Damage Detection in Structures using Artificial Neural Networks

Shilei Zhang, Huanding Wang, Wei Wang School of Civil Engineering

Harbin Institute of Technology Harbin, China

e-mail: lei202202@yahoo.com.cn, Hdwhrb@hit.edu.cn and wwang@hope.hit.edu.cn

Shaofeng Chen School of Transportation Science and Engineering

Harbin Institute of Technology Harbin, China

e-mail: Chensf@hit.edu.cn

Abstract—In order to select a sensitive input parameter for artificial neural networks in damage detection and construct an efficient and robust back propagation algorithm for damage assessment, the application of neural networks to damage detection in structures is summarized and analyzed in this paper. By discussing the use of natural frequency as a diagnostic parameter, natural frequency can rationally reflect damage location but not provide enough information about damage degree. Mode shape and transfer function include abundance information about damage degree compared with natural frequency but have a large measurement error. And three improved back propagation algorithms that are adaptive variable step-size algorithm, Levenberg-Marquart algorithm and homogeneous algorithm are introduced. The result shows that Levenberg-Marquart algorithm harmonizes Gauss- Newton method with steepest descent method and tunes gradually to Gauss-Newton method when the result can not converge to the minimum. Thus choosing complete vibration modal parameters and using Levenberg-Marquart algorithm, structural damage can be effectively detected.

Keywords-artificial neural networks; damage detection; input parameter; Levenberg-Marquart algorithm

I. INTRODUCTION Based on the characteristics of human brains, the

artificial neural network is a parallel distributed processing system by cell bodies. The artificial neural network is also called the connectionism mode, which can achieve the function of a nonlinear dynamical system. Now, many researches based on model updating and singnal processing are carried out in the field of the structural dynamical damage. But there are some problems for above researches such as easily influnce by the environment, the strong model dependence and the poor tolerance to the fault of the system and so on. However, in the field of structural damage detection, the artificial neural network is used widely due to its good nonlinear mapping ability, the ability of solving inverse problems and the strong robustness.

The realizing way to identify damage by artificial neural networks[1] is showed as follow. Firstly, based on the analysis of the finite element method, the learning sample of artificial neural networks is constructed by the structural dynamic characteristics on different damage states. Then, the artificial neural network is trained by the learning sample.

And the mapping between input parameters and damage states is established. At last, the artificial neural network is inputed by the parameter of the structural dynamic characteristics, which is obtained by measuring the damage structure. And damage is detected by the output of the artificial neural network.

Many scholars devote to study on the aspect. Kirkegaard and Rytter[2] use the back propagation algorithm to determine the damage location and degree of a steel beam by frequency variations before and after damage.Wu, Ghaboussi and Garrett[3] use the back propagation algorithm to identify the damage of a three layer frame by a response spectra. Anantha and Johnson[4] carry out the damage assessment of composite materials by a fuzzy neural network. Kaminski[5] compares the effectiveness among natural frequency, frequency variance and the ratio of regular frequency variances. Elkordy, Chang and Lee[6] use the variance of mode shapes to identify the damage of a five layer frame. Rhim and Lee[7] input the transfer function to detect the damage of a cantilever beam by a multilayer perceptron model. Pandy and Barai[8] use a multilayer perceptron model and the displacement under static loads to identify the damage of a steel truss bridge. Mitsuru Nakamura[9] chooses the relative displacement of interlayers as the input and the restoring force of interlayers as the output. And the valicity of the method is studied by the result of a seven layer steel structure before and after the Hanshin-Awaji-daishinsai earthquake in 1995. Using the acceleration of a steel truss, Chen and Kim train the neural network to identify the damage degree[10]. And the fuzzy neural network is combined to judge the damage of a beam[11]. Above results show that artificial neural networks have received increasing attention in damage detection. However, above researchers do not point out what kinds of input parameters is sensitive and which algorithm may lead to false or unreliable output results from such networks. In this paper, the application of artificial neural networks to damage detection will be summarized and analyzed. How to select input parameters for neural networks and some efficient and robust algorithm will be studied.

II. PARAMETRIC STUDY The selection of input parameters is very important to the

application of neural networks on damage detection. The damage should be reflected sensitively by input parameters.

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.50

207

2010 International Conference on Artificial Intelligence and Computational Intelligence

978-0-7695-4225-6/10 $26.00 © 2010 IEEE

DOI 10.1109/AICI.2010.50

207

In the past, mode data and static responses are choosed as input parameters. And they can be divided into several kinds as follow.

A. Two order frequency variation ratio Frequency is a sensitive parameter on damage detection.

And frequency has a good measuring accuracy. But frequency can not be used to identify the damage of symmetric positions on a symmetrical structure.

Cawley’s research shows that two order frequency variation ratio is only related to the damage location[12]. Based on the characteristics of structures,

( ),f K rδω δ= , (1) where r represents the damage location.

