS 169
First, two alternative approaches for the back analysis, referred to as “direct”
and “inverse” (with respect to the stress analysis) procedures are illustrated in
a deterministic context. Subsequently, the influence of the experimental error on
the back-calculated parameters is discussed on the basis of a probabilistic
Bayesian approach. Finally, two applications to actual tunnelling problems are
presented.
Notation: Upper and lower case underlined letters denote, respectively,
matrices and column vectors. The superscripts T and—1 mean transpose and
inverse.

An inverse method for elastic back analysis
The back analysis of mechanical parameters is a non-linear problem even in the
simple case of linear elastic material behaviour. In order to show this
characteristic, let us briefly recall the basic aspects of a technique for the back
analysis of elastic constants which is based on a finite element approach
originally proposed in [5], This method can be defined as “inverse”, with respect
to the corresponding stress analysis, since it requires the “inversion” of the
equations governing the linear elastic stress analysis problem.
To apply this procedure it is necessary to establish a linear relationship
between the stiffness matrix of each finite element K e and the unknown
mechanical parameters. In the case of isotropic material behaviour, such a
relationship can be obtained easily by describing the elastic behaviour in terms
of bulk B and shear G moduli,
(6.1)

The two matrices on the right-hand side of eq. (6.1) are, respectively, the
volumetric and the deviatoric stiffness matrices of the e-th element.
Consequently, the stiffness matrix K of the assembled finite element model can
be written in the following form,

(6.2)

where n is the number of different materials (2n being the number of unknown
elastic parameters) and K is the assembled stiffness matrix obtained by setting
i
all the parameters to zero, but with the i-th parameter set equal to 1.
Assuming that m displacement components are measured in the field, and that
the measurement points coincide with nodes of the finite element grid, the system
of linear equations describing the behaviour of the finite element discretization
can be partitioned as follows,