- Email: [email protected]
Parametric Study of Cable Vibration Effects on the Dynamic Response of Cable-stayed Bridges
Structural Engineering, Mechanics and Computation (Vol. 2) A. Zingoni (Editor) © 2001 Elsevier Science Ltd. All rights reserved.
1559
Parametric Study of Cable Vibration Effects on the D y n a m i c Response of Cable-stayed Bridges
S.H.Cheng
D.T.Lau
Ottawa-Carleton Bridge Research Institute, Department of Civil and Environmental Engineering, Carleton University, Ottawa, Ontario, Cananda, KIS 5B6
ABSTRACT: In this paper, a 3-node cable model is employed to investigate the cable vibration effects on the dynamic responses of cable-stayed bridges. The transverse vibration of the stay cables in cablestayed bridges under dynamic loads, and the interaction between the motions of the cables and the other parts of the bridge superstructure are considered. A comprehensive parametric study of eight example bridges is carried out. The parameters considered in the study include the span length, the pylon shape and the cable arrangement. The impact of considering the dynamic cable stiffness in the dynamic analysis of cable-stayed bridges is also discussed.
K E Y W O R D S : cable-stayed bridge, cable vibration, finite element models, dynamic cable stiffness, longspan structure, numerical, modal analysis
INTRODUCTION The common practice of modelling the stay cables in cable-stayed bridges by single-truss elements (Fleming Egeseli,1980; Parvez ~ Wieland, 1987; Wethyavivorn, 1987;Abdel-Ghaffar ~ Nazmy, 1991; Nazmy & Abdel-Ghaffar, 1992) cannot represent the lateral motions of the cables and the interaction effects with other parts of the bridge superstructure. The multi-link approach proposed in some recent studies (Baron & Lien, 1973; Yiu, 1982; Tuladhar & Brotton, 1987; Abdel-Ghaffar & Khalifa, 1991; Tuladhar et al, 1995) improves the accuracy in the modelling of the cable behaviour. However, by dividing each cable into multiple elements, the size of the analytical model is increased significantly. A 3-node cable model has been proposed in a recent study by Cheng and Lau (2001). The new cable model can simulate the transverse vibration of the cable and cable-bridge interaction effects. It is more computationally efficient than the multi-link model. A comprehensive parametric study using the 3-node cable model is carried out in the present study. Eight example bridges are analyzed to investigate the impact of considering cable vibration and the use of the dynamic cable stiffness on the dynamic responses of the cable-stayed bridges. Other parameters considered in the investigation include the span length, the pylon shape and the cable arrangement. CASE STUDY OF NANPU BRIDGE Nanpu Bridge is a long-span cable-stayed bridge located in Shanghai, China. It has a center span of 423m, as shown in Figure 1. The bridge girder has a composite deck cross-section comprised of a prestressed
1560
concrete slab supported by two /-shaped steel beams. Two different numerical models are employed to study the cable vibration effect on the dynamic characteristics of the Nanpu Bridge. In these two models, the main girder of the bridge and the two H-shaped pylon are modelled by the 3D beam finite elements (Cheng, 1999). The difference between these two models is in the way the stay cables are modelled. In the first model, the stay cables are modelled by single-truss elements, whereas in the second model, the cables are modelled by the proposed 3-node cable elements.
Modal analysis The modal analysis results of the Nanpu Bridge are presented in Table 1. The results show that considering the cable vibration effect in the dynamic model of the Nanpu Bridge, additional pure cable vibration modes and new cable-deck coupled vibration modes are identified in the modal analysis. These additional modes are found in the frequency spectrum between the previously determined modes of the bridge without considering the cable vibration effect. The new cable-deck coupled modes involve the transverse motions of the stay cables in the cable plane and the torsional vibration of the main girder in the center span, with the cable motions as the dominant component. These coupled vibration behaviour as well as the pure cable vibration modes cannot be identified by the single-truss model. Comparing the results obtained from the two models, it is noted that the natural frequencies of the corresponding bridge modes are close to each other, with those obtained from the 3-node cable model slightly lower. This is because in the 3-node cable model, the oscillations of the cables increase the vibrating mass of the bridge system. With approximately the same stiffness but a greater mass, the natural frequencies of the vibration modes obtained from the 3-node cable model are therefore lower. In the same frequency range up to 0.SHz, there are 14 modes extracted from the single-truss cable model, whereas more than 200 modes are identified by the 3-node cable model. The additional cable vibration modes may have a significant impact on the dynamic performance and long-term safety of the bridge system. The mode shapes of some typical bridge vibration modes obtained by using the 3-node cable model are shown in Figure 2. The interaction of the cable vibration behaviour with the motion of the main girder and the pylons is observed. The fundamental frequency of the longest cable in the Nanpu Bridge obtained from the modal analysis is 0.476Hz, whereas that of the shortest is 1.579Hz. The frequencies of the cable vibration modes fall in the frequency range between modes 5 to 45 of the bridge structure, i.e. between 0.426Hz and 1.607Hz. For dynamic motions of the bridge in this frequency range, the vibration of the cables and their interaction with the motions of the deck and the pylons can be significant, which can severely affect the response characteristics of the bridge.
