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Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Porto, Portugal, 30 June - 2 July 2014
A. Cunha, E. Caetano, P. Ribeiro, G. Müller (eds.)
ISSN: 2311-9020; ISBN: 978-972-752-165-4
Bayesian model updating based on modal flexibility for structural health monitoring
Z. Feng 1, L.S. Katafygiotis 2
Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water
Bay, Kowloon, Hong Kong, China
2
Department of Civil and Environmental Engineering, The Hong Kong University of Science and Technology, Clear Water
Bay, Kowloon, Hong Kong, China
Email: zqfeng@ust.hk, lambros@ust.hk
1
ABSTRACT: A Bayesian probabilistic model updating method based on modal flexibility for structural health monitoring is
presented. Firstly, the flexibility matrices are constructed by using a sequence of identified modal data sets including modal
frequencies and mode shapes. Then flexibility vectors are obtained by using vectorization of the lower triangular portion of the
flexibility matrices and their covariance matrix is also calculated. The obtained flexibility vector data sets and their covariance
matrix are incorporated into a formula of Bayesian updating and the most probable values of model stiffness parameters are
obtained by maximizing the posterior PDF of the model parameters. The associated uncertainties are also quantified by
calculating the covariance matrix of model parameters at the most probable values. By comparing the updated PDFs of stiffness
parameters of the structure before and after possible damage, the probability of damage in each sub-structure can be estimated.
The proposed method is illustrated with a numerical example.
KEY WORDS: Bayesian model updating; modal flexibility; Structural health monitoring.
1
INTRODUCTION
There has been a significant amount of research effort devoted
to vibration-based methods for structural health monitoring
during recent years [1]. Changes in the physical properties of
a structure result in changes in the modal properties
(frequencies and mode shapes). Many damage detection
methods are based on the changes in natural frequencies and
mode shapes occurring during damage. Doebling et al.
presented a review of the main methods for damage detection
based on modal parameters [2]. Methods based on changes in
natural frequencies are very attractive since this parameter can
be determined by measurement at only one point of the
structure [3]. However, changes in natural frequencies cannot
provide spatial information about structural changes. Mode
shapes which provide spatial information can be used for
localizing the damage [4]. Nonetheless, an accurate
characterization of these mode shapes requires measurements
at several locations and changes in mode shapes due to
structural damage are not very significant.
Besides modal frequency and mode shape based methods,
another class of damage identification methods uses the
dynamically measured modal flexibility matrix. Damage
identification based on modal flexibility matrix has been
recently shown to be promising. Since an inverse relationship
exists between the modal flexibility matrix and the square of
the modal frequencies, the modal flexibility matrix is not
sensitive to high frequency modes. This unique characteristic
allows the use of a small number of truncated modes to
construct a reasonably accurate representation of the
flexibility matrix. Toksoy and Aktan proposed a bridgecondition assessment method which is formulated based on
modal flexibility for evaluating the global state of bridge
health [5]. Pandey and Biswas employed the changes in modal
flexibility matrix to detect damage in structures [6]. More
recently, Bernal proposed a flexibility-based damage
localization method termed as DLV [7]. Jaishi and Ren
proposed a sensitivity-based finite element model updating
method using modal flexibility residual [8].
Damage identification techniques using modal data are
often based on methods of model updating. A nominal
parametric model of the structure is needed and the model
parameters are updated by minimization of some objective
function which reflects the errors between the measured data
and the predictions of the model. The success of the finite
