structural impedance based damage diagnosis by piezo-transducers

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Page 1: Structural impedance based damage diagnosis by piezo-transducers

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICSEarthquake Engng Struct. Dyn. 2003; 32:1897–1916 (DOI: 10.1002/eqe.307)

Structural impedance based damage diagnosis bypiezo-transducers

Suresh Bhalla and Chee Kiong Soh∗;†

Division of Structures and Mechanics; School of Civil and Environmental Engineering; Nanyang TechnologicalUniversity; 50 Nanyang Avenue; Singapore 639798

SUMMARY

Although structural mechanical impedance is a direct representation of the structural parameters, itsmeasurement is di�cult at high frequencies owing to practical considerations. This paper presentsa new method of damage diagnosis by means of changes in the structural mechanical impedanceat high frequencies. The mechanical impedance is extracted from the electro-mechanical admittancesignatures of piezoelectric-ceramic (PZT) patches surface bonded to the structure using the electro-mechanical impedance (EMI) technique. The main feature of the newly developed approach is thatboth the real as well as the imaginary component of the admittance signature is used in damagequanti�cation. A complex damage metric is proposed to quantify damage parametrically based on theextracted structural parameters, i.e. the equivalent single degree of freedom (SDOF) sti�ness, the mass,and the damping associated with the drive point of the PZT patch. The proposed scheme eliminates theneed for any a priori information about the phenomenological nature of the structure or any ‘model’of the structural system. As proof of concept, the paper reports a damage diagnosis study conducted ona model reinforced concrete (RC) frame subjected to base vibrations on a shaking table. The proposedmethodology was found to perform better than the existing damage quanti�cation approaches, i.e. thelow-frequency vibration methods as well as the traditional raw-signature based damage quanti�cationin the EMI technique. Copyright ? 2003 John Wiley & Sons, Ltd.

KEY WORDS: damage; electro-mechanical impedance (EMI) method; structural impedance; piezoelectric-ceramic (PZT) patch; structural health monitoring (SHM)

INTRODUCTION

Structures are assemblies of load carrying members capable of safely transferring the loadsto the foundations. To serve their designated purpose, the structures must satisfy strength andserviceability criteria throughout their stipulated design life. However, after a natural disaster,such as an earthquake, the strength as well as the serviceability of the structure becomesquestionable due to the possibility of ‘damage’. Even a minor incipient damage carries the

∗ Correspondence to: Chee Kiong Soh, Division of Structures and Mechanics, School of Civil and EnvironmentalEngineering, Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798.

† E-mail: [email protected]

Received 11 June 2002Revised 11 December 2002

Copyright ? 2003 John Wiley & Sons, Ltd. Accepted 29 January 2003

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1898 S. BHALLA AND C. K. SOH

potential to grow and lead to failure that could have a catastrophic impact on lives and prop-erties. Visual inspection by trained professionals alone cannot make a comprehensive healthassessment of the near in�nite and diverse infrastructure at short notice, such as immediatelyafter an earthquake. These considerations have motivated research for the development of auto-mated, real-time and online structural health monitoring (SHM) systems, with the help ofwhich structural integrity can be assessed as and when deemed necessary.Over the last two decades, many automated SHM techniques have been reported in the

literature. Global static response techniques, such as the static displacement measurementtechnique [1] or the static strain measurement technique [2] aim for structural system identi�-cation from the static response of structures. However, application of large loads and measur-ing corresponding de�ections or strains (which these techniques require) is not very practicalfor real-life structures. The global dynamic techniques [3–6] aim for similar system identi-�cation from the dynamic response of structures. However, these techniques rely heavily onmodal data pertaining to the �rst few modes only. These modes, being global, are not sensitiveenough to detect localized damage, such as incipient damage in the form of cracks. A commonlimitation of the global techniques (static=dynamic) is that they require a high-�delity modelof the structure to start with. They also demand intensive computation to process the measureddata. Many investigators have therefore integrated the global techniques with arti�cial neuralnetworks to non-parametrically predict the damage in an apparent attempt to minimize the apriori information about the structure [7–9].The conventional SHM approaches rely on the measured static stress or strain, or displace-

ment, or low-frequency vibration data for damage identi�cation. They employ conventionalsensors, which can only extract details such as load or strain histories [10]. However, the recentadvent of smart materials, such as piezoelectric materials, shape-memory alloys and optical�bres has added a new dimension to SHM. In particular, the electro-mechanical impedance(EMI) technique, which uses smart piezoelectric ceramic (PZT) materials, has emerged asa powerful technique for SHM [11–14]. In this technique, a PZT patch is bonded to thestructure and a high-�delity electro-mechanical admittance signature of the patch serves as adiagnostic signature of the structure. The impedance method has been shown to be extremelysensitive to incipient damage [14–16]. However, the technique still lacks a rigorous methodto interpret damage from the electrical signatures. To date, all damage quanti�cation methodsuse raw-signatures alone and are not able to correlate damage with a speci�c change in thestructural parameters. Presented in this paper is a new methodology which quanti�es dam-age based on the equivalent single degree of freedom (SDOF) parameters of the structure‘apparent’ at the actuator end points.

