# Fuzzified Data Based Neural Network Modeling for Health Assessment of Multistorey Shear Buildings

Authors: Deepti Moyi Sahoo; S. Chakraverty
Journal: Advances in Artificial Neural Systems
 (2013)

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## Abstract

The present study intends to propose identification methodologies for multistorey shear buildings using the powerful technique of Artificial Neural Network (ANN) models which can handle fuzzified data. Identification with crisp data is known, and also neural network method has already been used by various researchers for this case. Here, the input and output data may be in fuzzified form. This is because in general we may not get the corresponding input and output values exactly (in crisp form), but we have only the uncertain information of the data. This uncertain data is assumed in terms of fuzzy number, and the corresponding problem of system identification is investigated.


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## Body

## 1. Introduction

System identification methods in structural dynamics, in general, solve inverse vibration problems to identify properties of a structure from measured data. The rapid progress in the field of computer science and computational mathematics during recent decades has led to an increasing use of process computers and models to analyze, supervise, and control technical processes. The use of computers and efficient mathematical tools allows identification  of the process dynamics by evaluating the input and output signals of the system. The result of such a process identification is usually a mathematical model by which the dynamic behaviour can be estimated or predicted. The system identification problem has been nicely explained in a recent paper [1]. The same statements from [1] are reproduced below for the benefit of the readers.The study of structures dynamic behaviour may be categorized into two distinct activities: analytical and/or numerical modelling (e.g., finite element models) and vibration tests (e.g., experimental modal models). Due to different limitations and assumptions, each approach has its advantages and shortcomings. Therefore, in order to determine the dynamic properties of the structure, reconciliation processes including model correlation and/or model updating should be performed. Model updating can be defined as the adjustment of an existing analytical/numerical model in the light of measured vibration test. After adjustment, the updated model is expected to represent the dynamic behaviour of the structure more accurately as proposed by Friswell et al. [2]. With the recent advances in computing technology for data acquisition, signal processing, and analysis, the parameters of structural models may be updated from the measured responses under excitation of the structure. This procedure is achieved using system identification techniques as an inverse problem. The inverse problem may be defined as determination of the internal structure of a physical system from the system’s measured behaviour, or estimation of an unknown input that gives rise to a measured output signal according to Tanaka and Bui [3].Comprehensive literature surveys have been provided on the subject of model updating of the structural systems by Alvin et al. [4], and Time series methods for fault detection and identification in vibrating structures were presented by Fassois and Sakellariou [5]. Shear buildings are among the most widely studied structural systems. Previous works on model updating of shear buildings rely mostly on using modal parameter identification and physical or structural parameter identification to drive the corresponding update procedures. As regards the publications, Marsi et al. [6] gave various methodologies for different types of problems in system identification. Various techniques for improving structural dynamic models were reviewed in a review paper by Ibanez [7], and studies made by  Datta et al. [8] related to system identification of buildings done until that date were also surveyed. Some of the related publications may be mentioned as those of  Loh and Tou  [9] and Yuan et al. [10].It is known that, the systems which may be modeled as linear, the identification problem often turns in to a non-linear optimization problem. This requires an intelligent iterative scheme to have the required solution. There exists various online and offline methods, namely, the Gauss-Newton, Kalman filtering and probabilistic methods such as maximum likelihood estimation, and so forth. However, the identification problem for a large number of parameters, following two basic difficulties are faced often:(i)
objective function surface may have multiple maxima and minima, and the convergence to the correct parameters is possible only if the initial guess is considered as close to the parameters to be identified;(ii)
inverse problem in general gives nonunique parameter estimates.To overcome these difficulties, researchers have developed various identification methodologies for the said problem by using  powerful technique of  Artificial Neural Network (ANN). Chen [11] presented a neural network based method for determining the modal parameters of structures from field measurement. Using the observed dynamic responses, he trained the neural network based on back-propagation technique. He then directly identified the modal parameters of the structure using the weight matrices of the neural network. In particular, Huang et al.[12] presented a novel procedure for identifying the dynamic characteristics of a building using a back-propagation neural network technique. Another novel neural network based approach has been presented by Kao and Hung [13] for detecting structural damage. A decentralized stiffness identification method with neural networks for a multidegree of freedom structure has been developed by Wu et al. [14]. Localized damage detection and parametric identification method with direct use of earthquake responses for large-scale infrastructures has also been proposed by Xu et al. [15]. A neural network based strategy by Xu et al. [16] was developed for direct identification of structural parameters from the time domain dynamic responses of an object structure without anyeigen value analysis.System identification on the other hand tries to identify structural matrices of mass, damping and stiffness directly. Among various methodologies in this regard Chakraverty [17], Perry et al. [18], Wang [19], Yoshitomi and Takewaki [20], and Lu and Tu [21] developed different techniques to handle the system identification problems. Yuan et al. [10] developed a methodology that identifies the mass and stiffness matrices of a shear building from the first two orders of structural mode measurement. Koh et al. [22] proposed several Ga-based substructural identification methods, which work by solving parts of the structure at a time to improve the convergence of mass and stiffness estimates particularly for large systems. Chakraverty [17] proposed procedures to refine the methods of Yuan et al. [10] to identify the structural mass and stiffness matrices of shear buildings from the modal test data. The refinement was obtained using Holzer criteria. Tang et al. [23] utilized a differential evolution (DE) strategy for parameter estimation of the structural systems with limited output data, noise polluted signals, and no prior knowledge of mass, damping, or stiffness matrices. Recent works on model updating of multistory shear buildings for simultaneous identification of mass, stiffness, and damping matrices using two different soft-computing methods have been developed by Khanmirza et al. [1]. It may be seen from above that Artificial Neural Networks (ANNs) provide a fundamentally different approach to system identification. They have been successfully applied for identification and control of dynamics systems in various fields of engineering because of excellent learning capacity and high tolerance to partially inaccurate data.It is revealed from the above literature review that various authors developed different identification methodologies using ANN. They supposed that the data obtained are in exact or crisp form. But in actual practice the experimental data obtained from equipments are with errors that may be due to human or equipment error, thereby giving uncertain form of the data. Although one may also use probabilistic methods to handle such problems. Then, the probabilistic method requires huge quantity of data which may not be easy or feasible. Thus in this paper, a minimum number of data are taken in fuzzified form to have the essence of the uncertainty. Accordingly, in this paper, identification methodologies for multistorey shear buildings have been proposed using the powerful technique of Artificial Neural Network (ANN) models which can handle fuzzified data. It is already mentioned that identification with crisp data is known and also neural network method has already been used by various researchers for this case. Here, the input and output data may be in fuzzified form. This is because in general we may not get the corresponding input and output values exactly (in crisp form), but we have only the uncertain information of the data. This uncertain data has been assumed to be in terms of fuzzy numbers.In this paper, the initial design parameters, namely, stiffness and  mass and so the  frequency  of the said problem is known. But after a large span of time, the structure may be subjected to various manmade and natural calamities. Then, the engineers want to know the present health of the structure by system identification methods. It is assumed that only the stiffness is changed and the mass remains the same.  As such equipments are available to get the present values of the frequencies and using these one may get the present parameter values by ANN. But while doing the experiment, one may not get the exact values of the parameters. But we may get those values as uncertain, namely, in fuzzy form. So if sensors are placed to capture the frequency of the floors in fuzzy (uncertain) form, then those may be fed into the proposed new ANN model to get the present stiffness parameters in fuzzified form. In order to train the new ANN model, set of data are generated numerically beforehand. As such, converged ANN model gives the present stiffness parameter values in interval form for each floor. Thus, one may predict the health of the uncertain structure. Corresponding example problems have been solved, and related results are reported to show the reliability and powerfulness of the model.


