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NDT&E International 42 (2009) 741–747

Contents lists available at ScienceDirect

NDT&E International

0963-86

doi:10.1

� Corr

E-m

journal homepage: www.elsevier.com/locate/ndteint

Detection of vertical bending and torsional movements of a bridgeusing a coherent radar

Devis Dei a, Massimiliano Pieraccini a,�, Matteo Fratini a, Carlo Atzeni a, Gianni Bartoli b

a Department of Electronics and Telecommunications, University of Florence, Via Santa Marta 3, 50139 Florence, Italyb Department of Civil Engineering, University of Florence, Via Santa Marta 3, 50139 Florence, Italy

a r t i c l e i n f o

Article history:

Received 26 March 2009

Received in revised form

6 July 2009

Accepted 7 July 2009Available online 15 July 2009

Keywords:

Radar

Remote sensing

Dynamic monitoring

Accelerometer

Torsional movement

95/$ - see front matter & 2009 Elsevier Ltd. A

016/j.ndteint.2009.07.001

esponding author. Tel.: +39 0554796 273.

ail address: massimiliano.pieraccini@unifi.it (

a b s t r a c t

Interferometric radar has been recently proposed as a measurement instrument for static and dynamic

testing of large structures, such as bridges, towers, dams. At the state of the art, the equipment has been

demonstrated able to follow the displacements of a bridge provided that vertical bending movements

are predominant with respect to torsional ones. In this paper, the authors propose and experimentally

test an advancement of the interferometric technique able to overcome the previous assumption and to

retrieve the effective displacement of each single section of a bridge in case of both vertical bending and

torsional movements.

& 2009 Elsevier Ltd. All rights reserved.

1. Introduction

Dynamic tests of bridges, routinely carried out in order tovalidate structural specifications, or to provide diagnostic surveysfor planning maintenance and repair, are usually carried out usingnetworks of accelerometers. These sensors are accurate andreliable, but need to be positioned in contact with the surveyedstructure, and their placement is a time consuming and oftenunpractical operation. For these reasons, the use of non-contactsensors should be potentially of great interest. Laser sensors canbe used for remote monitoring of structures, but they can detectthe displacement of a single selected point, whereas detection ofthe whole deformation pattern is often required. Furthermore,laser sensors can result unpractical for in-field applications,because they are very sensitive to dust and abrupt changes ofatmospheric conditions [1]. A microwave sensor for non-contactdynamic measurement of the vibration of a structure, withoutimaging capability, was described by Farrar et al. [2]. Recently, theauthors designed and implemented an interferometric radar ableto provide dynamic images with sampling rate high enough totrack the movements of architectural structures, such as bridges[3] and towers [4,5]. This radar has been proven effective fordetecting the bending or rocking modes of the structures undertest, as for this kind of movement all the physical points at thesame distance from the radar can be considered moving

ll rights reserved.

M. Pieraccini).

coherently. The aim of this paper is to study the case when thestructure is moving with torsional modes too. This movementcannot be detected only by the phase of the interferometric signalbecause it involves also a signal amplitude modulation. Never-theless, it is demonstrated that significant information abouttorsional modes can be extracted by using both amplitude andphase information of the radar signal in addition to a radar crosssection (RCS) map of the structure.

2. The radar technique

The interferometric sensor is basically a coherent radartransceiver, operating at microwave frequency, based on thewell-known principle of electromagnetic interference betweencoherent waves.

The radar operates by transmitting a set of N continuous wavesat frequencies incremented by a constant step Df, according to atechnique referred as continuous-wave step-frequency (CW SF),scanning a bandwidth B ¼ NDf [3–5]. In particular, it radiates at16.75 GHz center frequency with about 400 MHz bandwidth. Theacquisition time rate can be as high as 100 Hz. Design specifica-tions on phase accuracy are suitable for measuring displacementswith accuracy better than 0.1 mm in operative conditions.

