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Investigation of dynamic cable-deck interaction in a physical model of a cable-stayed bridge. Part I

*Correspondence to:A.Cunha,Faculty of Engineering,University of Porto,Rua dos Bragas,4099Porto Codex,Portugal.

EARTHQUAKE ENGINEERING AND STRUCTURAL DYNAMICS

Earthquake Engng Struct .Dyn .2000;29:499}521

Investigation of dynamic cable }deck interaction in a physical

model of a cable-stayed bridge.Part II:seismic response

E.Caetano ,A.Cunha *and C.A.Taylor

Faculty of Engineering of Uni v ersity of Porto,Rua dos Bragas,4099Porto Codex,Portugal Earthquake Engineering Research Centre,Uni v ersity of Bristol,Queen +s Building,Uni v ersity Walk,Bristol BS81TR,U.K.

SUMMARY

The present paper describes an investigation of the e !ect of dynamic cable interaction with the deck and towers in the seismic response of a cable-stayed bridge.This study involved shaking table tests performed on a physical model of Jindo bridge,in order to validate two alternative numerical models,which di !er in terms of consideration of coupled cable/deck and towers modes.The response to arti "cial accelerograms was calculated and correlated with measured data.Additional numerical simulations are presented in order to clarify the role that cables play in the attenuation or ampli "cation of the structural response.It was found that the cable interference with global oscillations may cause a decrease of the bridge response.However,this &system damping 'may not develop in the case where a narrow-band excitation is applied,causing large amplitude of vibrations of some cables,with signi "cant non-linearity,and inducing higher-order modes.Copyright 2000John Wiley &Sons,Ltd.

KEY WORDS :cable-stayed bridges;physical models;seismic response;shaking table;cable dynamics.

1.INTRODUCTION

Cables are very e $cient structural elements widely used in many large span bridges,such as cable-stayed or suspension bridges.Since they are light,very #exible and lightly damped,cable structures can always face important dynamic problems under di !erent types of loads,e.g.wind,earthquake or tra $c loads,which requires appropriate modelling,in order correctly to predict and control the structural response.

Evidence of signi "cant stay cable oscillations,sometimes conjugated with simultaneous vibra-tions of the deck,has been made by long-term monitoring of several modern bridges.Although several reasons have been adduced to justify that behaviour,such as the direct turbulent wind excitation,eventually conjugated with rain,vortex-shedding phenomena and motion of the cable supports,the mechanism behind these oscillations is not yet fully explained.

However,it is sometimes suggested that cable vibrations can play a favourable role in terms of the dynamic behaviour of cable-stayed bridges,under wind or earthquake excitations,

500 E.CAETANO,A.CUNHA AND C.A.TAYLOR

contributing to the development of an additional damping(&system damping')in the structural response.This concept was"rst introduced by Leonhardt et al.[1],who attributed this peculiar behaviour of cable-stayed bridges both to the non-linear behaviour of the cables,associated with the sag e!ect,and to the interference of cable oscillations at di!erent natural frequencies.More recently,other researchers have newly de"ned a governing cause of system damping using the concept of internal resonance[2].

The most common practice of numerical analysis of cable-stayed bridges consists in the development of a"nite element model where the cables are represented by single truss elements with equivalent Young modulus[3].Such a procedure precludes lateral cable vibrations,thus leading to a separate treatment of local and global vibrations[4],the"rst referring to transverse oscillations of a cable between"xed supports,while the second corresponds to the motion of the girder,pylon and cables as an assemblage,the cables behaving as elastic tendons.The interaction between local and global vibrations has been investigated by several researchers,such as Maeda et al.[2],Causevic and Sreckovic[5],Kovacs[6],Abdel-Gha!ar and Khalifa[7],Fujino et al.

[4],and Tuladhar and Brotton[8].Causevic and Sreckovic modelled the cables as assemblages of linear springs and masses,and stressed the importance of the non-linearity of cable behaviour that results from the closeness between a cable natural frequency and a natural frequency of the global structure.Abdel-Gha!ar and Khalifa modelled the cables using a multiple link method previously used by Baron and Lien[9],Maeda et al.[2],Yiu and Brotton[10]and Tuladhar and Brotton[8],and showed that cable vibrations a!ect the mode shapes of the deck/towers system and the corresponding participation factors.The inadequacy of using single truss elements to model the cables and the necessity of considering a convenient discretization of the cables into several"nite elements was also stressed by Tuladhar et al.[11],who concluded that the interaction between cable vibrations and deck vibrations can have a signi"cant in#uence on the seismic response of the bridge,especially when the"rst natural frequencies of cables overlap with the"rst few frequencies of the bridge.