D y n a m i c cable s t i f f n e s s In the proposed 3-node cable model (Cheng, 1999; Cheng & Lau, 2001), the interaction effect between the cable vibration and that of the other parts of the bridge superstructure is represented by the dynamic cable stiffness K(w)=
K~ [1-~'~)2] [ ' ']
where Ke is the elastic stiffness, Kg is the geometric stiffness, w is r-------
the circular frequency of the structure, w0 = ~ ~/Tw--~is the fundamental frequency of the "taut wire". By m o d i f y i n g the longitudinal stiffness, the extensional behaviour of the cable is influenced. In recent years, a number of cases of cable replacements have been reported because of the corrosion and fatigue problems in the cables. These two phenomena occur simultaneously and reinforce each other. The tension in the cable can drop significantly due to the damage to the wires inside the stay cable. To study the effect of the tension loss on the dynamic behaviour of the stay cables, the dynamic stiffness of the longest and the shortest cables of the Nanpu Bridge are investigated by modifying the tension in the cable taken as 0.75TDL, 0.50TDL and 0.25TDL respectively, where TDL is the initial cable tension caused by the
1561 dead load. From the results presented in Table 2, it can be noticed that the value of the dynamic cable stiffness depends more on the tension in the cable than on the external exciting frequency of the bridge motion. A negative value of the extensional stiffness of the longest cable occurs under the condition when the tension force is reduced toO.25TDL and the oscillating frequency of the bridge is 1.0Hz. Physically, this negative stiffness indicates that instead of resisting the enlongation of the cable, the deformation tend to increase. The cable then is under an unstable condition. If the cable tension is reduced toO.5OTDL, when the bridge vibrates with a frequency of 0.2Hz, the extensional stiffness of the longest cable considering the dynamic interaction effect decreases by 15.18% to 0.837 x 107N/m. The effect of the dynamic interaction between the cables and the other parts of the superstructure on the extensional stiffness of the shortest cable is much less than that on the longest cable. Assuming there is no tension loss due to corrosion or fatigue problems of the cable, the dynamic cable stiffness of the longest and the shortest cables in the Nanpu Bridge are determined for the bridge natural frequencies at 0.1Hz, 0.2Hz, 0.5Hz and 1.0Hz. No significant changes in the extensional stiffness of the cables are found. If the dynamic interaction between the cable and the other parts of the superstructure is neglected, the extensional cable stiffness obtained is determined to be 0.9862 × 107N/m. Considering the dynamic stiffness for the bridge structural frequency of 0.5Hz, the extensional cable stiffness is changed to 0.9861 x 107N/m with only a difference of 0.01%. For the case of the extensional stiffness K(w) of the cable approaches the critical state of zero, the corresponding bridge frequency for the longest cable is found to be 7.76Hz, and for the shortest cable 105.6Hz. These frequencies are obviously beyond the bridge frequency range that are typical of cable-stayed bridges excited by earthequakes, wind and traffic. For the case of the shortest cable, the dynamic stiffness effect is obviously negligible. Thus, if there is no tension loss due to corrosion or fatigue problem,the high pre-tension and relatively low mass of the cables reduce the significance of impact of the dynamic stiffness on the extensional behaviour of the stay cables in cable-stayed bridges. The effect observed here is not as significant as that reported by Davenport and Steels(1965) on the behaviour of the guy cables, which typically have a lower initial tension and a greater massthan the stay cables in cable-stayed bridges. However, in reality the stay cables are usually subjected to deterioration due to the corrosion and fatigue problems, the tension in the cable may be reduced significantly. The dynamic stiffness effect may become more significant and thus should then be considered in the analysis, especially for the long stay cables in bridges.
PARAMETRIC
STUDY
To better understand the cable vibration behaviour on the dynamic responses of modern cable-stayed bridges, parametric studies are carried out to investigate its impact on bridges of different structural layouts in terms of span length, pylon shape and cable arrangement type. Four different center span lengths are considered, which are 300m, 423m ( span length of the Nanpu Bridge), 500m and 600m. The range covers the typical span length of modern medium- to long-span cable-stayed bridges. Two cases of typical cable arrangement in modern cable-stayed bridges are considered, i.e. the parallel cable-plane arrangement and the inclined cable-plane arrangement. Correspondingly, the associated pylon shapes are taken as the H shape as in the Nanpu Bridge, and the inverted-Y shape as shown in Figure 3. The combinations of these structural parameters yield eight different example bridges.
Pylon shape For the example bridge considered here, the span length, the cross-section of the main girder, the crosssectional area of the stay cables and the boundary conditions are taken the same as those of the Nanpu Bridge. In the parametric studies, the shape of the pylons is changed to an inverted-Y shape, as shown in Figure 4. Consequently, the parallel cable-plane arrangement of the Nanpu Bridge is changed to the inclined cable-plane arrangement in some of the parametric cases. Two sets of modal analysis results, of which the stay cables are modeled respectively by the single-truss element and the 3-node element, are compared. As discussed befored, numerous pure cable vibration modes
1562
are extracted by using the 3-node cable model. Corresponding to the first ten bridge structural vibration modes obtained from the single-truss cable model, 144 new pure cable vibration modes are identified by using the 3-node cable model. No cable-deck coupled mode is identified in this case. The torsional frequencies are increased when considering the influence of the cable vibration behaviour, whereas for the other bridge modes, the modal frequencies are slightly decreased. This is different from the case of the Nanpu Bridge having H-shaped pylons, of which the bridge natural frequencies, including the torsional modes, all decrease due to the influence of the transverse motion of the cables. Comparing the modal analysis results with those obtained from the Nanpu Bridge, it can be observed that by changing the pylon shape from H to the inverted-Y, the frequency of the floating mode is slightly decreased, whereas those for the heave modes are slightly increased. A 3.94% increase in the frequency of the sway mode is also observed. For the torsional modes, the impact is more significant. The frequency of the lowest symmetric torsional mode is increased by 32% from 0.495Hz to 0.654Hz, whereas that of the lowest anti-symmetric torsional mode is increased by 20.2% from 0.607Hz to 0.730Hz. This is because that the inverted-Y shape pylon has significantly higher torsional resistance than the H-shaped one, thus can suppress more effectively the torsional motion of the girder.