element (FE) model updating method depends on the accuracy
of the FE model, the quality of the modal data, the choice of
the objective function and the capability of the optimization
algorithm. Most model updating methods are based on a onestage optimization scheme. When these methods are applied
to large-scale structures with many unknowns, ill-conditioning
and non-uniqueness in the solution of such inverse problem
appear as inevitable difficulties. Furthermore, a large
computational effort is usually required. Generally speaking,
due to the limited number of sensors and the difficulty of
obtaining measurements for rotational DOFs and internal
DOFs of the structure, the number of DOFs in the FE model
usually exceeds that of the experimental model. Therefore, in
order to solve the problem of mismatch between the DOFs of
the FE model and those of the experimental model, model
reduction or modal expansion is needed. In addition to the
aforementioned concerns, accounting for measurement and
modeling errors and uncertainties is crucial when applying FE
model updating techniques. One possible approach to
incorporate uncertainty regarding measurement and modeling
errors into the FE model updating process is to adopt a
probabilistic scheme based on Bayesian inference [9, 10]. The
Bayesian inference approach has gained interest among
uncertainty quantification methods in recent years, mostly
because of its solid foundation on probability theory and its
rigorous treatment of uncertainties.
177
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
In this paper, a Bayesian model updating method based on
modal flexibility is proposed. Firstly, flexibility matrices are
constructed by using a sequence of identified modal data sets
including modal frequencies and mode shapes. Then
flexibility vectors are obtained by using vectorization of the
lower triangular portion of the flexibility matrices and their
covariance matrix is also calculated. The obtained flexibility
vector data sets and their covariance matrix are incorporated
into a formula of Bayesian updating and the most probable
values of model stiffness parameters are obtained by
maximizing the posterior PDF of the model parameters. The
associated uncertainties are also quantified by calculating the
covariance matrix of the model parameters at the most
probable values. By comparing the updated PDFs of stiffness
parameters of the structure before and after possible damage,
the probability of damage in each sub-structure can be
estimated. In order to facilitate the optimization problem, a
deterministic initial updating stage is added before the
Bayesian updating stage with the purpose of identifying an
initial estimate of the model stiffness parameters. The initial
estimates obtained in the preliminary updating stage can
facilitate the optimization problem in the Bayesian updating
stage, which makes the algorithm more efficient and robust.
Furthermore, the SEREP model reduction technique is
adopted to cope with the problem of mismatch between the
DOFs of the FE model and those of the experimental model
[11]. The proposed procedure is illustrated with a simulated
example.
2
MODAL FLEXIBILITY
The generalized eigenvalue equation for a linear dynamical
system with N degrees of freedom is:
K   M 
(1)
where M and K are the mass and stiffness matrix; Φ is the
eigenvector (mode shape) matrix and Λ is the diagonal
eigenvalue matrix with squared modal frequencies .
With the modal frequencies and the mass-normalized mode
shapes, the stiffness matrix K and flexibility matrix F of a
structure can be calculated by [12]
N
T
K  M T M  M ( i2
i i )M
Nm
F  Ft  
i 1
N
i 1
1
 T
2 i i
Nm
F m  Ft m  
i 1
where
i
It can be seen from Equation (2) that the modal contribution
of one mode to the stiffness matrix increases as the
corresponding modal frequency increases. To obtain an
accurate estimate of the stiffness matrix, one has to use highfrequency modes. Contrarily, it can be seen from Equation (3)
that the modal contribution of one mode to the flexibility
matrix is inversely proportional to the square of the
corresponding modal frequencies, implying that it can be
estimated with sufficient accuracy by using a few lowfrequency modes. Therefore, the Equation (3) can be
approximated as
178
iiT
(4)
1

