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CHAPTER 6
DEEP EXCAVATIONS: THE OBSERVATIONAL METHOD AND INVERSE ANALYSIS
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Predictions of the magnitude and distribution
of ground movements are used to estimate the tolerance of
structures
and utilities to the deformations associated with construction
of deep supported excavations. Many factors affect movements
associated with excavations, including soil properties (soil
type, presence of water), support system properties (wall
stiffness, support stiffness, preload) and construction activities
or workmanship (construction sequence, installation of support,
surcharge loads). In practice, when designers are faced with
an excavation where ground movements are a critical issue,
they can base their estimate of movements on semi-empirical
methods based in part on past performance data or on results
of finite element analyses. The main limitation of the first
approach is the variety of construction techniques of the
case studies used in developing them. These activities can
contribute significantly to the movements reported. Therefore
design approaches developed from these data should be considered
biased towards average construction practices.
The only way
to explicitly include the effects of the construction activities
in the analysis is to perform numerical simulations
of the problem. Indeed, if the exact construction procedure
is known, a finite element analysis conceptually allows
an engineer to model all aspects of excavation that cause
stress
change in soil: wall installation, dewatering, cycles of
excavation, bracing and brace removal, and preloading of
anchors. Finite element predictions, however, contain uncertainties
related to soil properties, support system details and
construction procedures. If one wants to predict and evaluate
the overall
performance of a design, a procedure that incorporates
an evaluation of the results of the analyses must be defined.
The
procedure to accomplish this task is usually referred to
as the "observational method" (see section
2.2.3). Morgenstern (1995), in his Casagrande Lecture, emphasizes
the importance of the observational method and stresses the
need to have plans to cope with possible eventualities. In
practice, however, it is very difficult to quantitatively
judge how well the work is proceeding especially considering
the time constraints associated with construction. Ad-hoc
mathematical tools are needed to compare observations and
predictions.
To improve the state-of-the-practice of controlling
ground movements associated with supported excavations, this
chapter
presents a procedure that "objectively" updates
design predictions of deformations for supported excavations
in clay. Monitoring data are used as observations in an inverse
analysis that calibrates the numerical model of the excavation
and thus supports, in an objective way, the engineering judgments
made during the construction of the excavation system. The
inverse analysis methodology was developed and tested using
data from a 39 ft deep excavation through soft clays in Chicago
(Finno et al. 2002).
6.2 MODEL CALIBRATION BY INVERSE ANALYSIS
In model calibration, various parts of the model are changed
so that the measured values are matched by equivalent computed
values until, hopefully, the resulting calibrated model accurately
represents the main aspects of the actual system. Despite
their apparent utility, inverse models are used for this
purpose much less than one would expect. In practice, numerical
models typically are calibrated using trial-and-error methods
because, perhaps, of the difficulties of implementing an
inverse analysis, the complexity of the simulated systems,
and/or the engineers' perception that automated estimation
of model parameters without "engineering judgment" is
impossible. One outcome of this work is to show that these
concerns are, in most cases, unjustified and inverse modeling
represents a valuable tool for geotechnical engineers.
With
an inverse modeling approach, a given model is calibrated
by interactively changing model input values until the simulated
output values match the observed data (i.e. observations). Figure
6-1 shows a schematic of an inverse analysis procedure.
The input parameters are initially estimated by conventional
means (e.g. using available laboratory and field test results).
Next a numerical simulation of the problem is run and the
results are stored in a file (generally in ASCII text format).
The simulated results are then compared to the field observations
and a regression analysis is performed to minimize an objective
function. The objective function quantifies the fit between
computed results and observations and its minimization is
reached by updating the set of input parameters needed to
perform the numerical simulation. If the model fit is not "optimal",
the procedure is repeated until the model is optimized.
6.2.1 An inverse analysis algorithm: UCODE
In the work described herein model calibration by inverse
analysis was conducted using UCODE (Poeter and Hill 1998),
a computer code designed to allow inverse modeling posed
as a parameter estimation problem. UCODE has been developed
for ground-water models, but it can be effectively used in
geotechnical modeling because it works with any application
software that can be executed in batch mode. Its model-independency
allows the chosen numerical code to be used as a "closed
box" in which modifications only involve model input
values. This is an important feature of UCODE, in that it
allows one to develop a procedure that can be easily employed
in practice and in which the engineer will not be asked to
use a particular finite element code or inversion algorithm.
In
UCODE the weighted least-squares objective function S(b)
is expressed by:
 (6.1)
where b is a vector containing values
of the number of parameters to be estimated; y is the vector
of the observations
being matched by the regression; y'(b) is the
vector of the computed values which correspond to observations; ω is
the weight matrix; and e is the vectors of residuals.
Non-linear
regression is an iterative process. The modified Gauss-Newton
method used by UCODE to update the input parameters is
expressed as:
 (6.2)
 (6.3)
where dr is the vector used to update
the parameter estimates b; r is the parameter estimation
iteration number; Xr is the sensitivity
matrix (Xij=∂ yi/∂bj)
evaluated at parameter estimate br; C is
a diagonal scaling matrix with elements cjj equal
to 1/√XTωX)jj;I
is the identity matrix; mr is a parameter used to improve
regression performance; and ρr is
a damping parameter.
6.2.2 Model fit statistics
Different quantities can be used to evaluate the model fit.
A commonly used indicator of the overall magnitude of the
weighted residuals is the model error variance, s2,
which equals:
 (6.4)
where S(b) is the objective function;
ND is the number of observations; and NP is the number of
estimated parameters.