V a r i a t i o n o f span length The impact of varying the span length on the natural frequencies of cable-stayed bridges with parallel cable-plane and the inclined cable-plane are shown in Figures 4 and 5, respectively. The frequencies of six typical low order modes are plotted against span length. The change of the frequencies of the fundamental floating mode (floating), the lowest symmetric sway mode (l-S-L), the lowest symmetric heave (l-S-V) and anti-symmetric heave (l-A-V) modes, and the lowest symmetric torsional (l-S-T) and anti-symmetric torsional (l-A-T) modes are presented. Two curves are drawn for each mode. The solid lines represent the results of neglecting the cable vibration behaviour and the dotted lines represent the results of considering the cable vibration behaviour. In all the cases analyzed, the frequencies drop more rapidly when the span length changes from a medium span of 300m to a long span of 423m. Changing from the span length of 423m to 600m, the changes in frequencies are more gentle. It is observed that the variation in the span length has a significant impact on the sway behaviour of the bridge girder. When the center span length is doubled from 300m to 600m, the sway frequency is reduced by 75.5%. The reduction percentage of the sway mode in the case of the inclined cable-plane arrangement is found to be almost the same. As shown in these two figures, the natural frequencies of the floating, the sway and the heave modes are hardly affected by the cable vibration behaviour, whereas the frequencies of the torsional modes are slightly changed due to the transverse motion of the cables. No obvious relationship is identified between the cable vibration effect on changes of the modal frequencies and the variation of the span length.
Cable a r r a n g e m e n t type Comparing the results shown in Figures 4 and 5, the frequencies of the floating, the sway and the heave modes are not significantly affected by the change in the cable arrangement type. However, the frequencies of the torsional modes are significantly increased when the type of cable arrangement is changed from the parallel to the inclined. For the lowest symmetric torsional mode, a 32.8% increase is observed for the case of span length of 300m, and a 37.58% increase is observed for the case of span length 600m. These increases in torsional frequencyare due to the fact that the inclined cable-plane arrangement type is more efficient in resisting the torsional response motion in the bridge system. The cable vibration behaviour hardly affects the frequencies of the floating, the sway and the heave modes of both types of cable arrangement. By taking into account the transverse cable motions, the obtained frequencies of these modes are slightly lower. For the torsional modes, the same trend of decrease in the frequency values due to the cable vibration effect is observed when the bridge has a parallel cable-plane
1563 arrangement, whereas the torsional frequencies are found to slightly increase in the cases when the inclined cable-plane arrangement is adopted for the bridge. SUMMARY
The effects of cable vibration behaviour on the dynamic characteristics of the cable-stayed bridges are studied in this paper. A 3-node cable model, which can represent accurately not only the transverse motion of the cable, but the interaction between the motion of the cables and the other parts of the superstructure by the dynamic stiffness in the longitudinal direction, is employed in the analysis. Results obtained from the case study of Nanpu Bridge show that the natural frequencies, the mode shapes and the modal order of the cable-stayed bridges are affected by considering the cable vibration behaviour. A large number of pure cable vibration modes as well as new cable-deck coupled modes are identified. The dynamic cable stiffness depends predominantly on the cable tension. Parametric study of eight example bridges show that the inverted-Y shape pylon and the inclined cable-plane arrangement result in higher torsional rigidity for the bridge system than the H-shaped pylon and the parallel cable-plane arrangement. For medium to long-span cable-stayed bridges, the frequencies of the floating, the sway and the heave modes are not much affected by the cable motion, whereas the torsional frequencies are slightly changed. The cable motions tend to decrease the modal frequencies, except for the torsional modes of the bridges having inclined cable-plane arrangement.
References 1. Abdel-Ghaffar, A.M. and Khalifa, M.A.(1991) Importance of Cable Vibrations in Dynamics of Cablestayed Bridges. Journal of Engineering Mechanics, ASCE. Vol. 117, No.11: 2571-2589. 2. Abdel-Ghaffar, A.M. and Nazmy, A.S.(1991). 3-D Non-linear Seismic Behaviour of cable-stayed bridges. Journal of Engineering Mechanics, ASCE Vol. 117, N o . l l : 3456-3476. 3. Baron, F., and Lien, S.(1973) Analytical Studies of a Cable-stayed Bridge. Computers and Structures Vol. 3: 443-465. 4. Cheng, S.H. (1999) Structural and Aerodynamic Stability Analysis of Long-span Cable-stayed Bridges.
Ph.D. thesis, Department of Civil and Environment Engineering, Carleton University, Ottawa, Canada. 5. Cheng, S.H., and Lau, D.T.(2001) Parametric Study of Cable Vibration Effects on the Dynamic Response of Cable-stayed Bridges. Proceedings of International Conference on Structural Engineering, Mechanics and Computation, Cape Town, South Africa, 2-~ April, Paper Ref: SEMC2001/211. 6. Davenport, A.G.(1994) A Simple Representation of the Dynamics of a Massive Stay Cable in Wind.
