1

EFFECT OF VARYING AXIAL LOAD UNDER FIXED

BOUNDARY CONDITION ON ADMITTANCE

SIGNATURES OF ELECTROMECHANICAL

IMPEDANCE TECHNIQUE

YEE YAN LIM 1 AND CHEE KIONG SOH

2

Division of Structures and Mechanics, School of Civil and Environmental Engineering,

Nanyang Technological University, 50 Nanyang Avenue, Singapore 639798.

1 Research Student, Email: limy0134@entu.edu.sg

2 Corresponding Author, Professor, Email: csohck@ntu.edu.sg, Tel: (65) 6790-5280.

ABSTRACT

In recent years, researchers in the field of structural health monitoring (SHM) have been

rigorously striving to replace the conventional NDE techniques with the smart material based

SHM techniques. These techniques possess distinct advantages over the conventional

techniques, such as capable of providing autonomous, real-time, reliable and cost efficient

monitoring. For instance, the electromechanical impedance (EMI) technique employing smart

piezo-impedance transducer (PZT) is known for its sensitivity in detecting damage. The

admittance signatures of EMI technique for SHM are susceptible to variation in the static load

applied on the host structure, which should not be ignored. This paper presents a series of

experimental, analytical and numerical studies conducted to investigate the effect of varying

axial load in the presence of fixed boundary condition on the admittance signatures.

Theoretically, extensional vibration is not affected by any static axial load acting in the same

direction. However, it is useful in identifying the effect of stiffening caused by the boundary

condition upon increase in the axial force, which contributes to the discrepancy between the

experiments and theoretical analytical models. This study suggests that the effect of boundary

2

condition is substantial, which should be carefully considered in real-life application of EMI

technique for SHM.

Keywords: Electromechanical impedance (EMI), piezoceramics (PZT), axial load, boundary

condition

This research received no specific grant from any funding agency in the public, commercial,

or not-for-profit sectors.

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INTRODUCTION

Non-destructive evaluation (NDE) techniques such as ultrasonic wave propagation, acoustic

emission, magnetic field analysis, dye penetrant test, eddy currents and X-ray radiography are

commonly used in the industry for damage detection. However, these techniques generally

depend heavily on visual inspection for gross assessment prior to applying the local

techniques to pinpoint the damage. These conventional techniques can sometimes be very

inefficient and tedious. For instance in large structures, the critical parts may not be readily

accessible. This situation is further aggravated by the requirement of bulky probes or

equipments to be carried along during inspection. Inspectors are also often exposed to

dangerous conditions or uncomfortable working environments. High dependency of visual

inspection on experience of inspector, high cost and unsystematic procedures render the

technique uneconomical and unreliable.

In recent years, researchers in the field of structural health monitoring (SHM) have been

rigorously striving to replace the conventional NDE techniques with the smart material based

SHM techniques, employing smart materials such as piezoelectric materials and fiber optics.

These techniques possess distinct advantages over the conventional techniques, such as

capable of providing autonomous, real-time, reliable and cost efficient monitoring. One of

these techniques, known as the electromechanical impedance (EMI) technique employing

smart piezo-impedance (lead zirconate titanate, PZT) transducer is known for its sensitivity in

detecting local damage. Being non-invasive, extremely light, cost effective, able to perform

self actuating and sensing, the EMI technique is constantly being developed in recent decades.

For instance, Park et al. (2000) reported significant proof-of-concept applications of the EMI

technique on civil-structural components such as composite reinforced masonry walls, steel

bridge joints and pipe joints. Soh et al. (2000) established the damage detection and

localization ability of piezo-impedance transducers on real-life RC structures. Giurgiutiu et al.

(2004) proved the feasibility of concurrent applications of the EMI technique and the wave

propagation technique by employing the same PZT patch attached on an aircraft structure.

Lim et al. (2006) attempted some parametric based damage detection using equivalent

structural parameters in characterizing the severity of damage in lab-sized aluminium and

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concrete structure. Park et al. (2008) adopted the PCA-data compression technique as a

pre-processing module to reduce the data dimensionality and eliminate the unwanted noises,

which is useful for wireless monitoring. Experimental study inspecting loose bolts in a

bolt-jointed aluminium structure was conducted. The damage detection capability is

significantly enhanced in comparison to the traditional root mean square deviation approach.

The EMI technique is also able to monitor the hydration process of concrete. For

instance, Shin et al. (2008) showed that the EMI signatures obtained from PZT patch surface

bonded on concrete gradually shifted to the right and subsided with curing time. They

attributed these behaviors to the stiffening action due to strength gain of concrete. Detailed

reviews considering various issues on the applications of EMI technique in SHM can be

obtained in publications by Park et al. (2003, 2008) and Annamdas and Soh (2010).

In practice, structures in service are often subjected to constant loading. Even with the

structure in standby mode, static load caused by self weight or dead load still exists.

