Substructural identification using neural networks

Chung-Bang Yun*, Eun Young Bahng

Department of Civil Engineering, Korea Advanced Institute of Science and Technology, Taejon, 305-701, South Korea

Received 18 February 1997; accepted 27 July 1999

Abstract

In relation to the problems of damage detection and safety assessment of existing structures, the estimation of theelement-level stiness parameters becomes an important issue. This study presents a method for estimating thestiness parameters of a complex structural system by using a backpropagation neural network. Several techniques

are employed to overcome the issues associated with many unknown parameters in a large structural system. Theyare the substructural identification and the submatrix scaling factor. The natural frequencies and mode shapes areused as input patterns to the neural network for eective element-level identification particularly for the case withincomplete measurements of the mode shapes. The Latin hypercube sampling and the component mode synthesis

methods are adapted for ecient generation of the patterns for training the neural network. Noise injectiontechnique is also employed during the learning process to reduce the deterioration of the estimation accuracy due tomeasurement errors. Two numerical example analyses on a truss and a frame structures are presented to

demonstrate the eectiveness of the present method. 7 2000 Elsevier Science Ltd. All rights reserved.

Keywords: Substructuring identification; Neural networks; Modal data; Noise injection learning; Latin hypercube sampling; Com-

ponent mode synthesis

1. Introduction

Structural identification has become increasingly an

important research topic in connection with damage

assessment and safety evaluation of existing structures[14]. The stiness parameters such as axial and flex-

ural rigidities can be identified based on the measured

responses and excitations using various system identifi-

cation techniques. Such inspection approaches providenondestructive means of safety assessment. Recent

development in this area is made possible by rapid

advances in computer technology for data acquisition,

signal processing and analysis. In general, the systemidentification methods inherently involve comprehen-

sive search processes such as extended Kalman filter,recursive least squares or modal perturbation. Most ofthem are computationally expensive and tend to be nu-

merically unstable for the case with a large-degrees-of-freedom system. In recent years, instead of the compre-hensive search processes, pattern-matching techniquesusing neural networks have drawn considerable atten-

tion in the field of damage assessment and structuralidentification [512]. Neural networks have uniquecapability to be trained to recognize given patterns and

to classify other untrained patterns. Hence, the neuralnetworks can be used for estimating the structural par-ameters with proper training.

Several researchers have dealt with neural networkapproaches for damage estimation of small degrees-of-

Computers and Structures 77 (2000) 4152

0045-7949/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved.PII: S0045-7949(99 )00199-6

www.elsevier.com/locate/compstruc

* Corresponding author. Tel.: +82-42-869-3612; fax: +82-

42-869-3610.

E-mail address: ycb@kaist.ac.kr (C.B. Yun).

freedom models. Wu et al. applied the neural networksto recognize the locations and the extent of individual

member damage of a simple three story frame [5].Szewczyk and Hajela proposed a modified counterpro-pagation neural network to estimate the stiness of in-

dividual structural elements based on observable staticdisplacements under prescribed loads [6]. Their pilotstudies showed that the neural-network-based

approach for damage assessment is promising. Tsouand Shen identified damage of a spring-mass systemwith eight degrees-of-freedom (DOF) that has closely

spaced eigenvalues [8]. They constructed three multi-layer subnets which performed three tasks of inputpattern generation, damage location identification, anddamage severity determination. Their study showed

good performance for on-line damage identification.Yagawa et al. applied neural network approach to esti-mate stable crack growth in welded specimens [9].

Yoshimura et al. proposed the Generalized-Space-Lat-tice transformation of training patterns to relax the ill-posedness of the problem. As the eect of transform-

ation, a training process converged fast, and estimationaccuracy was improved significantly [10].In this study, the neural network-based approach is

extended to the estimation of stiness parameters of acomplex structural system with many unknowns. Sev-eral techniques, such as substructural identification[1315] and submatrix scaling factor [16] are employed

to overcome the issues associated with manyunknowns. The concept of modal strain energy [17] isutilized for selection of modes to be used as input to

the neural networks, and Latin hypercube samplingtechnique [18] and the component mode synthesismethod [19] are employed for ecient generation of

training patterns. Two numerical example analyses ona truss and a frame structures are presented to demon-strate the eectiveness of the proposed method.

