Dynamic characteristics of the Crowchild TrailBridge

Carlos E. Ventura, Tuna Onur, and Pei-Chin Tsai

Abstract: This paper presents the results of a study on the dynamic characteristics of the Crowchild Trail Bridge inCalgary, Alberta. This bridge is currently being monitored by members of the Intelligent Sensing for Innovative Struc-tures Network (ISIS Canada). The effects of various modeling assumptions on the computed dynamic characteristics ofthe bridge are investigated and discussed in this paper. The dynamic characteristics of the bridge were determined bymeans of ambient vibration measurements and were used to calibrate a number of computer models developed by fourdifferent groups of engineers. The results of this study showed that in order to properly calibrate a model of the bridgefor structural dynamic analysis, it is necessary to have not only a good match of experimental and analytical naturalfrequencies but also a good match of experimental and analytical mode shapes. Improper determination of dynamiccharacteristics of a bridge using analytical models could lead to erroneous conclusions on its expected behaviour underdynamic loading.

Key words: ambient vibration testing, dynamic characteristics, bridge modeling, instrumentation, accelerometers, modalanalysis.

Résumé: Cet article présente les résultats d’une étude sur les caractéristiques dynamiques du pont Crowchild Trail àCalgary, Alberta. Ce pont est présentement sous la surveillance de membres de l’Intelligent Sensing for InnovativeStructure Network (ISIS Canada). Les effets de différentes hypothèses de modélisation pour le calcul des caractéristi-ques dynamiques du pont sont examinés et discutés dans cet article. Les caractéristiques dynamiques du pont ont étédéterminées au moyen de mesures de vibrations ambiantes et ont été utilisées pour calibrer plusieurs modèles de calculdéveloppés par quatre groupes d’ingénieurs. Les résultats de cette étude ont montré que, pour proprement calibrer unmodèle du pont pour une analyse dynamique structurale, il est nécessaire non seulement d’avoir une bonne correspon-dance des fréquences naturelles expérimentales et analytiques, mais aussi une bonne correspondance des formes desmodes expérimentaux et analytiques. Une détermination inadéquate des caractéristiques dynamiques d’un pont par lebiais de modèles analytiques pourrait aboutir à de fausse conclusions sur son comportement attendu sous un charge-ment dynamique.

Mots clés: examen de vibrations ambiantes, caractéristiques dynamiques, modélisation de pont, instrumentation, accélé-romètres, analyse modale.

[Traduit par la Rédaction] Ventura et al. 1056

Introduction and background

The objective of Theme 3 “Intelligent processing and re-mote monitoring” of the Intelligent Sensing for InnovativeStructures Network (ISIS Canada) is to develop the technol-ogy and software necessary to intelligently process andtransmit, via telephone or satellite, the structural sensing in-formation provided by integrated sensors. To accomplishthis, several research programs are being conducted by ISISmembers participating in this theme. One of these is the field

assessment of the behaviour of simply supported and contin-uous steel-free bridge decks. This paper presents the resultsof dynamic field testing studies conducted at one of thesebridges together with a comparison of the test results withanalytical models of this bridge.

The steel-free fibre-reinforced concrete (FRC) deck hasbeen used in several bridges in Canada (Bakht and Mufti1998). The main idea of using steel-free FRC deck inbridges is to solve the deterioration and corrosion problemsof steel reinforcement in a traditional concrete slab-on-girderdeck, which may be exposed to de-icing or marine-type en-vironment. The behaviour of a steel-free FRC deck bridgecan be idealized as an arching action, in which the steel-freeFRC deck acts as a compression member and the steel strapsbeneath the deck act as tension members. The removal of thesteel reinforcement inside a deck is made possible by addingtension steel straps beneath the deck as well as adding shortrandomly distributed polypropylene fibres reinforcement inthe concrete deck (Newhook and Mufti 1995). The straps areregularly spaced, typically 1 or 1.2 m, across the tops of ad-jacent girders to provide transverse restraint. Polypropylene

Can. J. Civ. Eng.27: 1046–1056 (2000) © 2000 NRC Canada

1046

Received August 9, 1999.Revised manuscript accepted January 5, 2000.