2
i
im  R No1 ( i  1,2,..., Nm )
im (im )T
(5)
are spatially incomplete
mode shapes and No denotes the number of measured DOFs at
the sensor locations. It should be noted that
im is
a vector
comprised by the components of the mass-normalized i in
Equation (4), corresponding to the measured DOFs.
As we known, the flexibility matrix is constructed from
modal frequencies and mass-normalized mode shapes, so the
key issue for the flexibility matrix construction is massnormalization of the experimentally identified mode shapes.
Numerous researchers have developed various methods for
mass normalization of the ambient vibration mode shapes. In
the present study, the mass-normalized mode shapes are
achieved with the aid of the finite element model (FEM). The
mass matrix is assumed to be known with reliable confidence.
If the measurements are complete, then the massnormalization is trivial. If the measurements are incomplete,
then the mass-normalization will be completed with the aid of
model reduction techniques. Herein, the SEREP reduction
method is adopted for this purpose [11]. The SEREP method
is first applied to the FEM mass matrix to obtain a reduced
mass matrix Mm with the same dimension as the number of
measured DOFs. Then the measured incomplete mode shape
vectors are normalized as
(2)
(3)

2
i
where Ft is the truncated flexibility matrix and Nm is the
number of selected lower modes. In practice, only modal
frequencies and mode shapes of a few lower modes are
actually obtained during vibration testing. In addition, usually
only spatially incomplete mode shapes, comprised of the
mode shape components corresponding to the measured DOFs,
which are less than the analytical DOFs, are available. Let Fm
denotes the flexibility submatrix with respect to the measured
DOFs, Ftm denotes the truncated flexibility submatrix, then
(m )T M mm  I
i 1
F   1T  
1
where
3
(6)
m  [1m , 2m ,..., Nmm ] .
MODEL REDUCTION
System equivalent reduction and expansion process (SEREP)
was proposed by O’Collahan et al. [11]. SEREP is a model
reduction or modal expansion technique, in which the reduced
system preserves the frequency and mode shapes of the
original system for selected modes of interest.
This method first partitions the mass and stiffness matrices
as follows,
M
M p   mm
 M sm
M ms  p  K mm
,K 
M ss 
 K sm
K ms 
K ss 
(7)
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
The subscripts m and s stand for the master (measured) and
slave (unmeasured) DOFs, respectively. Let Φm denotes the
retained mode shape matrix, which is a rectangular matrix of
size No×Nm, where No is the number of master DOFs and Nm is
the number of modes to be retained. In this method, the matrix
(Φm)† called the pseudo-inverse of Φm needs to be computed
to construct the SEREP transformation matrix.
m T
m
m T
1
M  T M T, K  T K T
T
p
m
T
p
e  T  m
Substituting Equation (19) into Equation (18), one obtains
(12)
The subscripts nom and exp stand for the nominal and updated
experimental structures, respectively. Let K and F be the
perturbation matrices such that the nominal model matrices
and the updated experimental model matrices are related as
follows:
Kexp  Knom  K
(13)
Fexp  Fnom  F
(14)
C p,n  ( Fnom Kn Fnom ) p
(15)
As the matrix K nom is full rank, then Equation (15) can be
(21)
and b is a vector of size (NoNo)1 with pth component
bp  Fp
(22)
where p is a new index with p = (i,j), that corresponds to the
(i,j) entry of the matrix. Because of the symmetry of the
modal flexibility matrix, only upper or lower triangular matrix
is useful. Thus the sizes of the matrix C and vector b are
reduced to [No(No+1)/2]Nθ and [No(No+1)/2]1
respectively. The stiffness factors  can be solved as
  (CT C )1 CT b
(23)
Using the above approximations, one can obtains initial
estimate of the model parameters.
5
BAYESIAN MODEL UPDATING BASED
MEASURED MODAL FLEXIBILITY DATA
5.1
ON
Formulation
Using model reduction as described in the previous section,
the number of DOFs for the analytical model can be equal to
the number of the measurements. For clarity, the size of
modal flexibility matrix for the analytical model is the same
as that for the experimental model. Because of the symmetry
of the modal flexibility matrix, only the upper or lower
triangular matrix is useful. Here we use the lower triangular
modal flexibility matrix Fl for model updating. We perform
vectorization on Fl as

f  F11 , F21 ,...FNo 1 ,..., Fii , F(i 1)i ,...FNoi ,..., FNo No
Substituting equations (13) and (14) into (12) yields
(20)
where C is a matrix of size (NoNo)Nθ with (p,n) entry
(11)
For the nominal and the updated experimental structures, the
global stiffness and flexibility matrices will satisfy the
following relationship
FKnom   Fexp K
C  b
(10)
INITIAL MODEL UPDATING BASED ON MODAL
FLEXIBILITY CHANGE
Fnom Knom  Fexp Kexp  I
(19)
n 1
(9)
Ignoring inherent numerical errors, this SEREP reduction
technique usually produces identical eigenvalues to those of
the full model and identical expanded eigenvectors to those of
the full model. Furthermore, this method can arbitrarily select
the modes preserved in the reduced model and the reduced
model accuracy does not depend on the selection of master
DOFs.
rewritten as
(18)
K   n K n
The retained mode shape matrix can also be expanded to full
mode shape matrix of size N×Nm using
4
F   Fnom KFnom
The stiffness perturbation matrix K is expressed as
The reduced mass matrix and stiffness matrix are then given
by
m
(17)
m T
( )  ( )  ( )  , N o  N m
m †
F  ( Fnom  F )KFnom  ( FnomKFnom  F KFnom )
N
1
( )  ( )   ( ) , N o  N m
m
Substituting Equation (14) into (16) yields
(8)
where the pseudo-inverse can be computed as
m T
(16)
Neglecting the high-order term in Equation (17), one obtains
†
 m 
T   s   m 
 
m †
F   Fexp K ( Knom )1   Fexp KFnom

T
(24)
The experimental data D from the structure are assumed to
consist of Nt sets of modal flexibility vector data,
D  { fˆ1, fˆ2 ,..., fˆNt } .
fˆt  f ( )  et
(25)
179
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
where f(θ) is the prediction from the analytical model and
et is a prediction error term. The choice for the probability
model of the prediction error is based on the maximum
entropy principle [13] which yields a multi-dimensional
Gaussian distribution with zero mean and covariance matrix ,
that is et ~ N (0, ) . The covariance matrix  is estimated by
computing the sample covariance matrix of multiple modal
flexibility data sets. Here it is computed based on Nt modal
flexibility vector data sets as