2. Neural network-based structural identification

Studies on neural networks have been motivated toimitate the way that the brain operates. It has recentlydrawn considerable attention in various fields of

science and technology, such as character recognition,electro-communication, image processing, and indus-trial control problems [20]. Many researchers have

developed various neural network models for dierentpurposes [21]. In this study, a model called backpropa-gation neural network (BPNN) or multilayer percep-

tron is used for structural identification. The BPNNconsists of an input layer, hidden layers, and an outputlayer as in Fig. 1. The input and output relationship ofa neural network can be nonlinear as well as linear,

and its characteristics are determined by the weightsassigned to the connections between nodes in two adja-cent layers. Changing these weights will change the

input/output behavior of the network. Systematic waysof determining the weights of the network to achieve adesired input/output relationship are referred to as

training or learning algorithm. In this study, the stan-dard backpropagation algorithm is used. The basicstrategy for developing a neural network-based

approach to identify a structural system is to train theBPNN to recognize the element-level structural par-ameters from the measurement data on the structuralbehavior, such as natural frequencies and mode

shapes.

2.1. Substructural identification

For the identification of a structure with manyunknowns, it is not practical to identify all of the par-

ameters in the structure at a same time, because mostof the identification techniques require expensive com-putation that would be even prohibitive, as the numberof unknown parameters increases. Several researches

were reported on identification of a part of a structureso as to reduce the size of the system under consider-ation [1315]. Those works were based on the reason-

ing that the expected damages of a structure occur atseveral critical locations, hence it is more reasonable toconcentrate the identification at critical locations of

the structure.The local-identification method is generally based on

substructuring, in which the structure is subdivided

Fig. 2. Substructuring for localized identification.Fig. 1. Architecture of backpropagation neural network.

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 415242

into several substructures and the identification is car-ried out on a substructure at a time. In the present

study, the substructure to be identified is called as theinternal substructure, while the others as the externalsubstructures as shown in Fig. 2. Since the parameters

to be estimated are limited to an internal substructure,it is expected that the numerical problems such asdivergence or falling into local minima may be

avoided. Another advantage is that this approachrequires measurement data only on the substructure ofinterest, instead of on the whole structure.

The two-stage identification is another idea toreduce the number of variables. The two-stage identifi-cation was also studied by the present authors [22]. Inthe study, Topole and Stubbss damage index method

[23] was employed to select probable damaged mem-bers in the first stage identification. Then the damageseverities of the selected members were estimated using

the neural network technique in the second stageidentification. However, there may be many caseswhere we can estimate which region shall be investi-

gated through visual inspection or knowledge ofexperts without carrying out the first stage identifi-cation. Then the substructural identification can be

performed on the interested part of the structure, whilethe parameters of the other parts are still unknown. Inthe worst case any information may not be obtained todetermine the substructure to be identified, then the

substructuring technique can be applied to each sub-structure in turn.

2.2. Submatrix scaling factors

The number of the parameters to be estimated shall

be kept reasonably small for successful identificationof a structure. Reduction of parameters will improvethe results of the identification and minimize the

required measurement data. For the purpose of e-cient parameterization of the structure, the submatrixscaling technique is adopted in this study [16]. Usingthe technique, the stiness matrix of the system can be

described by introducing the submatrix scaling factors(SSF) corresponding to the element level stinessmatrices as

K Xqj1

Kj Xqj1

sjK0j 1

where K is the stiness matrix of the structure in the

present condition, Kj and K0j are the present and the

reference values of the stiness submatrix for the jthelement transformed into the global coordinates. sj is

the j-th SSF, and q is the total number of submatrices.Then, the identification of the stiness matrix can beperformed by estimating the submatrix scaling factors

instead of all of the stiness matrix coecients. Directidentification of the system stiness matrix coecients

may face with problems which violate the basic proper-ties of the stiness matrix related to the member con-nectivity, boundary condition, and those as

symmetricity and positive definiteness. However, theuse of the SSFs described in Eq. (1) can avoid suchproblems.

