COMPUTATIONAL METHODS IN ENGINEERING AND SCIENCE
EPMESC X, Aug. 21-23, 2006, Sanya, Hainan, China
c
2006
Tsinghua University Press & Springer
An Extremely Efficient Finite-Element Model Updating Methodology
with Applications to Damage Detection
Ka-Veng Yuen*
Department of Civil and Environmental Engineering Engineering, University of Macau, Macao, China
Email: [email protected]
Abstract: This paper presents a finite-element model updating methodology using noisy incomplete measurement of
the natural frequencies and mode shapes with applications to damage detection. The proposed method does not require
matching between the measured and calculated modes from the finite-element model, which is in contrast to most
of the existing methods. Furthermore, the proposed iterative scheme is computationally efficient without nonlinear
optimization programming. A ten-story building and a three-dimensional braced frame are used to demonstrate the
proposed approach with applications to damage detection.
Key words: damage detection, modal testing, model correction, model updating, structural health monitoring
INTRODUCTION
Damage detection/structural health monitoring has been attracting much attention in the past two decades, including
several workshops, e.g., [1-3]; and special issues of journals, e.g., Journal of Engineering Mechanics (July 2000
and January 2004) and Computer-Aided Civil and Infrastructure Engineering (January 2001 and May 2006). Many
methods have been developed including the class of direct methods using pattern recognition techniques [4-6] and
the class of structural model-based inverse methods [7-11].
In the existing methods, they usually involve an optimization problem of a measure of the difference between the
measured quantities, modal frequencies and mode shapes, and those calculated from the finite-element model. A
generic form of this objective function can be written as follows:
m
m
i=1
i=1
J = ∑ αi (ω2i − ω̂2i )2 + ∑ βi ||φi − φ̂i ||2
(1)
where ω̂i and φ̂i are the measured natural frequency and mode shape of the ith mode; ωi and φi are the model natural
frequency and mode shape of the ith mode; and αi and βi are the weighting, where i = 1, 2, . . . , m. Note that one
of the major difficulties is on the mode matching, that is, it is necessary to determine which model mode matches
which measured mode. In the case when only measurements of partial mode shapes are available, this is not an easy
task. Another major difficulty is that the m measured modes might not necessary be the lowest m modes in practice.
Furthermore, in the case when there is possible damage in the structure, the order of the modes might switch, which
increases the difficulty of this problem. The first attempt for solving this model updating problem without mode
matching was made in [10] and [12] with the Bayesian probabilistic framework.
This paper presents a very efficient finite-element model updating method using incomplete measured frequencies and
mode shapes with applications to damage detection. The frequencies and mode shapes are measured. The proposed
method does not require any mode matching between the model modes and measured modes. Furthermore, the proposed method operates in an efficient iterative procedure instead of nonlinear optimization/searching.
First, let’s introduce the key idea of the proposed approach with a simple ideal case. Consider the eigenvalue problem
of a structure with d degrees-of-freedom (DOFs):
Kφ1 = ω21 Mφ1
(2)
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where M and K are the d × d mass and stiffness matrix of the structure, respectively; ω1 and φ1 are the natural frequency
and mode shape of the first mode.
The stiffness matrix K can be parameterized as follows:
n
K = K0 + ∑ θ j K j
(3)
j=1
where θ = [θ1 , θ2 , . . . , θn ]T is the uncertain/unknown parameter vector that governs the stiffness matrix and the matrices
K j , j = 0, 1, . . . , n, are known. Then, the eigenvalue problem becomes:
K1 φ1 , . . . , Kn φ1 θ = (ω21 M − K0 )φ1
(4)
Therefore, the stiffness parameter vector θ is readily obtained:
−1 2
θ = K1 φ1 , . . . , Kn φ1
(ω1 M − K0 )φ1
(5)
However, complete mode shape measurements are not available in practice. Furthermore, the measurements are contaminated by measurement noise. In the next section, the formulation of the problem is described. Then, an expanded
mode shape technique is introduced following with a method for fine tuning the noisy measurement of the natural frequencies. Then, the stiffness parameter updating methodology will be introduced. Finally, the proposed approach will