Proceeding of International Conference on Cable-Stayed and Suspension Bridges, Deauville. October Vol.2: 427-438. 7. Davenport, A.G., and Steels, G.N.(1965) Dynamic Behaviour of Massive Guy Cables. Journal of Structure Division, ASCE Vol.91.ST2: 43. 8. Fleming, J.F., and Egeseli, E.A.(1980). Dynamic behaviour of a cable-stayed bridge. Earthquake Engineering and Structural Dynamics 8(1): 1-16. 9. Nazmy, A.S., and Abdel-Ghaffar, A.M.(1992) Effects of ground motion spatial variability on the response of cable-stayed bridges. Earthquake Engineering and Structural Dynamics Vol. 21: 1-20. 10. Parvez, S.M., and Wieland, M.(1987). Earthquake behaviour of proposed multi-span cable-stayed bridge over river Jamuna in Bangladesh. Proceedings of International Conference on Cable-stayed Bridges, Bangkok, Thailand, 479-489. 11. Tuladhar R., and Brotton, D.M.(1987) A Computer Program for Nonlinear Dynamic Analysis of Cable-stayed Bridges under Seismic Loading. Proceedings of International Conference on Cablestayed Bridges, Bangkok, Thailand, 315-326. 12. Tuladhar R., Dilger, W.H. and Elbadry, M.M.(1995) Influence of Cable Vibration on Seismic Response of Cable-stayed Bridges. Canadian Journal of Civil Engineering Vol. 22: 1001-1020. 13. Wethyavivorn, B.(1987) Dynamic behaviour of Cable-stayed Bridges. Ph.D. thesis, Department of
Civil Engineering, The University of Pittsburgh, Pittsburgh, Pa. 14. Yiu, P.K.A.(1982) Static and Dynamic Behaviour of Cable Assisted Bridges. PhD. thesis, Department
of Civil Engineering, the University of Manchester, Manchester, United Kingdom.
1564
Table 1: IMPACT OF CABLE VIBRATION BEHAVIOUR ON THE DYNAMIC CHARACTERISTICS OF NANPU BRIDGE
Mode 1 2 3 4
Single-Truss Cable Element No. Frequency (Hz) 0.1216 0.3433 0.3687 0.4255
5 6
0.4992 0.5069
7
0.5392
8
0.6106
Proposed Three-Node Cable Element Mode No. Frequency (Hz) 1 0.1216 2 0.3422 3 0.3679 4 0.4246 5 0.4696 6-12 0.4758-0.4767 13 0.4829 14-20 0.4890-0.4901 21 0.4951 22 0.5042 23-30 0.5256-0.5262 31 0.5376 32-38 0.5407-0.5419 39 0.5448 40-47 0.5464-0.5474 48 0.5537 49-54 0.5571-0.5577 55 0.5599 56-63 0.6048-0.6058 64 0.6073
Mode Shape Floating(girder, Anti-Sym.) Heave(girder, Sym.) Sway(girder, Sym.) Heave(girder, Anti-Sym.) New mode * Pure cable motion (#C22) New mode * Pure cable motion (#C21) Torsion(girder, Sym.) Bending(pylon, Anti-Sym.) Pure cable motion (#$22) Bending(pylon, Sym.) Pure cable motion (#C20) New mode * Pure cable motion (#$21) New mode * Pure cable motion (#C19) New mode * Pure cable motion (#$20) Torsion (girder, Anti-Sym.)
* New mode: Dominant cable vibration coupled with torsion of main girder at the center span.
Table 2: I M P A C T OF C A B L E T E N S I O N VARIATION ON T H E DYNAMIC C A B L E S T I F F N E S S Tension
(N) 0.75TDL
0.50TDL
0.25TDL
w
(rad/s) 0.1 0.2 0.5 1.0 0.1 0.2 0.5 1.0
x 2r x 2r x 27r x 2~r X 2r x 2r x 2u x 2r
0.1 0.2 0.5 1.0
x 2r X 2r X 2r X 2~
Dynamic Cable Stiffness K(w) (N/m) Longest Cable Shortest Cable 0.9530x 107 3.0847 x 107 0.9529 x 107 3.0847 x 107 0.9524x 107 3.0847 x 107 0.9507x 107 3.0847 x 107 0.8370 X 107 3.0620 X 107 0.8365 x 107 3.0620 x 107 0.8323 x 107 3.0620 x 107 0.8156 x 107 3.0620 × 107 0.3766 x 107 2.8538 x 107 0.3689 X 107 2.8538 X 107 0.3092 × 107 2.8535 X 107 -0.3964 X 107 2.8525 X 107
1565 I _
171000
21 1 5 0 0
:
I (All dimensions in ram)
Figure 1" Structural layout of Nanpu Bridge(half)
•••••11 M o d e 2 F r e q u e n c y = 0.342Hz
M o d e 21 F r e q u e n c y = 0 . 4 9 5 H z
M o d e 4 F r e q u e n c y = 0.425Hz
M o d e 64 F r e q u e n c y = 0 . 6 0 7 H z
Figure 2: Mode shapes of the Nanpu Bridge (Three-node cable model)
(a)
f*)
Figure 3: Modification of the pylon shape(a) Pylon of the original Nanpu Bridge (b) Modified pylon of the example bridge
1566 1.0
------- Not consider cable vibration effect ation effect
0.8
~
0.6
-
0.4
-
ii;i
0.2
-
1-$-L
2;
floating 0.0 200
t
t
t
v
300
400
500
600
~
700
Span length L (m)
Figure 4: Impact of span length variation on the natural frequency of parallel cable-plane cable-stayed bridges
1.o
~ 0.8
]
l
NotconsidercableJvibrationeffect
-
N ~oe
-
ID ~ 0.4
-
I-A-V I-S-V
Z 0.2
-
I-S-L floating
0.0 200
=
300
v
t
t
400
500
600
700
Span length L (m)
Figure 5: Impact of span length variation on the natural frequency of inclined cable-plane cable-stayed bridges
1559
Parametric Study of Cable Vibration Effects on the D y n a m i c Response of Cable-stayed Bridges
S.H.Cheng
D.T.Lau
Ottawa-Carleton Bridge Research Institute, Department of Civil and Environmental Engineering, Carleton University, Ottawa, Ontario, Cananda, KIS 5B6
ABSTRACT: In this paper, a 3-node cable model is employed to investigate the cable vibration effects on the dynamic responses of cable-stayed bridges. The transverse vibration of the stay cables in cablestayed bridges under dynamic loads, and the interaction between the motions of the cables and the other parts of the bridge superstructure are considered. A comprehensive parametric study of eight example bridges is carried out. The parameters considered in the study include the span length, the pylon shape and the cable arrangement. The impact of considering the dynamic cable stiffness in the dynamic analysis of cable-stayed bridges is also discussed.