2.3. Input to neural networks

A way of choosing the patterns representing the

characteristics of the structure, which are to be used asthe input to neural networks, is one of the most im-portant subjects in this approach. Several researchershave used the various input patterns (input vectors)

suitable for their purpose. For example, Wu et al. usedthe frequency spectrum for each DOF of the examplestructure for damage estimation [5]. This input has an

advantage that does not need the modal parameteridentification from the measurements. Nonetheless, tocharacterize those spectral properties, many sampling

data are required. Accordingly, a large number ofinput neurons in the neural networks are needed,which may reduce the eciency and accuracy of the

training process. Tsou and Shen used the dynamic re-sidual vector that can be obtained from the modaldata [8]. It provides a simple and eective way todetect the damage and the length of the input pattern

can be reduced significantly compared with the spec-trum data. However, it still has the restriction that themodes should be measured at every finite element

DOF. In reality, the modes obtained from a typicalmodal survey are generally incomplete in the sensethat the mode vector coecients are available only at

the measured DOFs. Therefore, it is desirable to usethe input patterns which are more suitable for the casewith partial measurement data.

The natural frequencies and modes of a structureare used as the input patterns in this study. The choicehas been made based on the following advantages: (1)the length of the input pattern is limited to several

times of the number of the measured DOFs; (2) thenatural frequencies represent global behaviors, whilethe mode vectors do local characteristics; and (3) they

can be obtained from the measurements of the struc-tural behavior. The relationship between the modesand the stiness matrix can be written as

KF MFL 2where M is the mass matrix, F is the modal matrix,and L is the diagonal matrix containing the eigen-values. The mass matrix is assumed to be knownexactly in this study.Using the substructuring technique and referring to

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 4152 43

Fig. 2, the modal matrix F can be partitioned as

F 24FL1 FH1FL2 FH2FL3 F

H3

35 3where the subscripts 1, 2, and 3 denote substructures,

while Substructure 2 is the internal substructure, andthe superscripts L and H represent lower and highermodes, where only the lower modes are generally used

for the structural identification. Rewriting the ithmode in the partitioned modal matrix FL2 as ~ji for thesimplicity, the input pattern can be defined as

Input pattern vector f fi, ~j1i, . . . , ~jni , i 1, . . . ,

mg4

where fi is the ith natural frequency, ~jji denotes the jthcomponent of ~ji which is normalized as ~j

Ti ~ji 1, n is

the number of DOFs measured for Substructure 2, andm is the number of modes to be included in the identi-fication.

A useful relationship between the submatrix for theelement stiness and the modes may be obtained byexamining the strain energy of each submatrix withrespect to the modes [17]. The modal strain energy dis-

tribution among the stiness submatrices for the ithmode can be evaluated as

bji jTi K

0j ji

jTi Mji

1

o2i5

where bji is the modal strain energy (MSE) coecientof submatrix K0j in the ith mode, ji is the ith mode inthe modal matrix F, and oi is the ith circular naturalfrequency. The submatrices containing large values ofthe MSEs for a mode indicate that the corresponding

structural elements are major load carrying elementsfor the particular mode. Stiness changes in those el-ements will cause significant changes in the frequency

and mode shape. Hence, detection of the stinesschanges may be more eective by examining the modaldata for those elements. On the other hand, the struc-tural elements having negligible MSEs reflect that it

would be dicult to detect the stiness changes inthose elements using the corresponding modal data.Topole and Stubbs utilized the changes in the MSEs

from the undamaged to the damaged structures to pre-dict structural damage [23]. In the present study, how-ever, the MSEs are used as indicators in selecting

important vibration modes for the input patterns. TheMSEs distribution gives a good information for select-ing the modal data.