2.OBJECTIVES OF THE STUDY

To complement the above-mentioned numerical investigations,the authors conducted an experi-mental study on an existing physical model of a cable-stayed bridge[12],the Jindo bridge(in South Korea),which was modi"ed for the purpose of studying the dynamic behaviour of the cables(Figure1).The description of this bridge and of a series of modal analysis tests performed on the model is presented in a companion paper[13].The study con"rmed the existence of interaction between the cables and the deck/towers,which in this case is characterized by the appearance of several modes of vibration with very close natural frequencies and with similar mode shape con"gurations of the deck and towers,but involving di!erent movements of the cables.The appearance of these new mode shapes proved to be conditioned by the closeness between a natural frequency of the global structure and the natural frequencies of some cables. In the present paper,the authors attempt to evaluate the importance of the dynamic cable/deck interaction in terms of the response to seismic excitations.The study involved an experimental component that consisted of a series of shaking table tests,using di!erent types of arti"cial accelerograms as input.The measured response was then used to validate"nite element models previously developed,in which the cables were idealized either as simple truss elements,or as sets of several truss elements(multiple link method).

Figure 1.Physical model of Jindo Bridge on the shaking

table.

Figure 2.Structural discretization used in the MECS model.

The analysis and comparison of the experimental and calculated responses obtained under each of the two "nite element models developed,OECS (One-Element Cable System)and MECS (Multi-Element Cable System),showed,as will be demonstrated later,the following main aspects:(i)a good correlation between the experimental and calculated responses predicted by both numerical models;and (ii)some slight di !erences between the OECS and MECS responses,which DYNAMIC CABLE }DECK INTERACTION IN BRIDGES.PART II 501

502 E.CAETANO,A.CUNHA AND C.A.TAYLOR

did not reveal however any signi"cant&system damping'e!ect for the type of excitation con-sidered.

Two di!erent numerical simulations were subsequently performed,in order to enhance and better understand this situation.The"rst consisted of modifying the natural frequency of the fundamental mode of vibration to the range of the"rst frequency of the cables.The second corresponded to the analysis of the response of the Jindo model to a severe large-amplitude base excitation de"ned in a narrow frequency range,containing both a natural frequency of the structure,and the1st frequencies of some cables.It was found that the cable interference with global oscillations may cause a signi"cant decrease of the bridge response(with regard to standard OECS analysis,where the local behaviour of the cable is not modelled).However,this &system damping'may not develop in the case where a narrow-band excitation is applied.In this case,the large amplitude of vibration of some cables may cause signi"cant non-linearity and induce higher-order modes,thus causing an increase of the response.

3.MODAL PROPERTIES OF THE PHYSICAL MODEL OF JINDO BRIDGE According to the results already presented in the companion paper[13],two3-D"nite element models were developed and appropriately validated on the basis of the experimental data:the OECS and the MECS.The two models idealize the structure as an assemblage of beam and truss elements and di!er only in the number of truss elements used to represent the stay cables.The OECS employs a simple truss element to describe each stay cable,while the MECS idealizes each cable as a set of several truss elements.

The calculation of natural frequencies and mode shapes presented in Reference[13]was based on a tangent sti!ness matrix,obtained at the end of a geometric non-linear static analysis of the structure under permanent load,and on a lumped mass matrix.A subspace iteration algorithm, integrated in a structural analysis software,SOLVIA[14],was used to extract the"rst20modes associated with the OECS model,in the range0}46Hz,and the"rst150modes related with the MECS model,lying in the range0}21.3Hz.

A plot of the calculated frequencies obtained from the MECS analysis against the order of the mode is presented in Figure3.The frequencies associated with the OECS analysis are also represented in this"gure,in correspondence with the mode of closer characteristics obtained in the MECS analysis(VSYM and VASM*vertical symmetric and anti-symmetric bending modes; TSYM and TASM*transversal symmetric and anti-symmetric bending modes).It is clear from the"gure that the numerous modes of vibration that resulted from the modelling of local cable behaviour are separated by#at regions,which can generally be associated with a common con"guration of the deck and towers.These sets of modes involve di!erent movements of the cables,with a varying intensity level(relative to the girder/towers movement),and occur at an almost identical natural frequency.