K E Y W O R D S : cable-stayed bridge, cable vibration, finite element models, dynamic cable stiffness, longspan structure, numerical, modal analysis
INTRODUCTION The common practice of modelling the stay cables in cable-stayed bridges by single-truss elements (Fleming Egeseli,1980; Parvez ~ Wieland, 1987; Wethyavivorn, 1987;Abdel-Ghaffar ~ Nazmy, 1991; Nazmy & Abdel-Ghaffar, 1992) cannot represent the lateral motions of the cables and the interaction effects with other parts of the bridge superstructure. The multi-link approach proposed in some recent studies (Baron & Lien, 1973; Yiu, 1982; Tuladhar & Brotton, 1987; Abdel-Ghaffar & Khalifa, 1991; Tuladhar et al, 1995) improves the accuracy in the modelling of the cable behaviour. However, by dividing each cable into multiple elements, the size of the analytical model is increased significantly. A 3-node cable model has been proposed in a recent study by Cheng and Lau (2001). The new cable model can simulate the transverse vibration of the cable and cable-bridge interaction effects. It is more computationally efficient than the multi-link model. A comprehensive parametric study using the 3-node cable model is carried out in the present study. Eight example bridges are analyzed to investigate the impact of considering cable vibration and the use of the dynamic cable stiffness on the dynamic responses of the cable-stayed bridges. Other parameters considered in the investigation include the span length, the pylon shape and the cable arrangement. CASE STUDY OF NANPU BRIDGE Nanpu Bridge is a long-span cable-stayed bridge located in Shanghai, China. It has a center span of 423m, as shown in Figure 1. The bridge girder has a composite deck cross-section comprised of a prestressed
1560
concrete slab supported by two /-shaped steel beams. Two different numerical models are employed to study the cable vibration effect on the dynamic characteristics of the Nanpu Bridge. In these two models, the main girder of the bridge and the two H-shaped pylon are modelled by the 3D beam finite elements (Cheng, 1999). The difference between these two models is in the way the stay cables are modelled. In the first model, the stay cables are modelled by single-truss elements, whereas in the second model, the cables are modelled by the proposed 3-node cable elements.
Modal analysis The modal analysis results of the Nanpu Bridge are presented in Table 1. The results show that considering the cable vibration effect in the dynamic model of the Nanpu Bridge, additional pure cable vibration modes and new cable-deck coupled vibration modes are identified in the modal analysis. These additional modes are found in the frequency spectrum between the previously determined modes of the bridge without considering the cable vibration effect. The new cable-deck coupled modes involve the transverse motions of the stay cables in the cable plane and the torsional vibration of the main girder in the center span, with the cable motions as the dominant component. These coupled vibration behaviour as well as the pure cable vibration modes cannot be identified by the single-truss model. Comparing the results obtained from the two models, it is noted that the natural frequencies of the corresponding bridge modes are close to each other, with those obtained from the 3-node cable model slightly lower. This is because in the 3-node cable model, the oscillations of the cables increase the vibrating mass of the bridge system. With approximately the same stiffness but a greater mass, the natural frequencies of the vibration modes obtained from the 3-node cable model are therefore lower. In the same frequency range up to 0.SHz, there are 14 modes extracted from the single-truss cable model, whereas more than 200 modes are identified by the 3-node cable model. The additional cable vibration modes may have a significant impact on the dynamic performance and long-term safety of the bridge system. The mode shapes of some typical bridge vibration modes obtained by using the 3-node cable model are shown in Figure 2. The interaction of the cable vibration behaviour with the motion of the main girder and the pylons is observed. The fundamental frequency of the longest cable in the Nanpu Bridge obtained from the modal analysis is 0.476Hz, whereas that of the shortest is 1.579Hz. The frequencies of the cable vibration modes fall in the frequency range between modes 5 to 45 of the bridge structure, i.e. between 0.426Hz and 1.607Hz. For dynamic motions of the bridge in this frequency range, the vibration of the cables and their interaction with the motions of the deck and the pylons can be significant, which can severely affect the response characteristics of the bridge.