C.E. Ventura,1 T. Onur, and P.-C. Tsai. Department ofCivil Engineering, The University of British Columbia,2324 Main Mall, Vancouver, BC V6T 1Z4, Canada.

Written discussion of this article is welcomed and will bereceived by the Editor until February 28, 2001.

1Author to whom all correspondence should be addressed(e-mail: ventura@civil.ubc.ca).

fibres, in general, are mixed in the concrete to control theshrinkage and temperature cracks and also to provide somepost-crack ductility to the brittle concrete slab (Newhookand Mufti 1996; Newhook 1997). When a concentrated loadis applied in between the girders, the transverse forces aretrying to push the girders away from their positions in thetransverse direction. If the lateral restraint is sufficient to re-sist the transverse forces, an arching action takes place.

Many researchers, such as Newhook and Mufti (1995),have shown that it is an effective and alternative method tosolve corrosion problems of the steel reinforcement in a con-ventional deck. A steel-free FRC deck not only solves corro-sion problems in a conventional deck, but also has thefollowing advantages (Bakht and Mufti 1996): (a) increasedshear punching strength of the deck, (b) low maintenancecost and easy replacement of steel straps during the bridgeservice life, (c) reduced life-cycle costs of the deck bridge,and (d) reduction of labour costs during the placement ofdeck reinforcement meshes.

Recent studies on the concept of steel-free bridge deckcan be grouped into two main areas: dynamic behaviour andstatic behaviour. Research on dynamic behaviour of steel-free decks includes the work by Zhu et al. (1996), who ana-lyzed the dynamic behaviour of a bridge with steel-free FRCdeck by using the finite element method; Black et al. (1997)and Black and Ventura (1999) conducted ambient vibrationtests on two steel-free bridge decks to identify the dynamiccharacteristics of each bridge; Selvadurai and Bakht (1995)performed rolling wheel load tests on a deck to investigateits fatigue capacity; and Taing et al. (1999) performed dy-namic measurements under passing heavy truck on a steel-free bridge deck in Alberta to estimate the dynamic amplifi-cation factor and damping coefficient. Research on the be-haviour of steel-free bridge deck under static loadingincludes the work by Sargent et al. (1999), Taing et al.(1999), Afhami et al. (1998), Bakht and Mufti (1996),Newhook and Mufti (1996), Thorburn and Mufti (1995), andMufti et al. (1993), who have placed emphasis on load shar-ing over girders, temperature effects on strains, punchingshear strength study, optimization research, and static finiteelement method analysis. Newhook and Mufti (1995) devel-oped a rational model for predicting the behaviour of later-ally restrained concrete bridge decks without internalreinforcement. Wegner and Mufti (1994) developed a non-linear finite element analysis model to predict the failureload of polypropylene FRC deck slabs, and Hearn andGabrielli (1998) and Mufti et al. (1999) have investigated theformation of longitudinal cracks in existing bridges withsteel-free decks.

It is clear that significant laboratory and field work hasbeen, and is being, conducted on steel-free bridge decks, andvaluable information and lessons have been obtained fromsuch investigations. Theories and techniques are generallyproved or disproved first under idealized conditions in thelaboratory and are eventually validated by field testing of ex-isting structures that have been designed and constructedmaking use of these theories and techniques. No matter howmuch work is done in the laboratory, it is nearly impossible,and economically impractical, to replicate real life condi-tions that would occur during the lifetime of an existing

structure. Field testing and monitoring of existing structuresprovides valuable information that can be used to verify re-sults from analytical or laboratory studies. The emphasis ofthis paper is to present an example on how field test resultscan be used to assess the effectiveness of current bridgemodeling techniques.