## 2. Analysis and Modelling

System identification refers to the branch of numerical analysis which uses the experimental input and output data to develop mathematical models of systems which finally identify the parameters. The floor masses for this methodology are assumed to be[m1_,m1c,m1¯], [m2_,m2c,m2¯],…,[mn_,mnc,mn¯], and the stiffness [k1_,k1c,k1¯],[k2_,k2c,k2¯],…,[kn_,  knc,  kn¯] are the structural parameters which are to be identified. It may be seen that all the mass and stiffness parameters are taken in fuzzy form. As such here for each mass mi, we have mi_ as the left value, mic as the centre value, and mi¯ as the right value. Similarly for the stiffness parameter for each mass ki, we have ki_ as the left value, kic as the centre value, and ki¯ as the right value. The n-storey shear structure is shown in Figure 1. Corresponding dynamic equation of motion for n-storey (supposed as n degrees of freedom) shear structure without damping may be  written as
(1){M~}{X~¨}+{K~}{X~}={0~},
where {x~¨}={x¨_,x¨c,x¨-},{x~}=[x_,xc,x¯].Figure 1
Multistorey shear structure withn-levels having fuzzy structural parameters.{
M
~
}
=
[
M
_
,
M
c
,
M
¯
] is n×n  mass matrix of the structure and is given by(2){M~}=[[m1_,m1c,m1¯]0……00[m2_,m2c,m2¯]0…0…………………0[mn-1_,mn-1c,mn-1¯][mn_,mnc,mn¯]0……00].{K~}=[K_,Kc,K¯]isn×n stiffness matrix of the structure and may be written as
(3){K~}=[k~1+k~2k~20⋯0-k~2k~2+k~3-k~3⋯0⋯⋯⋯0⋯-k~n-1k~n-1+k~nk~n0⋯-k~nk~n],
and {X~}={x~1,x~2,…,x~n}T are the vectors of displacement.We will first solve the above free vibration equation for vibration characteristics, namely, for frequency and mode shapes of the said structural system in order to get the stiffness parameters in fuzzified form. Accordingly putting{X~}={ϕ~}ei(ω~)t in free vibration equation (1), we get
(4)({K~}-{M~}[ω~]2){ϕ~}={0~},
where {ω~}2=[ω_,ωc,ω¯]2=[λ_,λc,λ¯] are eigenvalues or the natural frequency  and {ϕ~} are mode shapes of the structure, respectively.