2.4. Generation of training and testing patterns

Since the neural network-based structural identifi-

cation is highly dependent on the training patterns, itis very important to prepare well-examined data sets.The number of training patterns must be large enough

to represent the structural system properly. However,for the computational eciency, it must be limited tobe reasonably small, because most of the compu-

tational time for this approach is required for prepar-ing the training patterns and training the networks.When the number of the unknown parameters is P

and each parameter has L sampling points, the size ofthe whole population is LP. This size increases expo-

nentially, as the number of the unknown parametersdoes. To reduce the number of training patterns eec-tively, a sparse sampling algorithm such as Latin

hypercube technique [18] is employed in this study,which makes the required number of samples reducedfrom LP to L.

The training patterns for the proposed neural net-work-based method consist of the modal data as inputand the corresponding SSFs as output. To generate

training patterns, a series of eigenanalyses are to beperformed. However, repeated computations of eigen-value problems for a complex structure can be very ex-

pensive. Thus, the fixed-interface-component-mode-synthesis (CMS) method [19] incorporating the sub-structuring technique is employed. In the component

mode synthesis method, each substructure is analyzedindependently for local mode shapes. Then, the com-

ponent mode shapes are assembled to construct Ritzvectors, which are subsequently used to compute themode shapes for the whole structure. When the num-

ber of DOFs is large, the computational eciency ofthe CMS method becomes more significant.The neural-network approach is an approximation

technique that handles non-unique cases by eitherreturning one of the possible solutions or an averagetaken over all possible solutions [6]. Although the

input information to the neural network is limited tothe components of the mode vectors for an internalsubstructure by applying the substructuring technique

to the structural identification, most of the informationregarding the stiness parameters of the internal sub-structure is included in the components of the mode

vectors of the internal substructure, rather than thoseof the external substructures. Thus, those stiness par-

ameters may be estimated with a reasonable accuracyfrom the components of mode vectors of the internalsubstructure as the input to the neural network. On

the other hand, as the range of the submatrix scalingfactors in the external substructure reduces, the esti-mation accuracy increases. If possible, the uncertainties

of the unknowns in the external substructure shall bereduced by using all the available information.

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 415244

In the process of computing the mode vectors foreach pattern, the mode shapes may be obtained in

dierent order compared with the baseline case, par-ticularly for the higher modes. In this study, the modeshapes of each pattern are rearranged to match the

baseline shapes using the modal assurance criteria(MAC) [24], in which the MAC is defined as

MACij j ~jTi ~jTj j2j ~jTi ~ji k ~jTj ~jj j

6

For instance, to select the mode matching to the ithbaseline shape ~ji , the mode ~jj corresponding to thelargest MACij shall be chosen. Additionally, the direc-

tion (sign) of the mode shape must be kept same asthe corresponding baseline shape. The condition canbe satisfied, if the inner product of the mode and its

baseline is always kept positive.

2.5. Noise injection learning

A crucial problem on the BPNN is its generalizationcapability. Usually, training patterns used for learningare only a limited number of samples selected from a

population of input and output patterns. Hence, a net-work successfully trained to a given set of samplesmay not provide the desired input and output associ-

ations for untrained patterns, particularly in the caseswith measurement noises and errors in the mode esti-mation. Concerning this problem, several researchers

reported that adding some noise to input patternsduring backpropagation learning process can remark-ably enhance the generalization capability of the resul-tant networks, if the mapping from the input space to

the output space is smooth [25,26]. Matsuoka foundthat the noise injected into the input reduces the sensi-tivity of the network to the variation of the input [25]:

that is, it makes the mapping from the input space tothe output space be smooth.In this study, noises are artificially added to natural

frequencies and mode shapes in generating the trainingand the testing patterns as

f ni fi1 vi 7

~jnji ~jji1 vji 8

where f ni and ~jnji are the ith natural frequency with

noise and the j-th component of the corresponding

modes with noise, and vi and vji are Gaussian randomnoises. The sequence of noise is generated by using asubroutine which returns a series of random numbers

normally distributed with zero mean and a specifiedstandard deviation. Based on Eqs. (7) and (8), thenoise level can be defined as the value of the standard

deviation. For instance, if the standard deviation is0.03, the noise level in the modal data can be referred

as 3% in the root-mean-square (RMS) level.