The value of the objective function
(Eq. 2.1) is also used
to indicate model fit informally, because its variation
indicates by how much an optimized model improves with respect
to the
initial simulation of a problem. The objective function
changes can be expressed through a new statistic, the fit
improvement
(FI), which indicates by what percentage the optimized
results improved compared to the initial fit between experimental
data and computed results. The fit improvement is defined
as:
 (6.5)
where S(b)initial is the initial value
of the objective function; and S(b) optimized is
the value of the objective function for the optimized set
of parameters.
6.2.3. Input parameters statistics
The relative importance of the input parameters
being simultaneously estimated can be defined using parameter
statistics, including
the sensitivity of the predictions to changes in parameters
values, the variance-covariance matrix, confidence intervals
and coefficients of variation.
Different quantities can be
used to evaluate the sensitivity of the predictions to
parameters changes. One percent sensitivities,
dssij, scaled sensitivities, ssij,
and composite scaled sensitivities, cssj, can
be used for the purpose. These sensitivities are defined
in Eq. (6.6), (6.7) and (6.8), respectively.
 (6.6)
 (6.7)
 (6.8)
where y'i is the ith
simulated value; yi/bj is the sensitivity
of the ith simulated
value with respect to the jth parameter; bj is
the jth estimated parameter; ωjj is
the weight of the ith observation.
One percent
scaled sensitivities represent the amount that the simulated
value would change if the parameter value increased
by one percent. Scaled sensitivities are dimensionless
quantities that can be used to compare the importance of
different observations
to the estimation of a single parameter or the importance
of different parameters to the calculation of a simulated
value. Composite scaled sensitivities indicate the total
amount of information provided by the observations for
the estimation of one parameter.
The reliability and correlation
of parameter estimates can be analyzed by using the variance-covariance
matrix, V(b'),
for the final estimated parameters, b', calculated
as:
 (Figure
6.9)
where s2 is the error variance; X is the sensitivity
matrix; and ω is
the weight matrix.
The diagonal elements of matrix V(b')
equal the parameter variances, the off-diagonal elements
equal
the parameter covariances. Parameter variances are most
useful when used to calculate two other statistics: confidence
intervals
for parameter values and coefficients of variation. Parameter
covariances can be used to calculate correlation coefficients.
Coefficients
of variation, covi, are equal to:
 (6.10)
where σi is
the standard deviation of parameter b1
Correlation coefficients are calculated by:
 (6.11)
where cor(i,j) indicate the correlation between the ith and
jth parameter; cov(i,j) equal the off-diagonal
elements of V(b'); and var(i) and var(j) refer
to the diagonal elements of V(b').
The coefficients
of variation provide dimensionless numbers with which the
relative accuracy of different parameter estimates
can be compared. Correlation coefficients close to –1.0
and 1.0 are indicative of parameters that cannot be uniquely
estimated with the observations used in the regression.
6.2.4 Observations weighting
The weights assigned to the observations are an important
part of the regression analysis because they influence the
value of the objective function, and thus the regression
results. UCODE uses a diagonal weight matrix. Weighting is
used to reduce the influence of observations that are less
accurate and increase the influence of observations that
are more accurate. For problems with more than one kind of
observation, weighting also produces weighted residuals that
have the same units, so that they can be squared and summed.
In UCODE the weight of every observation, ωii,
is equal to the inverse of its error variance, σ2:
 (6.12)
Users assign the weight of an observation by specifying
a value for its variance, standard deviation or coefficients
of variation. Assigning appropriate weight values to the
observations can be problematic. For regression methods to
produce parameter estimates with the smallest possible variance
Hill (1998) suggests that weighting needs to be proportional
to the inverse of the variance of the measurement errors.
At the end of the regression analysis, the value of the model
error variance, s2 (Eq. 2.4), can be used to evaluate
the consistency between the model fit and the measurement
errors, as expressed by the observations' weights.
Values larger than 1.0 indicate that the model fits the data
less well than would be accounted for by expected measurement
errors.
6.3 UPDATE DESIGN PREDICTIONS USING MONITORING DATA
BY INVERSE ANALYSIS
This section shows how inverse analysis based on field monitoring
data can be used to objectively update the predicted performance
of supported excavation systems. Movements of the soil surrounding
an excavation, measured to evaluate how well the actual construction
process is proceeding in relation to the predicted behavior,
can be recorded by inclinometers, which measure lateral deformations
at various depths at discrete locations around the construction
site, and survey points, which record ground movements and/or
displacements of structures adjacent to the excavation. With
an inverse analysis procedure (see Chapter 2), these recorded
displacements can be used to control the construction process
and update predictions of movements at early stages of construction.
Any time a new set of construction monitoring data are available,
the finite element model of an excavation can be "recalibrated" to
provide the best fit to the field observations.
Inverse analysis
algorithms allow the simultaneous calibration of multiple
input parameters. However, identifying the important
parameters to include in the inverse analysis can be problematic.
Indeed, it is not possible to use the regression analysis
to estimate every parameter of every soil model used in
the simulation. The number and type of input parameters that
one can expect to estimate simultaneously depend from many
factors, among which:
- Soil models used. The characteristics
of the soil models and the number and type of observations
used in the simulation determine the input parameters that
are expected to be successfully calibrated. Some model
parameters may be correlated to one another and thus not
likely to be estimated simultaneously.
- Aspects of the simulated system represented
by estimated parameters. In many instances, supported
excavations generally generate only small deformations
in the soil surrounding the excavation. In these instances,
stiffness parameters are expected to be more important
than failure parameters in defining the behavior of the
soil mass. Sensitivity analyses can be used to determine
the input parameters of a soil model that are most relevant
to the computed system response.
- Available observations. The number
of observation points used in the inverse analysis is related
to the maximum number of parameters that one can expect
to estimate by regression analysis. Their spatial distribution
influences the number of soil layers whose parameters can
be calibrated.