MECHANICAL IMPEDANCE OF STRUCTURES

The concept of mechanical impedance of structures is similar to the concept of electricalimpedance in electrical circuits [17]. The impedance method allows a simpli�ed analysis ofcomplex mechanical systems by reducing the di�erential equations of Newtonian mechan-ics into simple algebraic equations [18]. The analysis presented in this paper follows theimpedance approach to analyse the electro-mechanical admittance signatures of PZT patchesbonded to structures.

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:1897–1916

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1899

FResultant

Fd

Fi

(Fi - Fs)

Fs

θx x(t) = xocos(ωt -θ )

F(t) = Focosωt

k

c

m

(a) (b)

Figure 1. (a) A single degree of freedom (SDOF) system under dynamic excitation, (b) Phasorrepresentation of spring force (Fs), damping force (Fd) and inertial force (Fi).

Consider a single degree of freedom (SDOF) spring-mass-damper system subjected to adynamic excitation force Fo at an angular frequency !, as shown in Figure 1(a). Let theinstantaneous velocity response (which is the same for each component of the system due toparallel connection) be given by

x= xo cos(!t − �) (1)

where xo is the velocity amplitude and � is the phase lag of the velocity with respect to theforce. Displacement and acceleration can be obtained from Equation (1) by integration anddi�erentiation, respectively. Hence, the force associated with each structural element, i.e. thespring (the elastic force), the damper (the damping force) and the mass (the inertial force)can be calculated.

Damping force; Fd = cx= cxo cos(!t − �) (2)

Inertial force; Fi =m�x=mxo! cos(!t − �+ �

2

)(3)

Spring force; Fs = kx=(kxo!

)cos

(!t − �− �

2

)(4)

This system is analogous to a series LCR circuit in classical electricity [17]. The term xis analogous to the current (which is the same for each element of the LCR circuit) and themechanical force is analogous to the electro-motive force (voltage). The damper is analogousto the resistor since Fd is in phase with x (Equation (2)). The mass is analogous to the inductorsince Fi leads x by 90◦ (Equation (3)). Similarly, the spring is analogous to the capacitorsince Fs lags behind x by 90◦ (Equation (4)). These terms can be analogously representedby a phasor diagram as shown in Figure 1(b). Hence the amplitude of the resultant force(analogous to voltage across the entire LCR circuit) is given by

Fo =√F2do + (Fio − Fso)2 (5)

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:1897–1916

Page 4: Structural impedance based damage diagnosis by piezo-transducers

1900 S. BHALLA AND C. K. SOH

where the subscript ‘o’ denotes amplitude of the concerned force. Substituting expressions forthe amplitudes from Equations (2) to (4), we get

Foxo=

√c2 +

(m!2 − k!

)2(6)

The quantity Fo=xo is analogous to the electrical impedance (ratio of voltage to current) of anLCR circuit. It is called the ‘mechanical impedance’ of the system and is denoted by Z . Usingcomplex number notation, analogous to that used in classical electricity, it may be expressedeither in cartesian or polar coordinates as

Z = x + yj= c+(m!2 − k!

)j= |Z |ej� (7)

The phase lag � of the velocity x with respect to the resultant driving force F is given by(Figure 1(b)):

tan �=Fi − FsFd

=m!2 − kc!

(8)

Here, ‘x’ is the dissipative or real part and ‘y’ is the reactive or imaginary part of themechanical impedance and � is the phase angle. It should be noted that the damping couldequivalently be represented using a complex sti�ness given by

�k= k(1 + �j) (9)

where the term �, commonly known as the mechanical loss factor, is given by

�=c!k

(10)

The concept of mechanical impedance can be easily extended to complicated multiple degreeof freedom (MDOF) systems. Although Equations (7) and (8) have been derived for the SDOFsystem shown in Figure 1, the complex structural systems too have mechanical impedanceconsisting of both the real (dissipative) and imaginary (reactive) components. These two termscan be considered to represent a purely resistive element (such as a damper) connected inparallel to a purely reactive element (such as a spring or mass, or both). The two terms canbe considered to be the ‘equivalent SDOF’ representation of the actual system [18].However, analytical determination of mechanical impedance for complex MDOF systems is

very tedious. It can be measured experimentally by applying a sinusoidal force at a point andmeasuring the resulting velocity at that point in the direction of the force. Conventionally, thisis done by using impedance head, which consists of a force transducer and an accelerometer[18]. The force transducer is an electro-magnetic shaker, which produces a sinusoidal forceproportional to the input sinusoidal voltage. The accelerometer measures the acceleration ofthe point of interest, again in the form of a proportional sinusoidal voltage signal. Beingharmonic, velocity can be easily determined from acceleration by integration. The magnitudeof the mechanical impedance is calculated from the ratio of the measured force and thevelocity amplitudes, and the phase di�erence is calculated from the phase di�erence betweenthe corresponding measured voltage signals. However, the conventional impedance heads have

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:1897–1916

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1901

a small operational bandwidth, which prohibits their use for high frequencies. The same holdsequally true for the accelerometers. Even the high-tech miniaturized accelerometers share thedisadvantages of high cost and small operational bandwidth. The next sections show how thisdi�culty could be overcome through the use of ‘smart’ PZT transducers.