1 NT ˆ
( ft  f )( fˆt  f )T

Nt  1 t 1
Assuming independence between these Nt modal flexibility
vector data sets, the likelihood function for the whole data sets
D can be expressed as
 1 Nt

p( D  , M )  c2 exp   [ fˆt  f ( )]T 1[ fˆt  f ( )] 
 2 t 1

(28)
Assuming a non-informative uniform distribution for the prior
PDF p(θ), the posterior PDF p( D, M ) can be written in a
similar form as the likelihood function in Equation (28).
It should be noted that if the measurements are incomplete,
then the measured modal flexibility vector data are also
fˆt = fˆt ( ) . Since the
model reduction transformation matrix T is dependent on the
stiffness matrix K which is a function of model parameters θ,
then the reduced model mass Mm in Equation (10) will be
dependent on θ. The measured mode shapes are massnormalized with respect to the reduced model mass Mm.
Therefore the measured modal flexibility matrix, which is
constructed by measured modal frequencies and massnormalized mode shapes, will be dependent on θ.
Most probable value and uncertainty
In the case when sufficient amount of modal flexibility data
sets are available and the incompleteness of the modal
parameters is not significant, the model updating problem is
usually globally identifiable. By maximizing the posterior
PDF, the most probable stiffness parameter vector  can be
determined. Instead of maximizing the posterior PDF, one can
equivalently
minimize
the
objective
function
J ( )   ln p( D, M ) to obtain the optimal (most
180

.
centered at the optimal (most probable) parameters  with
covariance matrix  equal to the inverse of the Hessian of
parameters, i.e.,
(27)
probable) stiffness parameter vector
(29)
the function J ( )   ln p( D) calculated at the optimal
 1

p( fˆt  , M )  c1 exp   [ fˆt  f ( )]T  1[ fˆt  f ( )] 
 2

5.2
1 Nt ˆ
[ ft  f ( )]T  1[ fˆt  f ( )]

2 t 1
This can be done by using the built-in function “fminsearch”
or “fminunc” in MATLAB.
Using Laplace’s asymptotic approximation, the posterior
PDF can be well approximated by a Gaussian distribution
(26)
where f is the averaged experimental modal flexibility
vector.
For the tth set of modal flexibility vector data, the likelihood
function can be expressed as
function of the model parameters θ,
J ( ) 
  [ H ( )]1  [J ( )T
 
]1 . This
can be done by using finite difference.
It should be noted that if the model is locally identifiable or
unidentifiable, the approach presented above is not applicable
and sampling-based approaches may be helpful to find the set
of all optimal parameters.
6
SUMMARY OF
PROCEDURES
THE
MODEL
UPDATING
In the complete measurements case, the model updating flow
chart is shown in Figure 1. The procedures are summarized as
follows:
(1) A sequence of modal frequencies and complete mode
shapes data are identified and the mode shapes are mass
normalized.
(2) The modal flexibility matrices are constructed from the
modal frequencies and mass-normalized mode shapes. Then
the modal flexibility vectors are obtained by vectorization of
the lower triangular portion of the modal flexibility matrices.
The covariance matrix of the modal flexibility vectors is also
calculated.
(3) An initial estimate of the model parameters is obtained
by using the deterministic model updating technique described
in Section 4.
(4) The modal flexibility vectors and their covariance
matrix are incorporated in a Bayesian updating formula. The
most probable values of the model parameters are obtained by
maximizing the posterior PDF of the model parameters, where
the initial guess in this optimization problem is set to be equal
to the initial estimate of the model parameters obtained in the
deterministic updating stage. The covariance matrix of the
model parameters is calculated at the most probable values by
finite differences.
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
by vectorization of the lower triangular portion of the modal
flexibility matrices.
(3) An initial estimate of the model parameters is obtained
by using the deterministic model updating technique described
in Section 4.
(4) The modal flexibility vectors and their covariance
matrix are dependent on the updated model parameters when
the mode shapes are incomplete. The modal frequencies and
incomplete mode shapes data sets are directly incorporated in
a Bayesian updating formula. The most probable values of the
model parameters are obtained by maximizing the posterior
PDF of the model parameters, where the initial guess in this
optimization problem is set to be equal to the initial estimate
of the model parameters obtained in the deterministic
updating stage. The covariance matrix of the model
parameters is calculated at the most probable values by finite
differences.
7
Figure 1 Model updating with complete modal flexibility data.
7.1
ILLUSTRATION EXAMPLE
Model updating
In this example an eight-story shear building is considered. It
is assumed that this building has uniformly distributed floor
mass and uniform inter-storey stiffness. The mass per floor is
taken to be 2×104 kg, the nominal value of the inter-storey
stiffness of each floor is 15×106 N/m. The real inter-storey
stiffness of the second and fourth floor is assumed to have a
reduction due to damage of 20% and 40%, respectively. For
the simulated modal data, samples of zero-mean Gaussian
noise with covariance matrix  were added to the exact
modal frequencies and mode shapes. The covariance matrix
 is assumed to be diagonal with the variances
corresponding to 1% coefficient of variation for both the
modal frequencies and mode shapes for all modes, a
reasonable value for typical modal testing. The nominal
substructure stiffness matrices are given by
15 106
K1  
 071
017 