be presented which is an iterative process, that updates the mode shapes, natural frequencies and stiffness parameters
alternately. Examples with a ten-story building and a three-dimensional braced frame will be used to demonstrate the
proposed method with applications to damage detection.
PROBLEM DESCRIPTION
Assume that m modes are measured. Note that they are not necessary the first m modes. Consider the eigen-equation
of the ith measured mode of a structure with d DOFs:
Kφi = ω2i Mφi , i = 1, 2, . . . , m
(6)
where M and K are the d × d mass and stiffness matrix of the structure; and ωi and φi are the natural frequency and
mode shape of the ith measured mode.
Use ω̂i and φ̂i to denote the measurements of the natural frequency and mode shape for the ith measured mode, respectively. These measurements are contaminated by measurement noise, that is, there is a difference between the measured
and the actual frequency and mode shape. Let δ denote the difference operator. For example, δω2i is the difference, or
measurement noise, between the actual and measured natural frequency of the ith measured mode:
δω2i = ω2i − ω̂2i
(7)
Similarly, the difference between the actual and measured mode shape of the ith measured mode is given by
δφi = φi − φ̂i
(8)
In the case when a component of the mode shape is unmeasured, one can simply assign zero to its measured value.
Then, all these random differences are grouped into an error vector:
ε = [δω21 , δω22 , . . . , δω2m , δφT1 , δφT2 , . . . , δφTm ]T
(9)
For an unbiased estimate of the modal quantities and the stiffness parameters, this error vector has zero mean with
covariance matrix:
Σω2 Σω2 ,φ
T
(10)
E[εε ] = Σε = T
Σω2 ,φ Σφ
which can be obtained by Bayesian modal identification methods [13-16]. Note that the error of the modal frequencies/mode shapes of different modes might be correlated. For the unobserved components of the mode shapes, one can
simply assign a large value for their variance without correlation with other modal quantities.
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By using a substructuring approach, the stiffness matrix K can be parameterized as follows:
n
K = K0 + ∑ θ j K j
(11)
j=1
where θ = [θ1 , θ2 , . . . , θn ]T is the unknown parameter vector that governs the stiffness matrix and the substructural
stiffness matrices K j , j = 0, 1, . . . , n, are known up to a scaling θ j .
Then, in the similar fashion as in Eq. (7) and (8), the difference between the identified values and the nominal values θ̃
for the stiffness parameter vector is denoted as
δθ = θ − θ̃
(12)
Therefore, the difference between the actual and identified stiffness matrix depends on the difference between the actual
and identified stiffness parameters δθ as follows:
δK = K − K̃ =
n
∑ (θ j − θ̃ j )K j =
j=1
n
∑ δθ j K j
(13)
j=1
MODE SHAPES EXPANSION
If sensors are available at s DOFs, the mode shape components can be measured at these DOFs only. In order to recover the unmeasured components, many existing mode shape expansion techniques were proposed. A comprehensive
summary and comparison among them was reported in [17]. Note that in the existing methods, the measured mode
shape components are fixed at their observed values. Therefore, the expanded unobserved mode shape components are
sensitive to the measurement noise for the observed components.
In the following, a ‘flexible’ and computationally efficient method is presented. It does not completely constrain the
observed values. Instead, the measured components will be probabilistically constrained by the measurements. In other
words, the measured components can be different from their observed values and the allowable difference is controlled
by the covariance of the measurement uncertainty. This is expected to improve the accuracy of the expanded mode
shapes.
Now, the mode shape φi is updated using Eq. (6) but the stiffness parameters θ, and hence the stiffness matrix K, and
the natural frequency ω2i are uncertain. By using Eq. (7) and (13), Eq. (6) can be rewritten as
n
(K̃ − ω̃2i M)φi = δω2i Mφi − ∑ δθ j K j φi
(14)
j=1
where ∑nj=1 δθ j K j φi can be rewritten in terms of δθ:
n
∑ δθ j K j φi = [K1φi , . . . , Kn φi ]δθ
(15)
j=1
Define the equation error εeq,φ = δω2i Mφi − [K1 φi , . . . , Kn φi ]δθ, which is the right hand side of Eq. (14). Therefore, the
covariance matrix of the equation error Σeq,φ is given by
T
Σeq,φ = F(1) Σω2 F(1) + F(2) Σθ F(2)
T
(16)
where the matrix Σθ is the covariance matrix of the stiffness parameter vector by prior information. In the case when
there is no prior information, large values can be assigned to the variances. The matrices F(1) and F(2) are given by