Scope and objectives

A comparison of the dynamic properties of the CrowchildTrail Bridge in Calgary, Alberta, determined from ambientvibration measurements and several analytical models cre-ated by various engineers is presented and discussed. Thedynamic properties investigated include the natural frequen-cies and corresponding mode shapes in the vertical, trans-verse, and torsional degrees of freedom of the bridge. Theexperimental and analytical frequencies for each of the mod-els considered are compared. The comparison and correla-tion of the mode shapes are performed by making use ofmodal assurance criterion (MAC), modal scaling factors(MSF), and coordinate modal assurance criterion (COMAC).The models studied in this paper are 2-D and 3-D models ofthe bridge that were created utilizing commercially availablecomputer programs. The details and different aspects ofthese models are presented and discussed.

Description of the bridge

The University Drive/Crowchild Trail Bridge (Fig. 1) is a90-m three-span overpass located in Calgary, Alberta, whichcarries the southbound traffic of the Crowchild Trail overUniversity Drive. The original bridge was not capable of car-rying the load of increasing traffic and hence it was rebuiltreplacing the original deck with a 185 mm thick, steel-free,polypropylene fibre reinforced concrete slab. The re-construction of the bridge was completed in mid-August1997. The 9030 mm wide deck is supported by five 900 mmdeep steel girders with four cross beams in each span (Blacket al. 1997). Additional details of this bridge and of the per-manent instrumentation installed there are given by Afhamiet al. (1998) and Taing et al. (1999).

Ambient vibration measurements

In ambient vibration testing, the response of a structure toexcitations due to wind, nearby traffic, and human activity ismeasured. There is no controlled external force applied tothe structure. The duration of the measurements and the lo-cation of the instruments are arranged so as to ensure that allthe modes of interest are properly recorded. The first step inthe process of identification of modal properties of an exist-ing structure from ambient vibration measurements is thedetermination of all the possible natural frequencies of themodes participating in the vibration of the structure. A fastand effective technique to accomplish this involves the com-putation of averages of the spectral densities of all the ambi-ent vibration records obtained in each direction of interest.To this end, a spectral densities normalization and averagingfunction introduced by Felber (1993) and denoted as aver-aged normalized power spectral density (ANPSD) can be ef-fectively used to identify the most probable natural

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frequencies of a structure. The ANPSD permits a convenientway to display, in a single plot, the most significant frequen-cies existing in a series of recorded ground motions at a cer-tain direction. However, not all the peaks necessarilycorrespond to a natural frequency. Therefore, to confirmwhether or not a peak of the ANPSD is associated with anatural frequency, additional tools such as the transfer func-tion, coherence, and phase between pairs of records are re-quired for the system identification analysis. Also, a naturalfrequency may be in the vicinity of a peak rather than beingprecisely at the peak. This is because of the presence of

damping in the system, as well as the effect of averagingmultiple records.

The test on University Drive/Crowchild Trail Bridge wasconducted on Friday, August 15, 1997, by a team of re-searchers from The University of British Columbia (UBC).Since the test was performed before the re-constructedbridge was opened to the public, there was no traffic on thebridge. Therefore, the bridge vibrations were due to thewind, human activity, and traffic flowing below and besidethe bridge. The maximum levels of recorded vibration in thevertical and transverse directions were 1.7 × 10–3 g. Eight

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1048 Can. J. Civ. Eng. Vol. 27, 2000

Fig. 1. Overall view of the University Drive/Crowchild Trail Bridge, Calgary, Alberta.