Annamdas et al. (2007) showed that the electrical admittance signatures, especially its

imaginary component adopted in the EMI technique for SHM, is susceptible to variation in

the static load applied on the host structure. On the other hand, applied loads are associated

with the imposed boundary conditions. Annamdas and Soh (2010) reported in a review paper

that if the host structure is smaller than the sensing range of the PZT transducer bonded on it,

the boundary conditions can also influence the PZT-based signatures. Park et al. (2000)

claimed that the sensing radius of a typical PZT patch might vary from 0.4m on composite

reinforced structures to about 3m on simple metal beams. The boundary conditions in real life

structures are however, extremely difficult to characterize analytically, and tend to exhibit

poor repeatability between structures (David, 2006). Esteban (1996) concluded that structural

discontinuities (at boundaries) acting as the sources of multiple reflections cause maximum

attenuation to the propagating waves. Hu and Yang (2007) investigated the PZT sensing

region based on the elasticity solution of PZT generated wave propagation and PZT–structure

interaction effect. The effect of boundary condition, especially for metallic structure should be

carefully considered for effective SHM. They found that with an excitation voltage of 1V,

applied in the typical excitation frequency range of 100–200 kHz for the EMI technique, the

valid sensing region of PZT sensors is about 2–2.5 m, subjected to the sensing limit of PZT

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transducer of 0.01V. The sensing zone can be extended with the use of higher excitation

voltage.

Up to date, there are only a few studies (Ong et al. 2002 and Abe et al. 2000) conducted

to investigate the effect of static axial load on admittance signatures. Annamdas and Soh

(2007) presented briefly the effect of different boundary conditions on admittance signatures

without the presence of loading. However, the effect of boundary condition on varying load

has yet to be investigated.

This paper aims at bridging this gap by presenting a series of experimental, analytical

and numerical studies to investigate the effect of axial stress in the presence of fixed boundary

condition on the admittance signatures acquired through EMI technique. The problem is

approached by first revisiting the 1-D analytical model involving axially loaded simple beam

with surface-bonded PZT patch. Both axial and flexural modes of vibration are considered.

Experimental test and finite element (FE) simulation are carried out to verify the model. The

effects of loading and boundary condition are then investigated.

PRINCIPLES OF EMI TECHNIQUE

Application of the EMI technique, employing piezoceramics, in SHM was proposed by

Chaudhry et al. (1994) and Sun et al. (1995). The EMI technique shares similar working

principles as the conventional dynamic response techniques. However, the frequency range

employed in the EMI technique (30 - 1000 kHz) is much higher.

The EMI technique utilizes PZT transducer to simultaneously actuate the host structure

and to sense its response through electromechanical (EM) coupling ability of the piezoelectric

material. In other words, PZT transducer is employed as collocated actuator and sensor. When

a mechanically attached (surface-bonded or embedded) PZT patch is dynamically excited by

an alternating voltage, the vibrational force generated by the PZT patch can be transferred to

the host structure. The corresponding structural response at different excitation frequency will

modulate the electric current across the PZT patch.

The modulated current, in terms of complex electrical admittance (conductance and

susceptance signatures) is also measured and recorded by the impedance analyzer. This

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admittance signature is a function of the stiffness, mass and damping of the host structure.

The frequency response function (FRF) of the admittance signatures can be graphically

plotted to yield a spectrum that serves as an indication to the structural response. Any

subsequent damage or interference on the structure, which causes a change in structural

response, would directly affect the admittance signatures and can be reflected qualitatively

from the alteration in the spectrum.

Giurgiutiu and Zagrai (2000) proposed a model for transducer dynamics by an elastically

constrained boundary condition represented by a pair of springs (dynamic stiffness, Kstr) at

both ends of the patch as shown in Figure 1. The drive point dynamic stiffness represents the

host structure’s interaction with the PZT patch at its end points. This approach of modeling

allows for a complete range of transducer boundary condition to be investigated. The complex

electrical admittance, Y, can be expressed as:

RCjY

)cot(

111 2

31

, (1)

where2

al , and 1

33

a

T

aa hblC is the zero load capacitance. la, ba and ha are the length,

width and thickness of the transducer, respectively. is the wave number, ω is the angular

frequency and j denotes imaginary number. PZTstr KKR / is the stiffness ratio between

host structure and PZT patch, a

E

aPZT lYAK /11 and aA are the quasi-static stiffness and

the cross-sectional area of the patch, respectively. T

33 and EY11 are the complex electrical

permittivity and complex Young’s modulus of transducer. T

EYd

33

11

2

3131

is the EM cross

coupling coefficient of the PZT patch, and d31 is the piezoelectric strain coefficient.

Equation (1) presents a relationship between electrical admittance and structural stiffness.

All other terms are piezoelectric properties, implying that any change in structural stiffness

can directly be reflected in the electrical admittance.

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EFFECT OF AXIAL STRESS ON ADMITTANCE SIGNATURES

Changes in vibrational behavior of host structure, including damage, can be indirectly

measured from the electrical impedance/admittance signatures. Giurgiutiu (2007) modeled the

1-D PZT-beam interaction considering both axial and flexural vibrations. Close agreement

with the experimental counterparts, up to 30 kHz, under free-free boundary condition is

shown. The model, however, did not include the effect of static axial stress.