## 3. Basic Concept of Fuzzy Set Theory

Definition 1.
LetX be a universal set. Then, the fuzzy subset A of X is defined by its membership function
(5)μA:X→[0,1],
which assigns a real number μA(x) in the interval [0,1], to each element x∈X, where the value of μA(x) at x shows the grade of membership of x in A.Definition 2.
Given a fuzzy setA in X and any real number α∈[0,1], then, the α-cut or α-level or cut worthy set of A, denoted by Aα, is the crisp set
(6)Aα={x∈X∣μA(x)≥α}.
The strong α-cut, denoted by Aα+, is the crisp set
(7)Aα+={x∈X∣μA(x)≥α}.Definition 3.
A fuzzy number is a convex normalized fuzzy set of the real lineR whose membership function is piecewise continuous.Definition 4.
A triangular fuzzy numberA can be defined as a triplet [a1,a2,a3]. Its membership function is defined as
(8)μA(x)={0,x<a1x-a1a2-a1,a1≤x≤a2a3-xa3-a2,a2≤x≤a30,x>a3.
Above TFN may be transformed to an interval form Aα by α-cut  as
(9)Aα=[a1(α),a3(α)]=[(a2-a1)α+a1,-(a3-a2)α+a3].


## 4. Operation of Fuzzy Number

In this section, we consider arithmetic operation on fuzzy numbers and the result is expressed in membership function:(10)∀x,y,z∈R.(1)
Addition:A(+)B(11)μA(+)B(z)=⋁z=x+y(μA(x)∧μB(y)).(2)
Subtraction:A(-)B(12)μA(-)B(z)=⋁z=x-y(μA(x)∧μB(y)).(3)
Multiplication:A(·)B(13)μA(·)B(z)=⋁z=x·y(μA(x)∧μB(y)).(4)
Division:A(/)B(14)μA(/)B(z)=⋁z=x/y(μA(x)∧μB(y)).(5)
Minimum:A(∧)B(15)μA(∧)B(z)=⋁z=x∧y(μA(x)∧μB(y)).(6)
Maximum:A(∨)B(16)μA(∨)B(z)=⋁z=x∨y(μA(x)∧μB(y)).


## 5. Artificial Neural Network (ANN) and Error-Back Propagation Training Algorithm (EBPTA) for Fuzzified Data

Traditional ANN and EBPTA are well known, but here for the sake of completeness, those are developed for fuzzy case. In ANN, the first layer is considered to be input layer and the last layer is the output layer. Between the input and output layers, there may be more than one hidden layer. Each layer will contain number of neurons or nodes (processing elements) depending upon the problem. These processing elements operate in parallel and are arranged in patterns similar to the patterns found in biological neural nets. The processing elements are connected to each other by adjustable weights. The input/output behavior of the network changes if the weights are changed. So, the weights of the net may be chosen in such a way so as to achieve a desired output. To satisfy this goal, systematic ways of adjusting the weights have to be developed to handle the fuzzified data which are known as training or learning algorithm. Neural network basically depends upon the type of processing elements or nodes, the network topology, and the learning algorithm. Here, error back-propagation training algorithm and feedforward recall have been used but to handle the uncertain system. The typical network is given in  Figure2.Figure 2
Layered feed-forward fuzzy neural network.In this Figure,Zi, Pj, and Om are input, hidden, and output layers, respectively. The weights between input and hidden layers are denoted by vji, and the weights between hidden and output layers are denoted by Wkj. Here, Z~i=[λi_λicλi¯]  and O~k=[km_kmckm¯].GivenR training pairs {Z~1,d~1;Z~2,d~2;…,Z~R,d~R} where Z~i(I×1) are input and d~i(M×1) are desired values for the given inputs, the error value is computed as
(17)E~=12(d~m-O~m)2,m=1,2,…M,
for the present neural network as shown in Figure 2. The error signal terms of the output (δ~Om) and hidden layers (δ~pj) are written, respectively, as
(18)δ~Om=0.5*(d~m-O~m)(1-O~m2),m=1,2,…M,δ~Pj=0.5*(1-P~j2)∑m=1Mδ~OmW~Pj,j=1,2,…J.Consequently, output layer weights(W~mj) and hidden layer weights (v~ji) are adjusted as (19)W~mj(New)=W~mj(Old)+ηδ~OmPj,m=1,2,…M,j=1,2,…J,v~ji(New)=v~ji(Old)+ηδ~PjZi,j=1,2,...J,i=1,2,...I,where η is the learning constant.