Fig. 2. Sensor locations and directions of ambient vibration measurements: (a) plan view and (b) elevation view.

accelerometers with a total of 14 setups were used to capturethe natural frequencies and mode shapes. The 46 locationschosen for the measurements are shown in Fig. 2. Two ac-celerometers for the first eleven setups and three accelerom-eters for the remaining three setups were used as referencesensors. The filter cutoff was set to 50 Hz, which is just be-low the natural frequency of the accelerometers. Besidesbase line corrections and linearization, no other filtering wasperformed on the signals. The frequency range of interestwas 0–20 Hz. The tests were recorded at a sampling rate of100 samples per second per sample. The custom data acqui-sition program AVDA (Schuster 1994) was used to recordthe ambient vibrations, and the computer programs P2, U2,and V2 (EDI 1997) were used to identify the natural fre-quencies and mode shapes of the structure. Program P2 wasused to compute the ANPSD for the ambient vibration re-cords (Felber 1993). The ANPSD plots for the torsional,vertical, and transverse directions are presented in Fig. 3.Program U2 was used to view signals quickly in order to de-cide whether the data obtained were satisfactory or not. Itwas also used to calculate individual power spectral densi-ties and the potential modal ratios needed for the frequencyand mode shape estimations. Program V2 was developed inconjunction with program U2 to illustrate and animate modeshapes obtained from ambient vibration data (Black et al.1997).

Sixty-four records of vertical ambient vibration measure-ments were used to estimate the natural frequencies andmode shapes in the vertical direction, 40 records in thetransverse direction and seven records in the longitudinal di-rection. The torsional response was estimated from the dif-ference in the vertical motions obtained from opposite sidesof the deck, assuming the deck to be rigid. The power spec-tral densities of the measurements were averaged to producethe ANPSD. The transfer function, coherence, and phasewere determined from the records obtained to confirmwhether or not the peaks in the ANPSD corresponded to nat-ural mode shapes or to a mode of vibration at that frequency.Damping values were not evaluated, because the techniquesfor estimation of damping from ambient vibration recordsproduce unreliable results.

The natural frequencies obtained at the end of these pro-cesses and the description of the modes are given in Table 1.The corresponding mode shapes for some selected modesare shown in Fig. 4.

Analytical models

The models described in this section were developed byfour different groups of engineers with various levels ofbridge design experience. All groups created at least one 2-D and one 3-D model. Although only one 3-D model fromeach group is described in detail, results from selected 2-Dmodels are also presented. Views of the models developedby each group are shown in Fig. 5.

Engineer group 1The structural analysis software SAP90 was used by

Group 1 to prepare and analyze the models. The modulus ofelasticity for steel was taken to be 200 000 MPa. For con-crete in deck it was taken to be 28 288 MPa, which corre-sponds to 80% of the nominal design value. For concrete inpiers it was taken to be 14 790 MPa, which corresponds to50% of the nominal design value. The Poisson ratio for con-crete was taken to be 0.2. The models discussed here are(a) Model 2 (E1M2), a 2-D model with beam elements andlumped mass (lumped at 18 nodes), and (b) Model 4(E1M4), a 3-D model with shell and beam elements (shellelements for deck and piers; beam elements for girders).

The properties of Model 4 are summarized below:— Support conditions: The abutments and the column

footings were modelled as pinned and fixed supports,respectively. Bearings were not modelled.

— Girders: The steel girders were modelled as beam ele-ments with composite action between the deck and thegirders. The effective stiffness of the beam element wastaken as the summation of all five girders, including thestiffening effect of parapets.

— Deck and piers: The deck and the piers were modelledwith shell elements. The slab thickness was taken to be150 mm.

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Fig. 3. Averaged normalized power spectral densities in torsional, vertical, and transverse directions.

Engineer group 2The structural analysis software SAP2000 was used by

Group 2 to prepare and analyze the models. The modulus ofelasticity was taken to be 200 000 MPa for steel and35 355 MPa for concrete in deck and piers. The Poisson ra-tios of 0.3 and 0.2 were used for steel and concrete, respec-tively. The unit weights of steel and concrete were taken to

be 23.5 and 77.0 kN/m3, respectively. The models discussedhere are (a) Model 2 (E2M2), a 2-D model with beam ele-ments and distributed mass, and (b) Model 4 (E2M4), a 3-Dmodel with shell and beam elements (shell elements fordeck and piers; beam elements for girders).

