Influence of Cable Vibration on Seismic Response of Cable-stayed Bridges

lnfluence of cable vibration on seismic response of cable-stayed bridges

Raju Tuladhar, Walter H. Dilger, and Mamdouh M. Elbadry

Abstract: In cable-stayed bridges, modelling the cables is of particular significance for the seismic behaviour of the structure. The common practice of modelling a cable by a single truss element is inadequate for seismic response calculations because it essentially precludes the lateral cable vibration modes. The present paper studies the influence of cable vibrations on the seismic response of cable-stayed bridges. Three bridge examples with different spans and properties were used. Cable vibrations are accounted for through the use of multiple links for each cable. Cable vibration effects are found to be significant for seismic response calculations, particularly when the cable fundamental frequencies are overlapping with the first few frequencies of the bridge. Parametric studies are conducted with regard to the number of links per cable, the effect of the modulus of elasticity of the cables, and different earthquakes on the bridge response. Modelling the cables by two links per cable such that at least the fundamental modes of the cable vibrations are represented can significantly account for the effect of cable vibrations. It is also observed that the equivalent modulus method cannot in any way account for the cable vibration effects.

Key words: cable vibration, dynamic analysis, equivalent modulus, multiple links, seismic response, cable-stayed bridge.

RCsumC : En ce qui concerne les ponts i haubans, la modClisation des cibles est d'une importance particulikre pour dCterminer le comportement de la structure lorsque soumise i des sCismes. La pratique courante qui consiste h modCliser un cible B l'aide d'un seul treillis est inadCquate dans le calcul de la rCponse sismique parce qu'elle exclut les modes latCraux de vibration des cibles. Cet article examen l'influence des vibrations des cibles sur la rCponse sismique de ponts i haubans. Trois exemples de ponts ayant diverses travCes et caractkristiques ont servi a cette Ctude. Des sections multiples ont CtC utilisCes pour chaque cible afin de tenir compte des vibrations. Les effets de celles-ci se sont rCvClCs importants dans le calcul de la rCponse sismique, surtout lorsque les frCquences fondamentales des cibles co'incident avec les premikres friquences du pont. Des Ctudes paramktriques ont CtC rCalistes en ce qui a trait au nombre de sections par csble, i l'effet du module d'ClasticitC sur les cibles et de divers tremblements de terre sur la rCponse des ponts. La modClisation des cibles i l'aide de deux sections de manikre i ce que les modes fondamentaux de vibration soient au moins reprCsentCs permet de tenir compte de l'effet de la vibration des cibles. On a Cgalement observC que la mCthode du module equivalent ne permet d'aucune facon de tenir compte des effets de la vibration des cibles.

Mots clis : vibration des cibles, analyse dynamique, module Cquivalent, sections multiples, rtponse sismique, ponts i haubans. [Traduit par la redaction]

1. Introduction

In 1938, F. Dischinger revived the idea of cable-stayed sup- ports for bridge decks by using prestressed stays in the design of a railway suspension bridge over the river Elbe,

Received August 4, 1994. Revised manuscript accepted January 25, 1995.

R. Tuladhar and W.H. Dilger. Department of Civil Engineering, The University of Calgary, Calgary, AB T2N 1N4, Canada. M.M. Elbadry. Department of Civil Engineering, Concordia University, Montreal, PQ H3G 1M8, Canada.

Written discussion of this paper is welcomed and will be received by the Editor until February 29, 1996 (address inside front cover).

Hamburg. The first modern steel cable-stayed bridge was the Stromsund Bridge in Sweden designed by Dischinger and completed in 1955. Since then, numerous cable-stayed bridges have been built all over the world and record breaking spans are built almost every year. Recent examples are the Nor- mandie Bridge in France with an 856 m span and the Tatara Bridge in Japan with an 890 m span now under construction. Leonhardt and Zellner (1991) have predicted a great future for cable-stayed bridges and envisioned spans up to 1800 m. Aesthetics, ease of construction, and economic competitive- ness have been the main reasons for the success of cable- stayed bridges. Even with this great popularity, the experience with the behaviour of these bridges in major earthquakes is limited. Only a few cases of earthquake-induced damage have been reported and no earthquake-related collapse of cable-stayed bridges has been recorded to date. Recently,

Can. J. Civ. Eng. 22: 1001- 1020 (1995). Printed in Canada I Imprimt au Canada

Can. J. Civ. Eng. Vol. 22, 1995

Fig. 1. Geometry and mesh details of Bridge 1.