Mφ1
0


Mφ2


F(1) = 
(17)

.
.


.
0
−K1 φ1
 −K1 φ
2

F(2) =  .
 ..

Mφm
−K2 φ1
−K2 φ2
..
.
···
···
..
.
−K1 φm −K2 φm · · ·
dm×m

−Kn φ1
−Kn φ2 

.. 
. 
−Kn φm
(18)
dm×n
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Note that even though the eigen-equations of different modes are uncoupled, the equation errors are correlated.
Then, the expanded mode shapes are obtained by minimizing the following objective function:
 
T


T

φ1 − φ̂1
φ1 − φ̂1
(K̃ − ω̃21 M)φ1
(K̃ − ω̃21 M)φ1
 (K̃ − ω̃2 M)φ   φ − φ̂ 
 φ − φ̂ 
 (K̃ − ω̃2 M)φ 
2  2
2
2
2
2
2

−1  2
−1 
Jφ = 
Σ
+
Σ





..
..
..
.. 
eq,φ
φ 







.
.
.
. 

(K̃ − ω̃2m M)φm
(K̃ − ω̃2m M)φm
φm − φ̂m
(19)
φm − φ̂m
where φ̂i , i = 1, ..., m is the mode shape measurement of the ith measured mode. For those unmeasured components,
zero value can be assigned. Σφ is the covariance matrix of all the mode shapes. Minimizing this objective function
is equivalent to minimizing the equation error with probabilistic constraint for the measured components of the mode
shape. It allows the measured components to vary from the measured values but the variation will be constrained by a
scale of its standard deviation of the error in the measurement.
It can be easily shown that the updated mode shapes (recovering the unmeasured components and fine tuning the
measured components) is given by
 
 
φ̂1
φ̃1
 φ̃ 
 φ̂ 
 2
(3)
−1 −1  2 
(20)
+ Σ−1
 ..  = (F(3) Σ−1
eq,φ F
φ ) Σφ  .. 
 . 
 . 
φ̃m
φ̂m
where the (symmetric) matrix F(3) is given by
 2
ω̃1 M − K̃
0
2

ω̃2 M − K̃

F(3) = 
..

.





2
ω̃m M − K̃ dm×dm
0
(21)
FINE TUNING OF THE MEASURED NATURAL FREQUENCIES
Note that the frequencies are measured with noises so it is possible to further improve the prediction by the following
procedure. Now, the squared natural frequency ω2i is updated using Eq. (6) but the stiffness parameters θ (and so
the stiffness matrix K) and the mode shape φi are uncertain. By using Eq. (8) and (13), Eq. (6) can be rewritten by
neglecting the second order error:
n
(K̃ − ω2i M)φ̃i = (ω2i M − K̃)δφi − ∑ δθ j K j φ̃i
(22)
j=1
where ∑nj=1 δθ j K j φ̃i can be written in terms of δθ:
n
∑ δθ j K j φ̃i = [K1φ̃i , . . . , Kn φ̃i ]δθ
(23)
j=1
Define the equation error εeq,ω2 = (ω2i M − K̃)δφi − [K1 φ̃i , . . . , Kn φ̃i ]δθ, which is the right hand side of Eq. (22). Therefore, the covariance matrix of the equation error Σeq,ω2 is given by
T
Σeq,ω2 = G(1) Σω2 G(1) + G(2) Σθ G(2)
T
(24)
where the matrices G(1) and G(2) are given by
 2
ω1 M − K̃
0
2 M − K̃

ω
2

G(1) = 
..

.
0





2
ωm M − K̃ dm×dm
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(25)
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−K1 φ̃1
 −K1 φ̃
2

G(2) =  .
 ..