077 
(30)
for the first storey and
Figure 2 Model updating with incomplete modal flexibility
data
In the case of incomplete measurements, the model
updating flow chart is shown in Figure 2. The procedures are
summarized as follows:
(1) A sequence of modal frequencies and incomplete mode
shapes data are identified and the mode shapes are initially
mass normalized with respect to the nominal reduced model
mass.
(2) The initial modal flexibility matrices are constructed
from the modal frequencies and initial mass-normalized mode
shapes. Then the initial modal flexibility vectors are obtained
0(l 2)8

0
6
6
 1(l 2) 15 10 15 10
01(l 2) 15 106 15 106

0(8l )8


01(8l ) 
01(8l ) 


(31)
for other stories, i.e., l = 2, 3,…, 8.
For the first case, complete measurements are considered,
i.e., all the DOFs are assumed to be measured and the first
four modes are used to construct the modal flexibility matrix
for model updating. Figure 3 shows the identified most
probable values of the stiffness parameters. From the figure,
we can see that the second floor and the fourth floor have
damages of about 20% and 40%, respectively. Table 1 shows
the identified most probable values, the calculated standard
deviations (SD), coefficient of variation (CV) for each
parameter, and the value of a ‘normalized distance’ (ND) for
181
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
each parameter. The ‘normalized distance’ represents the
absolute value of difference between the identified value and
actual value, normalized with respect to the corresponding
calculated standard deviation. From the identified results, we
can see that the proposed method can update the model
successfully.
For the second case, incomplete measurements are
considered such that the 1st, 3rd, 5th, 7th and 8th DOFs are
measured, and the first four modes are used to construct the
modal flexibility matrix for model updating. In order to check
the efficacy of the model reduction technique for mode shape
mass-normalization, the first four mode shapes of the nominal
model are mass-normalized with respect to full model mass
and reduced model mass, respectively. The first four mode
shapes of the nominal model which are mass-normalized with
respect to full model mass are shown in Table 2, while those
mass-normalized with respect to reduced model mass are
shown in Table 3. Comparing Table 2 with Table 3, we can
see that they are identical, which validates the efficacy of the
model reduction technique for mode shape massnormalization. Figure 4 shows the identified most probable
values of the stiffness parameters. From the figure, we can see
that the second floor and the fourth floor have damages of
about 20% and 40%, respectively. Table 4 shows the
identified most probable values, the calculated standard
deviations (SD), coefficient of variation (CV) for each
parameter, and the value of a ‘normalized distance’ (ND) for
each parameter. From the identified results, we can see that
the proposed method can update the model successfully using
incomplete measurements. Comparing the standard deviations
identified from complete and incomplete measurement data as
shown in Table 1 and 4 respectively, we can see that the
standard deviations identified from incomplete measurement
data is larger, which indicates that larger uncertainties exist in
the case of incomplete data.
Figure 4 Identified most probable values of stiffness
parameters (incomplete measurement)
Table 1 Identification results of damaged structure (complete
measurement)
Parameter
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
true
0
-0.2
0
-0.4
0
0
0
0
identified
-0.0008
-0.1974
0.0024
-0.3999
0.0021
0.0040
0.0028
-0.0002
SD
0.0010
0.0012
0.0011
0.0007
0.0011
0.0019
0.0013
0.0011
CV
1.1935
0.0058
0.4477
0.0017
0.5347
0.4749
0.4756
5.1308
ND
0.8378
2.2761
2.2336
0.1683
1.8701
2.1059
2.1025
0.1949
Table 2 Mass-normalized mode shapes with respect to full
model mass
1st DOF
1st mode
2nd mode
3rd mode
4th mode
-0.0006
-0.0018
-0.0027
0.0033
rd
-0.0018
-0.0034
-0.0012
-0.0023
th
-0.0027
-0.0012
0.0034
0.0006
th
-0.0033
0.0023
-0.0006
0.0012
th
-0.0034
0.0033
-0.0031
-0.0027
3 DOF
5 DOF
7 DOF
8 DOF
Table 3 Mass-normalized mode shapes with respect to
reduced model mass
Figure 3 Identified most probable values of stiffness
parameters (complete measurement)
182
1st mode
2nd mode
3rd mode
4th mode
1st DOF
-0.0006
-0.0018
-0.0027
0.0033
3rd DOF
-0.0018
-0.0034
-0.0012
-0.0023
th
-0.0027
-0.0012
0.0034
0.0006
th
7 DOF
-0.0033
0.0023
-0.0006
0.0012
8th DOF
-0.0034
0.0033
-0.0031
-0.0027
5 DOF
Proceedings of the 9th International Conference on Structural Dynamics, EURODYN 2014
Table 4 Identification results of damaged structure
(incomplete measurement)
Parameter
θ1
θ2
θ3
θ4
θ5
θ6
θ7
θ8
7.2
true
0
-0.2
0
-0.4
0
0
0
0
identified
-0.0021
-0.1983
-0.0008
-0.395
-0.0129
-0.0093
0.0029
0.0015
SD
0.0013
0.0034
0.0049
0.0023
0.0052
0.0064
0.0065
0.0025
CV
0.6182
0.0174
5.9819
0.0059
0.4026
0.684
2.2631
1.6859
ND
1.6175
0.4877
0.1672
2.1464
2.4837
1.4619
0.4419
0.5932
Structural health monitoring
If the Bayesian model updating approach is used to update the
probability density function (PDF) of the stiffness parameters
of the structural model based on the measured modal
flexibility data corresponding to the undamaged and possibly
damaged states, possible structural damage due to stiffness
loss can be detected. The detection of damage is based on the
probability that a sub-structure stiffness parameter has a
fractional decrease compared to its value corresponding to the
undamaged structure.
In order to portray the damage, the identified most probable
values and their calculated standard deviations for the
stiffness parameters are used to find the probability that a
given stiffness parameter  n has been reduced by certain
fraction d compared to its value corresponding to the
undamaged state. An asymptotic Gaussian approximation is
used for the integrals involved to give:
Pndm (d )  P((1   npd )  (1  d )(1   nud ))
 (1  d )(1   ud )  (1   pd ) 
n
n

