EngineeringStructures,Vol.
PII: S0141-0296(97)00158-7
19, No. 10, pp. 857-867, 1997
@ 1997 Elsevier Science Ltd
All rights reserved. Printed in Great Britain
0141-0296/97 $17.00 + 0.00
ELSEVIER
Modal analysis of tower-cable
system of Tsing Ma long
suspension bridge
Y. L. Xu, J. M. Ko and Z. Yu
Department of Civil and Structural Engineering, The Hong Kong Polytechni~ Universi~,
Hung Horn, Kowloon, Hong Kong
(Received August 1996; revised version accepted November 1996)
A three-dimensional dynamic finite element model is established
for the tower-cable system of the Tsing Ma long suspension bridge
which is currently under construction. The two bridge towers,
made up of reinforced concrete columns and deep prestressed concrete beams, are modelled by three-dimensional Timoshenko
beam elements with rigid arms at the connections between columns and beams. The main span and side span cables are modelled by three-node cable elements accounting for geometric nonlinearity and large elastic deflection. The modal analysis is then
performed to determine the dynamic characteristics and dynamic
interaction between the towers and cables. The results show that
at lower natural frequencies, the modes of vibration of the system
can be reasonably separated into in-plane modes and out-of-plane
modes. Dynamic interactions between the towers and cables are
significant at global natural frequencies in either in-plane or outof-plane vibration. There are many local natural frequencies at
which the cables vibrate but the towers remain stationary or have
relatively small modal motion only. The dynamic interactions
between the main span and side span cables are also observed at
some local natural frequencies. The finite element model and the
analytical results presented in this paper have been verified by
measuring the dynamic properties of the system. @ 1997 Elsevier
Science Ltd.
Keywords: finite element modelling, tower-cable system, long suspension bridge, modal analysis, dynamic interaction
I.
the construction of the bridge is exposed to the risk of very
high wind speeds. The dynamic characteristics and windinduced vibration of the bridge, therefore, should be considered for various stages of construction. Figure 1 shows
the tower-cable system of the Tsing Ma Bridge under construction.
Dynamic characteristics and wind-induced vibration of
free-standing towers in suspension bridges have been studied by many researchers 2. For very slender and low-damped
free-standing towers, some temporary or permanent damping devices may have to be installed on the towers to mitigate wind-induced vibration 3. The dynamic characteristics
of a single horizontal or inclined cable with small sag confused researchers for many years, but now they are well
understood 4. Ambient vibration measurement and the finite
element technique are also widely used to determine the
Introduction
Hong Kong's new port and airport developments are
located on Lantau Island, the largest undeveloped area in
Hong Kong. The key section of the transportation between
the new facilities and the existing commercial centres of
Hong Kong Island and Kowloon is the Lantau Fixed Crossing, in which the Tsing Ma long suspension bridge is the
central structure. When completed in mid-1997, the Tsing
Ma Bridge will carry a dual three-lane highway on the
upper level of the bridge deck and two railway tracks and
two protected carriageways on the lower level within the
bridge deck. The bridge will span the main shipping channel between Tsing Yi Island and Ma Wan Island with a
main span of 1377 m and a total length of 2160 m ~.
Since the Tsing Ma Bridge is built in a typhoon region,
857
Modal analysis of Tsing Ma bridge: Y. L. Xu et al.
858
Figure 1 Tower-cable system of Tsing Ma Bridge under construction 1
dynamic characteristics of completed long suspension
bridges 5 8. However, much less attention has been paid to
the dynamic characteristics of tower-cable systems of long
suspension bridges during the construction stage, but this
does not detract from their importance to the construction
of a long suspension bridge in a typhoon region.
The tower-cable system of the Tsing Ma Bridge was,
therefore, measured by the Hong Kong Polytechnic University using ambient vibration measurement techniques under
the auspices of the Highways Department of Hong Kong
Government. The measured results have been detailed in
the literature 9. This paper focuses on the development of
a three-dimensional dynamic finite element model for the
tower-cable system of the Tsing Ma suspension bridge.
After validation of the finite element model by comparing
computed results with measured results, the modal properties of the system are compared with those of the freestanding towers and the isolated main span and side span
cables, from which dynamic interactions between the towers and cables are identified. The dynamic characteristics
of the towers with cables obtained in this study provide a
clear picture of the changes in dynamic characteristics from
the free-standing towers to the tower-cable system. They
also shed light on the computation of dynamic response of
tower-cable systems in long suspension bridges under
strong wind and the possible vibration control.
tower saddles to the mare anchorage on the ground, forming a 300 m Tsing Yi side span. At Ma Wan the main
cables extended from the Ma Wan tower are secured first
by pier saddles at the deck level and a horizontal distance
about 355.5 m from the Ma Wan tower and then by main
anchorage saddles on the ground.
The tower legs are made of reinforced concrete, and the
concrete cross-beams are prestressed. The centre-lines of
the tower legs are 40 m apart at the base level reducing to
36 m at the top tower (see Figure 3). The width of the leg
is constant at 6 m in the tower plane, but in the in-plane
of the bridge the leg tapers from 18 m at the base level to
9 m at the top tower. Two hollow shafts are symmetrically
arranged inside each leg from the topmost cross-beam level
to a point approximately 15 m above the base level. The
cross-beams measure 7 m deep for the topmost one to 12 m
deep for the lowest one just beneath the bridge deck. Each
cross-beam includes a steel truss cast in concrete enclosing
a narrow corridor for access between either leg. The towers
are built on massive reinforced concrete slabs found on
competent rock.
Each main cable consists of 91 strands of parallel galvanized steel wires in the main span and 97 strands in the side
spans. The number of wires per strand is 360 or 368, and
the diameter of each wire is 5.38 mm. The cables are formed by the traditional aerial spinning technique. The resultant cables have an overall diameter of approximately 1.1 m.