## 6. Results and Discussion

To investigate the present method here, examples of one- and two-storey shear structures are considered. So, for example, the floor masses for two-storey shear structure are[m1_,m1c,m1¯],  [m2_,m2c,m2¯]and the stiffnesses [k1_,k1c,k1¯],  [k2_,k2c,k2¯] are the structural parameters. Here, masses are assumed to be constant (as mentioned earlier). So, we will identify the stiffness parameter in fuzzy form using the fuzzy form of the frequency where frequency may be obtained from some experiments. In the following paragraphs, we have used the proposed method to identify the stiffness parameter for one-, two-, five-, and ten-storey frame structures. Here, we have considered the cases with crisp data for five- and ten-storeys and then fuzzified data for one- and two-storeys. The training data are also considered with the influence of noise, namely, in terms of triangular fuzzy number data. Accordingly we have considered the following four cases: Case(i):
Five-storey shear structure with crisp data,Case(ii):
Ten-storey shear structure with crisp data,Case(iii):
Single-storey shear structure with fuzzified data,Case(iv):
Double-storey shear structure with fuzzified data.Computer programs have been written and tested for variety of experiments for the above cases. For the first two cases, namely, Case(i) and Case(ii), the inputs are taken as the crisp frequency values and the outputs are the stiffness parameters which are also in crisp form. On the other hand, for Cases(iii) and (iv), the inputs are taken as the fuzzified frequency values and the outputs are the stiffness parameters again in fuzzified form in the developed Fuzzy Neural Network (FNN) algorithm.For the first case, an example of a five-storey shear structure is taken where the masses arem1=m2=m3=m4=m5=36000 and the stiffness parameters are within the range k1=[100000200000], k2=[50000100000], k3=[4000060000], k4=[3000050000], and k5=[2000030000]. A comparison between the desired and ANN values has been presented in Table 1. This table has been plotted in Figure 3.Table 1
Comparison between the desired and the ANN values ofk1, k2, k3, k4, and k5 for a five-storey shear structure.
Datanumber
k
1 (Ann)
k
1 (Des)
k
2 (Ann)
k
2 (Des)
k
3 (Ann)
k
3 (Des)
k
4 (Ann)
k
4 (Des)
k
5 (Ann)
k
5 (Des)
1
181277.7318
181472.3686
57722.68
57880.6541
53111.561
53114.814
44531.8138
44120.9218
24304.5301
24387.4436
2
195883.7223
190579.1937
98497.3362
98529.6391
40890.7789
40714.2336
30510.5884
30636.6569
24013.4116
23815.5846
3
110889.324
112698.6816
97540.2878
97858.3474
56939.1333
56982.5861
35467.0709
35538.4597
27905.0723
27655.1679
4
191334.6369
191337.5856
74367.0906
74268.7824
58151.3678
58679.865
30921.7444
30923.4278
27758.983
27951.999
5
162671.148
163235.9246
90080.9291
90014.0234
53507.0758
53574.7031
31851.263
31942.6356
21775.0509
21868.726
6
109757.8778
109754.0405
57034.3391
57094.3169
55073.904
55154.8026
46360.9403
46469.1566
24899.5619
24897.644
7
127671.411
127849.8219
71003.1282
71088.0641
54999.758
54862.6494
43688.9611
43896.5725
24554.2598
24455.862
8
154429.3614
154688.1519
98500.0073
95786.7763
48031.3148
47844.5404
36250.0047
36341.9896
27719.1389
26463.1301
9
196399.16
195750.6835
89751.9025
89610.3665
53100.0372
53109.5578
48498.0106
49004.441
27950.5364
27093.6483
10
193722.4668
196488.8535
98520.881
97974.6213
43228.929
43423.7338
30822.0268
30688.9216
26813.9637
27546.8668Figure 3
Comparison between the desired and the ANN values ofK~ for a single-storey shear structure.In Case(ii), an example for a ten-storey shear structure has been considered with constant masses similar to Case(i) and the stiffness parameters are in the rangek1=[100000200000], k2=k3=k4=k5=k6=k7=k8=k9=k10=[2000030000]. The desired and ANN values  for k1  to  k5 and k6  to  k10 are compared in Tables 2(a) and 2(b), respectively.Table 2
(a) Comparison between the desired and the ANN values ofk1, k2, k3, k4, and k5 for a ten-storey shear structure. (b) Comparison between the desired and the ANN values of k6, k7, k8, k9, and k10 for a ten-storey shear structure.