Abe et al. (2000) first proposed the use of EMI technique for identification of in-situ

stress in thin structural members. Both 1-D and 2-D analytical model based on wave

propagation theory for beam and plate structures were developed, incorporating the effect of

axial stress. They reported that the wave number and wave frequencies can be directly related

to the electrical admittance through the EM coupling equation suggested by Liang et al.

(1994). They showed that the tensile stress increased the resonance frequency in the electrical

impedance spectrum. However, they found that considerable discrepancy exist between the

measured and the predicted stress. No in-depth reasoning is provided regarding the difference.

Ong et al. (2002) investigated the effect of in-situ stress on the FRFs of beam structure

by applying pure bending actuation through a pair of symmetrically located surface-bonded

PZT transducers, activated in out-of phase mode. They simulated the axially loaded beam

structure with PZT patch using 1-D EMI model for an elastically constrained transducer.

Effect of bonding was neglected. Back calculation of the in-situ stress from the electrical

FRFs of the transducer was presented. However, experimental verification was not included.

Both of the abovementioned models concentrated only on flexural vibration of beam under

tension. The extensional (axial) vibration and effect of boundary condition were not

considered.

In the following sections, the model used by Ong et al. (2002) involving an axially

loaded Euler-Bernoulli beam is revisited, but with extensional vibration considered. The EMI

model (Equation 1) based on elastically constrained transducer conditions is adopted to

evaluate the electrical admittance from the PZT transducer. Theoretically, extensional

vibration is not affected by any static axial load acting in the same direction. It is however

useful in identifying the effect of boundary stiffening, which contributes to the discrepancy

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between experimental and theoretical analytical mode, as presented in the ensuing section.

1-D MODELING OF AXIALLY LOADED BEAM EXCITED BY

PZT TRANSDUCER

Consider a uniform thin beam with a surface bonded PZT patch being loaded axially by a

static force, N, at both ends of the beam in the x-direction as shown in Figure 2(a). To

investigate the effect of static axial force on the admittance signatures acquired from the EMI

technique, we consider the simplified dynamic PZT-structure interaction model as depicted in

Figure 2(b). The PZT patch’s actuation can be modeled as a pair of equivalent axial forces, Ns

and bending moments, Ms to generate the extensional and transverse vibration on the beam.

Rao (1999) suggested that both motions can be considered uncoupled if the amplitude of

vibration is small. Separate solutions for each mode can thus be derived and superimposed to

determine the complete solution.

Since the static axial force is acting in the direction of the extensional vibration, it will

not alter the extensional dynamic equation. Its dynamic response resembles a rod vibrating

axially, which can be expressed as (Rao, 1999):

),('' txfuAuAY nssss , (2)

where u is the displacement in x-direction. sY , As and ρs are the complex Young’s modulus,

cross sectional area and material density of the beam, respectively.

tjn

nn

nnn exFtxf

1

0

)(),( is the distributed loading function for extensional vibration. Fn is

the modal participation coefficient for extensional vibration, n = 1, 2, 3, 4 …… are the

extensional mode numbers, and )(xn is the nth mode shape of the beam in extensional

vibration. ()' and )(

denote spatial and temporal differential operator, respectively.

However, the effect of axial force is substantial in the case of transverse vibration. The

corresponding dynamic equation for forced transverse vibration can be expressed as

(Dukkipati, 2004):

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),(),(),(''),( txftxwAtxNwtxwIY mss

iv

ss , (3)

where N is the magnitude of the axial load, with positive value denoting tension. w is the

transverse displacement with respect to x, and sI is the moment of inertia of the beam.

tjm

mm

mmm exFtxf

1

0

)(),( is the distributed loading function for transverse vibration. Fm is

the modal participation coefficient for transverse vibration, m = 1, 2, 3,…… are the transverse

mode numbers, and )(xm is the mth mode shape of the beam in transverse vibration.

To solve for Equations (2) and (3), both displacements are assumed to be variable

separable. The solution process can be found in Ong et al. (2002) and Giurgiutiu (2007), and

is hereby omitted. Only significant outcomes are presented.

Considering simply supported boundary conditions, the total forced response of the

system in axial direction is given by:

tj

n

nn nss

nn eA

xFtxu

1

0

22

)(),( , (4)

The nth mode shape of beam in extensional vibration can be evaluated as

s

nl

xnx

cos)( .

The nth natural frequency of beam in axial vibration is:

s

s

n cl

n . (5)

The corresponding total response of the forced vibration in transverse direction is:

1

0

)(),(22

m

mm

tj

m

mss

m exA

Ftxw

, (6)

The mth mode shape of beam in transverse vibration is derived as

x

l

mx

s

m

sin)( , and

the mth natural frequency of the beam in transverse vibration is:

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1

ss

ss

ss

ml

mN

l

mIY

A

. (7)

On the other hand, the point-wise dynamic stiffness, defined as the ratio between the

excitation force to the displacement at that point can be evaluated by superimposing the

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displacement from the extensional and transverse vibrations:

1

22

21

22

121

0

1

0

)(')('

2

)()(

m

mm m

mmws

n

nn n

nnussstr

xxkhxxkAK

, (8)

where )()(2

12 xxl

k nn

s

u and

)(')(' 21 xxl

hhk mm

s

asw

respectively.

hs is the thickness of beam.