−K2 φ̃1
−K2 φ̃2
..
.
···
···
..
.
−K1 φ̃m −K2 φ̃m · · ·

−Kn φ̃1
−Kn φ̃2 

.. 
. 
−Kn φ̃m
(26)
dm×n
Then, the natural frequencies can be updated by minimizing the following objective function:
Jω2
  2
T
 2

T

(K̃ − ω21 M)φ̃1
(K̃ − ω21 M)φ̃1
ω1 − ω̂21
ω1 − ω̂21
 (K̃ − ω2 M)φ̃   ω2 − ω̂2 
 ω2 − ω̂2 
 (K̃ − ω2 M)φ̃ 
2
2  2
2
2
2
2

−1  2
−1 
=
 Σeq,ω2 
+
 Σω 2 

..
..
..
..
 






.
.
.
.

(K̃ − ω2m M)φ̃m
(K̃ − ω2m M)φ̃m
ω2m − ω̂2m
(27)
ω2m − ω̂2m
where ω̂i , i = 1, ..., m is the measurement of the natural frequency of the ith measured mode. Σω2 is the covariance
matrix of all the measured frequencies. Minimizing this objective function is equivalent to minimizing the equation
error with probabilistic constraint for the measured frequencies. It allows them to be different from the measured values
but the variation will be constrained by a scale of the standard deviation of the error in the measurement.
By minimizing Jω2 with respect the ω2i , the natural frequencies can be fine tuned as follows:

 2

 2
 T
ω̂1 !
v1
K̃φ̃1
ω̃1
 K̃φ̃ 
 ω̂2 
 ω̃2 
vT 
 2
 2
 2
−1  2 
)−1  .  Σ−1
 ..  = (G(3) + Σ−1
2  .  + Σω2  . 
ω2
 .. 
 ..  eq,ω  .. 
 . 
ω̃2m
ω̂2m
K̃φ̃m
vTm
where the matrix G(3) is given by
 T −1
v1 Σeq,ω2 v1 vT1 Σ−1
v
eq,ω2 2
vT Σ−1 v vT Σ−1 v
 2 eq,ω2 1 2 eq,ω2 2
G(3) = 
..
..

.
.

−1
−1
T
T
vm Σeq,ω2 v1 vm Σeq,ω2 v2
···
···
..
.
···
(28)

vT1 Σ−1
v
eq,ω2 m
vT2 Σ−1
v 
eq,ω2 m 

..

.

−1
T
vm Σeq,ω2 vm
(29)
and the vector vi , i = 1, 2, . . . , m, is given by
vi = [01×(i−1)d , (Mφ̃i )T , 01×(m−i)d ]T
(30)
that is, vi is a zero vector except that the sub-vector that includes the (i − 1)d + 1 to id elements is equal to Mφ̃i .
UPDATING OF THE STRUCTURAL PARAMETERS
Consider the eigen-equation for the ith measured mode in Eq. (6). Note that the natural frequencies and mode shapes
are uncertain. By using Eq. (7), (8) and (11), Eq. (6) can be rewritten by neglecting the second order error:
(31)
K1 φ̃i , . . . , Kn φ̃i θ = (ω̃2i M − K0 )φ̃i + δω2i Mφ̃i + (ω̃2i M − K)δφi
Here, the updated values are used for the natural frequencies and mode shapes instead of the measurements. If these
equations for all measured modes are grouped together, one can obtain the following equation:
Aθ = b + δb
(32)
where the matrix A is given by