,7

Bridae Geometry

Mesh Model - 1

'L+x f Left Pylon - Deck - Right Pylon

Mesh Model - 2 Connection Details

Table la. Properties of Bridge 1. the behaviour of a structure. The accuracy of such studies to

A I E Unit wt. Item (m2) (m4) (MPa) (kN/m3)

Deck 1.20 2.58 200 000 110.0 Pylon 1 14.2 30.0 31 600 24.0 Pylon 2 35.8 40.0 31 600 24.0

Table lb . Area of cables for Bridge 1.

Cable No. Area (mm2)

1 60 000 2 40 000 3 28 000 4 28 000 5 40 000 6 50 000

Notes: Cable elastic modulus, E = 165 000 MPa; unit wt. = 92.5 kN/m3.

Filiatrault et al. (1993a, 1993b) reported some damage to the anchor plates of the Shipshaw Bridge in Quebec caused by the 1988 Saguenay earthquake.

The recent trend in bridge construction indicates a very favorable future for cable-stayed bridges and, as a result, it can be safely predicted that more of these bridges will be built in zones of high seismic risk. This makes the seismic behaviour of these complicated structures an important topic of current research. Analytical studies can help to understand

predict the dynamic behaviour of a structure depends on many factors, and one of the most important aspects is the appropriate modelling of the structure for the computer anal- $is. The present paper discusses some modelling problems and their effects on the prediction of seismic behaviour of cable-stayed bridges.

It is common practice to use the equivalent modulus, Eeq, to account for the effects of cable sag in the analysis of cable- stayed bridges for static loads. The equivalent modulus has also been used by many researchers in the dynamic analysis for the prediction of seismic behaviour; Fleming and Egeseli (1980), Abdel-Ghaffar and Nazmy (1991), Nazmy and Abdel-Ghaffar (1992), Wethyavivorn (1987), and Parvez and Wieland (1987) used this approach. But the equivalent modulus approach is reasonably accurate only for a static or quasi-static analysis. However, for a seismic analysis, the equivalent modulus method has the following drawbacks:

- (a) The equivalent modulus of a cable depends upon the cable tension which varies rapidly during seismic excitations. Cables act as stiffening elements when the tension is increasing and softening elements when the tension is decreasing. Using the same equivalent modulus throughout the duration of seismic response is not valid.

(b) In the equivalent modulus method, each cable is represented by a single truss element. This essentially excludes the lateral vibration modes of the cables and, as a result, the interaction of the cable vibrations with the bridge response is not accounted for, even though the decrease in the cable stiffness due to the cable sag is considered.

It has been observed that cable vibrations have a more sig-

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Bridge Geometry

(dimensions in m)

3

Z1 t 20 ' L X 70 Mesh Model - 1

Mesh Model - 2 Connection Details

Table 2a. Properties of Bridge 2.

A I E Unit wt. Item (m2) (m4) (MPa) (kN/m3)

Deck 1.47 1.30 200 000 201.0 Pylon 1 19.68 66.43 29 580 24.0 Pylon 2 22.56 121.32 29 580 24.0 End pier 10.00 3.333 29 580 24.0 Bearing element 0.0301 3.763 29 580 -

nificant influence on the seismic response of the bridge than does the change in modulus of elasticity of the cables due to the sag effect (Tuladhar 1995). Also, the fundamental frequencies of cables in most cable-stayed bridges are very close to the first several frequencies of the bridge itself and these frequencies can also lie in the frequency content range of an earthquake. Therefore cable vibrations during an earthquake can be quite significant, and the interaction between cable vibrations and bridge vibrations can affect the seismic response of the bridge substantially. This interaction of cable vibrations has recently come to the attention of some researchers. Abdel-Ghaffar and Khalifa (1991) have shown that cable vibrations can affect deck modes and participation factors. They stressed the inadequacy of using single truss

Table 2b. Area and unit weight of cables for Bridge 2.