 
 (1  d ) 2 ( ud ) 2  ( pd ) 2 
n
n


where
(32)
() is the standard Gaussian cumulative distribution
function;
 nud and  npd denote the most probable values of the
stiffness parameters for the undamaged and (possibly)
damaged structure, respectively;
 nud
and
 npd
are the
corresponding standard deviations of the stiffness parameters.
The structural model used is the same as before. Incomplete
measurements are considered such that the 1st, 3rd, 5th, 7th
and 8th DOFs are measured, and the first four modes are used
to construct the modal flexibility matrix for model updating.
The stiffness parameters  n (n=1,2,…,8) for the undamaged
nd
Figure 5 Probability of damage curves
8
ACKNOWLEDGMENTS
This research has been supported by the Hong Kong Research
Grants Council under grants 613511, 613412 and 613713.
These supports are gratefully acknowledged.
REFERENCES
[1]
th
structure are assumed to be zero, while the 2 and 4 stories
are assumed to be damaged with a stiffness loss of 20% and
40% respectively. The probabilities of damage for the eight
stories are shown in Figure 5. It can be clearly seen that the
second storey and the fourth storey have damage with
probability almost unity and the mean of the damage
percentage is 20% and 40%, respectively.
CONCLUSION
A Bayesian structural model updating methodology based on
modal flexibility is presented with application to structural
health monitoring. The proposed method is based on modal
flexibility matrix which can be constructed from identified
modal frequencies and mode shapes. The most probable
values of the model stiffness parameters are obtained by
maximizing the posterior PDF of the model parameters. The
associated uncertainties are also quantified by calculating the
covariance matrix of the model parameters at the most
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technique is adopted to cope with the problem of mismatch
between the DOFs of the FE model and those of the
experimental model. The proposed approach is demonstrated
using a numerical example, which confirms its accuracy and
effectiveness.
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