Figure4presents the participation factors calculated for both the OECS and MECS models.It is evident,from the analysis of these"gures and of the mode shape con"gurations,that(i)the structural response along the vertical(Z)and transversal(>)directions is clearly dominated by one mode of vibration(the1st VSYM and the1st TSYM modes,respectively);and(ii)the structural response along the longitudinal(X)direction is strongly conditioned by two vertical anti-symmetric modes(the1st VASM and the2nd VASM).Another aspect to refer to is that although the highest participation factors associated with the OECS modes are in some cases Copyright 2000John Wiley&Sons,Ltd Earthquake Engng.Struct.Dyn.2000;29:499}521

Figure 3.OECS vs MECS natural

frequencies.

Figure 4.Participation factors associated with models OECS and MECS.

DYNAMIC CABLE }DECK INTERACTION IN BRIDGES.PART II 503

Figure 5.Example of an arti "cial record of seismic action.Acceleration time history along the

longitudinal direction X and Fourier spectrum.

slightly higher than that corresponding to modes obtained from the MECS analysis,the participation factors associated with the new mode shapes may have some signi "cance for the response evaluation.This fact justi "es the importance of the present investigation.

4.SEISMIC TESTS ON THE SHAKING TABLE

The shaking table tests of the Jindo bridge physical model were conducted at the Earthquake Engineering Research Centre of the University of Bristol.Three di !erent ground acceleration time histories (with about 30s duration for the prototype)were generated and scaled (the scale factor for time measurements is S R "(150,according to Caetano et al .[13])based on three di !erent target response spectra.The de "nition of these response spectra attempted,in the "rst instance (records RRS1),to excite predominantly the fundamental modes of the cables,whereas,in a second situation,the objective was to excite essentially the "rst mode shape of the structure.The non-stationarity of the seismic action was introduced in terms of amplitude by the applica-tion of a trapezoidal time modulation function,simulating the usual three phases of a common accelerogram.A third time history was generated introducing also a non-stationarity in terms of the frequency content.

Graphical representations of an acceleration time series measured on the shaking table platform and of the corresponding single-sided Fourier spectrum are presented in Figure 5.These correspond to a component of the record RRS1along the longitudinal direction X (i.e.along the axis of the deck)with about 10per cent g peak value.

The response was measured for three input directions,X (longitudinal),>(transversal)and Z (vertical),and for the combinations XZ and X >Z ,with about 5and 10per cent g peak values along the two horizontal directions,and about 3and 6per cent g in the vertical direction.A total of 13small piezoelectric accelerometers and 1non-contact displacement transducer were used to obtain simultaneous measurements along the bridge.Figures 6and 7present a few examples of the measured response at some important locations (see Figure 2).

5.ANALYSIS OF THE SEISMIC RESPONSE

Using the accelerograms measured on the seismic platform,the response of both OECS and MECS models to di !erent combinations of time series from the input records has been evaluated,

504 E.CAETANO,A.CUNHA AND C.A.TAYLOR

DYNAMIC CABLE}DECK INTERACTION IN BRIDGES.PART II505

Figure6.Examples of measured responses for input RRS1,10%g(X):(a)node D4-Z;(b)node D5-Z.

based on a direct integration algorithm(Newmark method)and on a geometric non-linear dynamic analysis.Table I summarizes some measured and calculated absolute peak values on some of the most signi"cant nodes of the structure(mid-span,node D3-Z;attachment of longest cable,node D4-Z;third of span,node D5-Z;top of left tower,node LT1-X),for an input de"ned as a XZ combination(10%g X,6%g Z)of time series from record RRS1.

Due to the di$culty of accurately reproducing the real damping characteristics of the physical model,damping was numerically modelled by means of a mass proportional damping matrix, which was formed specifying a modal damping factor "1.0per cent for the"rst vertical bending mode of vibration(f "6.20Hz).This value resulted from the analysis of the measured response.It is important to note that sensitivity studies developed to"x the value of this damping coe$cient showed that it has a strong in#uence both on the peak values of the response that occur in a"rst part of the records,particularly in terms of accelerations,and on the corresponding decay phase.So the value adopted represents a compromise in order to achieve a relatively good global agreement between the experimental and numerical responses,and not exclusively in terms of local response peak values.