D y n a m i c cable s t i f f n e s s In the proposed 3-node cable model (Cheng, 1999; Cheng & Lau, 2001), the interaction effect between the cable vibration and that of the other parts of the bridge superstructure is represented by the dynamic cable stiffness K(w)=
K~ [1-~'~)2] [ ' ']
where Ke is the elastic stiffness, Kg is the geometric stiffness, w is r-------
the circular frequency of the structure, w0 = ~ ~/Tw--~is the fundamental frequency of the "taut wire". By m o d i f y i n g the longitudinal stiffness, the extensional behaviour of the cable is influenced. In recent years, a number of cases of cable replacements have been reported because of the corrosion and fatigue problems in the cables. These two phenomena occur simultaneously and reinforce each other. The tension in the cable can drop significantly due to the damage to the wires inside the stay cable. To study the effect of the tension loss on the dynamic behaviour of the stay cables, the dynamic stiffness of the longest and the shortest cables of the Nanpu Bridge are investigated by modifying the tension in the cable taken as 0.75TDL, 0.50TDL and 0.25TDL respectively, where TDL is the initial cable tension caused by the
1561 dead load. From the results presented in Table 2, it can be noticed that the value of the dynamic cable stiffness depends more on the tension in the cable than on the external exciting frequency of the bridge motion. A negative value of the extensional stiffness of the longest cable occurs under the condition when the tension force is reduced toO.25TDL and the oscillating frequency of the bridge is 1.0Hz. Physically, this negative stiffness indicates that instead of resisting the enlongation of the cable, the deformation tend to increase. The cable then is under an unstable condition. If the cable tension is reduced toO.5OTDL, when the bridge vibrates with a frequency of 0.2Hz, the extensional stiffness of the longest cable considering the dynamic interaction effect decreases by 15.18% to 0.837 x 107N/m. The effect of the dynamic interaction between the cables and the other parts of the superstructure on the extensional stiffness of the shortest cable is much less than that on the longest cable. Assuming there is no tension loss due to corrosion or fatigue problems of the cable, the dynamic cable stiffness of the longest and the shortest cables in the Nanpu Bridge are determined for the bridge natural frequencies at 0.1Hz, 0.2Hz, 0.5Hz and 1.0Hz. No significant changes in the extensional stiffness of the cables are found. If the dynamic interaction between the cable and the other parts of the superstructure is neglected, the extensional cable stiffness obtained is determined to be 0.9862 × 107N/m. Considering the dynamic stiffness for the bridge structural frequency of 0.5Hz, the extensional cable stiffness is changed to 0.9861 x 107N/m with only a difference of 0.01%. For the case of the extensional stiffness K(w) of the cable approaches the critical state of zero, the corresponding bridge frequency for the longest cable is found to be 7.76Hz, and for the shortest cable 105.6Hz. These frequencies are obviously beyond the bridge frequency range that are typical of cable-stayed bridges excited by earthequakes, wind and traffic. For the case of the shortest cable, the dynamic stiffness effect is obviously negligible. Thus, if there is no tension loss due to corrosion or fatigue problem,the high pre-tension and relatively low mass of the cables reduce the significance of impact of the dynamic stiffness on the extensional behaviour of the stay cables in cable-stayed bridges. The effect observed here is not as significant as that reported by Davenport and Steels(1965) on the behaviour of the guy cables, which typically have a lower initial tension and a greater massthan the stay cables in cable-stayed bridges. However, in reality the stay cables are usually subjected to deterioration due to the corrosion and fatigue problems, the tension in the cable may be reduced significantly. The dynamic stiffness effect may become more significant and thus should then be considered in the analysis, especially for the long stay cables in bridges.
PARAMETRIC
STUDY
To better understand the cable vibration behaviour on the dynamic responses of modern cable-stayed bridges, parametric studies are carried out to investigate its impact on bridges of different structural layouts in terms of span length, pylon shape and cable arrangement type. Four different center span lengths are considered, which are 300m, 423m ( span length of the Nanpu Bridge), 500m and 600m. The range covers the typical span length of modern medium- to long-span cable-stayed bridges. Two cases of typical cable arrangement in modern cable-stayed bridges are considered, i.e. the parallel cable-plane arrangement and the inclined cable-plane arrangement. Correspondingly, the associated pylon shapes are taken as the H shape as in the Nanpu Bridge, and the inverted-Y shape as shown in Figure 3. The combinations of these structural parameters yield eight different example bridges.
Pylon shape For the example bridge considered here, the span length, the cross-section of the main girder, the crosssectional area of the stay cables and the boundary conditions are taken the same as those of the Nanpu Bridge. In the parametric studies, the shape of the pylons is changed to an inverted-Y shape, as shown in Figure 4. Consequently, the parallel cable-plane arrangement of the Nanpu Bridge is changed to the inclined cable-plane arrangement in some of the parametric cases. Two sets of modal analysis results, of which the stay cables are modeled respectively by the single-truss element and the 3-node element, are compared. As discussed befored, numerous pure cable vibration modes
1562
are extracted by using the 3-node cable model. Corresponding to the first ten bridge structural vibration modes obtained from the single-truss cable model, 144 new pure cable vibration modes are identified by using the 3-node cable model. No cable-deck coupled mode is identified in this case. The torsional frequencies are increased when considering the influence of the cable vibration behaviour, whereas for the other bridge modes, the modal frequencies are slightly decreased. This is different from the case of the Nanpu Bridge having H-shaped pylons, of which the bridge natural frequencies, including the torsional modes, all decrease due to the influence of the transverse motion of the cables. Comparing the modal analysis results with those obtained from the Nanpu Bridge, it can be observed that by changing the pylon shape from H to the inverted-Y, the frequency of the floating mode is slightly decreased, whereas those for the heave modes are slightly increased. A 3.94% increase in the frequency of the sway mode is also observed. For the torsional modes, the impact is more significant. The frequency of the lowest symmetric torsional mode is increased by 32% from 0.495Hz to 0.654Hz, whereas that of the lowest anti-symmetric torsional mode is increased by 20.2% from 0.607Hz to 0.730Hz. This is because that the inverted-Y shape pylon has significantly higher torsional resistance than the H-shaped one, thus can suppress more effectively the torsional motion of the girder.