The properties of Model 4 are summarized below:— Support conditions: The abutments were modelled as

roller supports except at the centerline of each abut-ment, where they were modelled as pinned. The pierfootings were modelled as pinned supports. Bearingswere modelled as vertical beam elements.

— Girders: Girders were modelled as beam elements act-ing compositely with the deck.

— Deck and piers: The deck and piers were modelled us-ing shell elements (an equivalent thickness for thehaunched slab was calculated to be 240 mm).

Engineer group 3The structural analysis software SAP90 was used by

Group 3 to prepare and analyze the models. The modulus ofelasticity was taken to be 200 000 MPa for steel and 42 000and 27 400 MPa for concrete in deck and piers, respectively.The Poisson ratio for concrete was taken to be 0.25. The unitweight of steel was taken to be 77.0 kN/m3, and for concretein deck and piers the unit weights were taken to be 22.5 and23.5 kN/m3, respectively. The three models discussed hereare (a) Model 1 (E3M1), a 2-D model with beam elementsand lumped mass (lumped at 5 nodes), (b) Model 2 (E3M2),a 2-D model with beam elements and distributed mass, and(c) Model 3 (E3M3), a 3-D model with shell and beam ele-ments (shell elements for deck and piers; beam elements forgirders).

The properties of Model 3 are summarized below:— Support conditions: North abutment and the pier foot-

ings were modelled as fixed supports. South abutmentwas modelled as pinned support. Bearings were notmodelled.

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1050 Can. J. Civ. Eng. Vol. 27, 2000

MODEFrequency(Hz)

Period(s) Description

1 2.78 0.3597 Fundamental vertical mode (1st vertical)2 3.13 0.3195 Fundamental torsional mode (1st torsional)3 3.76 0.2660 Coupled mode (vertical (2nd vertical*)/

torsional (2nd torsional*))4 4.05 0.2469 Torsional mode (3rd torsional)5 4.64 0.2155 Vertical mode (3rd vertical)6 5.18 0.1931 Coupled mode (torsional (4th tor-

sional*)/transverse (lst transverse*))7 7.13 0.1403 Coupled mode (torsional (5th torsional*)/

transverse (2nd transverse*))8 9.13 0.1095 Vertical mode (4th vertical*)9 10.74 0.0931 Torsional mode (6th torsional)

10 12.84 0.0779 Fundamental transverse mode(6th transverse)

11 15.77 0.0634 Torsional mode (7th torsional*)12 17.68 0.0566 Vertical mode (5th vertical)13 19.28 0.0519 Torsional mode (8th torsional)

Note: The asterisk (*) indicates the presence of modal coupling between vertical, horizontal, andtorsional modes.

Table 1. Summary of modes determined from measurements.

Fig. 4. Plan, elevation, and 3-D views of selected mode shapesobtained from the measurements.

— Girders: Girders were beam elements with no compos-ite action with the deck.

— Deck and piers: The deck was modelled using shell ele-ments with uniform thickness (185 mm). Piers werealso modelled as shell elements.

Engineer group 4The finite element analysis software ANSYS (version 5.3)

was used by Group 4 to prepare and analyze the models.Since the input and output files for the models are unavail-able, these models are discussed using the material availablein the group report. The modulus of elasticity was taken tobe 200 000 MPa for steel and 35 000 MPa for concrete indeck and piers. The Poisson ratios used for steel and con-crete were 0.3 and 0.2, respectively. The unit weights ofsteel and concrete were taken to be 77.0 and 23.0 kN/m3, re-spectively. The models that were prepared by this group are(a) Model 1 (E4M1), a 2-D model with beam elements anddistributed mass and (b) Model 2 (E4M2), a 3-D model withshell and beam elements (shell elements for deck and piers;beam elements for girders).