- Finite element implementation.
Computational time may constitute an important variable
for very complex simulations. The number of finite element
runs at any given iteration and the number of iterations
needed for the convergence of the regression analysis
are proportional to the number of estimated parameters,
NP.
Figure
6-2 shows a procedural flowchart that can be used for
the identification of the soil parameters to optimize by
inverse analysis. As subsequently described, the total
number of input parameters can be reduced, in four steps,
to the number of parameters that are likely to be successfully
optimized by inverse analysis.
Step 1: Model's
input parameters → Model's
uncorrelated parameters. The soil model chosen to simulate
the soil behavior determines the total number of input parameters
to estimate (e.g., the H-S model has 10 input parameters).
The number of parameters that can be estimated by inverse
analysis depends from the characteristics of the model and
from the type of observations available. Parameter correlation
coefficients (Eq. 6.11) can be used to evaluate which parameters
are correlated and are, therefore, not likely to be estimated
simultaneously by inverse analysis.
Step 2: Model's
uncorrelated parameters → Model's
relevant parameters. The parameters that most affect the
computed results are determined by the stress conditions
in the soil around the excavation. Composite scaled sensitivity
values (Eq. 2.9) can provide valuable information on the
relative importance of the different input parameters of
a given model.
Step 3: Model's relevant parameters → Total
relevant parameters. The number of soil layers to calibrate
and the type of soil model used to simulate the layers determines
the total number of relevant parameters of the simulation.
A new sensitivity analysis may be necessary to check for
correlations between parameters relative to different layers.
Step 4:
Total relevant parameters → Parameters
to optimize. The total number of observations available and
computational time considerations may prompt a final reduction
of the number of parameters to optimize simultaneously.
Figure
6-2 showed the key role that sensitivity analyses have
in determining the parameters that are important for
the finite element simulation of an excavation. Once the
parameters to optimize have been chosen, sensitivity results
continue to play an integral part in the regression analysis.
Indeed, a sensitivity matrix is computed at every regression
iteration. This is necessary because the simulation of
an excavation system by finite element methods is a highly
non-linear
problem. Thus, the sensitivity of the results to changes
in parameter values is not constant but depends on the
particular values at which the sensitivity matrix is computed.
The
design chart (Clough et al. 1989) given in Figure
6-3 will be used to explain the importance of this
approach. The graph is generally used to design retention
systems for supported excavations in soft to medium clays.
The curves show how the ratio between the maximum horizontal
movement of the wall and the height of excavation (δH/H)
is a function of the factor of safety against basal heave
(FSBS)and of the retaining system stiffness
(EI/h4γw),
a combination of wall stiffness and strut spacing.
The chart
can be considered a model of the excavation problem,
where FSBS and EI/h4γw are
the input parameters and δH /H
is a measure of the design performance. The tangent of the
design curves expresses the sensitivity of the movements
with respect to the system stiffness, the distance between
the curves is related to the sensitivity of the movements
with respect to FSBS. The graph shows that, if
the system is stiff or the factor of safety against basal
heave is high, the performance is less sensitive to either
parameter (i.e. a small value of the tangent to the curve)
than would be the case if the system is flexible or has a
low FSBS (i.e. a higher value of the tangent to
the curve).
6.4 PROCEDURE VALIDATION: THE CHICAGO & STATE
CASE STUDY
The proposed methodology was developed and tested using
data from a project in downtown Chicago (Finno et al. 2002),
the excavation/renovation of the Chicago & State CTA
subway station.
6.4.1 Finite element simulation of the problem
The finite element software PLAXIS was used to compute the
response of the soil around the excavation. Figure
6-4 shows a schematic of the PLAXIS input. Details about
the definition of the finite element problem, the calculation
phases and the model parameters used in the simulation described
herein can be found in Appendix C.
The problem was simulated
in plane-strain conditions. The soil stratigraphy was assumed
to be uniform across the site
(see Figure 4.4). The soil layers considered were 8: a
fill layer overlaying a clay crust, a compressible clay deposit
(in which 4 distinct clay layers were modeled) and two
relative
incompressible stiff silty clay strata. All elevations
in the figure refer to the Chicago City Datum (CCD). Note
that
the figure, for display purposes, does not show the side
boundaries of the mesh (600 ft x 94 ft), which was extended
beyond the zone of influence of the settlements induced
by the excavation (Hsien and Ou 1998 and Caspe 1966). The
finite
element mesh boundary conditions were set using horizontal
fixities, for the left and right boundaries, and total
fixities, for the bottom boundary.
6.4.1.1 Calculation phases
The tunnel tubes and the school adjacent to
the excavation were explicitly included in the finite element
simulation
of the problem to take into account the effect of their construction
on the soil surrounding the excavation. Table
6-1 shows the PLAXIS calculation phases of the simulation
described herein. The second column of the table shows the
calculation phase number, the third column explains the purpose
of the calculation phase, the fourth column indicates the
calculation type, and the last column specifies the loading
input condition. A plastic calculation indicates that an
elasto-plastic deformation analysis is carried out in either
fully drained or fully undrained conditions. For the simulation
described herein, plastic calculations are always associated
with staged construction loading conditions, which indicate
changes in the geometric configuration of the FE mesh, and
clay layers are always assumed to be in undrained conditions.
Consolidation calculations are used to analyze the development
and dissipation of excess pore pressures in the water-saturated
soil layers as a function of time. An "ultimate time" (loading
input condition) is specified to terminate a consolidation
calculation. Note that in PLAXIS it is not possible to perform
a staged construction calculation with simultaneous consolidation.
More details about calculation types and loading input conditions
can be found in the PLAXIS manual (Brinkgreve and Vermeer,
1998).