(a)
Data number
k
1 (Ann)
k
1 (Des)
k
2 (Ann)
k
2 (Des)
k
3 (Ann)
k
3 (Des)
k
4 (Ann)
k
4 (Des)
k
5 (Ann)
k
5 (Des)
1
116900.6563
114999.7254
23626.293
23947.0748
25255.905
24299.2141
29078.6982
29493.0391
27766.1413
28842.8102
2
137413.4659
135922.821
22400.1311
21970.538
22076.7651
22160.1892
28671.6078
29898.7215
23928.6799
23185.2425
3
174166.5579
171165.6706
26751.8243
27587.6627
28439.4836
28089.9027
28056.19
27636.7332
28909.5075
29349.7909
4
185745.6345
187147.6518
28203.5977
29952.1598
23288.5994
23565.0893
25414.2964
25588.2055
24175.8385
24794.8455
5
131376.8088
132868.9612
22531.4396
21865.7144
20887.9013
20732.4343
22490.5032
21838.4294
21747.4496
22317.9161
6
174687.251
165011.8025
25813.5863
27811.4527
25015.1015
25909.9146
25615.7328
24979.4882
24612.1573
23962.9025
7
185115.3655
197483.6148
24841.291
21957.9798
27352.9705
29101.8783
25184.1779
25178.456
27414.6225
27050.7748
8
107439.5782
107596.7361
28408.1614
29923.5897
22079.0745
21937.6594
29361.5732
29942.4301
25283.6014
25585.5903
9
164872.2954
158701.9167
25498.9862
28022.6157
26269.1879
24323.6779
28505.5766
28548.5168
27421.7863
27566.307
10
134496.3197
136428.6869
23984.6699
23091.3643
26528.5422
27288.6387
20735.4877
20391.8449
26590.0291
26789.4101
(b)
Data number
k
6 (Ann)
k
6 (Des)
k
7 (Ann)
k
7 (Des)
k
8 (Ann)
k
8 (Des)
k
9 (Ann)
k
9 (Des)
k
10 (Ann)
k
10 (Des)
1
21170.0974
20899.5068
25777.6655
25605.5953
28204.3487
29899.5021
27503.4171
25859.8704
27163.5463
25814.4649
2
22417.1364
20549.7415
27621.6492
28654.3859
27612.505
28451.7819
28303.9996
29823.0322
22093.3505
22094.0508
3
29672.9685
29638.7013
25998.8211
27124.1481
22438.2305
21982.2179
26315.5549
26153.251
27684.8546
29019.9081
4
19962.2582
19656.5635
21156.2588
20166.7471
22269.5196
21950.7153
23818.7821
23766.1108
27240.5745
27020.6645
5
20161.805
20514.4829
25387.6098
28009.2088
23922.4631
23268.3965
27519.8427
28771.8175
23219.6412
23774.551
6
22307.0935
23043.4895
21527.1166
21425.0932
27311.6444
28803.3786
27670.7454
27848.5243
27234.7985
27349.5593
7
27804.4452
25801.9183
24442.8169
24784.7447
25319.1653
24711.0187
25870.6554
24649.5428
27640.8021
29541.0279
8
25509.2344
25309.6445
22738.1369
22568.3535
23936.8813
24039.6937
28924.1116
28139.7693
25172.6749
25428.1311
9
28341.8519
29012.0809
25380.1928
23690.9169
23071.1828
21792.3148
28034.5751
28984.4414
26519.4395
25401.0583
10
27911.2912
29624.314
26475.8553
24319.8061
21306.3632
21696.0881
25327.5363
24074.5574
24485.2563
23343.2942For Case(iii), the first example is that of a single-storey shear structure with massesM~=36000 and the stiffness parameters lie within the range K_=[100000200000], Kc=[100010200010], and K¯=[100020200020]. A comparison between desired and the ANN values has been incorporated in Table 3. This table has been plotted in Figure 4. In the second example, a single-storey shear structure is considered with masses M~=36000 and the stiffness parameter varying within the range K_=[50000100000], Kc=[50010100010], and  K¯=[50020100020]. Comparison between the desired and the ANN values is tabulated in Table 4 and is plotted in Figure 5.Table 3
Comparison between the desired and the ANN values ofK~ for a single-storey shear structure.
Data  number
K
_ (Ann)
K
_ (Des)
K
c (Ann)
K
c (Des)
K
¯ (Ann)
K
¯ (Des)
1
124932.9319
124189.1286
135932.8213
135095.2381
191905.7285
190281.611
2
135763.4956
140411.2146
152464.2896
151324.954
191726.9669
194488.719
3
110291.5736
109665.4525
141591.1084
140180.8034
148514.9196
149096.4092
4
107394.0584
107596.6692
115134.7117
113217.3293
149119.797
148935.2638
5
124430.5911
123991.6154
134455.7107
133781.941
192947.0787
194225.0591
6
112574.5401
112331.8935
188853.0283
190015.3846
194542.4439
195633.454
7
120988.7913
118390.7788
138166.8698
136934.6781
157619.3944
157540.8595
8
108960.7756
105997.9543
113526.3083
111130.2755
126504.2017
123995.2526
9
124700.9011
123497.9913
143556.5197
141726.7069
179985.5092
178035.2068
10
105227.009
104965.443