Figure8presents a comparison between experimental and calculated responses for one speci"c node of the structure,D4-Z.Figure9presents the Fourier spectra associated with those experimental and calculated responses.The global peak response of the bridge in terms of maxima displacements,accelerations,bending moments and axial forces along the deck,the cables and one of the towers is depicted in Figures10}12.Note that only the dynamic component of the response is analysed here.

506 E.CAETANO,A.CUNHA AND C.A.TAYLOR

Figure7.Examples of measured responses for input RRS1,10%g(X),6%g(Z):(a)node D4-Z;

(b)node D5-Z;(c)node LT1-X.

Inspection of Table I and of Figures8}12suggest in particular the following comments: 1.The MECS model leads to a slight modi"cation of the response.The variation of the peak

response is relatively small,as can be observed in Figures10}12and in Table II,which shows some values of the most signi"cant changes in the negative and positive peak response that resulted from a MECS analsyis,with regard to a standard OECS analysis.

2.It is also possible to observe three distinct periods in the response analysis.During a"rst

period of about1s(12s in the prototype),i.e.from1.5to2.5s,the response obtained on the basis of the OECS model is similar to the corresponding response obtained using the MECS model.Then,the MECS response starts to deviate from the OECS response,su!ering

Table I.Measured and calculated peak response for input RRS1:10%g (X ),6%g (Z ).

Node component

Experimental OECS MECS D4!Z,displ.(m)

!0.0029/0.0029!0.0026/0.0025!0.0025/0.0026D3!Z,accel.(m/s )

!6.6/6.5!4.8/4.1!5.7/4.4D4!Z,accel.(m/s )

!4.2/4.6!4.5/4.0!5.3/4.3D5!Z,accel.(m/s )

!2.9/3.2!3.1/3.0!3.1/3.7LT1!X,accel.(m/s )!1.5/1.5!1.2/1.2!

1.0/0.9

Figure 8.Displacement at node D4-Z (a)calculated,OECS vs MECS,structural damping included;(b)

experimental vs calculated response,MECS,structural damping included.

DYNAMIC CABLE }DECK INTERACTION IN BRIDGES.PART II 507

508 E.CAETANO,A.CUNHA AND C.A.TAYLOR

Figure9.Fourier spectra at node D4-Z:(a)experimental vs MECS,structural damping included;

(b)experimental vs OECS,structural damping included;(c)experimental,segment analysis;(d)MECS,

segment analysis,structural damping included.

a signi"cant reduction during the next2s,after which the relative di!erence maintains

approximately constant.Figure8shows these phases for the vertical displacement at node D4.It can be observed that the MECS analysis leads to a displacement decrease at node D4-Z,of about50per cent.

3.A comparison with the experimental data shows that,during the"rst1s of excitation

(1.5}2.5s),both the OECS and MECS signals are slightly lower than the measured response.

Figure8(b)shows a second period,from about2.5s to about5s,where a gradual phase deviation between the experimental and the numerical response occurs.This corresponds,in practice,to changes of the fundamental frequency of the measured response.The deviations in relation to the numerical response start to reduce again in the"nal part of the records.The observation of Figure9(c)indicates that,during this second part of the motion,the funda-mental frequency of the measured response is lower than the corresponding frequency at the "nal part of the record.Neither the OECS nor the MECS analyses[Figure9(d)]were able to reproduce this behaviour,probably due to the practical di$culty of accurate numerical modelling of local particularities and slight imperfections of the physical model.

Figures8and9show that the structural response is strongly dominated by the"rst vertical mode shape,due to the important frequency content of the input excitation used in this analysis,

Figure 10.Peak values of the calculated response along the girder,OECS vs MECS analysis:displacements,

accelerations,bending moments and axial

forces.

Figure 11.Peak values of cable response:(a)tension,OECS vs MECS analysis;

(b)peak displacements at the midpoint of cables.

DYNAMIC CABLE }DECK INTERACTION IN BRIDGES.PART II 509

Table II.Change of peak response,MECS vs OECS analysis.