6.4.1.2 Hardening-Soil model initial calibration
The soil model used to characterize the clays in the PLAXIS
simulation of the excavation is the Hardening-Soil model
(Schanz et al. 1999). Table
6-2 shows the initial values of the H-S model parameters
for the five clay layers that will be calibrated by inverse
analysis. Layers 1 to 5 refer to the Upper Blodgett, Lower
Blodgett, Deerfield, Park Ridge and Tinley layers, respectively.
The model parameters of the soil layers that were not calibrated
by inverse analysis can be found in Appendix C.
The initial
estimates of the input parameters for layers 1 to 4 are
based on the results of the triaxial compression
tests(Calvello 2002). Because little laboratory data exists
for the layer 5 soil, the initial values of the parameters
for layer 5 are based on the following considerations:
(i) layer 5 failure parameters are assumed to have the same
values
of layer 4 failure parameters, and (ii) layer 5 stiffness
modules are assumed to be 1.5 times larger than layer 4
stiffness modules. For all layers, the value of parameter
E50refassumed
to be equal to 70% Eurref
The H-S stiffness parameters are defined with respect to
a reference pressure (pref=100 stress units). Thus, it is
difficult to relate the values of E50ref,
Eoedref, and Eurref to "typical" geotechnical
estimates of stiffness moduli. The following equations define
the stress dependent stiffness moduli used in the H-S model:
 (6.13)
 (6.14)
 (6.15)
where E50 is the secant Young modulus, Eoed is
the oedometric modulus, Eur is the unload-reload
elastic modulus,σ1'is
the major principal stress, σ3 ' is
the minor principal stress, φ is
the friction angle and c is the cohesion.
Figures
6-5, 6-6 and 6-7 show
the variation with the vertical stress of E50,
Eoedoed and Eurur. The curves
were computed, using the initial values of parameters c,φ and
m (see Table 6-2), according to Equations 5.1, 5.2 and
5.3, respectively. The vertical and horizontal directions
were assumed to be principal directions (i.e.σv'=σ1'and σ h'=σ3'=k0 σ v' ).
To
compare the stiffnesses of different layers, one has to
consider the effective stresses of the soil. Table
6-3 shows the initial vertical effective stress in
the middle of the five soil layers. These values can be
used
to compute, using Eq. 6.13 or Figure 6-5, the values of
the secant stiffness modulus at 50% shear strength, E50,
at the beginning of the simulation.
Table
6-4 shows, for all five layers, the initial E50refref
values, the computed E50 values, the estimated
undrained shear strength Su and the ratio
between E50 and Su. Note that the
Su values
were estimated from field vane results and correlations
based on water content data (Chung and Finno, 1992).
Note
that E50 represents a drained modulus. Nonetheless
the ratio E50/SuScan be used to judge
the "relative inherent stiffness" of the various
soil layers in undrained conditions. The initial /Su ratios
used in this simulation show that: (i) the Blodgett layers
(1 and 2) are assumed to have about the same relative stiffness,
and (ii) the other layers (3, 4 and 5) become, relative to
their undrained shear strength, progressively more deformable
with depth.
"Typical" values of Eu/Su ratios
are often presented in literature to evaluate the stress-strain
undrained response of clays. Lambe and Whitman (1969) report
Eu/Su values of about 500 and 1000
for soft and stiff clays, respectively. E50/Su values
are not generally quoted in literature. However, E50/Su ratios
can be related to typical Eu/Su ratios
if the initial undrained stiffness modulus Eu is
converted into an equivalent E50. The following
three steps describe a way of computing E50 from
a given value of Eu.
Step 1 (Eu→Gin)
The following relationship between elastic moduli can be
used to convert Eu (initial undrained stiffness modulus)
into an equivalent Gin (initial
shear stiffness modulus):
 (6.16)
Step 2 (Gin→ G50)
The value of the shear stiffness modulus of clays, G, decreases
with increasing shear strains:
G50<Gin<G0
The maximum stiffness, G0, only occurs at extremely
small strains (εsh<0.001%).
The initial undrained stiffness modulus Eu is
generally computed at higher strain levels (εsh=0.05-0.1%).
Thus, the initial shear stiffness modulus, Gin,
is smaller than G0. Based on published results
(Viggiani and Atkinson, 1995) the value of G0 is
assumed to be 0.5-0.75 times G0and G50 is
assumed to be 0.25-0.50 times Gin (i.e. G50 =
0.15-0.35 G0).
Step 3 (G50→ E50)
The following relationship between elastic modules can be
used to convert G50 (50%
shear stiffness modulus) into an equivalent E50 (50% secant stiffness
modulus):
 (6.18)
Finno and Chung (1992) reported Eu/Su values
of 400-600 for normally consolidated compressible Chicago
clays (Blodgett and Deerfield layers) sheared in triaxial
compression. Following the procedure outlined previously,
an equivalent E50/Su ratio can be computed.
Assuming Eu/Su=500, G50=Gin/3
and ν=0.2,
the ratio E50/S equals 133. This value
is slightly higher than the initial values used in this simulation
(see Table 6-4), suggesting that the initial estimates of
the H-S parameters defining the soil stiffness may be conservative.
6.4.2 Inverse analysis set-up
The optimization algorithm UCODE was used to calibrate,
by inverse analysis, the PLAXIS finite element simulation
of the excavation. A schematic of the interaction between
PLAXIS and UCODE was presented in Figure 4-7. Examples of
input and output files of the inverse problem analyzed in
this section can be found in Appendix C.
6.4.2.1 Observations and weighting
Table 6-5 shows the construction stages for which the model
predictions are updated. Lateral movements of the soil behind
the secant pile wall were recorded using five inclinometers.