ELECTRO-MECHANICAL IMPEDANCE (EMI) TECHNIQUE

Physical principles

The EMI technique uses PZT transducers for SHM. The PZT patches are made up of materi-als which exhibit the phenomenon of ‘piezoelectricity’ [19]. The characteristic feature of the‘piezoelectric’ materials is that they generate surface charge in response to an applied mechan-ical stress (direct e�ect); and conversely, they undergo mechanical deformation in response toan applied electric �eld (converse e�ect). This unique capability of the piezoelectric materialsto exhibit stimulus-response behaviour quali�es them to be members of the group of so-called‘smart materials’ [20]. Traditionally, the direct e�ect is used in numerous sensing applications[21; 22] and the converse e�ect is used in actuator applications (e.g. Kamada et al. [23]). TheEMI approach, on the other hand, uses these two smart features of the piezoelectric materialsin total synergy. The same patch performs the dual role as an actuator as well as a sensor.In the EMI technique, the PZT patch is bonded to the surface of the monitored structure

by high-strength epoxy adhesive. In this con�guration, the PZT patch essentially behaves as athin bar undergoing axial vibrations as shown in Figure 2(a). The patch expands and contractsdynamically in direction ‘1’ when an alternating electric �eld E3 is applied in direction ‘3’.The patch has half-length la, width wa and thickness ha. In the present analysis, the hoststructure is assumed to be a skeletal structure, that is, it is composed of one-dimensionalmembers with their sectional properties (area and moment of inertia) lumped along theirneutral axes. Therefore, the vibrations of the PZT patch in direction ‘2’ can be conveniently

Alternating electric field source

la la

Point ofmechanical fixity

PZT Patch

3 (z)

1 (x)

Host structure

2 (y)

PZT patch

Structural Impedance

la

ha

wa

E31

3 2

la

Z Z

(a) (b)

Figure 2. (a) A PZT patch bonded to a structure, (b) Interaction model of PZT patchand host structure.

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:1897–1916

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1902 S. BHALLA AND C. K. SOH

ignored. Under these conditions, the behaviour of the patch is governed by the followingconstitutive relationships [19]:

D3 = �T33E3 + d31T1 (11)

S1 =T1YE11

+ d31E3 (12)

where S1 is the strain in direction ‘1’, D3 is the electric displacement over the PZT patch, d31is the piezoelectric strain coe�cient and T1 is the axial stress in the PZT patch in direction‘1’. YE11 =Y

E11(1 + �j) is the complex Young’s modulus of the PZT patch at constant electric

�eld and �T33 = �T33(1− �j) is the complex electric permittivity of the PZT material at constant

stress. Here � and � denote, respectively, the mechanical loss factor and the dielectric lossfactor of the PZT patch.The vibrating patch is assumed in�nitesimally small as compared to the host structure and

has negligible mass and sti�ness. The structure can be assumed to possess uniform dynamicsti�ness over the entire bonded area. The two end points of the patch can therefore be assumedto encounter equal mechanical impedance Z by the structure, as shown in Figure 2(b). Underthis condition, the patch has zero displacement at the mid-point (x=0) irrespective of thelocation of the patch on the host structure. At the same time, we neglect the PZT loading indirection ‘3’ taking into account the fact that the PZT loading in direction ‘3’ is similar to anin�nitesimal inertial shaker vibrating over an in�nitely large base, therefore, with negligibletransmissibility e�ect. This is also supported by the fact that the typical resonant frequencyof the patch for thickness vibrations is much higher than the range of frequencies consideredpresently (100–150 kHz). The one-dimensional vibrations of the PZT patch are governed bythe following di�erential equation [24].

YE11@2u@x2

=�@2u@t2

(13)

where u is the displacement at any point of the patch in direction ‘1’. Solution of the governingdi�erential equation by the method of separation of variables yields

u=(A sin �x + B cos�x)ej!t (14)

where � is the wave number, related to the angular frequency of excitation ! by

�=!

√�

YE11(15)

Application of the mechanical boundary condition that at x=0; u=0 and using the PZTconstitutive relation Equation (12) yields

A=ZaVod31

ha� cos(�la)(Z + Za)and B=0

where Za is the short-circuited mechanical impedance of the PZT patch given by

Za=�wahaY E11

(j!) tan(�la)(16)

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:1897–1916

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1903

Making use of the PZT constitutive relation (Equation (11)) and integrating over the entiresurface of the PZT patch (−la to +la), we obtain an expression for the electromechanicaladmittance (the inverse of electro-mechanical impedance) as

