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298
IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 3
Behavior prediction models and control charts for safety control of concrete dams
(2)
Ʃ
f β(
[
[
(
0
,β
1
,…,β
n
=
y
i
i
-
(β
0
+ β
1
X
i1
+ β
2
X
i2
+…+ β
n
X
in
) ²
where
f
(
β
0
,β
1
,…,β
n
)
must be minimized with respect to
b
0
,
b
1
,...,
b
n
.The least squares estimates of
b
0
,
b
1
,...,
b
n
must satisfy:
(3)
ϑf β
0
,β
1
,…,β
n
ϑβ
n
= 2
Ʃ
y
i
- β
0
- β
1
X
i1
- β
2
X
i2
-…- β
n
X
in
(-X
in
) = 0
By rearranging the equations and putting them in matrix notation,
one obtains:
(4)
β
?
=(X
T
X)
-1
X
T
y
Where the i-th line of matrix X is formed by the vector
[1 X
i1
X
i2
X
i3
… X
in
]
, which corresponds to variables that influ-
ence the response
y
^
i
. Vector y corresponds to displacements to-
ward the flow direction of block TA-2.
For the estimate of least squares, errors are assumed to be statis-
tically independent, and to have zero mean and constant variance.
If these conditions are met, the coefficients
β
^ ^ ^ ^
0
,β
1
,β
2
,…,β
n
can be
considered unbiased estimators of the regression coefficients.
The adjustment of the model to the desired values is measured quanti-
tatively by the coefficient of multiple determination, R
2
. This coefficient
measures the ability of the regression model to explain the variation
of
y
^
i
. R
2
values close or equal to 1 mean that there is good adherence
between the model and the experimental data, while R
2
values close
to zero mean that the model is not suitable for the prediction. The
coefficient of multiple determination is obtained as follows:
(5)
R
2
= 1 –
SQE
SQT
(6)
Ʃ
SQE=
(
y
i
- y
i
)²
(7)
Ʃ
SQT=
(
y
i
-
–
y
)²
where:
SQE is the quadratic sum of the error;
SQT is the total quadratic sum of the deviations;
y is the mean of observations.
throughout its phases, but also to verify the assumptions made in
the design phase, making the construction works more economi-
cal. In the TA-2 block of the Tucuruí Hydroelectric Power Plant,
the auscultation instruments installed include the direct pendulum,
piezometers and triorthogonal joint gauges.
The readings provided by the instruments can describe dam be-
havior quite reliable. The data measured by the instruments may
show if the structure is being acted in such a way that its design
limits might be exceeded. Consequently, steps can be taken in
order to bring the dam back to normal operating condition.
According to historical readings, it is possible to notice that some
instruments begin to show anomalous readings over time. This of-
ten occurs because the instrument has suffered wear and fatigue,
i.e., it has exceeded its useful life.
In the case of the HPP Tucuruí, the frequency of readings is set
by EEGE (Eletronorte Geotechnical and Structures Management)
[2].The frequency of readings varies for each instrument, and can
be daily, weekly, biweekly, monthly, bimonthly, quarterly or every
six months.
According to [8], in terms of structural behavior, assessment of
safety conditions can be conducted by comparing the collected
readings to the estimated values based on predictive behavior
models. The results estimated by the prediction models can be
taken as an indication of the behavior considered normal for the
structure in the future.
The parameters set by the statistical models should be reevalu-
ated at regular time intervals, as changes in behavior might occur
due to changes in the characteristics of the structure or founda-
tion. Because of these behavior changes, it becomes difficult to
define normal and abnormal behavior, because according to [12],
as quoted in [5], one can always make two types of errors: the first
corresponds to a judgment of a false normality and the second cor-
responds to a false judgment of abnormality.
4. Methodology
4.1 Multiple regression
Multiple regression is used to develop an empirical model that as-
sociates a dependent variable Y to more than one independent
variable. The model equation is as follows:
(1)
^y
i
=β
0
+ β
1
X
i1
+ β
2
X
i2
+…+ β
n
X
in
+ ϵ
i
where:
n is the number of values for the historical series that was
analyzed;
y
^
i
is the dependent variable (i = 1,2, ... n);
X
ik
are the independent variables (k = 1, ... n);
β
0
,β
1
,β
2
,…,β
n
are the regression coefficients;
Є
i
is the random error.
Multiple regressions should provide a curve that has the best ad-
herence possible to the observed data. According to [13], this ad-
hesion can be obtained through the Least Squares Principle, which
aims to reduce the sum of squared deviations between the desired
values and the estimated values, as shown in equation [2]:
–