K1 φ̃1 K2 φ̃1 · · · Kn φ̃1
 K1 φ̃ K2 φ̃ · · · Kn φ̃ 
2
2
2

A= .
..
.. 
.
.
.
 .
.
.
. 
(33)
K1 φ̃m K2 φ̃m · · ·
Kn φ̃m
dm×n
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the vector b is given by
(34)
and the equation error vector δb is given by
(35)
where ε is the error vector given in Eq. (9) and the matrices
and
are given by
(36)
(37)
Therefore, the covariance matrix of the equation error is given by
(38)
where the matrix
is the covariance matrix of ε given in Eq. (10).
It is shown in the Appendix that the optimal estimate of the stiffness parameter vector θ is given by
(39)
is the same as
, except that it replaces the zero eigenvalues of
by a small value. The
where the matrix
matrix
is invertible if enough constraints (number of sensors and measured modes) are available. Otherwise,
more sensors or measured modes are needed.
Note that the least squares solution is a special case of Eq. (39) under the condition when
:
(40)
Uncertainty Estimation The uncertainty of the estimation can be quantified by its covariance matrix which is given
by
(41)
The diagonal elements give an estimation of the variance for each stiffness parameter.
PROPOSED ITERATIVE APPROACH
The proposed methodology updates the natural frequencies, mode shapes and stiffness parameters in an alternating
manner with the aforementioned methods. It is recommended to operate in the following order:
1) Assume an initial trial vector for
and calculate the stiffness matrix
with .
2) Expand and fine tune the mode shapes , i = 1, 2, . . . , m, using Eq. (20). Note that this mode shape expansion
technique can be used for reducing measurement noise even when complete measurements of mode shapes are
available.
3) Fine tune the squared natural frequencies
, i = 1, 2, . . . , m, using Eq. (28).
4) Update the estimation of the stiffness parameter vector
negative, replace it by the value in the last iteration.
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by using Eq. (39). In the case if a component becomes
__
5) Iterate the previous steps 2-4 until the stiffness parameter vector
converges.
6) Calculate the covariance matrix of the estimates of the stiffness parameters using Eq. (41).
It is recommended to skip step 3 in the first few, say 10, iterations in order to have faster convergence.
ILLUSTRATIVE EXAMPLES
1. Example 1: Ten-story Building In this example, a ten-story building is considered. It is assumed that this building
has uniformly distributed floor mass and stiffness across all floors. The mass per floor is taken to be 100 tons, while the
interstory stiffness is chosen to be k0 = 176.729 MN/m so that the first five modal frequencies are 1.00, 2.98, 4.89, 6.69
and 8.34 Hz. The coefficient of variation of the measurement error of the squared natural frequencies and mode shapes
are taken to be 1.0% for all modes.
The initial trials of the stiffness parameters are taken to be uniformly distributed from 2k0 to 3k0, where k0 is the actual
interstory stiffness. In this case, the initial trial is significantly overestimated and the variation between different
interstory stiffness is substantial. Table 1 shows the initial trial values and identified values of the stiffness parameters
using different number of measured modes, where the actual value is 176.729 MN/m for all these parameters. The
estimated standard deviations, calculated using Eq. (41), are shown in parenthesis. In these cases, complete
measurements at all DOFs are utilized. It is not surprising that the uncertainty reduces when the number of measured
modes increases.
Table 1 Identification results using different number of measured modes (Example 1)
Incomplete mode shape measurements: Consider only five sensors on the first, fourth, fifth, seventh and top floor.
Table 2 shows the initial trial values, identified values, standard deviation and the coefficient of variation (C.O.V.) of
the stiffness parameters. Fig. 1 shows the iterative history of the identification process. It converges with 50
is smaller than 1.0×10−6 compared to those estimated after 1000 iterations. The CPU
iterations. The difference in
time is about one second with a normal personal computer with 2.4 GHz under the MATLAB environment [18]. It is
seen that the iteration virtually converges after ten iterations.
Table 2 Identification results with measurements of five sensors and five modes (Example 1)
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Figure 1: Iteration history for the stiffness parameters with incomplete measurements of mode shapes (Example 1)
2. Example 2: Three-dimensional braced frame The proposed method is applied to update the finite-element model
of a three-dimensional five-story braced frame. It has a square section with width a = 4 m. There are four columns for
each floor, one at each corner. Each of them have interstory stiffness 10 MN/m and 15 MN/m in the x and y direction,
respectively. Furthermore, each face in each floor is stiffened by a brace and its stiffness is taken to be 20 MN/m. As a