Cable No. Area (mm2) Unit wt. (kN/m3)

elements to model the cables and recommended the use of multiple elements for each cable. This multiple link method has been used previously by Baron and Lien (1973), Yiu



85 Deck centre no 0 Cable number

1 1 1 o (dimensions in m)

CT77777rmm7TTn

1 12.0 + 1 Deck section-2

Typical deck- tower pylon sectibn-2 Pylon section-3 Pylon section-4 connection

at each tower

Mesh Model -1

Table 3. Area of cables for Bridge 3.

Cable No. Area (mm2)

1, 22 8960 2, 21 8960 3, 20 7280 4, 19 7280 5, 18 6160 6, 17 6160 7, 16 5040 8, 15 5040 9, 14 4480

10, 13 4480 11, 12 3360

Notes: Cable elastic modulus, E =

190000 MPa; unit wt. = 90.0 kN/m3.

L_) Mesh Model -2

(1982), and Tuladhar and Brotton (1987) in dynamic behaviour studies for wind and earthquakes. However, these authors did not compare their results with data from bridges where multiple links were not used for the cables, and consequently they did not stress the need of using multiple links. A different approach to represent cable vibrations has been

presented by Causevic and Sreckovic (1987). In their method, cables are replaced by linear springs and lumped masses. However, the multiple link method is found to be the more convenient approach.

The influence of cable vibrations on seismic response of cable-stayed bridges is studied in this paper by comparing the responses of three bridge examples using two different mesh models. In mesh model 1, the cables are divided into multiple segments and therefore cable vibrations are properly accounted for. In mesh model 2, the cables are modelled as single truss elements with equivalent modulus, E,,, and therefore cable vibration modes are not present.

2. Bridge examples and earthquake loading

The three bridge examples studied in this paper have spans ranging from 250 to 465 m. Details are described below:

Bridge I. A 371 m main span bridge is considered. The deck is steel and the towers are concrete. The main span and the deck properties are similar to that of the Luling Bridge in New Orleans, but the towers and cables have been modified. A three-dimensional model with similar main span and deck properties has been used by Wethyavivorn (1987) to study seismic behavior. The geometry and mesh details of

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this bridge model are shown in Fig. 1 and relevant bridge properties are given in Tables l a nd lb.

Bridge 2. This is a 465 m main span bridge with composite deck and concrete towers. This bridge is derived from the Annacis Bridge in Vancouver, but the number of cables has been reduced to 16 from the original 48 per tower and correspondingly the deck inertia had to be increased. The geometry and mesh details of this bridge are shown in Fig. 2 and relevant bridge properties are listed in Tables 2a and 2b.

Bridge 3 . One typical module of a multi-span bridge proposed for the Northumberland Strait Crossing in eastern Canada (see Dilger et al. 1992) is selected. The main span is 250 m and both deck and tower are concrete. The geometry and mesh details of this bridge are shown in Fig. 3 and relevant bridge properties are given in Table 3.

For each bridge, two mesh models are considered as mentioned earlier. For the analysis of mesh model 1 with multiple links, it is necessary to consider the geometric stiffness of cables in order to avoid slack cable condition. For mesh model 2, the equivalent modulus, Eeq, of the cables is used and it is obtained using the modified form of Ernst's equation (Walther et al. 1988) given below:

[2] a = a, - [(l l:;)4] In

In these equations, Eo, w, and L are respectively the original material modulus, the unit weight, and the chord length of the cable, and a is the tensile stress in the cable. With slow and a,, denoting respectively the lower and upper limits of the cable tensile stress, the parameters a, and 77 are defined in [3].

For earthquake loading, the 1940 El Centro earthquake is used. The analysis duration is 15 s, out of which the first 10 s are for forced motion and the latter 5 s are for free motion. The accelerations are input in both the longitudinal and the vertical directions.

3. Frequency range of cables in cable-stayed bridges

Before studying the differences in modal properties and seismic response, the range of frequencies of stay cables needs some discussion. Consider a simple vibrating string fixed at two ends. A segment of string of length Ax is shown in Fig. 4. In the figure, T is the tensile force in the string and v is the lateral displacement. With p as the mass per unit length of string, at any time t , the equilibrium is maintained by the transverse component of T with the inertial force as

This equation can be solved with the boundary condition of a string fixed at two ends to give the following equation for

Fig. 4. A segment of a vibrating string.