The properties of Model 2 are summarized below:— Support conditions: The abutments were modelled as

pinned supports.— Girders: The steel girders were modelled as beam ele-

ments with no composite action.— Deck and piers: Deck was modelled as shell elements

of uniform thickness (185 mm). The piers were notmodelled, and pinned supports were introduced instead.

Comparison of natural frequencies

The natural frequencies obtained from the 2-D lumpedmass models (E1M2 and E3M1) are compared with those

obtained from the ambient vibration measurements inTable 2. Only the frequencies for the first 10 modes are pre-sented, since none of the 2-D lumped mass models couldpredict higher modes. Similarly, the frequencies obtainedfrom the 2-D distributed mass models (E2M2, E3M2, andE4M1) and 3-D models (E1M4, E2M4, E3M3, and E4M2)are compared with those from the ambient vibration mea-surements in Tables 3 and 4, respectively. In Table 3, modes2, 4, 9, 11, and 13 are not presented, and in Table 4, mode11 is not presented, because the associated models (2-D dis-tributed mass models in Table 3 and 3-D models in Table 4)could not predict those modes. The natural frequencies ob-tained from the 3-D models are also compared graphically inFig. 6.

Mode shape comparison and correlationmethods

The mode shapes obtained from computer models are of-ten checked against the on-site measurements (usually theambient vibration measurements, as they are easy to obtain)to be able to rely on them in more detailed analyses, such asresponse spectrum analysis and linear or nonlinear time his-tory analyses. There are several comparison techniquesamong which modal assurance criterion (MAC), coordinatemodal assurance criterion (COMAC), and modal scaling fac-tors (MSF) will be discussed here. The first two of these in-dices quantify the correlation of the mode shapes betweendifferent analytical models and measurement results. Forthese indices, a value of 1.0 indicates that the two modes areperfectly correlated, whereas a value less than 0.05 generallyimplies that there is no correlation between the two modes.The MSF represents the slope of a best-fit line; therefore avalue of 1.0 indicates a perfect agreement, and the correla-

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Fig. 5. Views of 2-D and 3-D computer models considered in this study.

Mode

1 2 3 4 5 6 7 8 9 10

Model E1M2 2.77 — 3.92 — 5.47 5.76 7.96 — — 12.47Model E3M1 2.29 — 2.95 — 3.06 8.16 14.65 — — 24.87Measured 2.78 3.13 3.76 4.05 4.64 5.18 7.13 9.13 10.74 12.84

Table 2. Frequencies (Hz) for 2-D lumped mass models.

tion gets worse as the values diverge from 1.0. The advan-tage of these three correlation methods is that they are easyto apply, since they do not require the mass or stiffness ma-trix of the structure.

The modal assurance criterion (MAC) is commonly usedto correlate analytical and experimental mode shape vectors,while it can also be used to compare two sets of analyticalmode shape vectors. It provides a qualitative way of assess-ing the correspondence between two mode shapes. In gen-eral, a MAC value of greater than 0.80 was considered agood match and a MAC value of less than 0.40 was consid-ered as a poor match. The vector representation of MAC canbe formulated as

[1]| |

MACA

TX

AT

A XT

X

=×

× × ×

{ } { }

{ } { } { } { }

F F

F F F F

i j

i i j j

2

where {FA} i is the analytical mode shape vector for modeiand {FX} j is the experimental mode shape vector for modej(Blaschke and Ewins 1997).

The coordinate modal assurance criterion (COMAC) indi-cates the correlation between the mode shapes at a selectedmeasurement point on the structure. Before using theCOMAC, other correlation tools should be used to identifythe pre-correlated mode pairs. After constructing a set ofLcorrelated mode pairs, a correlation value is calculated foreach coordinate over all correlated mode pairs as follows:

[2]

| |COMAC( )

A X

A

i

ir irr

L

r

L

ir

=

×æ

è

çç

ö

ø

÷÷

×

=

=

å

å

{ } { }

{ } {

F F

F

1

2

1

2

r

L

ir=å

1

2FX}

where {FA} ir is the element of the analytical mode shapevector in correlated mode pairr at measurement locationi.Similarly {FX} ir is the element of the experimental modeshape vector in correlated mode pairr at measurement loca-tion i (Blaschke and Ewins 1997).

