Page 225 - Numerical Analysis and Modelling in Geomechanics
P. 225

206 I.-M.LEE AND D.-H.KIM
            values utilizing the EBM (the posterior estimation). Unfortunately, the posterior
            distribution is not a simple normal distribution with respect to θ due to the fact
            that  u k  is  a  non-linear  function  of  θ,  which  causes  the  covariance  matrix  to  be
            non-linear.  In  order  to  resolve  this  problem,  with  u k  the  estimated  value,  θ  is
                                      k
            linearized. With the linearized u , it is possible to obtain the posterior covariance
            matrix  with  the  conventional  Bayesian  theory  (Honjo,  Wen-Tsung  and  Guha
            1994: 709):


                                                                        (7.18)


            As shown in Equation (7.18), the covariance matrix of the posterior estimation
            (Σ )  is  composed  of  the  prior  estimation  (V )  and  the  covariance  matrix  of
                                                 p
              p
            measurements  (V ).  It  is  interesting  to  observe  that  when  the  first  term  in
                          u
            Equation  (7.18)  is  added  to  the  second  term,  it  reduces  the  variance  of  the
            posterior estimation since the covariance matrix of measurements is added to the
            prior estimation in inverse matrix form.

                             Numerical method for back analysis
            For  problems  involving  geometries,  external  loadings,  and  engineering
            properties, it is generally not possible to obtain an analytical solution. Hence, we
            need to rely on numerical methods, such as the finite element method, to predict
            the  ground  motion  caused  by  underground  excavation.  For  the  finite  element
            program, we can either use conventional software or write our own program. The
            existing  elastoplastic  finite  element  program,  developed  by  Owen  and  Hinton
            (1980), was modified by the authors to simulate a tunnel excavation and support
            system.  The  Mohr-Coulomb  failure  criterion  was  used  to  represent  the  plastic
            behavior.


                             Framework of the feedback system


                                         General
            This  section  introduces  the  implementation  of  the  extended  Bayesian  method,
            combined  with  the  finite  element  method,  for  parameter  estimation  in
            underground  structures.  The  procedure  for  the  proposed  feedback  analysis
            technique is summarized as a flowchart in Figure 7.1.
              The initially estimated values of ground parameters (the prior information) as
            well  as  measured  data  are  utilized  for  the  feedback  analysis.  Geotechnical
            parameters  to  be  estimated  in  the  current  analysis  are  chosen  and  formulated
            statistically.  In  situ  measured  values  also  have  observation  errors  and  may
            contain  bias  terms  due  to  initial  ground  movement  that  occurred  before  the
   220   221   222   223   224   225   226   227   228   229   230