/ \ /

Table 4. First mode frequencies of cables of Bridge 1 (main span = 371 m).

Cable TDL I.1 L f, (Hz)

No. (MN) (kglm) (m) From [6] Modal analysis*

*Range given to indicate more than one local mode appears for cables on left and right pylons.

the nth natural frequency:

The first mode frequency is -

The situation of cables in cable-stayed bridges is slightly different as the support points themselves are moving. How- ever, [6] can still be used to estimate the fundamental frequency of the cables in cable-stayed bridges with reasonable accuracy. This statement is supported by the comparison of the fundamental frequencies based on [6] with the frequencies obtained from local cable vibration modes of Bridge 1 during modal analysis (Table 4).

Equation [6] for the natural frequency of a cable can be written in a form that directly relates the stress level in the cable with the frequency. With a , p , and A representing respectively the tensile stress, the mass density, and the area of the cable, the tension force, T, and the mass per unit length, p, can be expressed as T I = aA and p = PA. With these relationships, [6] can be written as - From [7] the frequency range of the fundamental modes of stay cables can be estimated. The stress, a , can be taken as dead load stress which can vary from 300 to 700 MPa.


Table 5. Comparison of bridge frequencies versus cable frequencies

Bridge frequencies (Hz) Cable frequencies (Hz)*

Bridge name First mode Tenth mode Longest cable Shortest cable

Bridge 1 0.348 1.930 0.522 1.279 Bridge 2 0.226 0.910 0.495 1.845 Bridge 3 0.290 0.909 1.057 2.669

*Frequency obtained from eq. [6].

Fig. 5. Comparison of natural frequencies and participation ratios for Bridge 1 using mesh models I and 2.

20 30 40 50 60 70 80 90 Mode number

0 10 20 30 40 50 60 70 80 90 Mode number

Similarly, depending upon the protective covering used for the cables, the net density, p, can vary from 8000 to 13 000 kg/m3. This density applies to the cable area and includes the weight of the protective cover and grout, if any. Considering a main span range of the bridge between 200 and 900 m and a tower height of 0.2 times the main span, the approximate cable length can range from 30 to 460 m. With these approximate values, the range of the first fundamental frequency of stay cables can be estimated by using the extreme combinations.

Highest fundamental frequency = - 2 X 30

Lowest fundamental frequency = -

Therefore the first mode frequency of the stay cables in most cable-stayed bridges will fall in the range of 0.16 to 5 Hz. In practice, this range is narrower, as the cable stresses are normally higher than 300 MPa and, considering the protective material of the cables, the apparent cable density is higher than 8000 kg/m3. The frequency range of the first several modes of most cable-stayed bridges also falls within this range. As mentioned before, the frequency content of most earthquakes also falls in this range. These combinations

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Fig. 6. Some significant mode shapes of Bridge 1, mesh model 1.

mode number 1; f = 0.347 Hz. mode number 2; f = 0.457 Hz.








make cable vibrations in cable-stayed bridges quite promi- nent during seismic excitations, and their interaction with the bridge response can be significant in some cases. In Table 5, the frequencies of the shortest and longest cables of the three bridges are compared with the frequency range of the first 10 modes of the bridge itself (basic bridge modes). It can be observed that for Bridge 3 with a concrete deck, the cable frequencies are higher than the first 10 mode frequencies of the bridge. This means that for concrete bridges the cable frequencies are less likely to match with the first few frequencies of the bridge itself than for the steel or composite deck bridges.

In some long-span bridges, such as the Normandie Bridge (Virlogeux 1993), damping ropes interconnecting the cables are present. Since the frequency of each cable is different, the presence of such damping ropes will lead to interference

of vibration of cables with each other. And if the damping ropes are connected to the deck, the cable frequencies will increase. This will change the overall cable frequencies and will influence the extent of interaction between the cable vibration and the bridge vibration. But this is not further discussed here.