The excavation, however, was modeled in plane strain conditions.
Thus, only two of them (incl. 1 on the east and incl. 4 on
the west) were used to compare field data and computed displacements.
The measured settlements were not used as observations because
the finite element predictions of the ground settlement induced
by excavation are generally not as good as those of the horizontal
movements of the soil.
Table 6-5 Excavation stages considered for updating model
predictions
Figure
6-8 shows the observation points retrieved from the
field readings of inclinometers 1 and 4 (the data in
the plot refer to stage 1). The soil profile and a schematic
of the support system are also shown in the figure. Inclinometer
readings were taken in the field every two feet. Not
every
reading, however, could be used as an observation for
the inverse analysis because the finite element displacements
were computed only at the intersection between the finite
element mesh and the inclinometer location. Thus, 13
observation
points were used for the east side and 11 observation
points for the west side.
The inverse of the variance
of the measurement errors was used to assign weights
to the observations (i.e.
inclinometer data). Table 6-6 shows the values of
the observation points used for the five construction stages and their
measurement errors. See Appendix C for details about the
inclinometer probe used to monitor
the movements at the excavation site, its accuracy and the computed measurement
errors. The measurement error of the horizontal displacement inclinometer
data is not constant. Its value increases as one moves
further away from the bottom
of the casing because the inclinometer probe measures tilt and not displacements,
thus errors become larger as one gets closer to the ground surface. Note
that inclinometer data are available, for the east
side, at all 5 construction stages
considered. On the west side, however, the inclinometer was damaged by
construction activities after stage 3. That is why the west
side inclinometer readings
are not shown, in subsequent figures, for the last
two stages of construction.
Table 6-6 Values of observations on the east side and west
side and their measurement errors
6.4.2.2 Parameterization
The soil layers calibrated by inverse analysis are the upper
Blodgett, lower Blodgett, Deerfield, Park Ridge and Tinley
strata. In the analysis described herein they are referred
to as layer 1, 2, 3, 4 and 5, respectively. All the layers
are modeled using the Hardening-Soil model. The initial estimates
of the H-S input parameters were presented in section 6.4.1.2..
The input parameters optimized by inverse analysis were chosen
following the procedure described in section 6.3 Note that
the first two steps of the procedure (see Figure 6-2) refer
to the selection of the "model parameters" (e.g.,
H-S model) that are relevant to the problem under study,
the last two steps refer to the selection of the total number
of "simulation parameters" (e.g., 5 soil layers
calibrated simultaneously) to optimize by inverse analysis.
The
H-S model features 10 input parameters. The characteristics
of the model determine the number of uncorrelated parameters
that one can expect to successfully optimize by inverse analysis.
The H-S parameters that can be effectively estimated from
laboratory data using an automated optimization algorithm
are E50ref, m and φ (Calvello
2002). For the simulation discussed herein the values of
the other model parameters are either kept constant at their
initial value (parameters c, ψ, ν and
Rf), or are assumed to be related to one of the
other parameters (Eoedref = 0.7 E50ref,
Eurref = 3.0 E50ref and
k0 = 1 - sinφ)
The sensitivity of the observations to changes in values
of Eref,
m and φ determines
the parameters that are relevant to the problem simulated herein. Figures
5.12 and 5.13 show
the composite scaled sensitivities of the three parameters for layers 1 to
5. In the first figure the bar chart refers to sensitivities computed using
all the observations, and the line charts refer to sensitivities computed from
the observations of the different layers. In the second figure the sensitivities
are grouped by construction stages. Both figures show that all three parameters
(i.e. E50ref, m and φ)
are important, from a model perspective, in affecting the outcome of the analysis.
From a simulation perspective, results show that the parameters that most influence
the simulation are the ones relative to layers 1 and 4. Layer 1 is the softest
soil layer, thus its major influence on the displacement results is expected.
Layer 4 is the stiff clay layer below the bottom of the excavation, into which
the wall is tipped. The high sensitivity values relative to this layer indicate
that the strength and the stiffness of the clay below the excavation have significant
impact on movements. As one would expect Figure
6-9 also shows that the observations relative to a soil layer are mainly
influenced by changes in that soil layer's parameters. For instance,
the values of the sensitivities from layer 4 and layer 5 observations show
a clear "peak", respectively, at layer 4 and layer 5 input parameters.
Likewise, Figure
6-10 shows that the parameters of the deeper layers become more important
at later construction stages (i.e. deeper excavation depths).
Other important
parameter statistics resulting from a sensitivity analysis are the correlation
coefficients. The sensitivity
analysis performed on E50ref, m and φ for
layers 1 to 5 indicated that high correlation values occur
between parameters E50ref and m.
Table 6-7 shows the correlation coefficients between the
three
parameters at every layer. Results indicate that the two
stiffness parameters (i.e. E50ref and
m) cannot be simultaneously and uniquely optimized, even
though the results of the analysis are sensitive to both
parameters. Parameter E50ref, rather
than parameter m, was chosen to "represent" the
stiffness of the H-S model. The reasons behind this choice
are: (i) m values are bounded between 0 and 1.0, thus they
would require the use of a "mapping function" (see
section 2.4) to avoid possible problems with unreasonable
updated values during the regression iterations, and (ii)
changes in E50ref values also produce
changes in the values of parameters Eoedref(equal
to 0.7 times E50ref) and Eurref(equal
to 3 times E50ref), thus their calibration
can be considered as "representative" of the
calibration all three H-S stiffness parameters.
Table 6-7 Highest values of correlation coefficients
The
results of the sensitivity analysis seem to indicate that
the total number of relevant parameters is 10 (i.e.
E50ref an φ for
layers 1 to 5). However a final reduction of the parameters
to optimize was necessary to establish a "well-posed" problem.