Equation (8) can be substituted into Equation (1) to derive the electrical admittance. The

difference between the load free beam and the axially loaded beam is highlighted in their

natural frequency, as shown in Equation (7). Note that the mechanical and electrical damping

effects are incorporated in the imaginary component of the complex Young’s modulus and

complex electrical permittivity, respectively.

Effect of Axial Force on Natural Frequency

As discussed in the previous section, the applied axial force will not affect the extensional

behavior but only the natural frequency of the beam. So, when the axial force is increased

from 0 to N1, we can assume a change in the natural frequency (only for the transverse mode),

ΔΩm, and Equation (7) can be reduced to:

mssss AIYN 41 , (9)

In fact, the outcome suggests that a tensile stress will cause an increase in natural

frequency while a compressive stress will lead to a reduction. The variation in natural

frequency is directly proportional to the variation in axial force, regardless of the mode

number and range of frequency. Equation (9) is similar to the outcome presented by Abe et al.

(2000).

Consider three rectangular aluminium beams of the same length (236mm) and width

(26mm) but with different thickness, (hs = 2mm, 4mm and 8mm) being subjected to varying

axial stress, from -100 MPa to 100 MPa. The shifts in natural frequency against axial force

stress are calculated and plotted in Figure 3. The figure shows an interesting observation

whereby the shift in natural frequency is not only affected by the magnitude of stress applied

but also the cross-sectional area of the beam. Doubling or halving the thickness at constant

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stress will resulted in halving and doubling the amount of shift in natural frequency,

respectively (according to Equation 9). Annamdas et al. (2007) also presented similar findings

in their experimental study. This observation gives meaningful implication for the real-life

application of the EMI technique. In practice, the cross-sectional area of a structure can be

much larger than the beam adopted in this study, thus minimizing the contamination of axial

loading on the admittance signaturesacquired for SHM.

Effect of Axial Force on Admittance Signatures

To investigate the effect of axial force on admittance signatures, we first derived the

theoretical electrical admittance signatures of a PZT transducer surface-bonded at the centre

of an axially loaded aluminium beam (236mm x 26mm x 4mm). The material properties used

for deriving the admittance signatures are tabulated in Table 1. In this study, discussion is

focused on the real part of the admittance signatures (conductance) as they are more useful for

SHM application (Sun et al. 1995). The typical frequency range employed by EMI technique,

between 10 to 100 kHz, is adopted (Yang et al. 2008a, 2008b). Figure 4 depicts the

conductance signatures evaluated at varying tensile forces, N up to 10kN, in the frequency

range of 10 to 50 kHz.

In general, the slope and magnitude of the signatures are unaffected by the axial force.

Five out of seven structural resonance peaks shift progressively to the right (increase in

resonance frequency) with increase in tensile force. These peaks are identified as the peaks

induced by transverse modes of vibration of the beam. On the other hand, two of the

resonance peaks (circled in red) with lower amplitudes, excited by the extensional modes, are

unaffected by the change in axial force. In the following sections, both types of peaks are

separately studied, supported with numerical and experimental verifications.

EXPERIMENTAL SETUP AND TEST

For the experiment on tensile test, an aluminium (grade Al6061-T6) beam specimen of size

236mm x 26mm x 4mm was prepared. One piece of PZT patch (type PIC 151, manufactured

by PI Ceramics) measured 10mm x 10mm x 0.3mm was surface-bonded at the centre of the

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specimen using two parts high strength epoxy. Wayne Kerr precision impedance analyzer

6420 (Figure 5a) was used to supply alternating voltage and to measure the corresponding

admittance signatures across the PZT patch. A notebook computer with customized software

was used to record the admittance signatures.

In the experiment, tensile force was applied using a Universal Test Machine (UTM) as

shown in Figure 5(b). Baseline signatures were recorded after the specimen was clamped on

the UTM, before application of load. Axial force was applied progressively at 1kN step, up to

a maximum of 10kN, i.e., a maximum applied stress of 96.2MPa which is equivalent to

one-third of the specimen yield strength. Admittance signatures were acquired after each load

step. The load was allowed to stabilize before initiating the signature acquisition process. Two

pieces of strain gauges (each 3mm in length), type FLA-3-11-5LT manufactured by TML,

were installed one at the center of the beam (opposite to face with PZT patch) and another at

5cm from the center of the beam, serving as controllers for load monitoring. The values of

strain measured were consistent with the readings from the UTM (in terms of stress divided

by Young’s modulus of elasticity).

Giurgiutiu et al. (2007) reported that for a PZT patch surface bonded on aluminium beam,

which failed at about 7200 micro-strain, the impedance signatures were minimally affected

when the strain does not exceed 3000 micro-strain. In this study, the maximum strain

measured was 1360 micro which is well below the suggested threshold.

Resonance Peaks Induced by Transverse Vibration

Zooming into the resonance peak induced by transverse mode of vibration, occurring between

46.5 to 48.5 kHz of Figure 4, we can observe a progressive rightward shift in peak against

increase in tensile force as shown in Figure 6.