135988.057
135335.8571
140612.991
138983.8837Table 4
Comparison between the desired and the ANN values ofK~ for a single-storey shear structure.
Data    number
k
_ (Ann)
k
_ (Des)
k
c (Ann)
k
c (Des)
k
¯ (Ann)
k
¯ (Des)
1
62365.0341
62104.5643
67644.4378
67547.619
95458.8463
95145.8055
2
67613.8959
70215.6073
75826.931
75662.477
95893.1612
97249.3595
3
55005.0087
54842.7263
70556.3936
70090.4017
74368.7997
74553.2046
4
53128.7186
53798.3346
56947.542
56618.6646
74261.1768
74472.6319
5
62246.1626
61995.8077
66881.2194
66895.9705
96278.4523
97122.5295
6
55957.2602
56165.9467
94391.1219
95012.6923
97271.6645
97826.727
7
60056.7259
59195.3894
68223.7328
68472.3391
78385.7211
78780.4298
8
53587.3374
53008.9771
55747.5298
55570.1378
61968.3882
61997.6263
9
62126.6388
61758.9957
71196.546
70863.3535
89991.9346
89022.6034
10
52238.7913
52482.7215
67702.5259
67677.9286
69679.1387
69496.9418Figure 4
Comparison between the desired and the ANN values ofK~ for a single-storey shear structure.Figure 5
Comparison between the desired and the ANN values ofK~ for a single-storey shear.In Case(iv), the first example of a double-storey shear structure is considered where the masses arem~1=m~2=36000 and the stiffness parameters varying within the range  k1_=[100000200000], k1c=[100010200010], k1¯=[100020200020] and k2_=[2000030000], k2c=[2001030010], and k2¯=[2002030020]. The desired and ANN values have been compared in Tables 5(a) and 5(b). This table has also been shown in Figures 6(a) and 6(b). In the second example, a double-storey shear structure is implemented with masses m~1=m~2=36000 and the stiffness parameters having the range k1_=[50000100000], k1c=[50010100010], k1¯=[50020100020] and k2_=[2000030000], k2c=[2001030010], and k2¯=[2002030020]. Comparison between the desired and ANN values are again incorporated in Tables 6(a) and 6(b). This table is plotted in Figures 7(a) and 7(b).Table 5
(a) Comparison between the desired and the ANN values ofk~1 for a double-storey shear structure. (b) Comparison between the desired and the ANN values of k~2 for a double-storey shear structure.
(a)
Data number
k
1
_ (Ann)
k
1
_ (Des)
k
1
c (Ann)
k
1
c (Des)
k
1
¯ (Ann)
k
1
¯ (Des)
1
113258.5422
113317.1008
133858.2555
133969.3413
162741.2326
162807.3359
2
116526.66
117338.8613
128622.4646
129208.408
193447.2534
195183.0465
3
139119.0131
139093.7802
143501.0894
143175.117
191801.5675
192053.204
4
101316.7594
101558.7126
106512.8803
105287.6998
183040.5812
183137.9743
5
173554.7972
173805.8096
180269.037
180336.4392
198199.4731
198416.3724
6
105985.8857
106047.1179
117592.2043
116726.841
127688.8972
126931.9426
7
110472.1362
110631.6345
140132.9188
139925.7771
142461.6007
142303.5615
8
137120.3743
137250.974
152856.9037
152687.5831
155005.6994
154807.0901
9
119783.849
119821.8403
142480.6438
141679.9468
195824.6624
194293.6984
10
141708.5217
141794.4104
148654.6797
148978.7638
165370.4902
165685.9891
(b)
Data number
k
2
_ (Ann)
k
2
_ (Des)
k
2
c (Ann)
k
2
c (Des)
k
2
¯ (Ann)
k
2
¯ (Des)
1
58488.0205
58566.0533
91472.5908
92796.1403
99954.016
99152.6233
2
51977.6459
51640.041
65679.199
65072.7474
85856.2114
82258.2268
3
69213.7504
68833.6105
77764.8854
78069.9896
85939.2478
85054.9378
4
60124.2902
59566.1848
83319.0924
83316.9426
93794.9834
94103.325
5
71317.3378
71432.6496
76748.5343
76956.3233
83930.3883
83468.7652
6
59384.591
59531.6634
73687.8221
74121.1031
85087.5853
84905.276
7
56031.7629
56050.5807
68215.603
68455.8273
83728.3652
83326.3957
8
58849.1438
58906.6227
72741.8224
73046.2969
79796.3231
79495.3742
9
57077.8681
56400.72
61035.9761
61329.384
95110.8992
99091.8975
10
57858.7023
57830.2476
69404.8867
69250.9562
99433.354
99954.0197Table 6
(a) Comparison between the desired and the ANN values ofk~1 for a double-storey shear structure. (b) Comparison between the desired and the ANN values of k~2 for a double-storey shear structure.
(a)
Data number
k
1
_  (Ann)
k
1
_ (Des)
k
1
c (Ann)
k
1
c (Des)
k
1
¯ (Ann)
k
1
¯ (Des)
1
62193.4503
62149.2679
65871.2249
66562.894
69219.2648
69672.8181