Table 6-8 shows the parameters that are optimized by inverse
analysis for the simulation described herein.
Table 6-8 Parameters optimized by inverse analysis
The parameters to optimize were chosen based on the following
considerations:
1. Hill (1998) suggests applying the "principle
of parsimony." Thus, to begin calibration estimating
very few parameters that together represent most of the
features of interest and to increase the complexity of
the parameterization slowly.
2. Poeter and Hill (1997) warn
against "spreading the data too thin." If
all 10 relevant parameters were to be optimized simultaneously, the ratio
between the number of observations available at the first construction stage
(24 displacement observation points) and the number of parameters estimated
(NP=10) would be too low.
3. The first construction stage, which refers
to the excavation of the 60'-deep secant pile wall, causes
movements in all 5 clay layers. Thus, at least one
parameter per layer had to be considered for optimization.
4. The stiffness
parameters (E50ref) were chosen over
the failure parameters (φ)
because the excavation-induced stress conditions in the soil around the excavation
were, for the most part, far from failure. Thus, the stiffness parameters
were perceived to be more relevant to the simulated problem.
5. Results of
the parametric study conducted on the number of input parameters considered
for optimization by inverse analysis (section 3.3.2) suggest that
a model can be successfully calibrated even if some "relevant" parameters
are excluded from the optimization, provided that their initial estimate
is "reasonable."
6.4.3 Results
This section presents the results of the inverse analysis
performed on the finite element simulation of the Chicago & State
excavation project. Visual examination of the horizontal
displacement distributions at the inclinometer locations
plots provides the simplest way to evaluate the fit between
computed and measured field response. Figure
6-11 shows the comparison between the measured field
data and the computed horizontal displacements when the initial
estimates of the parameters are used (see Table 6-2). The
comparison shows that the finite element model computes significantly
larger displacements at every construction stage. The maximum
computed horizontal displacements are about two times the
measured ones and the computed displacement profiles result
in significant and unrealistic movements in the lower clay
layers (Deerfield and Park Ridge). Results indicate that
the stiffness properties for the clay layers have been underpredicted,
as was suggested in 6.4.1.2.
6.4.3.1 Optimization based on observations from
construction stage 1
Table 6-9 shows the initial values of the 5 input parameters
considered in the analysis and the values of the parameters
that best fit the observations relative to stage 1. Results
show that the optimized values of the parameters are, at
all layers, higher than their initial estimates.
Table 6-9 Initial and optimized input parameters at stage
1
Note that: (i) parameters E1 and E2 were
optimized together (i.e. layer 1 and layer 2 were considered
as a single layer) and converged to a value slightly higher
that the initial estimate of parameter E2, and
(ii) parameter E5 was not optimized by inverse
analysis, instead its changes were "tied" to
changes in the value of E4 (at every iteration
E5 was set equal to 1.5 times E4).
This approach was employed after various unsuccessful attempts
at optimizing all five parameters simultaneously and independently.
The exclusion of parameters E2 and E5 from
the regression was based on the results of the sensitivity
analysis presented in section 6.4.2.2, which showed that
the parameters relative to layer 2 and layer 5 did not affect
the computed results as "significantly" as the
parameters relative to layers 1, 3 and 4.
Figure
6-12 shows the comparison between the measured field
data and the computed horizontal displacements when parameters
are optimized based on stage 1 observations. The improvement
on the fit between the computed and measured response is
significant. Despite the fact that the optimized set of
parameters is calculated using only stage 1 observations,
the positive influence on the calculated response is substantial
for all construction stages. At the end of the construction
(i.e. stage 5) the maximum computed displacement exceeds
the measured data by only about 15%. These results are
significant in that a successful recalibration of the model
at an early construction stage positively affects subsequent "predictions" of
the soil behavior throughout construction.
Table 6-10 shows
the value of the following statistics that indicate the
model fit to the field observations: S(b)in (initial
objective function), S(b)fin (final objective
function), s2fin (final error variance),
CCfin (final correlation coefficient) and FI (fit
improvement). The values of these statistics prove that the
calibration of the finite element simulation by inverse analysis
based on stage 1 observations was extremely effective. The
improvement over the initial predicted response is considerable
(FI = 99.6%) and the final computed response fits the observations
better than one would expect if the inclinometer measurement
errors are taken into account (s2fin < 1.0).
Table 6-10 Model fit statistics at optimization stage
1
Note that the statistics presented in Table 6-10 refer to
stage 1 observations only. Thus, they cannot be used to assess
how the calibrated simulation improves the fit between measured
and computed response for the other construction stages.
To quantify the "predictive" improvement of the
calibration shown in Figure
6-12, one needs to consider two new model statistics:
the global objective function, OF, and the predictive fit
improvement, PFI.
OF= [y-y'*]T ω[y*-yi*]
(6.19)
 (6.20)
where y* is the vector of all available
observations (from stage 1 to stage 5); yi*is
the vector of the composed values which correspond to the
observations;
and ω is
the weight matrix used in regression, PFI, is the predictive
fit improvement at the end of stage 1, OFbase is
the global objective function relative the initial estimates
of the parameters and OF is global objective function relative
to the parameters on the optimization up to stage i.
The global
objective function, OF, is computed according to Eq. 6.1
using all the observations available even if they
are not used during the regression analysis of the simulation.
The predictive fit movement, PFIi, is computed
using the value of OF after the optimization (at stage
i) and the value of OF relative to the intial estimates
of the
parameters. Table 6-11 shows the values of the intial OF,
the global objective function at the end of stage 1 and
the predictive fit movement at the end of stage 1.