�Y =2!jwalaha

[(�T33 − d231YE11) +

(Za

Z + Za

)d231YE11

(tan �la�la

)](17)

or YE =12�Y =!j

walaha

[(�T33 − d231YE11) +

(Za

Z + Za

)d231YE11

(tan �la�la

)](18)

where YE can be termed as ‘e�ective-electromechanical admittance’. Any damage to the struc-ture will modify the drive point mechanical impedance Z , that will in turn a�ect �Y , therebygiving an indication of damage.The complex admittance �Y (units Siemens or ohm−1) consists of real and imaginary parts,

the conductance (G) and the susceptance (B), respectively. These can be measured by com-mercially available impedance analysers. The impedance analyser imposes an alternating volt-age signal of 1 Volt rms (root mean square) to the bonded PZT transducer and records themagnitude and the phase of the steady state current in the form of G and B. The excitationfrequency should be high enough to generate a stress wave of wavelength of the order of the�aw size to be detected [25]. A plot of G over a su�ciently high frequency range, typicallyof the order of kHz is called the ‘conductance signature’ or simply ‘signature’ of the structure.The EMI technique is shown to have the leading edge over the existing conventional

methods [14; 15]. The PZT patches are bonded non-intrusively, typically exhibit low-cost,demand low power supply, are immune from noise, and are robust under harsh environments[26]. Whereas the existing NDE methods have a very limited scope of application, the EMItechnique has evolved as a universal method suitable for almost all engineering materials andstructures such as steel [12; 13], reinforced concrete (RC) [14], pipeline structures [16], jetengine components [26] and composites [27].

Traditional signal processing and damage quanti�cation approach

The prominent e�ects of damage (to the host structure) on the conductance signature are theappearance of new peaks in the signature and the lateral and vertical shifts of the existingpeaks [11]. For example, Figure 3 shows the e�ect of damage (simulated by drilling a 5mm-diameter hole along axis AA) on the conductance signature of a PZT patch bonded to asteel beam. The beam was made up of structural steel of grade 43C conforming to BS4360:1990 (density =7850kg=m3, Young’s modulus=205kN=mm2, Poisson ratio=0:3). Suchrolled sections are commonly used in the construction industry. The signature was acquiredthrough an HP 4192A impedance analyser [28] in the frequency range of 140–150 kHz atan interval of 100 Hz. Although the induced damage was small (amounting to 0.015% massloss only), the e�ect on the signature is clearly visible (Figure 3(b)). This demonstrates thehigh order of sensitivity of the technique. More details of a damage progression study on thebeam can be found in Reference [13].Traditionally, statistical pattern recognition techniques have been employed to quantify

changes occurring in the conductance signature due to damage; such as relative deviationor RD [12], root mean square deviation or RMSD [13; 25], and mean absolute percentage

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:1897–1916

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1904 S. BHALLA AND C. K. SOH

0.4

0.5

0.6

0.7

0.8

0.9

140 142 144 146 148 150

Pristine State After damage

730mm= =

PZT Patch

12x12x0.3mm

35mm

100m

m

A

A

55mm

5mmHole PZT

Section A-A

Con

duct

ance

(mS

)

Frequency (kHz)

(a) (b)

Figure 3. (a) A steel beam instrumented with a PZT patch, (b) E�ect of damage onconductance signature.

deviation or MPAD [29]. The RMSD index, for example, is de�ned as

RMSD (%)=

√√√√√√√√N∑j=1(G1j −Goj )2

N∑j=1(Goj )2

× 100 (19)

where G1j is the post-damage conductance at the j-th measurement point and Goj is the corre-

sponding pre-damage value. For the damage induced on the steel beam, the RMSD index wasworked out to be 12% using Equation (19). Other indices such as RD or MPAD similarlymeasure the deviation between the current signature with respect to the baseline signature,and are similarly de�ned.Although the statistical methods are easy to implement and have the advantage of being

non-parametric [8; 14], their major shortcoming is that they are unable to correlate change inelectrical conductance with any speci�c change in the structural properties. RMSD or RD hasno upper limit, the magnitude demanding an alarm could vary from case to case. Besides, allthe existing damage interpretation techniques rely only on the real part of the signature. Theinformation possessed by the imaginary part is thereby lost.The majority of the published work related to the EMI technique has been focused on rel-

atively light structures, mainly aerospace structures and machine components. In the majorityof the reported investigations, the damage was typically simulated non-destructively such asby loosening bolts or similar components [11; 12; 16]. Only a few destructive tests on thestructures instrumented with PZT patches have been reported [14; 15]. In many structures,simply the ‘detection’ of damage might be more than su�cient, which can be done conve-niently by means of conventional statistical indices. However, in the civil-structures, we oftenneed to �nd out whether the damage is ‘incipient’ or ‘severe’. We might even tolerate an

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:1897–1916

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1905

incipient damage without endangering lives or properties. This fact has motivated us to extractthe drive point structural impedance from the measured signatures for damage quanti�cation.Studies for calibrating the changes in drive point impedances with damages for typical civilengineering structures and materials will be reported in the future.

DECOMPOSITION OF ELECTRO-MECHANICAL ADMITTANCE SIGNATURES

Equation (18), which represents the mechatronic coupling between the structure and the PZTpatch, can be rewritten as

YE =YP + YA (20)

where

YP=!jwalaha[�T33 − d231YE11] and YA=!j

walaha

Za(Z + Za)

d231YE11(tan �la�la

):

The �rst part, YP, depends solely on the parameters of the PZT patch. It is a passive componentsince it is not a�ected by any damage to the structure. The second part, YA, represents thecoupled interaction between the structure and the PZT patch. It is an active component andwill be a�ected by any damage to the structure (i.e. any change in Z) in the vicinity of thepatch, thereby actively diagnosing the damage.So far all the reported studies have used the raw-conductance signatures alone for damage

prediction. However, if the PZT parameters can be predicted, the passive component can be�ltered o�. From Equation (20),