As the equipment is a coherent radar, phase information ispreserved. Thus, considering a single target, a displacement Dr

along the radar line-of-sight is detected by the related phase shiftDj impressed to the reflected signal according to the following

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D. Dei et al. / NDT&E International 42 (2009) 741–747742

equation:

Dr ¼c

4pfcDf ð1Þ

with fc central frequency. The calculated interferograms betweensubsequent radar images give Dj, pixel by pixel, providing mapsof the displacements Dr in the illuminated scenario.

The radar can be operated in two different modalities:providing only range resolution or providing both range andcross-range resolution.

2.1. Range resolution modality

When the radar front is not moving, for example is justmounted on a tripod, the equipment is capable of only rangeresolution, i.e. it is unable to distinguish objects into the sameresolution cell (see Fig. 1). The inverse discrete Fourier transform(IDFT) of the received CW signals provides a radar trace withrange resolution DR given by

DR ¼c

2Bð2Þ

with c speed of light.Caution must be paid about possible range measurement

ambiguity due to the CW SF technique. As the signal bandwidth B

is sampled at frequency step interval Df, the sampling theoremstates that unambiguous measurements can be carried out only ifall detectable targets are closer than a maximum range given bythe following equation:

Rmax ¼c

2Dfð3Þ

Thus, the frequency step Df must be chosen according to thedesired unambiguous range.

2.2. Range and cross-range resolution modality

In order to obtain cross-range resolution, the radar front-endhas to be operated as a synthetic aperture radar (SAR). At this aim

Fig. 1. Targets within the antenna cone are represented by peaks in the radar range

map. Those within the same range resolution cell, contribute to the same peak so

that they cannot be distinguished independently from their looking angles.

the radar front-end can be mounted on a linear mechanical guide,along which the antennas can be moved step-by-step withsubmillimetric accuracy (see Fig. 2) transmitting and receivingat incremental M positions. The cross-range resolution angle Db isobtained by the following equation:

Db ¼c

2Lfc cosðb0Þð4Þ

where L is the length of the mechanical guide and b the anglebetween the direction of the target and the direction orthogonalto the mechanical guide. Such an arrangement has been widelyused by the authors to monitor structural deformations andunstable slopes [8], but it does not permit to detect fastmovement because of the time necessary to complete the scanalong the mechanical guide. A focusing algorithm [6,7] has to beapplied in order to synthesize two-dimensional radar images.With reference to Fig. 2, let Si,k be the measured signal at the i-thfrequency step and k-th radar position on the guide. The complexvalue F of the point m of the focusing image is calculated byadding all signal contributions, taking into account the phasedepending on the distance Rk,m between the point m and the radarin the k-th position:

Fm ¼Xk;i

Si;kejð4pfi=cÞRk;m ¼X

k

ejð4pfi=cÞRk;m IDFTnðSi;kÞ ð5Þ

where fi is the i-th transmitted frequency, fmin is the lowesttransmitted frequency and IDFT is the inverse discrete Fouriertransform of the CWSF signal, transmitted at the position k. Theelement n of the IDFT vector that is closer to the geometric Rk,m

distance is assumed to be the best approximation of the signalcoming from the point m, as shown in Fig. 2, therefore it isconsidered into the sum over k index.

3. Detection of vertical bending modes

When the radar operated in only range resolution modality, theresolution cell has length equal to the range bin with an angularwidth determined by the antenna’s lobe (see Fig. 1). In the case ofcomplex targets, a resolution cell can contain several physicalscatterers. The radar response of each scatterer can be representedas a phasor, i.e. a vector with length given by the amplitude of thereflected wave and angle equal to the phase of the received wavewith respect to the transmitted one. The detected radar response

Fig. 2. Sketch of the focusing algorithm: the m-th point of the RCS map is obtained

by sum all the contributes of the k positions of the radar on the mechanical guide.

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Fig. 3. Sketch of the measurement geometry under a bridge using an interfero-

metric radar.