Table 6-11 Model statistics quantifying the predictive
improvement achieved by the calibration
6.4.3.2 Model fit for all construction stages
Section 6.4.3.2 presented the optimization results relative
to the calibration of the simulation after construction stage
1. Parameters were recalibrated at every stage until the
final construction stage (i.e. stage 5). At every new construction
stage, the inclinometer data relative to that stage were
added to the observations already available. Figure 6-12
showed the visual fit comparison between inclinometer data
and computed displacements when observations relative to
stage 1 were used in the regression analysis. Figures 6-13, 6-14, 6-15 and 6-16 show
the comparison between experimental and computed results
for the remaining 4 stages (i.e. stage 2 to stage 5, respectively).
When
stage 2 observations are used for the regression analysis
(Figure 6-13) results do not change significantly. This
indicates that the inclinometer readings relative to this
construction
stage do not provide significant extra information to improve
the model calibration. When inclinometer data from stage
3 are added to the observations (Figure 6-14), the fit
between the field and the computed results relative to this
stage
improves, especially in the upper soil layer. These results
also improve the fit of stages 4 and 5. That is why observations
from these last two construction stages (Figures 5-18 and
5-19) do not produce any change in the model. The calibrated
model cannot be improved any further after stage 3.
Overall,
the comparison between the measured field data and the computed
horizontal displacements at the different
optimization stages shows that the fit between the two sets
of data is always remarkable. The inverse analysis performed
after the first construction stage significantly improves
the initial prediction of displacements and "recalibrates" the
soil models in a way that allows them to capture the main
behavior of the soil layers throughout construction. By the
end of construction stage 3, the evolution of the horizontal
movements at the inclinometer locations is predicted with
great accuracy at all stages.
These results demonstrate that
using inverse modeling techniques enhances the observational
method practice. An engineer with
detailed knowledge of finite element procedures and constitutive
modeling, instead of performing a regression analysis, could
modify the input soil parameters by trial-and-error. However,
processing displacement data and using them to calibrate
the results of the finite element model is an extremely time-consuming
and highly subjective task. Figure
6-17 shows the comparison between the measured field
data and the computed horizontal displacements in a simulation
for which the "optimal" parameter values were
estimated by trial-and-error. This research task was performed
before developing the procedure described in this thesis.
The fit between computed and observed data is relatively
good, yet not every stage of construction is simulated with
the same accuracy reached by the inverse analysis. Computed
results match the maximum movements well, but they tend to
overpredict the displacements of the stiffer layers.
6.4.3.3 Best-fit parameters
Table 6-12 shows the initial and optimized values of the
5 input parameters at the different optimization stages.
Results show that, after construction stage 3, the values
of the optimized input parameters do not change. This indicates
that the observations at stages 4 and 5 "match" the
computed results of the model calibrated at stage 3 and therefore
further optimization of the model is neither possible nor
necessary.
Table 6-12 Best-fit values of input parameters at various
optimization stages
In Figure
6-18 the variation of the input parameters at the various
optimization stages is shown above a bar chart representing
the progress of the excavation. The excavation depth is
normalized with respect to the excavation width and plotted
for the 5 construction stages. Results show that the maximum
changes in parameter values occur at stage 1, when the
observations relative to the installation of the secant-pile
wall are used. Note that the excavation was 74 ft wide,
the secant pile wall was 60 ft deep and the maximum excavation
depth was 39 ft below ground surface.
From the results presented
in Table 6-12 and Figure 6-18 the following can be inferred:
1. The initial estimates of the stiffness
parameters are significantly lower than the optimized values
of the parameters.
This is due to the fact that the initial values are based
on triaxial compression test results, whereas most of the
soil around the excavation experiences different stress
paths (mainly undrained reduction of stresses).
2. The parameters
that vary the most are the ones corresponding to the
stiffer clay layers. The stiffer the clay, the more the
laboratory test results are
affected by sample disturbance and by the underestimation of stiffness
values due to global measures of strains in a triaxial
apparatus.
3. The simulated excavation is a 3D problem
modeled in plane strain. When the excavated depth is small,
most
of the wall can be adequately modeled
as plane strain and hence little changes in parameters are noted between
stage 1 and 2. As the excavation deepens (stage 3), the ratio between
excavation depth and excavation width increases and higher
parameter values compensate
for the lack of constraints in the out-of-plane direction.
4. The highest
changes in parameter values occur at stage 1. The fit
between computed and field displacement after stage 1 is
extremely
satisfactory.
Thus, the calibrated parameters only need to change slightly at later
stages.
By the end of stage 3 the model essentially is calibrated.
6.4.3.4 Model statistics for all construction
stages
The inverse analysis procedure produces very important by-products
that allow one to quantify the effectiveness of the optimization
procedure and assess the reliability of the predictions.
Table 6-13 shows the values of a series of model fit statistics
for the 5 optimization stages. Most of the statistics presented
have been already defined elsewhere in this work, yet for
convenience their expressions will be presented again in
this section. Results show that the calibration improved
the fit between observations and computed results up to stage
3, after which there was no improvement.
Table 6-13 Model statistics of inverse analysis at various
optimization stages
 (6.21)
 (6.22)
 (6.23)
 (6.24)
 (6.25)
 (6.26)
 (6.27)
 (6.28)
 (6.29)
where b is the vector of the estimated parameters, y i is
the vector of the observations used at stage i, y'(b)base_i is
the vector of the results at stage i computed using the initial
(base) estimates of the parameters, ω is
the weight matrix, y'(b)in_i is
the vector of the results at stage i computed using the estimates
of the parameters at the beginning of that stage, y'(b)fin_i is
the vector of the results at stage i computed using the optimized
estimates of the parameters at that stage, NDi is
the number of observations used at stage i, NP is the number
of estimated parameters, y all is the vector
of all observations, y'(b)all_i is
the vector of all results computed using the optimized estimates
of the parameters at stage i, and OFbase is the
global objective function computed using the initial (base)
estimates of the parameters.