In addition to experimental test, a 3-D FE model was developed using the commercial

FE software ANSYS 12.1 to verify the theoretical analytical model. Coupled field element

(Solid 5) was adopted to simulate the PZT patch. Brick element (Solid 45) was used to model

the aluminium beam. The FE simulation was performed in two stages. Firstly, a static analysis

was performed in the presence of axial load. The outcome of this static analysis was then

brought forward to perform harmonic analysis under excitation of the PZT transducer. The

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properties and basic assumptions used in the theoretical analytical model were applied in the

FE model. The details of simulation can be found in Yang et al. (2008b).

The admittance signatures acquired from the experimental test and numerical analyses

are plotted in Figures 7(a) and 7(b) respectively. The frequencies of resonance peak at zero

load were recorded as 44.4 kHz and 43.5 kHz respectively. Both of them behave in the same

manner as the theoretical analytical model where the peaks shift progressively to the right

with increase in tensile force. Note that the selected resonance peaks were excited by

transverse mode of vibration. Similar behavior can be observed in the other peaks actuated by

this mode of vibration. It is worth mentioning that the variations between the magnitudes of

peaks from experimental, numerical and analytical results are attributed to the uncertainties

involved in determining the damping ratios, which are highly depend on trial and error. In this

case, the damping ratios are obtained from previous study without performing rigorous trial

and error. This is because the magnitude of peak does not affect the resonance frequency,

which is the major parameter used in the EMI technique (Yang et al. 2008).

For ease of comparison, the increase in resonance frequency obtained from the numerical

simulation, the analytical model (Equation 1 and 8) and the experimental test (as shown in

Figures 6 and 7) as well as the theoretical increase in natural frequency of beam (Equation 9)

are plotted against the axial load as shown in Figure 8.

The results presented demonstrate the capability of the numerical and the analytical

models in describing the interaction between PZT patch and beam substrate under varying

axial load. All of them exhibit a linear relationship. This conforms to the theoretical change in

beam natural frequency under axial load, as predicted by Equation (9). This result also shows

that the resonance peak obtained from the EMI technique is useful in identifying the structural

resonance frequency. The analytical and numerical results match seamlessly. The

experimental results, however, is consistently larger than those obtained by the models. The

amount of shift at 10kN is equal to 0.8 kHz in the experimental test and 0.5 kHz in both the

models. These variations are mainly attributed to the difficulties involved in simulating the

stiffening effect caused by the boundary conditions present in the experimental test. This issue

will be further discussed in the ensuing sections.

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Resonance Peak Induced by Extensional Vibration

Theoretical derivation of the extensional mode of vibration indicates that the corresponding

resonance peaks are unaffected by externally applied axial load, as exemplified in Figure 9(a)

(zoom into frequency range 31.5 to 33 kHz of Figure 4). It is apparent that the admittance

signatures for all 11 cases (from 0kN to 10kN) are exactly the same, overlapping each other

and resembling only one curve. To verify the results, FE simulation and experimental test

were carried out with their outcome presented in Figures 9(b) and 9(c) respectively. In the FE

model, only three scenarios were simulated, without load, with 5kN and with 10kN. The

peaks for all three cases turn out to be at the same frequency, thus verifying the analytical

counterpart. The other signatures ranging in between 10kN and 0kN can therefore be omitted.

In the experiment, a progressive rightward shift upon increase in tensile load occurs. The

amount of shift from 0 to 10kN is relatively small, about 0.3 kHz. This value is less than half

of the transverse mode induced peak’s shift, which is 0.8 kHz. Both transverse and

extensional induced peaks exhibited an additional 0.3 kHz of rightward shift in comparison to

the models. We can thus infer that the shift is caused by a factor omitted in both FE and

theoretical analytical models.

Revisiting both the theoretical analytical and FE models, the beams were assumed to be

simply supported with forces acting at both ends, without considering any contact with the

loading mechanism. In reality, this can never be achieved because the application of axial load

on the beam specimen mandates certain interaction with the loading instrument. The

additional peak shift observed in the experimental test, but absent from the theoretical

analytical and FE analysis, is most likely caused by the interaction at the boundary condition

during the application of axial load.

Figure 5(b) illustrates the actual boundary conditions during the tensile test. The

specimen is tightly held by the upper and lower jaws throughout the test. Despite maintaining

the clamping force, increase in tensile load increases the shear stress required to hold the

beam specimen. These increases in surface shear stress stiffened the portion of beam around

the clamping zone, resulting in about 0.3 kHz increase in resonance frequency upon loading

to 10kN. This observation could provide an explanation to Figure 8, in which the shift of

resonance frequency for the peak induced by transverse vibration is 0.3 kHz more than the

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theoretical analytical and numerical predictions.

Thus, the extensional mode induced peak’s shift as shown in Figure 9(c) is attributed to

the stiffening effect of boundary condition. These values can be used to compensate for the

frequency increment of experimental results in Figure 8. Upon compensation, ng the

stiffening effect caused by the boundary condition, Figure 8 is reproduced and presented in

Figure 10. A much closer agreement between the experimental test and the models is achieved.