4. Modal analysis comparison

Before comparing the seismic response, the modal frequencies and the mode shapes of Bridge 1 for two different mesh models are compared. Figures 5a and 5b show, respectively, the comparison of the mode frequency and participation ratio results for Bridge 1, for the two mesh models. The modal analyses are performed following nonlinear static analyses under the effects of dead loads, and all degrees of freedom

Can. J . Civ. Eng. Vol. 22, 1995

Fig. 7. Some significant mode shapes of Bridge 1, mesh model 2.









(two translations and one rotation) are considered. These modal analyses and the subsequent mode superposition analyses were done using ANSYS program (1989, 1993). The sub- space iteration option in the ANSYS program was used to extract about 50% of the modes present in the system. Figures 6 and 7 show the mode shapes of Bridge 1 using mesh models I and 2, respectively. For this modal analysis of mesh model 2 and the subsequent mode superposition analyses, the equivalent modulus of [I] is obtained for a = UDL, i.e., for dead load stress without considering dynamic variation. The number of links per cable in mesh model 1 is shown in Figs. 1-3.

As 'the cables of mesh model 1 have been divided into multiple links, cable vibrations develop, as is evident in Fig. 6 which shows the mode shapes of Bridge 1. The local cable vibration modes appear in between the basic modes of

the bridge. The flat regions in the frequency versus mode number plots in Fig. 5a are zones of local cable vibration modes. The frequencies of these local cable vibration modes for Bridge 1 are in close agreement with the cable frequencies obtained from [6], as shown in Table 4. As a result of the local cable vibration modes, the significant modes of the bridge appear later and, consequently, it becomes necessary to consider a much larger number of modes in the mode superposition or response spectrum analysis. Similar obser- vations are made for bridges 2 and 3. Other effects of the local cable vibration modes are the following:

The number of modes with significant participation ratio is increased.

The vibration modes of the longer (outer) cables are found to have more significant participation ratios than those of the shorter (inner) cable vibration modes.

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Fig. 8. Comparison of time histories for mesh models 1 and 2 of Bridge 1; for undamped linear analysis case.

Time (s)

4 8

Time (s)

Time (s)

101 0 Can. J. Civ. Eng. Vol. 22, 1995

Fig. 9. Comparison of extreme values of the undamped linear response of mesh I I

models 1 and 2 of Bridge 1.

Tower bending moment exuema (MN.m)

The basic frequencies, participation ratios, and mode shapes of the bridge are also slightly affected. The higher mode frequencies are found to be more affected than the lower mode frequencies.

In the vertical direction, some change in the order of modes with the highest participation ratios are found. For example, mode 35 in mesh model 2, with extension of right tower, and shortening combined with bending of left tower, has the highest participation ratio. The same mode appears as mode 73 in mesh model 1.

5. Seismic response comparison

As the dynamic characteristics of the bridge change, the seismic response also changes. The interaction effects of cable vibrations are demonstrated by comparing the seismic response of the two mesh models of the three bridges described earlier. The undamped linear dynamic analyses are performed by mode superposition using the modal analysis results mentioned earlier and the 1940 El Centro earthquake for seismic input. For all three bridges, lumped mass models~ are used. For the mode superposition analysis of Bridge 1, 80 modes and 42 modes are used for the mesh models 1 and

2 3 4 5

Cable number

2, respectively. These ranges cover all the modes with similar frequency ranges and with participation ratios of about 0.10 or higher. The seismic response using 42 modes in the mode superposition analysis of mesh model 2 of Bridge 1 was almost identical to the seismic response obtained from the direct integration method. This demonstrates the accuracy of the modal analysis and the adequacy of the number of modes considered. Similarly, 151 modes and 52 modes were used respectively for mesh models 1 and 2 of Bridge 2; and 188 modes and 104 modes were used respectively for mesh models 1 and 2 of Bridge 3.