result, the interstory stiffness is 80 MN/m and 100 MN/m in the x and y direction, respectively. The floor mass is taken
to be 10 tons for each floor. As a result, the first five natural frequencies of the structure are 4.05, 4.53, 7.44, 11.83 and
13.22 Hz.
Figure 2: Floor plan for the 12-DOF model for identification (Example 2)
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In order to locate the face(s) that sustain damage, four stiffness parameters are used for each story to give twenty
and
, l = 1, 2, . . . , 5,
stiffness parameters,
where the index l represents the story number and ‘+x’, ‘+y’, ‘−x’ and ‘−y’ represent the direction of the outward
MN/m and
MN/m, l = 1,
normal of a face. The actual values of these stiffness are
2, . . . , 5. In other words,
MN/m and
MN/m. The floor plan is shown
is the stiffness center of story l, where and , l = 1, 2, . . . , 5, are given by
in Fig. 2. The point
(42)
The stiffness matrix for story l with respect to the
coordinates is
(43)
(44)
Here, the first three DOFs and the last three DOFs correspond to the lower floor and the upper floor of the story,
respectively. Each of these sets of three DOFs correspond to the x-translational, y-translational and torsional motion.
The stiffness matrix for story l with respect to the (1), (2) snd (3) DOFs in Fig. 2 of the upper and lower floor is given
by
(45)
where
is given by
(46)
The stiffness matrix for the 12-DOF structural model is assembled from those of the floors. The DOFs for this stiffness
matrix are the (1), (2) and (3) shown in Fig. 2 for each floor. However, this stiffness matrix is not linear to the
stiffness parameters . In this case, one can linearize the relationship between the stiffness matrix and the stiffness
parameter:
(47)
where
(48)
and
(49)
where the matrix K0 will need to be updated in every iteration.
To increase the difficulty, it is assumed that only first three x-directional and y-directional modes are measured but not
any of the torsional modes. This is done deliberately to simulate the reality that some of the modes might not be excited
so they are not able to be measured. In the identification, it is assumed that we do not know there are missing modes.
These measured modes correspond to the 1st (4.05 Hz), 2nd (4.53 Hz), 4th (11.83 Hz), 5th (13.22 Hz), 7th (20.84 Hz)
and 9th (23.95 Hz) mode. Sensors are placed at the +y, −x and −y face of the 1st , 3rd and 4th floor to measure the
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natural frequencies and mode shapes. The coefficient of variation of the modal data is taken to be 0.5%. Initial trials are
taken to be 100 MN/m, which overestimates by 100% and 150% for the ±x and ±y faces, respectively. The iteration
history is shown in Fig. 3 and it virtually converges after 20 iterations. It took about 3.1 seconds after 50 iterations.
These stiffness parameters converges into two branches, one for the ±x faces (approximately 50 MN/m) and the other
for the ±y faces (approximately 40 MN/m).
Table 3 shows the actual values, identified values, standard deviations and coefficients of variation (C.O.V.) of the
stiffness parameters. The standard deviations are computed using Eq. (41) and they are up to about 1%. The estimates
are all close to the actual values. The difference between the actual and identified values are of similar order to the
corresponding calculated standard deviations.
Figure 3: Iteration history for the stiffness parameters of the undamaged structure (Example 2)
Application to damage detection: The structure is assumed to be damaged on the +y face of the first story and the +x
face of the third story. One third and a quarter of the stiffness reduction of the corresponding brace is imposed to these
two faces, respectively. It corresponds to 16.67% and 10% stiffness reduction of these faces, respectively. It is noted
that there is no sensor at the +x face. The first five natural frequencies of the damaged structures are 3.98, 4.50, 7.37,
11.67 and 13.19 Hz. Note that these damages alter the order of the modes. Furthermore, the translational and torsional
modes are mixed, especially for the high modes. In this case, the first six translational modes become the 1st (3.98 Hz),
2nd (4.50 Hz), 4th (11.67 Hz), 5th (13.19 Hz), 6th (18.47 Hz), and 7th (20.67 Hz) mode. Initial trials are again taken to
be 100 MN/m. Independent modal data is used to identify this damaged structure. The iteration history is shown in
Fig. 4. It took about 3.3 seconds for 50 iterations. These stiffness parameters converges into four branches, one for
the +y face of the first story, one for the other ±y faces, one for the +x face of the third story and the last one for the
other ±x faces. Identification results are shown in Table 4, which is similar to the one in Table 3. It is clearly seen that
the +y face of the first story and the +x face of the third story has substantial stiffness reduction compared to the
undamaged structure.
In order to visualize the damages, the means and standard deviations for the stiffness parameters are used to find the