YA=YE − YP (21)

Thus the active-conductance GA and the active-susceptance BA, which are respectively the realand the imaginary components of YA, can be determined.Let the structural system depicted in Figure 2(b) (and represented by structural impedance

Z) be a SDOF system with a mass of m=200kg, sti�ness of k=1:974× 109N=m, and damp-ing constant c=12566:4 Ns=m, connected in parallel. This system has a natural frequency of500Hz and damping ratio of 1%. The PZT patch is assumed to possess the parameters shownin Table I and is assumed to be 50:8× 25:4× 0:254 mm in size. Let damage be simulated inthis structural system by sequentially modifying the structural parameters, i.e. �rst ‘k’, then‘c’ and then ‘m’. Figure 4(a) shows the e�ect of simulated damage on the raw-susceptance(B) plot. It can be observed that the induced damage has no visible e�ect on the B-plot.Figure 4(b), on the other hand, shows the plots of the active susceptance, BA, for the samesystem. It can be seen that the plot of BA not only identi�es the resonant frequency of thesystem (seen as a sharp kink around 500Hz), it also exhibits appreciable sensitivity to damage.All the induced damages have observable e�ects on the signatures (lateral=vertical shifts). Thisanalysis shows that it is possible to derive useful information from the imaginary part of thesignature which was discarded in the previous works. Therefore, this has motivated us tomake use of the real as well as the imaginary part in damage prediction. A more detailedanalysis of signature decomposition is presented in Reference [30].

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1906 S. BHALLA AND C. K. SOH

-0.000005

-0.000003

-0.000001

0.000001

0.000003

0.000005

400

450

500

550

600

650

Frequency (Hz)

BA (

S)

Pristine State

20% increase in ‘c’

(a) (b)

0

0.0001

0.0002

0.0003

0.0004

0.0005

400

450

500

550

600

650

Frequency (Hz)

B (

S)

Pristine20% increase in 'c'20% decrease in 'k'20% increase in 'm'

20% reduction in ‘k’20% increase in ‘m’

Figure 4. E�ect of simulated damage on susceptance signatures of a SDOF system.(a) Raw-susceptance, (b) active-susceptance.

DETERMINATION OF STRUCTURAL MECHANICAL IMPEDANCEFROM ELECTRO-MECHANICAL SIGNATURES

Extraction of apparent structural impedance

The active complex admittance, YA, was derived in the previous section as a function of thestructural parameters, the PZT parameters and the frequency as

YA=GA + BAj=!jwalaha

Za(Z + Za)

d231YE11

(tan �la�la

)(22)

Substituting Za= xa + yaj (actuator impedance), Z = x + yj (structure impedance), tan �la�la

=

r + tj; Y E11 =YE11(1 + �j), and after eliminating the imaginary term from the denominator, we

obtain

YA=!jwalaha

(xa + yaj)[(x + xa)− (y + ya)j][(x + xa)2 + (y + ya)2]

d231YE11[(r − �t) + (t + �r)j] (23)

Denoting (x + xa) by xT and (y + ya) by yT , this equation can be rewritten in a simpli�edform as

YA=K!j(P +Qj)(R+ Tj)

(x2T + y2T )

(24)

where K; P; Q; R and T are de�ned as

K =walaha

d231YE11 (25)

P= xaxT + yayT Q=yaxT − xayT (26)

R= r − �t T = t + �r (27)

Copyright ? 2003 John Wiley & Sons, Ltd. Earthquake Engng Struct. Dyn. 2003; 32:1897–1916

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1907

YA can now be decomposed into the real and imaginary parts, GA and BA respectively as

GA =−K(QR+ PT )!

(x2T + y2T )

(28)

BA =K(PR−QT )!(x2T + y

2T )

(29)

Solving Equations (28) and (29), we obtain

xT =−K!(cR+ T )(yaCt + xa)

GA(1 + C2t ); and yT =CtxT (30)

where

Ct =ya − cxacya + xa

(31)

and

c=(GA=BA)R+ T(GA=BA)T − R (32)

Here, xa and ya can be determined from the PZT parameters, as given by Equation (16).From xT and yT ; x and y can be determined as

x= xT − xa and y=yT − ya (33)

It should be noted that ‘x’ and ‘y’ are determined from the measured conductance andsusceptance signatures. Only the PZT parameters are assumed known. No a priori informationabout the structure is needed. It is important to predict the PZT mechanical impedances xaand ya accurately. Here, this is done using Equation (16) on the basis of the data provided bythe manufacturer. Methods for more accurate predictions will be published in another moredetailed paper. It should also be noted that in these calculations, the quantity tan �la

�la= r+tj must

be determined exactly using the theory of complex algebra [31]. This term was approximatedby Liang et al. [24] as unity under the assumption that the operational frequency is muchlower than the �rst resonant frequency of the PZT patch (or in other words the quasi-staticsensor approximation [32]). However, this is not the case in SHM applications where thefrequency range is typically a few hundred kHz.