D. Dei et al. / NDT&E International 42 (2009) 741–747 743

is the summation of all the phasors relative to the physicalscatterers in the resolution cell. If all the scatterers of a singleresolution cell are shifted by Dr along the radar line-of-sight(LOS), as in the case of vertical bending movement of a bridge,their summation phasor, corresponding to the complex summa-tion of the received signal and can be expressed as

SðtÞ ¼ Ae�jjðtÞ ð6Þ

where the amplitude A can be assumed constant in time for smallvalues of Dr, and the rotation angle j is linearly related to Dr

according to

jðtÞ ¼ 4pfc

cðR0 þDrðtÞÞ ð7Þ

with R0 the distance that corresponds to the phase of S at timet ¼ 0.

The typical measurement arrangement for testing a bridgeusing an interferometric radar is shown in Fig. 3. In order to obtainthe radar image of the whole bridge, every section must be seen ata different range. This condition is usually fulfilled by placing theinstrument at the base of a pillar. If the lower surface of the bridgepresents discontinuities such as corners, crossbars, solders, bolts,these discontinuities act as scattering centers, producing peakecho responses. Measurement of bridge displacement is possiblejust in correspondence of these scattering points. Assuming thevertical shift Dz orthogonal to the surface, its actual value can beestimated by measuring the displacement Dr along the LOS. Byexciting the bridge in such a manner to give rise to verticalbending deformations prevalently, the interferometric radar hasbeen demonstrated very effective for detecting the static anddynamic displacements. The results have been exhaustivelyvalidated by comparison with static measurements performedthrough theodolites [6] and with dynamic measurementsobtained from accelerometers [9].

Fig. 4. Sketch of a section of the bridge. Axes orientation, Dz and Dy movements

are indicated.

Fig. 5. Flow charts of the direct and the inverse problems.

4. Detection of torsional modes: forward and inverseproblems

In case of rotation or deformation of the set of scatterers (as inthe case of the torsional movement), the phase does not followstrictly the relationship (7), and the phasor is no longer constant,but its amplitude is modulated. The basic idea of the techniqueproposed in this paper is to use both phase and amplitudeinformation to detect both vertical shift and torsional movements.

A transversal section of a bridge moves as a rigid body withtwo degrees of freedom, as shown in Fig. 4: Dz(t) is the verticaldisplacement of the center of the section, and Dy(t) is its rotationin the x–z plane. Combining these two degrees of freedom, everytorsional and vertical movement can be mathematicallyrepresented. From the point of view of the radar sensor, thesection can be modeled as a suitable number of point scattererswith different RCS. How to obtain this scatterers distribution isexplained in Section 2. By assuming that the RCS (si) and thedistance of every i-th point scatterer from radar are known, thecomplex radar response I(t)+jQ(t) of a single range resolution celldepends on Dz(t) and Dy(t) and can be calculated by adding thecontributions of all the point scatterers:

SðtÞ ¼ IðtÞ þ jQ ðtÞ ¼X

i

siejð4p=lÞRiðDzðtÞ;DyðtÞÞ ð8Þ

with si and Ri respectively RCS and distance from the radar of thei-th point scatterer. The computation of radar response from thetarget characteristics is the so-called forward problem (see Fig. 5a).In engineering practice it is more important the inverse problem

(see Fig. 5b), i.e. how to obtain Dz(t) and Dy(t) from I(t) and Q(t).

The simplest approach to the inverse problem, when the forwardone can be easily solved, is an iterative procedure that minimizesthe distance D between the measured radar response and theestimated response, by changing Dz(t̄) and Dy(t̄) at the fixed time(t̄). The distance D is calculated as

DðDzðtÞ;DWðtÞÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi�IMEASðtÞ � IEST ðDzðtÞ;DyðtÞÞ

�2þ ðQMEASðtÞ � QEST ðDzðtÞ;DyðtÞÞÞ2

r

ð9Þ

where IMEAS and QMEAS are the in-phase and quadraturecomponents of the measured signal and the IEST and QEST arecalculated by solving the forward problem. The starting solution

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Fig. 6. The radar equipment: the electronic core and the antennas installed on the rail moving system.

Fig. 7. Sketch of the measurement arrangement and information about the bridge dimensions. The radar is placed on the basement of a pillar and the two accelerometers

are fastened at the edges of a section of the bridge.