3. Illustrative numerical examples

Two dierent types of structures were selected for

example analyses to illustrate the applicability of theproposed approach. They are a two-span planar trussand a multi-storey frame structure.

3.1. Truss structure

A truss example is the two-span planar truss whichcontains 55 truss elements, 24 nodes and 44 nodal

DOFs as shown in Fig. 3. It was assumed that thebaseline parameters were known. Values for the ma-terial and geometric properties are as follows: the elas-

tic modulus=200 GPa; the cross-sectionalarea=0.03 m2; and the mass density=8000 kg/m3. Asthe first step of the identification procedure, the sti-

ness matrix of the structure is represented as

K X55j1

sjK0j 9

Then, the problem is to identify 55 unknown subma-trix scaling factors, sj, j = 1, . . . , 55, based on the

measured modal data. Instead of identifying the wholeSSFs simultaneously, the structure was subdivided intothree substructures; an internal substructure and two

external substructures as shown in Fig. 3. It wasassumed that the unknown SSFs for the elements inthe internal substructure were between 0.3 and 1.7,while those for the external substructures were between

0.5 and 1.5. The mode shapes were assumed to bemeasured only at 10 test DOFs, which include the dis-placements in the x- and y-directions at five nodes in

the internal-substructure.

3.1.1. Modal data for input to BPNN

For ecient training process, appropriate modaldata shall be selected as the input vector. It wasassumed that the first five modes of the internal-sub-structure were available at the test DOFs. The natural

frequencies and mode shapes of the reference structure,in which all the SSFs are unity, are shown in Fig. 4.The first, second and fifth modes are found to be verti-

cal, i.e., flexural modes, the third is a mixed mode withthe vertical and horizontal motions, and the fourth is ahorizontal mode. The MSE distribution in the first five

modes of the reference structure is shown in Fig. 5. Itshows that the Elements 1, 2, 4, 5, 7, and 10 are carry-ing large fractions of the strain energy in most of

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 4152 45

modes. Especially in the third and fourth modes whichcontain the horizontal motions, the horizontal elementssuch as the Elements 2, 5, 7, and 10 have relativelylarge values of MSE. On the other hand, the MSE

coecients associated with the Elements 3, 6, 8, 9, and11 are relatively small for all of the five modes, whichindicates that the accuracy of the SSF estimation for

the elements using the first five modal data may bepoor. Based on the above observation, the first fourmodal data were selected as an input pattern for this

example. However, other cases with dierent combi-nations of modes were also tested for the purpose ofcomparison.

3.1.2. Training and testing patternsFor the generation of training patterns using the

Latin hypercube sampling, 15 sampling points between

0.3 and 1.7 were selected with an equal interval of 0.1for each SSF in the internal substructures, assumingthat each SSF is uniformly distributed. On the other

hand, 11 sampling points between 0.5 and 1.5 weretaken with an interval of 0.1 for the SSFs in the exter-

nal substructures. Consequently, the number of thepossible sample cases is 165 (=15 11), which wasnot judged to be enough to represent the system ade-

quately. Hence, 10 iterations of the Latin hypercubesampling process were taken, and the 1650 training

patterns were sampled. For the cases of testing pat-

terns, the sampling points for the internal substructurewere taken as the same with those for the training

data, however the number of sampling points for theexternal substructures was taken to be 10 in the range

of 0.551.45 with an interval of 0.1. The dierences in

sampling points were imposed, because the real testingpatterns are not necessarily the same as the training

patterns. Thus, the number of possible sample cases

for testing was 150 (15 10).Using the CMS method, natural frequencies and

mode shapes were computed for sampled values of the

SSFs. In the process of computing the mode vectorsfor the sampled training and testing patterns and re-

arranging those in order of the baseline, the mode

shapes of some patterns may be very dierent withthose of the baseline. Accordingly, those patterns were

eliminated from the training and testing patterns.