To expand (1) and ignore the higher order term,

( ) ( )( )

0,0,

f rf r K

Kδω δ

δ∂

= +∂

(2)

can be obtained. Because ( )0, 0f r = ，

( )( ) ( )0,

i

f rK Kg r

Kδω δ δ

δ∂

= =∂

(3)

can be deduced. Assuming that the variation of the stiffness is

independent to frequency, ( ) ( ) ( )/ /i j i jg r g r h rδω δω = = (4)

can be obtained. So, according to (4), two order frequency variation ratio

is only the function of the damage location.

B. The square of the frequency variation The square of the frequency variation is related to the

damage location and the damage degree. However, the ratio of the frequency variation square is only related to the damage location [13].

The characteristic equation of the structural motion is ( )2 0K Mω− Φ = . (5)

The perturbation equation of (5) is ( ) ( )( )( )( )2 2 0K K M Mω ω+ Δ − + Δ + Δ Φ + ΔΦ = . (6)

Let ΔM equal to zero.To expand (6) and ignore the quadratic term,

( ) ( )2 /T TK MωΔ = Φ Φ Φ Φ (7) can be gotten.

The overall stiffness matrix can be decomposed to the element stiffness matrix. And the element deformation is related to the structural mode shape. And

( ) ( )m fν Φ = Φ . (8) For the ith mode, there are

( ) ( )1

MT Ti i m i m m i

mK kυ υ

=Φ Φ = Φ Φ∑ , (9)

where M is the amount of elements. To take (9) into (7),

( ) ( ) ( )2

1/

JT T

i j i j j ij

k Mω υ υ=

⎡ ⎤Δ = Φ Δ Φ Φ Φ⎢ ⎥

⎣ ⎦∑ (10)

can be deduced, where J represents the amount of damage elements.

For the Nth damage element, ( ) ( ) ( )2 /T T

i N i N N i i ik Mω υ υ⎡ ⎤Δ = Φ Δ Φ Φ Φ⎣ ⎦ (11) can be obtained by (10).

Assuming that N N Nk kαΔ = , ( ) ( ) ( )2 /T T

i N N i N N i i ik Mω α υ υ⎡ ⎤Δ = Φ Φ Φ Φ⎣ ⎦ (12) can be deduced.

The square of the frequency variation is related to the damage degree and location of the element. If the ith frequency and the jth frequency are used,

( ) ( ) ( ) ( )2

2 /TTN j N N jN i N N ii

T Tj i i j j

kkM M

υ υυ υωω

⎡ ⎤Φ Φ⎡ ⎤Φ ΦΔ⎢ ⎥= ⎢ ⎥Δ Φ Φ Φ Φ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦

(13)

can be gotten. It shows as in (13) that the ratio of the frequency

variation square is the function of damage locations, when damage degrees are near.

C. Frequency variation ratio Kaminski compares the effectiveness among the natural

frequency, the frequency variation and the regular frequency variation ratio to damage detection[5]. The result shows that the ratio of the frequency variation

( ) /iFC ui di uiF f f f= − (14)

is related to the damage degree and the damage location, but the regular frequency variation ratio

1

/i i j

p

FCR FC FCj

N F F=

⎛ ⎞= ⎜ ⎟

⎝ ⎠∑ (15)

is only related to the damage location.

D. Mode shape, mode curvature and transfer function According to the relationship between displacements and

strains, each displacement mode shape is corresponding to a strain mode shape. When the damage happens, there is a prominent stress redistribution nearby the damage location. That will cause a significant change of the strain mode shape. The damage can be located by comparison with every order strain mode shape before and after the damage. According to the theoretical analysis as in [14], high order mode shapes are easier to identify the structural damage than low order mode shapes. But exact high order mode shapes actually are not easy to be obtained. The error of mode shapes has an enormous implication to the result. Some researchers point out that the measurement error of mode shapes matches with the mode shape variation caused by the structural local change[15] .

The relationship between curvature and bending moment is /v M EI′′ = . When the damage happens, EI decreases and v′′ increases. Therefore, the damage can be located by the variation of v′′ . Chance, Worden and Tomlinson[16]

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prove that mode curvatures are better than mode shapes to damage detection. The mode curvature is a very sensitive parameter for the damage. But the measurement mode shape usually can not reach a high precision so that limits the application of mode curvatures.

The adventage of using dynamic responses and modes to identify the damage is the input parameter can be obtained easily. But there are also several shortcomings. Firstly, only using frequency information is difficult to identify the damage of symmetric positions on a symmetrical structure. Secondly, high order mode shapes can not be measured easily. In order to detect damage degrees, a large of comparisons between the new and the old model have to be carried out. At last, there may produce a blurring result unless using modes with a high strain energy. However, the transfer function has complete information and reduces the number of sensors. During the phase of training, the structural model is classified by the damage location and the damage degree. On the identification stage, the unknown damage structure is classified into the most similar category[17]. But for an actual structure, it is difficult to apply enough control forces or excitations as inputs. And the environmental random vibration is difficult to be measured. The input is difficult to be determined so that is not easy to get the transfer function.