D. Dei et al. / NDT&E International 42 (2009) 741–747744

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D. Dei et al. / NDT&E International 42 (2009) 741–747 745

(Dzstart, Dystart) is obtained calculating Dzstart directly from thephase relationship (7) and searching Dystart by minimizingD(Dzstart, Dy). The iterative method is described by the followingequation:

Dzðnþ 1Þ

Dyðnþ 1Þ

" #¼

DzðnÞ

DyðnÞ

" #�

gzðnÞ

gyðnÞ

" # @DðDzðnÞ;DyðnÞÞ@Dz

@DðDzðnÞ;DyðnÞÞ@Dy

2664

3775 ð10Þ

The iteration stops when a relative minimum of D is found. Theindex n41 represents the incremental step of the processing andthe (gz, gy) couple is a positive constant small enough compatiblywith a reasonable processing time. Both the forward and inverseproblems can be solved if the point scatterers model of eachsection of the bridge is known. For this aim, a two-dimensionalradar image of the bridge, as viewed by the radar, has to beacquired. In order to obtain this image, the radar has beenconverted into a SAR by moving the front-end along a mechanicalguide transversal to the LOS. The resulting image is a distributionof peaks in the bridge plane (x–y plane of Fig. 4), where each peakcan be modeled as a single scatterer. The lowest peaks aredischarged by setting a suitable threshold in amplitude.

Fig. 9. (a) Two-dimensional image of the whole bridge; (b) zoom of the section

under test and highlighting of the selected peaks used as point scatterers into the

processing.

5. Experimental results

5.1. Measurement arrangement

In order to verify the technique described above, an experi-mental campaign was carried out on a cable-stayed steelpedestrian bridge, located in the town of Poggibonsi (Siena, Italy).Eight pairs of stay-cables, fixed to two pairs of inclined piers,symmetrically support the steel bridge, which has a width ofabout 3.4 m and a central span of 45 m. The deck exhibits a set oftransversal steel beams every 2.5 m, which supports a corrugatedsteel plate, thus it appears as a composition of several highreflecting corners and edges along its whole lower surface. TheSAR equipment was mounted on the basement of a pier, in such amanner that the central position of the mechanical guide was atthe center of the bridge width (Fig. 6). A couple of accelerometerswas placed at the two extremities of the bridge section about 13 mapart from the radar. A sketch of the bridge and the instrumentsarrangement is shown in Fig. 7. Integrating and combining theinformation of the accelerometers 1 and 2, vertical bending (Dz(t))and torsional (Dy(t)) movements has been obtained. These datahave been used in the forward problem and also for comparingthe results of the inverse problem.

Fig. 8. Radar range map of the signal backscattered by t

5.2. Dynamic measurement

The radar was placed at the central position of the guide, insuch a manner to see the major part of the bridge extension, and itwas set to generate 920 equally spaced frequency tones samplinga 300 MHz bandwidth. This setting provides an unambiguousrange of 460 m (see Eq. (2)), and a, range resolution of 0.5 m (seeEq. (1)), adequate to separate the backscattered signals comingfrom the different transversal steel beams of the bridge. Thefrequency sweep was repeated every 10 ms, providing an imagerate of 100 Hz. The accelerometers acquisition chain (acceler-ometers 393C and amplifier 482A20, produced by PCB Piezo-tronics; acquisition system Spider8 produced by HBM) provided asampling rate of 300 Hz. Data were collected for 130 s by radar andaccelerometers simultaneously, while the bridge was solicited bya man walking on the bridge and by pulling by hand a stay-cable.

he bridge, projected on its longitudinal dimension.

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Using these solicitations, both vertical bending and torsionaldeflections were generated. By means of IDFT processing, theradar response of the section equipped with the accelerometerswas calculated. The amplitude versus range image of the signalbackscattered by the bridge is plotted in Fig. 8.