The modal scale factor (MSF) is the slope of the beststraight line through the data points of two correlated modes.For perfectly correlated data, this slope is 1.0.

[3] MSF AT

X

AT

A

={ } { }

{ } { }

F F

F Fi j

i j

where {FA} is the analytical mode shape vector for modeiand {FX} is the experimental mode shape vector for modej(Blaschke and Ewins 1997).

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1052 Can. J. Civ. Eng. Vol. 27, 2000

Mode

1 2 3 4 5 6 7 8 9 10 12 13

Model E1M4 2.78 2.72 3.56 — 4.57 6.20 8.05 — — 12.48 — —Model E2M4 2.78 3.38 3.72 4.05 4.65 6.13 7.58 9.71 9.80 — — —Model E3M3 2.59 2.73 3.65 3.91 4.21 4.42 7.91 7.99 8.24 — 9.95 13.77Model E4M2 2.49 3.48 4.65 5.11 5.46 6.03 6.16 9.69 9.91 10.80 — —Measured 2.78 3.13 3.76 4.05 4.64 5.18 7.13 9.13 10.74 12.84 17.68 19.28

Table 4. Frequencies (Hz) and errors (%) for 3-D models.

Mode

1 3 5 6 7 8 10 12

Model E2M2 3.69 4.82 6.12 — — 12.75 — 14.72Model E3M2 2.15 2.87 3.00 6.93 14.67 9.27 28.22 —Model E4M1 5.40 7.62 10.62 — — 21.25 — 26.61Measured 2.78 3.76 4.64 5.18 7.13 9.13 12.84 17.68

Table 3. Frequencies (Hz) for 2-D distributed mass models.

Fig. 6. Comparison of experimental and analytical natural fre-quencies.

Correlation of mode shapes

The mode shapes obtained from the four 3-D models(E1M4, E2M4, E3M3, and E4M2) are summarized in Fig. 7for the first three modes.

The MAC values for these models versus the measure-ments are presented in Table 5 and Fig. 8. For the 3-Dmodel developed by Group 4 (E4M2), only the first twomode shapes were available. The MSF values computed forthe two selected 3-D models (E2M4 and E3M3) are pre-sented in Table 6 and Fig. 9. The COMAC values were also

computed for these two models (Fig. 10). For E2M4, onlythe first three modes were considered in the calculation ofCOMAC, since the MAC value for mode 4 is quite low. ForE3M3, although the first six modes have good correlations,the best COMAC values were obtained when only the firsttwo modes were considered.

Discussion and conclusions

Comparison of the natural frequencies obtained from 2-Dand 3-D models with those obtained from the ambient vibra-

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Fig. 7. Plan, elevation, and 3-D views of selected mode shapes obtained from the computer models.

Mode

1 2 3 4 5 6 7 8 9 10 11 12 13

Model E1M4 0.871 0.249 0.913 — 0.902 0.584 0.649 — — 0.012 — — —Model E2M4 0.876 0.949 0.908 0.578 0.001 0.004 0.001 0.000 0.016 — — — —Model E3M3 0.873 0.936 0.888 0.865 0.907 0.882 0.160 0.599 0.245 — — 0.167 0.151Model E4M2 0.906 0.807 na na na na na na na na — — —Measured 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Table 5. Summary of the modal assurance criteria for 3-D models.