E. Other input parameters Pandy and Bara[8] identify the damage of a truss bridge

by the structural node static response. Their research shows that the static measured data usually has a high precision. But some parts of a large structure can not be measured. Based on the time domain theory, Mitsuru Nakamura[9] puts forward to use the structural story drift and story speed as input parameters and resiliences as output parameters. The sensibility of six kinds of input parameters to damage detection is compared. And the results from low to high are respectively displacement mode, natural frequency, displacement, transfer function, mode curvature and strain function[18].

III. IMPROVED BACK PROPAGATION ALGORITHMS Above studies mainly focus on the sensitivity of input

parameters of neural networks. Beside the influence of input parameters, the algorithm also has an impact on the result of damage detection. Now, the back propagation algorithm is used widely in damage detection. The standard back propagation algorithm uses the law of chain derivative to calculate the gradient of the error square sum[19-23]. The steepest descent method with a regular step is used to modify the weight of artificial neural networks to minimize errors[24]. The algorithm is

( ) ( )( 1)ij ij ijn n nω ω ω+ = + Δ (16)

( ) ( ) ( )ij i in n y nω ηδΔ = , (17) where ( )ij nω is the weight from the ith node to the jth node during the nth iteration; ( )ij nωΔ is the increment of the weight; η is the learning step; ( )i tδ and ( )iy t are respectively the local gradient and the output of the ith node.

A. Adaptive variable step-size algorithm The difference between adaptive variable step-size

algorithm and the standard back propagation algorithm lies in that the learning step is rectified by the change of the error curve. Due to the nonuniform of the gradient for an approximate error curve on the standard back propagation algorithm, the convergence becomes slow, when η is a small value. When η is a large value, an oscillation takes place at the valley region. The adaptive variable step-size algorithm can correct the problem effectively. The theoretical basis of the adaptive variable step-size algorithm is the advance and retreat method on evolutionism, which computes two successive training errors[25]. If ( ) minE k E er> ∗ ,

then ( ) ( )( ) ( )

1 0

1

k k de

k k in

η αη α α

+ = ∗ =⎧⎪⎨

+ = ∗ =⎪⎩ (18)

where minE is the smallest error on the kth iteration; er is

the learning rate based error rebound; de and in are respectively the decrease rate and increase rate of learning steps; α is the coeffcient of momentum terms.

B. Levenberg-Marquart algorithm During the process of solving a nonlinear equation, the

Gauss-Newton method has two order convergence speed. But the Hessian matrix may become an odd matrix so that the iteration can not be continued. Therefore, the stability of the Gauss-Newton method is not good. The Levenberg-Marquart algorithm is the harmonization between the Gauss-Newton method and the steepest descent method. When the result keep away from the minimum, the Levenberg-Marquart algorithm tunes gradually to the Gauss-Newton method[26].

The equation of the Levenberg-Marquart algorithm is ( ) ( )1 kW k W k P+ = + (19)

( ) ( )( )12k HP H D E W kμ −= − + ∇ , (20) where H is the Hessian matrix of the energy function E; DH is a diagonal matrix which is composed by the diagonal element of H; ▽E is the derivative matrix of E. If

( )( ) ( )( )1E W k E W k+ ≥ , then 10μ μ= . Else, /10μ μ= .

C. Homogeneous algorithm The homogeneous algorithm is an efficiency method on

the theory of the differential topology. The homogeneous back propagation algorithm brings the idea of advancements in a regular order into the back propagation algorithm , which is helpful for solving the minimum of the energy function. The global minimum can be obtained by tracking the change of the energy curvature and the curvature trajectoty of the least value[27]. The first step of the linear homogeneous algorithm with a teacher is to make sure the teacher homogeneous function T,which is

( ) ( )1T t t Tb t Te= − + ∗ , (21) where t is the variation to form the transition function and transits from 0 to 1, t∈[0,1]; Tb and Te are respectively the initial comformed teacher and the given teacher.

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The learning procedure of the homogeneous algorithm is

( ) ( ) ( ) 2, ,i i iE W t T t O W I= − (22)

( ) ( )1i i i iT t t Tb t Te= − + ∗ (23)

( )1,i iTb O W I−= (24)

where ( ),O W I is the actual output of the neural networks.

IV. CONCLUSIONS Damage detection based on artificial neural networks is

a subject with a widespread application. How to select the input parameter of artificial neural networks is very important to damage detection. The input parameter should be sensitive and obtained easily. Natural frequency is easy to be acquired. It is reasonable to use natural frequency as input parameters. But information is not sufficiency ,if only using natural frequency. Mode shape has abundance information but a relatively large measurement error. The transfer function includes complete information but not easy to be obtained. Meanwhile, the algorithm and the structure of the artificial neural network also have a large influence to damage detection. Results show that Levenberg-Marquart algorithm harmonizes the Gauss-Newton method with the steepest descent method and tunes gradually to the Gauss-Newton method when results keep away from the minimum. Thus choosing complete vibration modal parameters as input and using Levenberg-Marquart algorithm, structural damage can be effectively detected.

ACKNOWLEDGMENT This work is supported by the health monitoring project

of Yitong river bridge and the safety assessment project of Qiandaohu 1# bridge.

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