Physical interpretation of drive point impedance

As mentioned before, all the previous reported works used the real part of the raw com-plex admittance alone to quantify damage. However, in our newly developed methodology,both the real and imaginary parts are used. They are �rst �ltered to remove the PZT contri-bution to obtain the active signatures. They are further processed to determine the real andthe imaginary parts of the drive point mechanical impedance, which are direct functions of thestructural parameters. The real part, x, is the equivalent SDOF damping of the structureand the imaginary part, y, is the equivalent SDOF mass-sti�ness factor (Equation (7)) ofthe structure at the ‘drive point’ of the PZT patch. In other words they are the structural

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1908 S. BHALLA AND C. K. SOH

parameters ‘apparent’ to the PZT patch at its ends. Being direct structural parameters, theseare more sensitive to damage than stresses or strains which are secondary e�ects. Hence thePZT patch identi�es the parameters of the ‘black-box’ structure in the form of the equivalentSDOF damping and sti�ness-mass factors.

DEFINITION OF DAMAGE METRIC BASED ON STRUCTURAL IMPEDANCE

As pointed out above, a damage index based on the drive point structural impedance shouldbe more sensitive as well as more reliable than those based on the raw-conductance signature(i.e. the conventional approaches). This has motivated us to de�ne a complex damage metricas follows

�D=Dx +Dyj (34)

where Dx denotes the damage metric of the real part of the structural impedance and Dy de-notes the corresponding value for the imaginary part. Hence, Dx signi�es change in equivalentSDOF damping associated with the drive point of the PZT patch and Dy signi�es the variationin the equivalent mass-sti�ness factor. Being based on the extracted impedance rather thanthe raw-signatures, this method quanti�es the damage parametrically.Dx is de�ned as the average of Dxj, which is the value of the metric at the j-th frequency

point, de�ned as follows. If xjxo¡1, then Dxj=1 − xj

xjo, else, Dxj=1 − xjo

xj. Here, xjo is the

baseline value at the j-th frequency point and xj is the value at the current state. The othercomponent Dy is similarly de�ned using ‘y’ in place of ‘x’. This de�nition of the damagemetric quanti�es the damage on a uniform 0–100 scale.

PROOF OF CONCEPT APPLICATION: DIAGNOSIS OF VIBRATIONINDUCED DAMAGES

The proposed structural impedance based damage detection methodology was veri�ed usingthe test data obtained from a model RC frame subjected to base vibrations. The test structurewas a two-storey portal frame, made of reinforced concrete. The details of the scaled structuralmodel are shown in Figure 5. This model represented a prototype frame with storey height of2:9m and span length of 3:3m, at a scale of 1:10. The shaker was an electromagnetic shakingtable, rated to a maximum acceleration of 120 g and a maximum frequency of 3000 Hz. Thetest frame was instrumented with two PZT patches, shown in Figure 5 as Patch #1 and Patch#2, which were bonded to the structure using the RS 850-940 epoxy adhesive [33]. Patch #1was instrumented on the �rst �oor beam, very close to the beam-column joint, a location verycritical from the point of view of shear cracks. Patch #2 was instrumented at the bottom faceof the second �oor beam, near the mid span, a location very critical from the point of view of�exural cracks. Both the patches were 10mm square and 0:2mm thick (material designation:PI 151), and manufactured by PI Ceramic [34]. The electrical and mechanical parametersof the patches are as listed in Table I. The test frame was a typical skeletal structure andhence the PZT-structure force interaction analysis presented in the previous sections can beconveniently applied.

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1909

(a) (b)

130

Patch #2

Patch #1

20

1233.5

25

30

COLUMN

BEAM

Figure 5. (a) Details of test frame (all dimensions are in mm), (b) Test frame just before the test.

Table I. Key properties of PZT patches (PI Ceramic [34]).

Physical parameter Value

Density (kg=m3) 7800Electric permittivity, �T33 (farad=m) 2:124× 10−8Piezoelectric strain coe�cient, d31 (m=V) −2:10× 10−10Young’s modulus, YE11 (N=m

2) 6:667× 1010Dielectric loss factor, � 0.015

The test loads were applied in the form of vertical base motions of varying frequencies andamplitudes. The buildings are normally subjected to such base motions during earthquakesand underground explosions [35]. The test was performed in eight phases divided accordingto the range of the base motion frequencies, and the velocity and acceleration amplitudes. Theinduced base motions are graphically presented in Table II. After each excitation, the patcheswere scanned to acquire the raw-signatures in the frequency range of 100–150 kHz at aninterval of 100 Hz by an HP 4192A impedance analyser [28]. The signatures were processedand the damage metric was determined by the procedure outlined in the previous sections.The test structure was also instrumented with conventional sensors such as accelerometers,

LVDTs and strain gauges. This part of the instrumentation was done by another researchgroup [36], which was interested in monitoring the condition of the frame by low-frequencyvibration techniques.

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1910 S. BHALLA AND C. K. SOH

Table II. Typical base motions and time-histories to which test frame was subjected.