Change of peak response (MECS/OECS)(%)

Type of response

Deck Node no.Towers Node no.Displacement

105.9/115.7D4-Z 96.4/99.8LT1-X Acceleration

122.1/106.2D4-Z 91.4/78.6LT4-X Bending moment

100.0/115.4D498.9/100.4LT3Axial force 103.4/89.5D5102.4/98.6

LT3

Figure 12.Peak values of the calculated response along one tower,OECS vs MECS analysis:displacements,

accelerations,bending moments and axial forces.

in the vicinity of the corresponding natural frequency.This mode does not involve a signi "cant cable interaction,and so the di !erences obtained between responses calculated on the basis of the OECS and MECS models are relatively small.In fact,the analysis points to the existence of a certain amount of vibration damping provided by the stay cables,which leads to a decrease of the response only a few seconds after the beginning of the excitation.However,this damping is rather small,as the amplitude of the cable movements is not induced to a great extent.Moreover,as this damping does not occur immediately after the structure starts vibrating,the e !ect on the reduction of peak response to seismic action is not signi "cant.

510 E.CAETANO,A.CUNHA AND C.A.TAYLOR

Table III.Calculated natural frequencies of modi "ed models

Mode

number

MECS natural frequency (Hz)Associated OECS frequency (Hz)Min.ratio of cable/beam max.displ.Type of mode 1

6.62 6.72 2.21st transv.SYM 3

7.789.01/6.72 4.91st vert.SYM #1st transv.ASM 4

7.789.01/11.71 4.31st vert.SYM #1st transv.ASM 8

8.0211.7115.61st transv.ASM 9

8.03 6.7213.71st transv.SYM 11

8.469.01 6.01st vert.SYM 12

8.469.017.51st vert.SYM 13

8.469.01 6.21st vert.SYM 14

8.469.01 6.11st vert.SYM 15

8.469.01 6.01st vert.SYM 16

8.469.01 6.21st vert.SYM 17

8.479.01 6.11st vert.SYM 21

8.729.0116.21st vert.SYM 25

8.83 6.7217.01st transv.SYM 27

8.989.0112.01st vert.SYM 28

8.989.0111.91st vert.SYM 29

8.989.0111.71st vert.SYM 44

11.29.01/6.7213.01st vert.SYM #1st transv.SYM 45

11.29.01/6.7210.51st vert.SYM #1st transv.SYM 53

12.013.7212.51st vert.ASM 54

12.013.7212.01st vert.ASM 55

12.013.7214.91st vert.ASM 56

12.113.7217.71st vert.ASM 57

12.2 5.9tranversal 59

13.613.7211.81st vert.ASM 61

13.613.7213.91st vert.ASM 63

13.613.7214.01st vert.ASM 67

14.113.7212.01st vert.ASM 77

15.813.7210.21st vert.ASM 79

15.813.72/6.7210.11st vert.ASM #1st transv.SYM 85

16.413.72/11.7121.21st vert.ASM #1st transv.ASM 8716.413.72/11.7125.01st vert.ASM #1st transv.ASM

6.NUMERICAL INVESTIGATION OF DYNAMIC CABLE INTERACTION WITH

DECK AND TOWERS

The seismic tests and numerical analysis described above evidenced the following particular aspects:

(a)The frequency of the fundamental vertical bending mode of vibration of the bridge (6.21Hz)

lies outside the range of the "rst frequency of the cables (6.81}18.92Hz,according to Irvine theory,see Table III in the companion paper [13]).This fact,accompanied by the signi "cant z -participation factor associated to this mode,may have contributed to an attenuation of the damping e !ect induced in the response by the stay cables.

DYNAMIC CABLE }DECK INTERACTION IN BRIDGES.PART II 511

512 E.CAETANO,A.CUNHA AND C.A.TAYLOR

(b)Some of the cables(cables6,10and12,with fundamental frequencies of13.11,9.39and

6.81Hz,respectively)experienced higher levels of vibration than the others.Considering that

the"rst three natural frequencies of the structure,obtained on the basis of the OECS analysis, associated with vertical bending modes,are6.21,9.12and13.74Hz,it seems clear that major cable e!ects occur when a global natural frequency lies in the range of the"rst natural frequencies of some cables.