The discrepancy between them at 10kN is reduced to approximately 0.05 kHz. This suggests

that, in practical application, the effect of boundary condition must be considered along with

the applied axial load. The large discrepancy between theoretical analytical value and

experimental measurement reported by Abe et al. (2002) could be caused by boundary

stiffening. Similar compensation could be achieved by using the other peaks induced by

extensional vibration.

It is worth mentioning that the resonance peaks acquired from experimental test and FE

model match each other closer than with the analytical model. This is mainly attributed to the

fact that the theoretical analytical model is 1-D in nature, being less accurate than the 3-D FE

model. Exact matching is however very difficult due to the difficulty in realistically

simulating the boundary condition used in the experiment. The effect of stiffening will be

further studied in the following section involving compression for beam.

STIFFENING EFFECT INDUCED BY BOUNDARY CONDITION

FOR BEAM UNDER COMPRESSION

Two aluminium beams were prepared to experimentally investigate the effect of compression

under fixed boundary condition on the admittance signatures. A short beam (reusing the

specimen from tensile test, 236mm x 26mm x 4mm) and a long beam (786mm x 26mm x

4mm) having identical cross sectional area were loaded on a strut rig (TQ SM 105 Mk II) as

illustrated in Figure 11. One piece of PZT patch was surface bonded at the center of each

specimen. A maximum load of 600N was exerted on the short beam. Both ends of the beam

were left unconstrained. A certain amount of compressive load was initially applied for

securing the beam on the strut rig, which served as the baseline, because when the applied

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load was below 100N, the boundary condition was very unstable.

Compression Test on Short Beam

The conductance signatures acquired at various compressive loads for the short beam are

plotted in Figure 12. Two of the resonance peaks occurring at 32.1 kHz and 45.0 kHz are

selected for illustration. Note that they are not exactly the same as for the previous test

because of different initial boundary conditions.

Figure 12 shows that both resonance peaks move to the right upon increase in

compressive load, contradicting the theoretical outcome from Equation (9) and Figure 3, in

which compression will soften the host structure resulting in leftward shift. Analyzing the

peaks’ movements, we found that both peaks shifted about 0.3 kHz when loaded to 600N

(400N increase in compressive load). The corresponding rates of shift averaged at 0.75 kHz /

kN in comparison to -0.048 kHz / kN as calculated from Equation (9). The large variation in

magnitude (15 times difference) and opposite in direction suggest that the rightward shift of

peak must be induced by some other factors, rather than by the compressive load. The

stiffening effect from boundary condition is again suspected to be the main factor.

Compression Test on Long Beam

The conductance signatures acquired at varying compressive load for the long beam are

plotted in Figure 13. The peak once again moved to the right when loaded to 200N and 300N.

However, upon loading to 400N, the peak movement became stagnant. At 500N, the peak

shifted back to the left. Examining the specimen, found obvious curvature was found in the

beam which could have been caused by the P-delta effect at 500N. In other words, the beam

failed in buckling at 500N.

This phenomenon leads to two important observations. Firstly, the effect of local

stiffening remains dominant (though slightly less than that for the short beam) over the effect

of compressive stress in spite of the length of the beam (more than three times of short beam).

The PZT patch remains sensitive to the end conditions. Secondly, buckling of beam (a severe

damage) induced a leftward shift indicating stiffness reduction.

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Stiffening Induced by Boundary Condition

A simple test was conducted to separately investigate the stiffening effect caused by the

boundary condition. The short beam specimen used previously was clamped with a bench vise

at one end as shown in Figure 14. An attempt was made to minimize the area being clamped

to simulate the boundary condition in previous compression test. The clamping force was

increased progressively by tightening the screw. Admittance signatures were acquired after

each stage. The degree of clamping force applied was labeled as “Wrench” and classified by

the amount of rotation on the screw as tabulated in Table 2. Note that no axial force was

applied.

Figure 15(a) plots the conductance signatures against frequency with different levels of

clamping forces applied. Using resonance peak at 32.2 kHz as baseline, the peak shifted

progressively to the right, behaving in a similar manner as observed in the compression test.

The increase in resonance frequency is plotted against the relative rotations in Figure 15(b).

This result clearly shows that the increase in clamping force caused a rightward shift in the

resonance peaks. With increased clamping force, local compressive stress was build up in the

vicinity of the clamp resulting in local stiffening, thus inducing a rightward shift in the

resonance peaks. This observation qualitatively proves that the leftward shift caused by the

compressive force was overwhelmed by the local stiffening due to the boundary condition.

It is worth mentioning that the stiffening effect caused by the boundary condition in the

compression test differed from that in the tensile test. In the compression test, surface

compressive pressures were increasingly built up at both ends by direct compression, whereas

in the tensile test, surface shear stress increased without any increase in surface compressive

pressure. For example, 10kN of shear force in the tensile test and 400N of surface

compressive force both induced approximately 0.3 kHz of stiffening. Obviously, the effect of

surface compressive pressure is more dominant than the surface shear stress as well as the

axial stress itself. Note that this study only considered the axial and transverse mode of

vibration. Resonances induced by the other modes of vibration (width, thickness, torsion etc.)

may behave in different manner.