Figure
6-19 shows the base and the final values of the error
variance at the various stages. The graph allows one
to compare the overall magnitude of weighted residuals
relative
to the initial estimates of the parameters with the fit
resulting from the calibrated models. Results show that
error variance values decrease by more than an order
of magnitude at every stage. This indicates a significant
improvement in the fit between observed and computed
displacements.
The final error variance values are always very close
to 1.0. This indicates that the fit is consistent with
the
weighting of the observations, and thus with the measurement
errors associated with the inclinometer data.
Figure
6-20 shows the values of relative fit improvement,
RFI, at the various stages and the values of the initial
and final objective functions from which it is computed.
The RFI indicates by what percentage the optimized results
improved compared to the predictions at the beginning of
a given stage. Results show that stage 1 observations improved
the predictions by 2 orders of magnitude (i.e. RFI equal
99%) and that, by the end of stage 3, the "recalibration" of
the model is complete.
Figure
6-21 shows the values of the global objective function,
OFi, and the values of the predicted fit improvement,
PFIi, at the various stages. These statistics
can be used to quantify the "predictive" capabilities
of the analysis at the end of a given optimization stage.
As one would expect, the global objective function values
decrease as more observations become available (i.e. from
stage 1 to stage 5). However, the PFI values indicate than
most of the improvement "happens" at stage
1 and nothing is "gained" by including the
observations of stages 4 and 5. These results are extremely
promising because they indicate that early stage observations
are able to recalibrate the finite element simulation in
a way that is beneficial to the predictions at later stages.
6.4.3.5 Comments on the calibrated model
In section 6.4.3.5 a procedure was presented to compare
the initial stiffness-to-strength ratios, E50/Su,
to typical values of undrained E/Su ratios. Figure
6-22 shows the values of the initial and optimized E50/Su ratios
of the 5 soil layers at different optimization stages. In
Table 6-14 the values of the final equivalent undrained ratios,
(Eu/Su)equivalent, are presented. Note
that: (i) the stiffness E50 was computed from
the values of parameter E50ref (see
Table 6-12) according to Eq. 6.13 and considering the stress
conditions in the middle of the layers on the two sides of
the excavation at the inclinometer locations, and (ii) the
equivalent undrained ratios, (Eu/Su)equivalent,
were computed from the optimized values of E50/Su following
the procedure described in section 6.4.1.2 and assuming G50=Gin/3
and ν=0.2.
Table 6-14 Stiffness-to-strength ratios for the best-fit
values of the input parameters
Table 6-14 and Figure 6-22 showed that the optimized values
of the stiffness-to-strength ratios increase with depth and
are significantly higher than the initial values of the ratios.
Note that the final values of (Eu/Su)equivalent
for layers 1, 2 and 3 (i.e. Bloldgett and Deerfield layers)
are close to values reported by Finno and Chung (1992) for
normally consolidated compressible Chicago clays (i.e. 400-600).
In Figure
6-23 the E50/Su ratios are plotted
as a function of Su/σ'1.
The values in the figure refer to the optimized values
of the parameters (see Table 6-14). Results show that the
value of E50/Su increases linearly
with Su/σ'1.
The following expression fits the data extremely well:
 (6.30)
Eq. (6.30) can be conveniently used to estimate, for a given
clay layer, the value of E50 from estimates of
Su and σ'1,
which are generally more easily available.
In Figure
6-24 the optimized E50/Su ratios
are plotted as a function of (σ'1-σ'3)/2Su,
an approximate measure of the relative shear stress of
the layers. Results show that the value of E52/Su decreases
as (σ'1-σ'3/25u increases,
until the latter reaches the value of 1.0. Note that the
slightly higher than 1.0 (σ'1-σ'3/2Su)
values reported in the graph are possible because the undrained
shear strength, Su , is not an input parameter
of the H-S model, thus its estimate does not influence
the "exact" relative shear stress (whose maximum
value is 1.0).
A linear function can be used to interpolate
the data:
 (6.31)
If
we assume that, initially, the vertical and horizontal directions
are principal directions the (σ1-σ3)/Su ratio
can be written as:
 (6.32)
where σν is
the vertical effective stress and ko the coefficient
of earth pressure at rest.
Equations 6.31 and 6.32 can be combined
to produce an equation to use for estimating the stiffness
modulus E50 from
values of Su, k0 andσν:
 (6.33)
6.5 SUMMARY
This chapter presented a numerical procedure that uses construction
monitoring data to "recalibrate", by inverse
analysis, the input parameters of a finite element model
of a supported excavation. The inverse analysis methodology
was developed and tested using data from a 39 ft deep excavation
through soft clays in Chicago. The inverse analysis algorithm
UCODE was used to optimize the PLAXIS plane-strain finite
element simulation of the excavation. Five clay layers, modeled
using the Hardening-Soil model, were calibrated using inclinometer
data that measured lateral movements of the soil behind the
supporting walls. One stiffness parameter per layer was optimized
in the regression analysis. The other model parameters were
either kept constant at their initial value or related to
the updated optimized parameters.
For the first three construction
stages, the set of parameters converged to a new set of optimized
values based on the observations
available up to that stage. The improvement on the fit between
the computed and measured response was always significant.
When monitoring data relative to the first construction stage
(i.e. installation of the wall) were used, the recalibrated
model proved able to adequately "predict" the
behavior of the soil for all construction stages. These results
are significant in that a successful recalibration of the
model at an early construction stage positively affected
the predictions of the soil behavior throughout construction.
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