tion measurements clearly shows the superiority of 3-Dmodels in predicting the dynamic properties. The torsionalmodes or torsional components of vertical modes obviouslycannot be captured by 2-D models, and the frequencies ofthe other modes have a relatively poor match with the mea-sured frequencies. Introducing additional nodes in a lumpedmass model improves the correlation of experimental andanalytical frequencies considerably. In the 3-D models, ver-tical modes are the most accurately captured modes in termsof both mode shapes and frequencies. The first few torsionalmodes are also captured reasonably well. The transversemodes have the poorest match with the measurement results.Modes 9 through 13 have very low correlations in modeshapes and high errors in natural frequencies. It should benoted that higher modes are also difficult to capture by mea-surements, since the parameters are generally set to measurelow-frequency motions.

The mode shape correlation indices (MAC, COMAC, andMSF) were calculated using only the vertical componentsfor all modes. Making use of all the components of the cal-culated analytical and experimental modes would possiblyprovide a better correlation for the higher modes (especiallythe transverse modes). This is particularly important forCOMAC, since it correlates the coordinates of the modeshapes.

The MAC analysis results definitely bring more insight toevaluating the correlation of the modes. A close match offrequencies does not always assure a good match of themode, as the mode shape might be practically uncorrelated.This is the case for a number of modes; for example, al-though the 4th mode in E2M4 has only 6.3% error in the

natural frequency, it has a mode shape correlation of 0.022(less than 5%, i.e., uncorrelated).

The MSF results are in good agreement with the MAC re-sults. The correlation of the first three mode shapes fromE2M4 is high, but tends to decrease in the higher modes interms of both MAC and MSF. Similarly, for E3M3, bothMAC and MSF values indicate strong correlation for thefirst six modes and lower correlation in the higher modes.

The COMAC values are difficult to interpret when higher(torsional and transverse) modes are also included, since it isa coordinate correlation tool and only vertical components ofthe mode shapes were utilized. For the first two or threemodes (vertical and torsional modes), COMAC values arehigh, whereas when the higher modes (vertical, torsional, ortransverse) are considered, the COMAC values drop signifi-cantly. Therefore, to make use of COMAC properly, all com-ponents of the mode shapes should be utilized.

In some cases, a good match in frequency does not neces-sarily correspond to a good match in the mode shape. There-fore, the use of a mode shape correlation technique inaddition to the comparison of natural frequencies is essentialin the calibration of analytical models with experimentaldata.

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1054 Can. J. Civ. Eng. Vol. 27, 2000

Mode

1 2 3 4 5 6 7 8 9 10 11 12 13

Model E2M4 1.522 1.387 1.068 0.170 –0.054 0.050 –0.055 –0.069 0.082 — — — —Model E3M3 0.609 0.777 0.641 0.806 0.910 0.661 0.297 1.806 0.686 — — 0.859 0.538Measured 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0 1.0

Table 6. Summary of the modal scaling factors for selected 3-D models.

Fig. 8. Comparison of the modal assurance criteria for 3-D mod-els.

Fig. 9. Comparison of the modal scaling factors for the selected3-D models.

Acknowledgements

The authors are members of the Intelligent Sensing for In-novative Structures Network (ISIS Canada) and wish to ac-knowledge the support of the Networks of Centres ofExcellence Program of the Government of Canada and theNatural Sciences and Engineering Research Council. The fi-nancial support of the British Columbia Ministry of Trans-portation and Highways (MOTH) for the instrumentationand monitoring of Waterloo Creek Bridge is also acknowl-edged with thanks. Special thanks are also extended toMr. Peter Brett and Mr. Ron Mathieson of MOTH and toDr. Aftab Mufti, Dr. Roger Cheng, and Dr. Gamil Tadros ofISIS Canada for their interest and collaboration in carryingout the studies reported here. Mr. Cameron Black, formerM.A.Sc.student at The University of British Columbia par-ticipated in the field test and was responsible for analyzingthe ambient vibration data.

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Fig. 10. Comparison of the coordinate modal assurance criteria for the selected 3-D models.

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