The results are presented in Fig. 8 to show the earthquake- induced time histories of the vertical mid-span displacement and bending moment of the deck, and the tension in cable 6 for mesh models 1 and 2 of Bridge 1. In the plot for cable tension, the change of cable tension is given in meganewton (MN) and as a percentage of dead load tension. Similarly, Fig. 9 compares the extreme values of the response (vertical displacement and bending moment of the deck, tower bending moment, and cable tensions). Nonlinear dynamic analyses considering geometric nonlinearities were done for the two mesh models of Bridge 1 using a nonlinear dynamic analysis program based on iterative methods (see Tuladhar

Tuladhar et al. 1011

Fig. 10. Comparison of extreme values of the undamped nonlinear response of mesh models 1 and 2 of Bridge 1.

Tower bending moment extrema (MN.m) Cable number

1995). Figure 10 shows the extreme values of the response of the two mesh models of Bridge 1 for the case of undamped nonlinear dynamic analysis using direct integration method. Similar differences between the two models are observed as for the linear dynamic analysis case using the mode superposition method. Therefore, it can be stated that the differences in seismic resuonse of the two mesh models are not due to the exclusion of the modes not considered in the mode superposition analyses.

Damping reduces the response of the bridge; however, it does not eliminate the effect of cable vibration. This can be seen from the difference in extreme values of the response shown in Fig. 11 for Bridge 1 for which the damped mode superposition analysis is done using 2 % damping for all the modes.

Similar comparisons of time histories and extreme values of the responses are made in Figs. 12 and 13 respectively for the two mesh models for the undamped linear response of Bridge 2. The results for extreme values of the responses of Bridge 3 are presented in Fig. 14.

The influence of the cable vibrations on the bridge response is evident from the differences in results for the two mesh models. Differences can be noticed in the extreme

values as well as the time histories of deck displacements, bending moments, and cable tensions shown in Figs. 8- 14; and differences in time histories become more obvious in the latter stages of the response.

For Bridge 1, the interaction of cable vibrations generally caused a reduction in the response, this can be seen from a comparison of the extreme values in Figs. 9 - 1 1. For Bridge 2, the seismic response of mesh model 1 is generally smaller than the response of mesh model 2, even though at some nodes, the deck and tower bending moments are higher for mesh model 1. This can be seen from the extreme value comparison in Fig. 13. However, for Bridge 3, the differences in extreme values or the time history plots for the two mesh models as shown in Fig. 14 are less significant than for bridges 1 and 2.

The influence of cable vibrations can be significant and should be accounted for at least for those bridges in which the first fundamental frequency of the longer (outer) cables fall in the vicinity of the first few frequencies of the bridge. This is more likely to happen in steel deck bridges or composite deck bridges. Concrete bridges, which have main spans under 300 m, will have shorter cables but the deck mass is greater. Therefore, the first fundamental frequencies of

101 2 Can. J . Civ. Eng. Vol. 22, 1995

Fig. 11. Comparison of extreme values of the damped linear response of mesh models 1 and 2 of Bridge 1.

8 z 2 6 w

c .g 4 FJ .- 9 2 C)

2 0 2 C) 2 -2 - Model # l 0

"0 4

-6 $ * -8

-200 -100 0 100 200 1 2 3 4 5 6


cables may not fall into the vicinity of the first few frequen- include the modes with the same frequency range. The origi- cies of the bridge and the interaction of cable vibrations with nal elastic modulus of the cables is used for all cases. the bridge response will be less. Figures 15 and 16 show the comparison of time histories

and extreme values for this parametric study. From the 6. Parametric studies figures, it can be seen that the bridge model with 2 links per

Number of links per cable To account for cable vibrations, cables need to be divided into multiple links. However, dividing cables into a large number of links, the degrees of freedom in the system increases. This will increase the problem size and the cost of the analysis whether it is based on mode superposition method or on the direct integration method. Therefore, cables should be divided into optimum number of links so that, while cable vibrations are adequately represented, the problem size is not increased too much. A parametric study is now conducted to establish the optimum number of links to be used to model each cable.

For Bridge 1, two additional mesh models 3 and 4 are created, in which each cable is divided into 2 links and 8 links, respectively. Linear dynamic analyses are performed as before, using mode superposition. The number of modes considered in these analyses are 42, 57, 80, and 129 respectively, for the models with 1, 2, 4, and 8 links per cable to

cable significantly accounts for the effect of cable vibrations. Only small differences are observed between the responses of the 4-link and 8-link cable models. Therefore it can be concluded that 2 to 4 links per cable are sufficient to account for cable vibrations. Shorter cables can be modelled with 2 links and longer cables should be modelled with 3 or 4 links. A larger number of links per cable will unnecessarily increase the size and the cost of the analysis without extra benefits.