Actually 1200 of the sampled training patterns wereused for training process and 100 of the sampled test-

ing patterns were used for testing process. The elapsetime for generating whole training data set was about

53 sec by the Alpastation 2004/233 workstation. The

variations of the SSFs in the external substructureswere considered in preparation for the training data

set, though they were not included in the output vec-

tor. Therefore, the output vector of the BPNN consistsof the SSFs in the internal substructure, while the

input vector consists of the first four natural frequen-

cies and the mode shape components for the internalstructure. The training and testing data set were pre-

pared for the cases with 0, 3, 5, and 7% noise in RMSlevel. An example of training patterns was shown in

Table 1.

Fig. 3. Substructuring for localized identification.

Fig. 4. First five mode shapes of a truss structure.

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 415246

3.1.3. Training and testing processes

It is important to choose the proper network size. If

the network is too small, it may not be able to rep-

resent the system adequately. On the other hand, if the

network is too big, it becomes over-trained and may

provide erroneous results for untrained patterns. In

general, it is not straight forward to determine the best

size of the networks for a given system. It may be

found through trial and error process using knowledge

about the system. A four-layer neural network as

shown in Fig. 1 was selected for the present example.

The number of the neurons in the input layers varies

depending on the number of modes included: i.e. 11

neurons for each mode. However, the numbers of the

neurons in the first hidden, the second hidden and the

output layers were taken to be fixed to 17, 13, and 11,

respectively as shown in Fig. 1. Numerical investi-

gations show that small changes of the numbers of the

neurons in the hidden layers have little eect on the

estimation results.

Fig. 5. Modal strain energy distribution for internal substructure of a truss structure.

Table 1

One example of training patterns

(a) Input pattern without noise

5.352 0.362 0.340 0.050 0.268 0.113 0.002 0.299 0.290 0.495 0.4977.594 0.045 0.080 0.076 0.064 0.071 0.187 0.386 0.374 0.572 0.57110.415 0.347 0.345 0.321 0.418 0.468 0.121 0.175 0.196 0.283 0.31712.913 0.032 0.242 0.358 0.637 0.591 0.208 0.086 0.022 0.036 0.071

(b) Input pattern with 3% noise in RMS

5.727 0.358 0.333 0.050 0.268 0.105 0.003 0.299 0.279 0.496 0.5117.637 0.049 0.080 0.074 0.064 0.069 0.207 0.357 0.371 0.576 0.58110.967 0.327 0.328 0.318 0.430 0.485 0.122 0.159 0.209 0.305 0.29512.349 0.028 0.259 0.334 0.615 0.622 0.212 0.075 0.021 0.034 0.067

(c) Output pattern (SSFs, sIj )

0.5 1.5 1.7 0.3 0.9 1.1 0.6 1.6 1.1 1.1 1.6

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 4152 47

The architecture of the neural network and therequired number of training patterns are inter-related.

However, there is no established method to determinethe relationship. According to the VapnikChervonen-kis dimension theory [21], the required number of

training patterns may be roughly taken as two times ofthe number of weights. In this study, however, thenumber of the training patterns was taken as 1200

through trial and error, which is approximately thenumber of weights including threshold (1153) as shownin Fig. 1. It is found that the values of the weightsafter the learning process are well-distributed without

severe saturation. It means that the present architec-ture is suitable and the number of the training patternsis sucient.