Phase Load Description

Typical base motion time histories

Baseline

Phase 1 Freq. (850−200) Hz

Acceleration 12.48 g / Velocity 0.027 m/s

-10.0

0.0

10.0

0.00 0.05 0.10 0.15 0.20

State 1

Phase 2 (150−15) Hz3.016 g / 0.057 m/s

-2.0

0.0

2.0

0.00 0.20 0.40 0.60 0.80

State 2

Phase 3 700 Hz 20.36 g / 0.131 m/s

-50

0

50

0.00 0.10 0.20 0.30

State 3

Phase 4 700 Hz 25.62 g / 0.203 m/s

-50

0

50

0.00 0.10 0.20 0.30

State 4

Phase 5 200 Hz 23.67 g / 0.443 m/s

-50

0

50

0.00 0.20 0.40 0.60

State 5

Phase 6 200 Hz 13.46 g / 0.376 m/s

State 6

Phase 7 200 Hz 25.12 g / 0.744 m/s -50

0

50

State 7

Phase 8 200 Hz 25.12 g / 0.744 m/s -50

0

50

State 8

Base Acceleration

Base Acceleration

Base Acceleration

Base Acceleration

Base Acceleration

Base Acceleration

Base Acceleration

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

Time (s)

-50

0

50

0.00 0.20 0.40 0.60

0.00 0.20 0.40 0.60

0.00 0.20 0.40 0.60

Base Acceleration

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1911

(a) (b)

0.002

0.003

0.004

0.005

0.006

0.007

0.008

100 110 120 130 140 150

6

0.00025

0.00045

0.00065

0.00085

0.00105

0.00125

100 110 120 130 140 150

45 6

21

Baseline

3

Co

nd

uct

an

ce (

S)

Su

sce

pta

nce

(S

)

Frequency (kHz) Frequency (kHz)

4, 5

1,2,3

Figure 6. Raw-signatures of PZT Patch #2 at various damage states (1; 2; 3; : : : ; 6).(a) Raw-conductance, (b) raw-susceptance.

(a)

0

20

40

60

80

100

120

140

0 1 2 3 4 5 6 7 8

Visible cracks Patch found damaged

(b)

0

20

40

60

80

1 2 3 4 5 6

Dx

Dy

Dx,

Dy

RM

SD

(%

)

Damage States Damage States

Figure 7. Damage prediction by Patch #2. (a) Real and imaginary parts of complex damage metric,(b) RMSD (%) in raw-conductance.

Damage prediction by PZT Patch #2

The raw-signatures of the PZT Patch #2 are shown in Figure 6 and Figure 7 shows thecomponents Dx and Dy of the complex damage metric at various states. It also shows theRMSD index (conventional index in the EMI technique) for comparison. From State 1 toState 3, only minor deviations could be noticed in the raw-signatures. This observation wasconsistent with the previous prediction [36] that cracks will start from State 4 onwards. AtState 4, a prominent shift was noticed in the conductance signature (Figure 6(a)). The inherentcause of the shift can be correlated with damage indices shown in Figure 7(a). This shift inthe signature is accompanied by a prominent rise in the value of Dx. This signi�es a changein the equivalent SDOF damping associated with the drive point impedance of the PZT patch.An increase in damping is an expected phenomenon associated with crack development. AtState 5, further upward shift of the conductance signature (Figure 6(a)) as well as increaseof Dx (Figure 7(a)) can be observed. No major change in Dy is observed from State 1

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1912 S. BHALLA AND C. K. SOH

0.003

0.004

0.005

0.006

0.007

100 110 120 130 140 150

7

8

(a) (b)

0.0002

0.0004

0.0006

0.0008

0.001

100 110 120 130 140 150

8

76

Baseline,

Frequency (kHz) Frequency (kHz)

Su

sce

pta

nce

(S

)

Con

duct

ance

(S

)

1,2,3,4,5

Baseline, 1,2,3,4,5,6

Figure 8. Raw-signatures of PZT Patch #1 at various damage states (1; 2; 3; : : : ; 8).(a) Raw-conductance, (b) raw-susceptance.

to State 5. This is because damping is much more sensitive to damage at high frequenciesas compared to the sti�ness or inertia related e�ects [37]. The area around the patch wascontinuously monitored and observable cracks could only be detected at State 6. The patchhowever provided warning much earlier at State 4 itself. State 6 was accompanied by areduction of Dx and a rise in Dy. A reduction in the ‘apparent damping’ could be due to thedevelopment of disbonding between the patch and the host structure and possible damage tothe patch itself. This is also re�ected in the equivalent sti�ness-mass factor since the associatedindex Dy shows an abrupt rise in magnitude due to reduction in the equivalent spring sti�ness.This is also supported by the fact that the patch was found to be damaged at State 7 (a crackwas detected running through the patch). However, the patch provided the necessary warningmuch earlier, at State 4. The conventional processing approach, the RMSD, failed to respondto damage to the patch itself at State 6 and continued to show a rising tendency.

Damage prediction by PZT Patch #1

Figure 8 shows the raw-signatures of PZT Patch #1 and Figure 9 shows the components ofthe complex damage index and the RMSD index at various states. From the Baseline State toState 6, the raw-conductance signature of Patch #1 did not undergo any substantial change.The indices Dx and Dy also did not display any prominent change (Figure 9(a)). Again,higher sensitivity of the associated equivalent damping to incipient damage was con�rmed bythe relatively large magnitude of Dx as compared to Dy (State 1 to State 6). At State 7, anobservable shift was observed in the conductance signature (Figure 8(a)). This can be seento be accompanied by a rise of Dx (Figure 9(a)). At State 8, a sudden and more prominentvertical shift of the signature was observed. From Figure 9(a), it is observed that both Dxand Dy exhibit large values, suggesting severe nature of damage. Close examination of theregion surrounding the patch in fact showed the presence of a hairline shear crack near thebeam-column joint. The patch however provided the information of the imminent damage atState 7 itself.