Value of equivalent modulus The equivalent modulus of a cable depends on the tension in the cable; therefore, its value should be rapidly changing during seismic response. If cables are modelled as single links, the question is what value of the equivalent modulus should be used for the seismic analysis? To address this question, the effect of variation in cable modulus on seismic response is studied for Bridge 1 using three values of cable

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Fig. 12.

0.12

Comparison of time histories for mesh models 1 and 2 of Bridge 2; for undamped linear analysis case.

4 8

Time (s)

8 Time (s)

4 8 Time (s)

1014 Can. J. Civ. Eng. Vol. 22, 1995

Fig. 13. Comparison of extreme values of the undamped linear response of mesh models 1 and 2 of Bridge 2.

Tower bending moment extrema ( M N d Cable number

elastic modulus, namely Eo, E l , and E2. The equivalent modulus Eo is the original material modulus and its value is assumed to be 165 GPa. The equivalent modulus El is obtained from [I] for the dead load stress (a = aDL) without considering the dynamic stress variation (i.e., 7 = 1 in [3]). The E2 value is established considering the dynamic stress variation with the upper and lower limits of a obtained as (aDL + adyn), where ad,, is the extreme value of the dynamic seismic stress obtained in the analysis with E l . The values of equivalent modulus used for Bridge 1 are listed in Table 6.

Figure 17 shows the comparison of the extreme values of the vertical displacement and bending moment of the deck, the tower bending moment, and the change in cable tension for the three values of E mentioned above. Using different values of E changes the dynamic properties (mode frequencies, etc.) and the seismic response. It appears that the higher mode frequencies are not as much affected as the lower mode frequencies. The first mode frequency of the bridge changed from 0.3563 to 0.3481 and 0.3377 Hz when the values of Eeq were changed from Eo to El and E2, respectively. The use of the equivalent modulus results in a bridge response different from the model which properly accounts for the

cable vibration. For example, considering the response of mesh model 2 with the original elastic modulus for cables as the reference, the deck displacements are higher when the equivalent modulus E2 is used, whereas the deck displacements are lower when mesh model 1 is used. This can be observed by comparing the vertical displacement extreme values of deck nodes 8 - 1 1 in Figs. 16 and 17. The extreme values of deck and tower bending moments are also significantly different. When cables are modelled as single links, it is reasonable to consider the effect of dynamic stresses in evaluating the equivalent modulus. However, if the cable vibration effects are pronounced, the use of equivalent modulus cannot properly model the effect of cable vibrations.

Seismic response comparison for other earthquakes The influence of cable vibration has been demonstrated for three different bridges subjected to the 1940 El Centro earthquake. To show that the effects are similar for other earthquakes, Bridge 1 was also analysed for the 1966 Park Field (PF) and the 1971 San Fernando (SF) earthquakes. The two mesh models discussed before were used and the undamped linear seismic analysis was performed for 15 s with the first 10 s as forced motion and the latter 5 s as free motion.

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Fig. 14. Comparison of extreme values and time history of the undamped linear response of mesh models 1 and 2 of Bridge 3.

4 8 Time (s)

101 6 Can. J. Civ. Eng. Vol. 22, 1995

Fig. 15. Comparison of time histories for Bridge 1 using different number of links per cable.

0.5

Time (s)

4 8 Time (s)

31.5

21.0

8 . - 10.5 $ -

-2 0.0 2

-2 al

-10.5 z 0

s -21.0

-31.5 0 4 8 12

Time (s)

Tuladhar et al. 101 7

Fig. 16. Comparison of extreme values of response of Bridge 1 using different number of links per cable.