The training process took about 1000 epochs tolearn the pattern representation using the standard BP

algorithm, in which the elapse time was about 220 s byan Alpastation computer system. Generally the train-ing process is a time consuming work. However, once

the training is completed, the neural networks can beused very eciently and repeatedly for identifying thesystem parameters. Hence, it can be a very useful tool

for structural health monitoring.Each of the training patterns was used once for

training at an epoch, then the order of the trainingpatterns is randomly altered at the next epoch. An

example of the tested outputs was listed in the Table 2.Fig. 6 shows the averages of the relative errors in theestimated SSFs for various cases with dierent number

Table 2

One example of test outputs

NN Input 5.28 0.35 0.33 0.03 0.22 0.12 0.00 0.30 0.29 0.50 0.50( fj, ~jji) 8.01 0.03 0.14 0.07 0.11 0.07 0.08 0.30 0.27 0.62 0.61

10.63 0.24 0.28 0.27 0.56 0.59 0.00 0.05 0.07 0.20 0.2311.92 0.15 0.12 0.20 0.76 0.52 0.04 0.06 0.20 0.02 0.09

NN output sIj ) 1.47 0.89 1.44 0.90 1.13 0.37 0.35 1.27 0.45 0.62 1.52Target output 1.60 0.90 1.40 0.90 1.10 0.40 0.30 1.20 0.50 0.60 1.70

Relative error (%)a 7.65 0.26 2.98 0.22 3.06 5.32 19.67 5.89 8.10 4.79 10.44

a Relativeerror 100 jNN outputTarget outputTarget output

j.

Fig. 6. Average estimation errors for dierent modes used in input patterns (for testing data set without noise).

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 415248

of modes included in the input patterns. It can be

found that the accuracy of the estimation improveswith increasing number of the modes included until thefourth mode. But adding the fifth mode did not

improve the estimates. Hence, the first four modeswere included in the input to the BPNN.The eect of the measurement errors in the testing

data set and the eectiveness of the noise injection

learning were also investigated. Fig. 7 shows the esti-mation errors for the SSFs for dierent cases with var-ious noise levels in the training and testing data sets.

The noises in the training patterns mean the noisesinjected to the input patterns during the training pro-cess, and those in the testing patterns indicate the

measurement errors. Fig. 7 shows that the average esti-mation error for the noise-free testing data wasobtained as 8.9%, when the neural network has beentrained using the noise-free training data set. But, the

average error for the testing data with 5% noise usingthe same neural network was found to be 18.3%,which implies that the measurement error causes de-

terioration in the estimation. On the other hand, whenthe network has been trained using the training data

set with 5% noise, the error for the testing data with5% noise was obtained as 13.5%, which is much smal-ler than the error using the noise-free training data.

This indicates that it is desirable to use the 5% noisedata for training, if the measurement error is expectedto be 5%. As expected from the MSE distribution, the

estimation errors for the Elements 3, 6, 8, 9, and 11are larger than those for the other elements. However,the results generally lie in a satisfactory range. Table 3shows the similar results for the cases with 3 and 7%

measurement noises. The results also indicate that theestimation accuracy can be improved significantly, ifnoise with an intensity similar to the measurement

Fig. 7. Estimation error for dierent noise injection learning.

Table 3

Average estimation errors (%) for dierent noise injection

levels

Noise levels in RMS In testing data

0% 3% 5% 7%

In training data

0% 8.9 12.8 18.3 22.4

3% 9.7 11.1 14.0 15.8

5% 10.4 11.4 13.5 14.8

7% 12.0 12.6 13.7 14.7

Fig. 8. Learning curves without and with noise injection in

training patterns (for testing data set with 7% noise).

Fig. 9. Two-bay and ten-storey frame structure.

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 4152 49

noise level is imposed in the training process. Fig. 8 is

the learning curves to depict how the noise injectionlearning works. The learning rate is relatively high atthe epochs from 1 to 100, and after that, it becomes

lower. It can be observed from the curves that thenoise injection improves the results of estimationduring the learning process. Generally, the back-propa-

gation algorithm requires some reasonable criteria tostop the weight adjustment. The number of learningepochs was used as the termination criterion in this

paper, and it was taken as 1000 epochs, because thelearning curves showed that the decreasing rate of thetesting error became very flat after the epoch as inFig. 8. The level of the testing error may be used as a

practical termination criterion, too.