In order better to understand these points,and taking into account the existence of a signi"cant number of stay cables with fundamental frequencies close to the frequency of the"rst vertical anti-symmetric bending mode(9.12Hz),two di!erent numerical simulations were performed.The "rst consisted of modifying the mechanical characteristics of the structure,in order to increase the natural frequency of the"rst mode of vibration to the range of the"rst natural frequency of those cables.The second consisted of the application of a new arti"cially generated input signal,based on an almost rectangular power spectrum de"ned in a narrow frequency band that contains both the frequency of the third global mode of vibration(1st vertical anti-symmetric)and the"rst natural frequency of some cables.

With these tests,the authors intended to analyse:(a)the dynamic behaviour of a cable-stayed bridge in a situation where the fundamental natural frequency is in the vicinity of the1st natural frequency of some stay cables;and(b)the e!ect of a narrow band excitation in a frequency range that contains both the"rst natural frequency of some cables and a global natural frequency of the bridge.

6.1.E w ect of cable-deck/towers resonance at the fundamental mode of v ibration

The numerical models of the Jindo bridge physical model were modi"ed,by increasing the Young's modulus of the materials that constitute the deck/towers and stay cables,by factors of2.8 and2,respectively.This lead to an increase of the frequency of the"rst vertical bending mode from6.21to9.01Hz,while the fundamental frequencies of the cables increased from6.81}18.92to 7.73}23.69Hz.

The analysis of mode shape con"gurations for the new OECS and MECS models shows that groups of symmetric mode shapes alternate with groups of anti-symmetric modes(note that this designation is applied to describe only the con"guration of the deck and towers).It is also evident, according to Table III,which presents the mimimum ratio between the maximum normalised modal displacement components(along the three orthogonal directions x,y and z)of the group of cables and of the deck/towers,that the MECS analysis produced many new modes of vibration associated with the same"rst symmetric vertical con"guration.These modes involve signi"cant interference with cables.

The participation factors along the longitudinal(X)and vertical(Z)directions,presented in Figure13,show the contribution of a signi"cant number of modes(from the MECS analysis)to the response.

The calculation of the response of the OECS and MECS models of the new structure to the combination XZ(10%g(X),6%g(Z))of accelerograms from record RRS1above described,was based on a geometric non-linear formulation,using the direct integration method of Newmark and the same mass-proportional damping matrix(f"6.20Hz, "1.0per cent).Figures14and 15show the peak dynamic response along the deck and one of the towers,and along the cables, respectively,expressed in terms of displacements,accelerations and bending moments. Copyright 2000John Wiley&Sons,Ltd Earthquake Engng.Struct.Dyn.2000;29:499}521

DYNAMIC CABLE}DECK INTERACTION IN BRIDGES.PART II513

Figure13.Participation factors associated with models OECS and MECS.

Figure16represents the time history of the bending moment response at node D4and the corresponding Fourier spectrum.The relative di!erence between the peak response calculated at some signi"cant locations,based on the OECS and MECS analyses,is presented in Table IV.

The analysis of this table and of Figures14}16illustrates that,except for small extensions along the deck,the response obtained on the basis of the MECS analysis is much lower than the corresponding response obtained from the OECS analysis.This e!ect constitutes the so called &system damping'.The damping of the response is due mostly to the contribution of the several new modes of vibration associated with the1st symmetric vertical con"guration.These modes, occurring at close frequencies(7.78,8.46,8.47,8.72,8.98Hz)involve exclusively the movement of cables that have similar natural frequencies(e.g.mode17,freq."8.47Hz,involves vibration of cables8,9,10and11,whose"rst natural frequencies are9.39,8.94,8.29and7.75Hz,respectively).

6.2.E w ect of se v ere cable excitation

Using again the3-D OECS and MECS numerical models de"ned initially,the response of the physical model of Jindo bridge to a new input signal was calculated and analysed.The new accelerogram was generated arti"cially,based on a narrow-band target power spectrum,de"ned in the range8.5}12Hz,that includes the frequency of the second vertical mode of vibration (according to the OECS analysis)and the"rst natural frequency of a few stay cables that participate in the modes of vibration obtained from the MECS analysis.Figure17presents the generated time history and the corresponding Fourier spectrum.

