Page 214 - Numerical Analysis and Modelling in Geomechanics

BACK ANALYSIS OF GEOTECHNICAL PROBLEMS 195
• The diagram in Figure 6.18 indicates a reduction of the friction angle
approaching section 6, where large inward displacements are observed. This
reduction is probably associated to a similar decrease of cohesion; both
effects should be accounted for in subsequent analyses.

On the basis of these observations a new calibration problem was formulated,
concerning the determination of the variation of both c and φ along the tunnel
axis that, under the assumption of elasto-plastic behaviour of the support,
corresponds to the best approximation of the displacements δ for all sections.
m
The law governing the variation of cohesion and friction angle for this second
analysis should not involve more than seven parameters. In fact, in a
deterministic back analysis the number of free variables cannot exceed the
number of input data that, in the present case, is equal to 7 (excluding, as
previously observed, the displacements measured at sections 1 and 6).
The following bi-linear law was adopted which fulfils the above requirement,
and takes into account two different variations of the shear strength parameters
(suggested by the diagram of Figure 6.18) for sections 2 to 5 and for sections 7, 8
and 9:

• The values of c and φ are expressed as the product of reference quantities, C 0
and φ , by a non-dimensional function f of the distance d,
0
(6.31a)

(6.3 lb)

• Two expressions for f are introduced holding, respectively, for sections 2, 3 4,
5 and 7, 8, 9,

The back analysis is based on the direct approach, hence its solution requires the
minimisation of the discrepancy (or error) between the input data δ (cf. eq.(6.30))
m
for all sections and the corresponding numerically evaluated displacements δ.
The error is a non-linear function of six variables, since the numerical results
depend on c and φ, which in turn are functions of six unknown parameters c , φ ,
0
0
a ,…a .
1
4
The results of the back analysis are presented in Figure 6.19 by means of the
diagrams of cohesion and friction angle vs. the distance d, obtained by assuming
α=1.5 (curve I) and α=1.0 (curve II). The φ vs. d diagram reports also the peak, φ ,
p