- a2links B-----• 4 links - 8 links

Tower bending moment extrema (MN.m)

Figure 18 shows the comparison of the extreme values of the vertical deck displacement, deck and tower bending moments, and the change in cable tension for the two mesh models of Bridge 1 under earthquakes PF and SF. Under these earthquakes, the effects of cable vibration in mesh model 1 caused an increase or decrease in the seismic response relative to the response of mesh model 2, which can be observed in Fig. 18. For example, under earthquake PF, the tower bending moment, the deck displacements near mid- span, and the cable tension extreme values are smaller while the deck bending moments are higher for mesh model 1 than for mesh model 2. The differences in the seismic response of the two mesh models of Bridge 1 subjected to PF and SF earthquakes further assert that the influence of cable vibrations are not limited to one particular earthquake such as the El Centro earthquake.

7. Summary and conclusions

For complicated structures such as cable-stayed bridges, dynamic analyses are essential to understand the response of the bridge subjected to earthquake loading. Before starting the dynamic analysis, a structural engineer has to make many

Cable number

decisions regarding the analysis approach, modelling tech- niques, etc. The issue of modelling the cables and its impact on the seismic response are addressed in this paper. Three bridges with different spans and properties have been analysed under the 1940 El Centro earthquake to demonstrate the effect of cable vibrations on the seismic response. To show that the cable vibration effects are present for other earthquakes as well, Bridge 1 was analysed for the 1966 Park Field and the 1971 San Fernando earthquakes.

The influence of cable vibration was demonstrated by comparing the undamped linear seismic response from mode superposition analysis for two kinds of mesh models for each of the bridges. To show that the difference in seismic response was not due to the difference in number of modes considered in the analysis, one case of nonlinear dynamic analysis using direct integration method was also done and similar effects of cable vibrations were observed. Damping reduces the response of the bridge; however, it does not eliminate the effect of cable vibration. This was demonstrated by performing one case of damped analysis for Bridge 1.

Based on the results of the present study, the following conclusions are made:

1018 Can. J. Civ. Eng. Vol. 22, 1995

Fig. 17. Comparison of extreme values of response of Bridge 1 using different cable elastic modulus.

Tower bending moment extrema (MNern) Cable number

Table 6 . Equivalent modulus of cables for Bridge 1 .

Cable No. E, (GPa) E, (GPa)

The bridge response due to cable vibrations can decrease in some parts of the structure and increases in other parts of the structure.

The interaction of cable vibrations with bridge vibrations affects the modal characteristics and the seismic response. It is found that cable vibration can have a significant influence on seismic response of the bridge especially when the first mode frequencies of cables overlap with the first few frequencies of the bridge. This overlap is less likely

to happen for concrete deck bridges than for steel or composite deck bridges.

The vibration modes of the longer (or outer) cables have higher participation ratios than the vibration modes of the shorter (or inner) cables.

The equivalent modulus cannot appropriately account for the effects of cable vibrations. For example, referring to the deck displacement extreme values at nodes 8 to 11 of Bridge 1 shown in Figs. 16a and 17a, the extreme values of displacements are less when cable vibrations are accounted for and higher when Eeq is used, compared with the response of the bridge without cable vibrations and using the original modulus.

To account for cable vibrations, the cables need to be divided into multiple links. It is recommended to divide the cables into two links for the shorter cables and three to four links for the longer ones. A larger number of links per cable will significantly increase the problem size without any extra benefit.

When multiple links per cable are used, the original modulus of elasticity can be used without loss of accuracy. It is also necessary to consider the geometric stiffness of the cables to avoid slack cables.

Tuladhar et al. 1019

Fig. 18. Comparison of extreme values of the undamped linear response of mesh models 1 and 2 of Bridge 1 for the 1966 Park Field (PF) and the 1971 San Fernando (SF) earthquakes. - PF, Model#l

A------A PF, Model#2 ..A .... &-.-.-A PF, Model#2


Acknowledgement

The current research is financially supported by the Natural Sciences and Engineering Research Council of Canada. This support is gratefully acknowledged.

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List of symbols

cross-sectional area original material modulus of cables equivalent modulus of cables first mode (fundamental) frequency nth mode frequency chord length of cables time coordinate tensile force lateral displacement unit weight length coordinate ratio of lower limit to upper limit of cable tensile stress mass per unit length of cables mass density tensile stress in cables lower limit of cable tensile stress upper limit of cable tensile stress mean stress

Documents

Influence of Cable Vibration on Seismic Response of Cable-stayed Bridges