probability that a given stiffness parameter θj has been reduced by certain fraction d, compared to the undamaged state
of the structure. An asymptotic Gaussian approximation [19] is used for the integrals involved to give:
(50)
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Table 3 Identification results of the undamaged structure (Example 2)
Figure 4: Iteration history for the stiffness parameters of the damaged structure (Example 2)
where Φ(·) is the standard Gaussian cumulative distribution function;
and
denote the most probable values of
the stiffness parameters for the undamaged and (possibly) damaged structure, respectively; and
corresponding standard deviations of the stiffness parameters.
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and
are the
Table 4 Identification results of the damaged structure (Example 2)
Figure 5: Probability of damage (Example 2)
The probabilities of damage for the twenty θj are shown in Fig. 5. It can be clearly seen that the +y face of the first
story and the +x face of the third story certainly have damage with probability almost unity. The mean of the damages
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are 16.1% and 9.5% and their target values are 16.67% and 10%. Furthermore, uncertainty of these estimates are 0.6%
and 0.9%. The figure can be interpreted as follows. Consider the probability of damage curve of the +y face of the first
story. Damage is 15% damage or more with probability 0.95. All the other eighteen faces are undamaged. The stiffness
parameters difference between the undamaged and damaged structure is within the uncertainty level. This probability
of damage can be used directly to set up the confidence interval for damage with a given percentage of confidence.
CONCLUDING REMARKS
A finite-element model updating methodology is presented with applications to structural damage detection. The
proposed method does not require any matching between the measured and calculated modes from the finite-element
model as it does not require any calculation of the natural frequencies and mode shapes of the finite-element model.
Furthermore, it operates in a computationally efficient iterative manner without nonlinear optimization programming.
The illustrative examples confirm the efficiency and effectiveness of the proposed approach.
Acknowledgements
This work was supported by University of Macau under research grants RG097/03-04S/YKV/FST and RG068/0405S/YKV/FST. These grants are gratefully acknowledged.
APPENDIX
Consider a set of linear algebraic equations:
(51)
where δb is a random vector that describes that the equation error with zero mean and covariance matrix
is symmetric, it is diagonalizable:
. Since
(52)
where the diagonal matrix D contains the eigenvalues of
and the matrix V contains the corresponding eigenvectors.
can be chosen since
With a proper choice of the scaling factors of the eigenvectors, a matrix that satisfies
is symmetric.
By pre-multiplying Eq. (51) with VT , one obtains:
(53)
has zero mean and covariance matrix
, which is diagonal. This implies that the
where
are uncorrelated. Note that the covariance matrix
is in general singular. Therefore,
random components in
some of the diagonal elements of D are zero. The equations associated with these zero eigenvalues will have zero error,
that is, they give exact constraints to the unknown parameters. One possible solution is to consider those constraints to
eliminate some unknown parameters. Then, one can solve the remaining unknown parameters by considering the
remaining equations weighted by the non-zero entities in the diagonal of the matrix D. However, it is suggested to
solve this problem with the following approximated procedure.
The values on the diagonal elements of D indicate the variance of the error of each equation. A larger value implies
larger uncertainty. For those diagonal elements that are zero or close to zero, one can replace them by a small number,
, where
denotes the average of the eigenvalues of
. This means that the weighting for those
say
equations are still large compared to the others. There are two reasons for this procedure: 1) The matrix D becomes
invertible; 2) Instead of completely rely on those equations with zero eigenvalues, it just gives large weighting for
might not be exact, e.g., due to the fact that the second order errors are
them because the covariance matrix
neglected.
to denote that the diagonal matrix that replaces the zero diagonal elements by small values in D. Then, by
Use
pre-multiplying
, one obtains
(54)
where
has zero mean and unity covariance matrix I. Then, all the equations have independent errors
with the same variance. Therefore, the optimal solution of the unknown parameter vector θ is the least squares solution:
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(55)
Then, the unknown parameter vector θ can be estimated as follows:
(56)
where the matrix
is the same as
, except that the zero eigenvalues are replaced by a small value.
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