V a r i a t i o n o f span length The impact of varying the span length on the natural frequencies of cable-stayed bridges with parallel cable-plane and the inclined cable-plane are shown in Figures 4 and 5, respectively. The frequencies of six typical low order modes are plotted against span length. The change of the frequencies of the fundamental floating mode (floating), the lowest symmetric sway mode (l-S-L), the lowest symmetric heave (l-S-V) and anti-symmetric heave (l-A-V) modes, and the lowest symmetric torsional (l-S-T) and anti-symmetric torsional (l-A-T) modes are presented. Two curves are drawn for each mode. The solid lines represent the results of neglecting the cable vibration behaviour and the dotted lines represent the results of considering the cable vibration behaviour. In all the cases analyzed, the frequencies drop more rapidly when the span length changes from a medium span of 300m to a long span of 423m. Changing from the span length of 423m to 600m, the changes in frequencies are more gentle. It is observed that the variation in the span length has a significant impact on the sway behaviour of the bridge girder. When the center span length is doubled from 300m to 600m, the sway frequency is reduced by 75.5%. The reduction percentage of the sway mode in the case of the inclined cable-plane arrangement is found to be almost the same. As shown in these two figures, the natural frequencies of the floating, the sway and the heave modes are hardly affected by the cable vibration behaviour, whereas the frequencies of the torsional modes are slightly changed due to the transverse motion of the cables. No obvious relationship is identified between the cable vibration effect on changes of the modal frequencies and the variation of the span length.
Cable a r r a n g e m e n t type Comparing the results shown in Figures 4 and 5, the frequencies of the floating, the sway and the heave modes are not significantly affected by the change in the cable arrangement type. However, the frequencies of the torsional modes are significantly increased when the type of cable arrangement is changed from the parallel to the inclined. For the lowest symmetric torsional mode, a 32.8% increase is observed for the case of span length of 300m, and a 37.58% increase is observed for the case of span length 600m. These increases in torsional frequencyare due to the fact that the inclined cable-plane arrangement type is more efficient in resisting the torsional response motion in the bridge system. The cable vibration behaviour hardly affects the frequencies of the floating, the sway and the heave modes of both types of cable arrangement. By taking into account the transverse cable motions, the obtained frequencies of these modes are slightly lower. For the torsional modes, the same trend of decrease in the frequency values due to the cable vibration effect is observed when the bridge has a parallel cable-plane
1563 arrangement, whereas the torsional frequencies are found to slightly increase in the cases when the inclined cable-plane arrangement is adopted for the bridge. SUMMARY
The effects of cable vibration behaviour on the dynamic characteristics of the cable-stayed bridges are studied in this paper. A 3-node cable model, which can represent accurately not only the transverse motion of the cable, but the interaction between the motion of the cables and the other parts of the superstructure by the dynamic stiffness in the longitudinal direction, is employed in the analysis. Results obtained from the case study of Nanpu Bridge show that the natural frequencies, the mode shapes and the modal order of the cable-stayed bridges are affected by considering the cable vibration behaviour. A large number of pure cable vibration modes as well as new cable-deck coupled modes are identified. The dynamic cable stiffness depends predominantly on the cable tension. Parametric study of eight example bridges show that the inverted-Y shape pylon and the inclined cable-plane arrangement result in higher torsional rigidity for the bridge system than the H-shaped pylon and the parallel cable-plane arrangement. For medium to long-span cable-stayed bridges, the frequencies of the floating, the sway and the heave modes are not much affected by the cable motion, whereas the torsional frequencies are slightly changed. The cable motions tend to decrease the modal frequencies, except for the torsional modes of the bridges having inclined cable-plane arrangement.
References 1. Abdel-Ghaffar, A.M. and Khalifa, M.A.(1991) Importance of Cable Vibrations in Dynamics of Cablestayed Bridges. Journal of Engineering Mechanics, ASCE. Vol. 117, No.11: 2571-2589. 2. Abdel-Ghaffar, A.M. and Nazmy, A.S.(1991). 3-D Non-linear Seismic Behaviour of cable-stayed bridges. Journal of Engineering Mechanics, ASCE Vol. 117, N o . l l : 3456-3476. 3. Baron, F., and Lien, S.(1973) Analytical Studies of a Cable-stayed Bridge. Computers and Structures Vol. 3: 443-465. 4. Cheng, S.H. (1999) Structural and Aerodynamic Stability Analysis of Long-span Cable-stayed Bridges.
Ph.D. thesis, Department of Civil and Environment Engineering, Carleton University, Ottawa, Canada. 5. Cheng, S.H., and Lau, D.T.(2001) Parametric Study of Cable Vibration Effects on the Dynamic Response of Cable-stayed Bridges. Proceedings of International Conference on Structural Engineering, Mechanics and Computation, Cape Town, South Africa, 2-~ April, Paper Ref: SEMC2001/211. 6. Davenport, A.G.(1994) A Simple Representation of the Dynamics of a Massive Stay Cable in Wind.