2
71578.6175
71235.4748
71816.4848
72130.1157
83311.5254
83571.557
3
63514.5116
63533.5212
84447.6191
84399.8043
86708.0123
87062.8972
4
59459.7849
59872.6899
67907.0231
67971.4105
75148.6421
76002.6234
5
67809.323
67385.6336
85655.9802
86827.0037
91460.6917
91106.0592
6
57879.7498
57499.8627
69206.8111
69745.3738
71387.5577
71516.0705
7
77679.4222
79304.6034
83870.4195
84180.7933
93380.175
94408.5477
8
63484.2106
63107.2659
69028.8134
69579.1498
85313.0398
85212.3715
9
52329.6174
52222.7046
72316.9085
72125.2707
88399.7894
88475.7194
10
50895.5951
50988.8812
69532.938
69859.5759
87547.575
87746.6634
(b)
Data number
k
2
_ (Ann)
k
2
_ (Des)
k
2
c (Ann)
k
2
c (Des)
k
2
¯ (Ann)
k
2
¯ (Des)
1
27396.6322
27507.0572
27776.8241
27698.5425
28078.4211
28085.141
2
21439.7109
21682.5355
27548.5904
27550.771
29081.9724
28275.8382
3
23717.5546
23773.9554
27948.7301
27919.6303
28617.4807
28629.8048
4
22090.0439
22160.1892
23180.7111
23205.2425
28962.7934
29908.7215
5
25086.5697
25154.2346
25465.6455
25360.6413
27961.1253
27904.0722
6
20724.6386
20919.5068
28645.592
28852.8102
29353.9707
29493.0391
7
20706.4881
21137.0574
23381.0249
23275.6543
25716.8788
25890.2606
8
21056.3736
21382.9255
21574.7879
21557.5235
26851.9289
26712.6437
9
22075.9397
22008.6282
24479.3131
24386.4498
27062.0237
26806.523
10
24070.1411
24079.5484
25009.5776
24971.7702
28009.5803
28335.006(a) Comparison between the desired and the ANN values ofk~1 for a double-storey shear structure. (b) Comparison between the desired and the ANN values of   k~2 for a double-storey shear structure.
(a)
(b)(a) Comparison between the desired and the ANN values ofk~1 for a double-storey shear structure. (b) Comparison between the desired and the ANN values of   k~2 for a double-storey shear structure.
(a)
(b)The training data with the influence of noise for two-storey shear structure in TFN form for five sets of data have been presented here. Accordingly, Figures8(a) and 8(b) refer the fuzzy plot of frequency. Moreover, the Triangular Fuzzy Number (TFN) plots of identified stiffness are cited in Figures 9(a) and 9(b). Also for different alpha values such as α=0.3,α=0.5, and α=0.8, the comparison between the desired and ANN values with another five sets of data has been given in Tables 7(a), and 7(b), and 7(c).Table 7
(a) Comparison between the desired and the ANN values ofk1_, k1¯ and k2_, k2¯ for a double-storey shear structure for α=0.3. (b) Comparison between the desired and the ANN values of k1_, k1¯ and k2_, k2¯ for a double-storey shear structure for α=0.5. (c) Comparison between the desired and the ANN values of k1_, k1¯ and k2_, k2¯ for a double-storey shear structure for α=0.8.
(a)
Data number
k
1
_ (Ann)
k
1
_ (Des)
k
1
¯ (Ann)
k
1
¯ (Des)
k
2
_ (Ann)
k
1
_ (Des)
k
2
¯ (Ann)
k
2
¯ (Des)
1
109470
109250
124660
123870
63676
63908
81668
81670
2
119370
119420
141760
141590
59687
59772
79075
78865
3
141840
141880
154360
154170
63017
63149
77680
77561
4
126590
126380
179820
178510
58265
57879
84888
87763
5
143790
143950
160360
160670
61323
61256
90425
90743
(b)
Data number
k
1
_ (Ann)
k
1
_ (Des)
k
1
¯ (Ann)
k
1
¯ (Des)
k
2
_ (Ann)
k
1
_ (Des)
k
2
¯ (Ann)
k
2
¯ (Des)
1
111790
111390
122640
121830
66536
66826
79388
79513
2
125300
125280
141300
141110
62124
62253
75972
75891
3
144990
144970
153930
153750
65795
65976
76269
76271
4
131130
130750
169150
167990
59057
58865
78073
80211
5
145180
145390
157010
157330
63632
63541
84419
84602
(c)
Data number
k
1
_ (Ann)
k
1
_ (Des)
k
1
¯ (Ann)
k
1
¯ (Des)
k
2
_ (Ann)
k
1
_ (Des)
k
2
¯ (Ann)
k
2
¯ (Des)
1
115270
114590
119610
118770
70827
71203
75968
76278
2
134200
134070
140600
140400
65779
65975
71318
71430
3
149710
149600
153290
153110
69963
70218
74153
74336
4
137940
137310
153150
152200
60244
60344
67851
68882
5
147270
147540
152000
152320
67096
66967
75411
75392(a) Comparison ofλ1_ and λ1¯ with respect to α. (b) Comparison of   λ2_ and λ2¯   with respect to α.
(a)
(b)(a) Comparison ofk1_andk1¯ with respect to α. (b) Comparison of   k2_andk2¯ with respect to α.
(a)
(b)