From the three tests presented in this section, it is apparent that the EMI technique is

very sensitive to local stress induced by the boundary condition upon variations in axial stress,

18

even with the boundary condition fixed. In addition, the admittance signatures acquired from

the tests presented are repeatable for each load step upon loading and unloading as long as the

initial boundary condition is not disturbed. The stiffening effect may also diminish with

increase in the length and size of the structure; but this deserves further investigation.

In practice, SHM could still be performed by acquiring admittance signatures at different

load as baseline. Certain compensation algorithm could also be used to prevent false alarm,

which demands further study.

CONCLUSIONS

This paper investigates the effects of varying axial loadings and boundary conditions on the

admittance signatures acquired from PZT patch surface-bonded on lab-sized beam structure.

An theoretical analytical model was constructed to simulate the interaction between PZT

patch and uniform beam structure in the presence of axial loading. Both extensional and

transverse modes of vibrations were considered. Theoretically, axial load is influential to the

transverse mode of vibration but not the extensional mode. Tensile stress will induce

stiffening effect, resulting in increase in natural frequency as well as resonance frequency of

peaks in the admittance signature spectrum. The effect of stiffening by boundary condition is

investigated using extensional mode of vibration, explaining the variations of resonance peak

(transverse mode) between experimental test and models.

Compression test was also conducted but the expected softening effect, derived

theoretically, cannot be observed because it was overwhelmed by the local stiffening around

the boundary. The EMI technique is also found to be valuable in detecting buckling. This

study suggests that the effect of boundary condition is substantial, which should be carefully

considered in real-life application of EMI technique for SHM.

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22

Table 1(a): Material properties of aluminium beam (T6061-T6).

Parameters Symbols Values Units

Density ρs 2715 kg/m3

Young’s modulus Ys 68.95x109 N/m

2

Mechanical loss factor μ 0.005

Table 1(b): Material properties of PZT patch.

Parameters Symbols Values Units

Piezoelectric constant d31 -2.10 x 10-10

m/V

Electric Permittivity T

33 2.12 x 10-8

F/m

Density ρa 7800 kg/m3

Young’s modulus EY11 62.1x109 N/m

2

Mechanical loss factor η 0.003

Dielectric loss factor δ 0.02

Table 2: Different levels of clamping forces applied on aluminium beam specimen (236mm x

26mm x 4mm).

Labels Relative

rotations (0)

Descriptions Increase in resonance

frequency (kHz)

Wrench 0 0 Very loose (Touching) 0

Wrench 1 10 Loose 0.05

Wrench 2 20 Medium 0.1

Wrench 3 40 Tight 0.15

Wrench 4 60 Very tight 0.18

Wrench 5 64 Maximum tightness 0.2

23

Figure 1: Elastically constrained 1-D PZT-structure interaction model. (Giurgiutiu and Zagrai,

2000).

Figure 2(a): Pictorial illustration of PZT patch surface bonded on axially loaded beam

structure.

Figure 2(b): Simplified PZT-structure interaction with axial load.

sN sN

sM sM

x1 x2

N N

N N x

z

ls

x1

x2

la PZT patch

Aluminium Beam hs

xa

ya

2Kstr 2Kstr

la

PZT patch

24

-1.0

-0.8

-0.6

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0

-100 -80 -60 -40 -20 0 20 40 60 80 100

Axial stress (MPa)

Sh

ift

in n

atu

ral

freq

uen

cy (

kH

z)1

hs = 2mm

hs = 4mm

hs = 8mm

Figure 3: Variations in natural frequency against stress with varying beam thickness.

Figure 4: Plot of admittance signatures versus frequency (10 ~ 50 kHz) derived theoretically

analytically from PZT patch surface bonded on aluminium beam (236mm x 26mm x 4mm) at

varying tensile forces.

0

0.0001

0.0002

0.0003

0.0004

10 15 20 25 30 35 40 45 50

Frequency (kHz)

Co

nd

ucta

nce (

S)

N = 0kN N = 2kN N = 4kN N = 6kN

N = 8kN N = 10kN

Peaks excited by extensional

vibration

25

Figure 5: Pictorial illustration of experimental setup.

(a) Precision impedance analyzer.

(b) Aluminium beam monitored by PZT transducer and strain gauges tested on UTM.

Analytical

0

0.0001

0.0002

0.0003

0.0004

0.0005

0.0006

46.5 46.7 46.9 47.1 47.3 47.5 47.7 47.9 48.1 48.3 48.5

Frequency (kHz)

Co

nd

ucta

nce (

S)

N = 0kN N = 1kN N = 2kN N = 3kN

N = 4kN N = 5kN N = 6kN N = 7kN

N = 8kN N = 9kN N = 10kN

Figure 6: Plot of resonance peak (transverse mode) versus frequency (46.5 ~ 48.5 kHz)

derived theoretically analytically from PZT patch surface-bonded on aluminium beam

(236mm x 26mm x 4mm) under varying tensile force.