Considering a combination of two time histories from the generated record with60and30per cent g peak values along X and Z directions,respectively,and imposing a damping factor "1.0per cent for the"rst mode of vibration(f "6.20Hz)in order to generate a mass-proportional damping matrix,the system response has been calculated for the OECS and MECS models(using a geometric non-linear formulation and the direct integration method of New-mark).Figures18}20summarize the global response of the bridge in terms of the following extreme values:displacements,accelerations,bending moments and axial forces.Figures21}23

514 E.CAETANO,A.CUNHA AND C.A.TAYLOR

Figure14.Peak response along the deck and left tower:displacements,accelerations and bending moments.

represent the time and frequency response in terms of displacement,acceleration and bending moment at nodes D4-Z,D5-Z and LT4,respectively.Table V summarizes the negative and positive peak values obtained on some of the most representative nodes of the structure. Inspection of this table and these"gures shows that,except for the axial force,the inclusion of the local cable behaviour in the analysis(MECS)leads to a signi"cant increase of the peak response.This occurs in consequence of a high spectral content of the response at high frequencies (Figures21}23),which develops only about2s after the excitation has been applied.

Figure 15.Peak response along the cables:displacements and

tensions.

Figure 16.Bending moment at node D4,MECS vs OECS analysis.Time history and Fourier spectrum.

Table IV.Change of peak response,MECS vs OECS analysis.

Change of peak response (MECS/OECS)

Type of response

Deck Node no.Towers Node no.Displacement

48.2%D4-Z 54.8%LT1-X Acceleration

51.9%D4-Z 72.8%LT1-X Bending moment 54.5%D467.4%LT2

It was also possible to observe that,during the excitation period,cables 8and 9experienced relatively high levels of vibration.Figure 24,representing the ratio between the maximum amplitude of displacement at the cable mid-point and the corresponding length for the cables attached to the left tower,illustrates the relative importance of cable motion for the three analyses performed.The signi "cant cable movement associated with the narrow-band excitation (input DYNAMIC CABLE }DECK INTERACTION IN BRIDGES.PART II 515

Figure 17.Arti "cial record of ground motion.Acceleration time history and Fourier

spectrum.Figure 18.Peak responses along the deck:OECS vs MECS displacements,accelerations,

bending moments and axial forces.

RRS4)may be responsible for a marked non-linear character of the oscillations,evidenced by the translation upwards of the curves that represent peak displacements along the deck,and by a certain loss of regularity of the curves that represent the peak response along the deck (Figure 18).

516 E.CAETANO,A.CUNHA AND C.A.TAYLOR

Table V.Peak response for input RRS4*60%g (X ),30%g (Z ).

Response

OECS MECS Change (MECS/OECS)(%)D4!Z,accel.(m/s )

!8.97/8.29!11.6/7.41129.3/89.4D5!Z,accel.(m/s )

!8.00/8.24!8.81/8.52110.1/103.4LT1!X,accel.(m/s )

!0.80/0.76!2.81/2.16351.2/284.2D4!Z,displ.(m)

!0.00237/0.00238!0.00184/0.0025177.6/105.5D5!Z,displ.(m)

!0.00219/0.00199!0.00154/0.0021270.3/106.5LT1!X,displ.(m)

!0.000239/0.000231!0.000276/0.000221115.5/95.7D4,bend.mom.(N m)

!2.15/2.08!2.72/2.78126.5/133.7D5,bend.mom.(N m)

!1.56/1.47!1.73/2.00110.9/136.0LT3,bend.mom.(N m)

!0.591/0.559!0.733/0.622124.0/112.7D5,axial force (N)

!165.1/169.9!163.1/163.698.8/96.3LT3,axial force (N)

!36.2/39.4!37.1/40.6102.5/103.0Cable 1,left,tension (N)

!19.1/18.6!21.0/18.8109.9/101.1Cable 8,left,tension (N)

!5.8/5.5!5.7/8.198.3/122.7Cable 9,left,tension (N)!3.9/3.7!4.2/8.1

107.7/218.9

Figure 19.Peak responses along the left tower:OECS vs MECS displacements,accelerations,

bending moments and axial forces.

DYNAMIC CABLE }DECK INTERACTION IN BRIDGES.PART II 517

Figure 20.Peak responses along the cables:OECS vs MECS displacements at midpoint and

tensions.

Figure 21.Input RRS4*displacement calculated at node D4-Z and corresponding

Fourier spectrum,OECS vs

MECS.

Figure 22.Input RRS4*acceleration calculated at node D5-Z and corresponding

Fourier spectrum,OECS vs MECS.

518 E.CAETANO,A.CUNHA AND C.A.TAYLOR