## 7. Conclusion

Here, the procedure is demonstrated to identify stiffness parameters for multistorey shear structure using fuzzified data in ANN. The present study considers example problems of one-, two-, five-, and ten-storey shear structures. Identification study for five- and ten-storey shear structures has been done with crisp data. Then, fuzzified data has been considered for one- and two-storey shear structures for the present identification procedure. Initial design parameters, namely, stiffness and mass and so the frequency of the said problem is known in term of fuzzy numbers. The engineers want to know the present health of the structure by system identification methods. It is assumed that only the stiffness is changed and the mass remains the same. The present values of the frequencies may be obtained by available equipments, and using these, one may get the present parameter values by ANN. So, if sensors are placed to capture the frequency of the floors in fuzzy (uncertain) form, then, those may be fed into the proposed new ANN model to get the present stiffness parameters. The methods of one- and two-storey shear structures with fuzzified data may very well be extended for higher storey structures following the present procedure. As regards the influence of noise, it may be seen that the input and output data for two-storey shear structure are actually in terms of Triangular Fuzzy Number (TFN) which themselves dictate the noise in both monotonic increasing and decreasing senses. In order to train the new ANN model, set of data are generated numerically beforehand. As such, converged ANN model gives the present stiffness parameter values in fuzzified form for each floor. Thus, one may predict the health of the structure. Corresponding example problems (as mentioned) have been solved, and related results are reported to show the reliability and powerfulness of the model.



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*Source: 962734-2013-03-26.xml*