The IDFT of a point scatterer at a distance R from the radar is asinc function centered at R. In order to reduce the interference ofsidelobes between echoes of scatterers at different distances, thefrequency response of the scenario was weighted by a Kaiserwindow [10]. Windowing lowers the sidelobes amplitude butincreases the width of the mainlobe, thus reducing resolution. Byusing a Kaiser window with the parameter a ¼ 5.48, �40 dBsidelobes and a mainlobe width of about 2 m were obtained. Thiswidth is still adequate to separate the contributions of signalsbackscattered by different transversal beams of the bridge frame.

5.3. SAR measurement

A two-dimensional SAR image was obtained by moving theradar along a 2 m long mechanical guide at 401 steps incrementedby 5 mm. A Kaiser window (a ¼ 5.48) has been applied at the datasampled along the mechanical guide. The resulting cross-rangemainlobe width was about 12 cm at the distance of 13 m from theradar, where the accelerometers were installed. By applying thefocalization formula in Eq. (5), a two-dimensional complex RCS

Fig. 10. Comparison of modulus (a) and phase (b) between the signal measured by

radar and that estimated by means of the direct problem.

Fig. 11. Comparison of Dy (torsional movement) (a) and Dz (vertical shift) (b)

between the signal measured by accelerometers and that estimated by means of

the inverse problem, using only the radar data.

image on the plane x–y, sampled with increments Dx ¼ 2.5 mmand Dy ¼ 0.1 mm, was obtained. The modulus of the image of thewhole bridge is shown in Fig. 9a, and an enlarged image of thesection equipped with accelerometers is shown in Fig. 9b, wherethe top of the peaks, highlighted with asterisks, correspond to thepoint scatterers of that section. Thus, complex RCS andcoordinates of every scatterer are known.

5.4. Forward and inverse problem solutions

IMEAS(t) and QMEAS(t) have been obtained by radar in thedynamic measurement; DzMEAS(t) and DyMEAS(t) have beenobtained by accelerometer data; RCS and scatterers positions arededuced by SAR measurements.

The comparison between the complex signal measured by theradar and that estimated by the forward problem algorithm(shown in Fig. 5a) is visible in Fig. 10, for a time interval of 10 s.The close agreement of the signals validates the simulation model.By means of the inverse problem algorithm (shown in Fig. 5b),DzEST(t) and DyEST(t) has been calculated and compared withDzMEAS(t) and DyMEAS(t) in Fig. 11, within the same 10 s. Anestimation of algorithm convergence can be given by the D

residual that was of 0.32% of mean value of phasor amplitudealong the section. An error estimation is provided by calculatingthe standard deviation of the differences DzEST�DzMEAS and

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D. Dei et al. / NDT&E International 42 (2009) 741–747 747

DyEST�DyMEAS for all the dataset of 130 s. These errors result0.98 mm for vertical displacement and 0.13 mrad for angulartorsion. It is worth to note that error in vertical displacement is abit higher than the value that typically can be achieved by usingthe same radar technique with only bending movement (see forexample [9]). As it is reasonable, the torsion of the section in someway affects slightly the vertical displacement measurement.

6. Conclusion

This paper has reported a method to detect both verticalbending and torsional movements of the sections of a bridge usingan interferometric radar, so far used merely to detect verticaldisplacements when bending modes are predominant withrespect to torsional ones. The method works provided that atwo-dimensional RCS image of the structure is supplied. It hasbeen shown how this image can be obtained by using the radarequipment as a SAR system capable of cross-range resolution. Theeffectiveness of the technique has been demonstrated by the goodagreement with data obtained from two accelerometers placed onthe bridge. Even though the method can be not practical in somecases, because of size and weight of equipment in SAR configura-tion, the good experimental results confirm the potential of radarinterferometry also for detecting the torsional movements of abridge.

Acknowledgments

The interferometric equipment used in this work has beendesigned and constructed with the support of IDS—Ingegneria dei

Sistemi SpA, Pisa (Italy), and it remains the property of thiscompany. The collaboration of Giulia Bernardini of IDS company,for the measurements in site, is gratefully acknowledged. Finallythe authors would like thank to Saverio Giordano for his help inprocessing accelerometric data.

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