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1913

(a) (b)

0

20

40

60

80

1 2 3 4 5 6 7 8

Dx

Dy

0

20

40

60

80

100

0 1 2 3 4 5 6 7 8

Dx,

Dy

Damage States Damage States

RM

SD

(%)

Figure 9. Damage prediction by Patch #1. (a) Real and imaginary components of complex damage index,(b) RMSD (%) in raw-conductance.

0 1 2 3 4 5 6 7 80

50

100

150

200

Freq

uenc

y (H

z)

0

50

100

150

200

250

1 2 3 4 5

%Reduction in natural frequencyRMSD based on GRMSD based on x

RM

SD (

%)

Damage States Damage States(a) (b)

Figure 10. (a) Natural frequency of vibration of �oor #2 beam at various damage states, (b) Evaluationof damage based on natural frequency, raw-conductance and extracted mechanical impedance.

Damage sensitivity of the proposed methodology

It is worthwhile to compare the proposed damage diagnosis methodology with the low-frequency vibration based methods as well as the conventional approach based on raw-conductance signatures using statistical quanti�ers. Shown in Figure 10(a) are the reduction inthe natural frequency associated with the vibrations of the second �oor beam (on which PZTPatch #2 was instrumented). These are compared with the RMSD of the raw-conductance(traditional approach in the EMI technique) as well as with the RMSD of the extracted realpart of structural impedance ‘x’ in Figure 10(b). The higher sensitivity of ‘x’ to damageas compared to the low-frequency vibration techniques as well as the conventional damagequanti�cation approach in the EMI technique is clearly evident from Figure 10(b).

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1914 S. BHALLA AND C. K. SOH

Thus the new methodology enables us to obtain information about the nature of the damageoccurring in the vicinity of the PZT patch, viz. the equivalent SDOF sti�ness, the dampingand the mass associated with the drive point of the PZT patch. It predicts the damage ona uniform 0–100 scale. It is therefore more pragmatic than the previously reported non-parametric statistical approaches. It is recommended that (Dx +Dy)620% indicates incipientnature of damage and (Dx+Dy)¿50% indicates severe nature of damage. More tests will bereported in the future for calibrating damage with speci�c changes in ‘x’ and ‘y’.

DISCUSSION

The proposed methodology provides a simple approach to quantify damage from the measuredsignatures of the PZT patches in RC structures. This can be used to quickly assess theintegrity of the critical parts of the structure after a disaster such as an earthquake; especially,the inaccessible parts of the structure, which are not exposed to visual check, can be easilymonitored using the PZT patches. Since the PZT patches demand external bonding, they canbe installed on already constructed facilities as well. This gives a de�nite edge to the PZTpatches over other sensors which need to be embedded inside the concrete and cannot beinstalled on existing infrastructure.In the proposed derivation, only one-dimensional vibrations have been considered. The

electro-mechanical coupling in the other direction has been neglected. This is justi�able inthe present case due to the skeletal nature of the test structure. In other structures, signi�cantcoupling could be present in the other directions. However, consideration of vibrations inthe other directions will need the use of models such as the one developed by Zhou et al.[38; 39] which will only render the computational procedure highly tedious without much gainin the insight into the associated damage mechanism. Analysis for structures in which two-dimensional interaction is dominant will be presented in our subsequent papers. Nonetheless,the present method can still be applied to structures where 2D coupling is signi�cant. In thiscase, the extracted parameters will represent the ‘equivalent one-dimensional parameters’. Theone-dimensional analysis, as presented in this paper, o�ers a simple and convenient approachto make meaningful interpretations about damage.

CONCLUSIONS

In this paper, a new method of analysing the electro-mechanical admittance signatures obtainedfrom the PZT patches bonded to structures has been presented. The proposed method extractsthe ‘apparent’ drive point structural impedance associated with the bonded PZT patch. Acomplex damage metric has been proposed to quantify the damage based on the drive pointmechanical impedance of the structure. The real part of the damage metric represents thechange in the equivalent SDOF damping caused by damage, and the imaginary part representsthe change in the equivalent SDOF sti�ness-mass factor associated with the drive point of thePZT patch. The proposed method was tested on a model frame structure that was subjectedto base vibrations on a shaking table. The instrumented PZT patches were found to givemeaningful insight into the changes taking place in the structural parameters as a result ofdamage. The patches were successful in identifying �exural and shear cracks, two prominent

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STRUCTURAL IMPEDANCE BASED DAMAGE DIAGNOSIS 1915

types of incipient damage in RC frames. The proposed method has higher sensitivity to damageas compared to the existing approaches.

ACKNOWLEDGEMENTS

We gratefully acknowledge the reviewers of our paper for their invaluable comments and suggestionsand for the encouragement provided. Their invaluable comments helped us immensely in improving theoverall presentation of the paper.

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