(a) (b)

Tension: Rightward shift

26

Experimental

0

0.0001

0.0002

0.0003

44 44.2 44.4 44.6 44.8 45 45.2 45.4 45.6

Frequency (kHz)

Co

nd

ucta

nce (

S)

0kN 1kN 2kN 3kN 4kN 5kN

6kN 7kN 8kN 9kN 10kN

Numerical

0

0.0005

0.001

0.0015

0.002

43.2 43.4 43.6 43.8 44 44.2

Frequency (kHz)

Co

nd

ucta

nce (

S)

0N 1kN 2kN 3kN 4kN 5kN

6kN 7kN 8kN 9kN 10kN

Figure 7: Plot of resonance peak (transverse mode) versus frequency (43 ~ 46 kHz) acquired

from PZT patch surface-bonded on aluminium beam (236mm x 26mm x 4mm) under varying

tensile force. (a) Experimental, and (b) Numerical

Tension: Rightward shift

Tension: Rightward shift

(a)

(b)

27

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0 1 2 3 4 5 6 7 8 9 10

Incre

ase

in

fre

qu

en

cy (

kH

z)

Axial force (kN)

Numerical

Analytical

Experimental

Natural Frequency (Eqn 9)

Figure 8: Plot of frequency increment (resonance peak at 44.4 kHz) versus axial load (0 ~

10kN) obtained from numerical model, analytical model, experimental test and theoretical

beam’s natural frequency.

Analytical

0

0.00005

0.0001

0.00015

0.0002

31.5 31.7 31.9 32.1 32.3 32.5 32.7 32.9

Frequency (kHz)

Co

nd

ucta

nce (

S)

0kN 1kN 2kN 3kN 4kN 5kN

6kN 7kN 8kN 9kN 10kN

(a)

28

Numerical

0

0.001

0.002

0.003

0.004

0.005

31.5 31.7 31.9 32.1 32.3 32.5 32.7 32.9

Frequency (kHz)

Co

nd

ucta

nce (

S)

0N 5kN 10kN

Experimental

0

0.0001

0.0002

0.0003

31.5 31.7 31.9 32.1 32.3 32.5 32.7 32.9

Frequency (kHz)

Co

nd

ucta

nce (

S)

0kN 1kN 2kN 3kN

4kN 5kN 6kN 7kN

8kN 9kN 10kN

Figure 9: Plot of resonance peak (extensional mode) versus frequency (31.5 ~ 33 kHz)

acquired from PZT patch surface-bonded on aluminium beam (236mm x 26mm x 4mm)

under varying tensile load. (a) Analytical, (b) Numerical, and (c) Experimental.

(b)

(c)

29

Figure 10: Plot of frequency increment (resonance peak at 44.4 kHz) versus axial load (0 ~

10kN) obtained from numerical model, analytical model, experimental test and theoretical

beam’s natural frequency, after compensating the stiffening effect from boundary condition.

Figure 11: Pictorial representation of compression test where an aluminium beam specimen is

loaded on strut rig.

30

0

0.0004

0.0008

0.0012

0.0016

31.6 31.8 32 32.2 32.4 32.6

Frequency (kHz)

Co

nd

uc

tan

ce

(S

)200N 300N 400N

500N 600N

0

0.0002

0.0004

0.0006

44.5 44.7 44.9 45.1 45.3 45.5

Frequency (kHz)

Co

nd

uc

tan

ce

(S

)

200N 300N 400N

500N 600N

Figure 12: Plot of conductance signatures versus frequency acquired experimentally from

PZT patch surface-bonded on short aluminum beam (236mm x 26mm x 4mm) at varying

compressive load. (a) 31.6 ~ 32.6 kHz, and (b) 44.5 ~ 45.5 kHz

(a)

(b)

31

0

0.0002

0.0004

0.0006

0.0008

49.6 49.8 50 50.2 50.4 50.6

Frequency (kHz)

Co

nd

ucta

nce (

S)

100N

200N

300N

400N

500N

Figure 13: Plot of conductance signatures versus frequency (49.6 ~ 50.6 kHz) acquired

experimentally from PZT patch surface-bonded on aluminium beam (786mm x 26mm x 4mm)

at varying compressive load.

Figure 14: Pictorial illustration of short beam being clamped on bench vise.

32

0

0.0002

0.0004

0.0006

0.0008

0.001

31.7 31.9 32.1 32.3 32.5 32.7Frequency (kHz)

Co

nd

ucta

nce (

S)

Wrench 0

Wrench 1

Wrench 2

Wrench 3

Wrench 4

Wrench 5

0

0.05

0.1

0.15

0.2

0.25

0 10 20 30 40 50 60 70

Relative rotation (degree)

Incre

ase i

n f

req

uen

cy (

kH

z)

Figure 15: Effect of local stress on admittance signatures acquired from PZT patch

surface-bonded on aluminium beam (236mm x 26mm x 4mm) with one end clamped on

bench vice.

(a) Conductance signatures against frequency (31.7 ~ 32.7 kHz).

(b) Increase in resonance frequency against rotation.

(a)

(b)