3.2. Frame structure

The second example is a frame structure as shown inFig. 9. The planar frame structure contains 50 el-

ements, 33 nodes and 90 nodal DOFs. The baselineparameters were listed in Table 4. The SSFs of the in-ternal substructure were estimated for twelve elements,

marked in Fig. 9. The natural frequencies and modeshapes of the frame structure are shown in Fig. 10.The four measurement cases in Fig. 11 were investi-

gated on the eect of rotational and translational com-ponents of mode shapes. The first four modes wereused as input patterns to the neural networks; 1200training and 100 testing patterns were actually used

during the learning process, similarly with the trussexample.Fig. 12 shows the average estimation errors for four

cases with dierent measured components of the mode

vectors. When all the DOFs at 9 nodes were used asthe input (Case 1), the estimation accuracy was good.However, when only the x- and y-DOFs were included

in the input pattern (Case 2), the estimation accuracybecame poor. On the other hand, when the transla-tional DOF at the mid point of each element in the in-

ternal substructure was added into the input pattern(Case 3), the estimation accuracy was found to beimproved significantly, which indicates that the transla-

tional data at the additional nodes may make up forthe information on the unmeasured rotational DOFs,which are usually dicult to be measured in the fieldexperiment. Additionally when the translational DOFs

at two quarter points of each element were added intothe input pattern (Case 4), the estimation accuracy wasfound to be better than the case using the rotational

DOFs (Case 1). The average relative errors for thetraining and testing data set with 5% noise were 8.8and 9.8%, respectively. It means that the generaliz-

ation performance of the present neural networks isvery good.

4. Conclusions

A neural network-based substructural identificationwas presented for the estimation of the stiness par-ameters of a complex structural system, particularly

for the case with noisy and incomplete measurement ofthe modal data. The proposed approach does notrequire any complicated formulation for model re-

duction of the system, which is indispensable to theconventional approaches for substructural identifi-cation. On the other hand, this approach requires gen-eration of enough training patterns which can

adequately represent the relationship between the sti-ness parameters and the modal information. The Latinhypercube sampling and the component-mode-syn-

thesis methods were used for the ecient generation ofsuch training patterns.The numerical example analyses were carried out

on a two-span truss and a multi-storey frame, andthe results were summarized as follows: (1) The sub-structuring technique and the concept of the subma-Fig. 10. First four mode shapes of a frame structure.

Table 4

Baseline parameters for frame examplea

Shape Area (m2) I (m4) Density (kg/m3)

Beams W24 15 1.04 102 5.62 104 7850Columns W14 145 2.78 102 7.12 104 7850

a I is moment of inertia of the cross-sectional area; E is modulus of elasticity at 210 GPa.

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 415250

trix scaling factor were found to be very ecient to

reduce the number of unknown stiness parameters

to be estimated. (2) The modal strain energy coe-

cients were eective indicators in selecting vibration

modes for the input to the neural networks. (3) In

the truss structure, the horizontal elements were

major load carrying members for lower modes,

hence the accuracy of the parameter estimation for

the elements was better than the others. (4) For the

case of the frame structure, the estimates became

deteriorated, when the rotational DOFs at the nodes

were not included in the input to the neural net-

works. However, the estimation accuracy could be

improved significantly, when additional measurement

information on the translational DOFs within the el-

ements was utilized. (5) The noise injection learning

using noise at similar level to the measurement error

was found to be very eective to improve the esti-

mation accuracy. (6) The average relative estimation

errors for testing data set with various noise intensi-

ties were found to be in the range of 915%, which

shows the applicability of the present method for the

identification of large structural systems.

Fig. 11. Cases of measured DOFs (sensor locations).

Fig. 12. Average estimation error for dierent measured

DOFs with 5% measurement noise (w: Case 1, +: Case 2, Q:Case 3, R: Case 4).

C.-B. Yun, E.Y. Bahng / Computers and Structures 77 (2000) 4152 51

Acknowledgements

The authors would like to thank to the KoreaEarthquake Engineering Research Center and theKorea Science and Engineering Foundation for their

financial support for this study.

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