Proceeding of International Conference on Cable-Stayed and Suspension Bridges, Deauville. October Vol.2: 427-438. 7. Davenport, A.G., and Steels, G.N.(1965) Dynamic Behaviour of Massive Guy Cables. Journal of Structure Division, ASCE Vol.91.ST2: 43. 8. Fleming, J.F., and Egeseli, E.A.(1980). Dynamic behaviour of a cable-stayed bridge. Earthquake Engineering and Structural Dynamics 8(1): 1-16. 9. Nazmy, A.S., and Abdel-Ghaffar, A.M.(1992) Effects of ground motion spatial variability on the response of cable-stayed bridges. Earthquake Engineering and Structural Dynamics Vol. 21: 1-20. 10. Parvez, S.M., and Wieland, M.(1987). Earthquake behaviour of proposed multi-span cable-stayed bridge over river Jamuna in Bangladesh. Proceedings of International Conference on Cable-stayed Bridges, Bangkok, Thailand, 479-489. 11. Tuladhar R., and Brotton, D.M.(1987) A Computer Program for Nonlinear Dynamic Analysis of Cable-stayed Bridges under Seismic Loading. Proceedings of International Conference on Cablestayed Bridges, Bangkok, Thailand, 315-326. 12. Tuladhar R., Dilger, W.H. and Elbadry, M.M.(1995) Influence of Cable Vibration on Seismic Response of Cable-stayed Bridges. Canadian Journal of Civil Engineering Vol. 22: 1001-1020. 13. Wethyavivorn, B.(1987) Dynamic behaviour of Cable-stayed Bridges. Ph.D. thesis, Department of
Civil Engineering, The University of Pittsburgh, Pittsburgh, Pa. 14. Yiu, P.K.A.(1982) Static and Dynamic Behaviour of Cable Assisted Bridges. PhD. thesis, Department
of Civil Engineering, the University of Manchester, Manchester, United Kingdom.
1564
Table 1: IMPACT OF CABLE VIBRATION BEHAVIOUR ON THE DYNAMIC CHARACTERISTICS OF NANPU BRIDGE
Mode 1 2 3 4
Single-Truss Cable Element No. Frequency (Hz) 0.1216 0.3433 0.3687 0.4255
5 6
0.4992 0.5069
7
0.5392
8
0.6106
Proposed Three-Node Cable Element Mode No. Frequency (Hz) 1 0.1216 2 0.3422 3 0.3679 4 0.4246 5 0.4696 6-12 0.4758-0.4767 13 0.4829 14-20 0.4890-0.4901 21 0.4951 22 0.5042 23-30 0.5256-0.5262 31 0.5376 32-38 0.5407-0.5419 39 0.5448 40-47 0.5464-0.5474 48 0.5537 49-54 0.5571-0.5577 55 0.5599 56-63 0.6048-0.6058 64 0.6073
Mode Shape Floating(girder, Anti-Sym.) Heave(girder, Sym.) Sway(girder, Sym.) Heave(girder, Anti-Sym.) New mode * Pure cable motion (#C22) New mode * Pure cable motion (#C21) Torsion(girder, Sym.) Bending(pylon, Anti-Sym.) Pure cable motion (#$22) Bending(pylon, Sym.) Pure cable motion (#C20) New mode * Pure cable motion (#$21) New mode * Pure cable motion (#C19) New mode * Pure cable motion (#$20) Torsion (girder, Anti-Sym.)
* New mode: Dominant cable vibration coupled with torsion of main girder at the center span.
Table 2: I M P A C T OF C A B L E T E N S I O N VARIATION ON T H E DYNAMIC C A B L E S T I F F N E S S Tension
(N) 0.75TDL
0.50TDL
0.25TDL
w
(rad/s) 0.1 0.2 0.5 1.0 0.1 0.2 0.5 1.0
x 2r x 2r x 27r x 2~r X 2r x 2r x 2u x 2r
0.1 0.2 0.5 1.0
x 2r X 2r X 2r X 2~
Dynamic Cable Stiffness K(w) (N/m) Longest Cable Shortest Cable 0.9530x 107 3.0847 x 107 0.9529 x 107 3.0847 x 107 0.9524x 107 3.0847 x 107 0.9507x 107 3.0847 x 107 0.8370 X 107 3.0620 X 107 0.8365 x 107 3.0620 x 107 0.8323 x 107 3.0620 x 107 0.8156 x 107 3.0620 × 107 0.3766 x 107 2.8538 x 107 0.3689 X 107 2.8538 X 107 0.3092 × 107 2.8535 X 107 -0.3964 X 107 2.8525 X 107
1565 I _
171000
21 1 5 0 0
:
I (All dimensions in ram)
Figure 1" Structural layout of Nanpu Bridge(half)
•••••11 M o d e 2 F r e q u e n c y = 0.342Hz
M o d e 21 F r e q u e n c y = 0 . 4 9 5 H z
M o d e 4 F r e q u e n c y = 0.425Hz
M o d e 64 F r e q u e n c y = 0 . 6 0 7 H z
Figure 2: Mode shapes of the Nanpu Bridge (Three-node cable model)
(a)
f*)
Figure 3: Modification of the pylon shape(a) Pylon of the original Nanpu Bridge (b) Modified pylon of the example bridge
1566 1.0
------- Not consider cable vibration effect ation effect
0.8
~
0.6
-
0.4
-
ii;i
0.2
-
1-$-L
2;
floating 0.0 200
t
t
t
v
300
400
500
600
~
700
Span length L (m)
Figure 4: Impact of span length variation on the natural frequency of parallel cable-plane cable-stayed bridges
1.o
~ 0.8
]
l
NotconsidercableJvibrationeffect
-
N ~oe
-
ID ~ 0.4
-
I-A-V I-S-V
Z 0.2
-
I-S-L floating
0.0 200
=
300
v
t
t
400
500
600
700
Span length L (m)
Figure 5: Impact of span length variation on the natural frequency of inclined cable-plane cable-stayed bridges