2.
2.2. Finite element modelling of towers
Finite element modelling
2.1. Features of the tower-cable system
The Tsing Ma Bridge, stretching from Tsing Yi Island to
Ma Wan Island, has a main span of 1377 m between the
Tsing Yi tower in the east and the Ma Wan tower in the
west (see Figure 2). Each tower is made up of two legs
connected by four horizontal cross-beams (see Figure 3).
The height of the towers is 206.4 m measured from the base
level to the top saddle. The two main cables of 36 m apart
in the north and south are accommodated by the four
saddles located at the top of the tower legs in the main
span. Each tower saddle weighs approximately 500 t. On
the Tsing Yi side, the main cables are extended from the
The details of the finite element modelling of the bridge
towers have been described elsewhere ~°. The bridge towers
were represented by three-dimensional multilevel portal
frames with the two legs fixed at the base. The soilstructure interaction was not considered since the towers
are built on massive reinforced concrete slabs found on
competent rock. As the tower legs and the cross-beams are
relatively thick, the shear deformation and rotatory inertia
of the members were considered together with the bending
deformation.
Three-dimensional
Timoshenko
beam
elements with six degrees of freedom at one end were,
therefore, used to model the towers. Also due to the considerable size of the tower legs and cross-beams, the mem-
859
Modal analysis of Tsing Ma bridge: Y. L. Xu et al.
Ma Wan tower
t)q .~
355.5
1377
TsingYitower
300
i
2(16.4V Towersaddle
Side H~an cable J
IJ]
Main span cable
~
Towe~
/
!
4
~
1
cable
V +2.65
Main anchorage
All dimensions in m
West
._/-~ Mainanchorag/e
- - - - -
•
East
Figure 2 Configuration of tower-cable system of Tsing Ma Bridge
1311OO
IO0
M
36O0O
-~
1
I
70IX)
1
8lxlO
1
--
+201.009Ma Wan
+200.859
! TslngYi
+191.650Ma W u
] 2359
i .... i
2.3. Finite element modelling of cables
l
i
--
1
Rain levelat tower
+70.1XlO
+631gg6
121ll)-I----"~-
().(IIXl P D
+y~x)
+2.(11X1 I
,
,
:i
,
t
i
I
I
I
I 1+2.~AP
f
Front elevation
Figure3
considered insignificant for predicting the global structural
behaviour. Forty elements were used to model each tower
in the present analysis.
Section
Configurationof bridgetowers
ber length for calculating the flexural rigidities of the beam
element should be shorter than the length taken centre to
centre between the joints connecting the tower legs and the
cross-beams. Rigid arms were accordingly assigned to the
beam elements modelling the members at the joints.
The density of reinforced concrete was estimated to be
2530 kg/m 3 based on calculations from the amount of concrete and steel bars used in constructing the towers. The
Poisson ratio was taken as 0.2 for the reinforced concrete.
The short-term modulus of elasticity of the reinforced concrete was chosen as 30 GPa.
The geometric properties of the three-dimensional Timoshenko beam elements of the towers were calculated from
the design drawings. Since the cross-section of the tower
legs decreases from the base to the top, the sectional
properties of the beam elements for the tower legs were
assumed uniform along their axes with an average value.
The effective shear area of the elements was estimated by
considering the web area, and the torsional constant for the
tower legs was estimated by using the Salnt-Venant
expression. The effect of prestress in the cross-beams was
Three-node curved cable elements with nine degrees of
freedom, i.e., three translational movements in horizontal,
vertical, and transverse directions for each node, developed
by Henghold and Russell '~ were slightly modified and used
to model the main cables. Both geometric nonlinearity and
large elastic deflection were taken into account in the cable
elements. The 9 x 9 tangential stiffness matrix included linear elastic stiffness, static tension stiffness, and large
deflection stiffness. When deriving the stiffness matrix, a
parabolic displacement field was used, and the sectional
area of the cable element, the modulus of elasticity, and
the horizontal component of cable tension were assumed to
be constant. The nonlinear strain-displacement relationship
during deformation of the cable, which will lead to quadratic and cubic nonlinearities, was not considered. The consistent mass matrix of size 9 x 9 was derived using the same
parabolic displacement field as for the stiffness matrix.
Twenty four cable elements were used to model each main
span cable and eight elements were used to model each side
cable. These numbers were determined after a comparative
study of natural frequencies with the linear vibration theory
(Irvine's theory) within the interested frequency range.
Both material and geometric properties of the main
cables were taken from the design drawings. The crosssectional area is 0.759 m 2 for main span cables and
0.801 m 2 for side span cables. Under the free cable condition, the modulus of elasticity of the material is
200 kN/mm 2 at a temperature of 23°C. The mass density
per unit of cable length is, respectively, 5832 kg/m in the
main span and 6150 kg/m in the side spans. The horizontal
tension of the main cables under the free cable condition
was calculated to be 122.64 MN.
A computer program has been developed to carry out
modal analysis of the tower-cable system of the Tsing Ma
Bridge. The program can handle three-dimensional Timoshenko beam elements and three-node curved cable
elements. The accuracy of the program has been verified
through a comparison with the results given by some commercial software packages, such as SAP 9012, for the freestanding towers and by Irvine's theory for the main cables.
2.4. Conditions of construction
The properties of the tower-cable system described above
were slightly modified in consideration of the actual construction conditions of the system at the time of measuring
so that a reasonable comparison between the measured and
860
Modal analysis of Tsing Ma bridge: Y. L. Xu e t al.
numerical results could be conducted. For instance, the
masses of the working cranes and platforms fixed at the
top of the towers and the topmost cross-beams were added
to the corresponding nodes as additional lumped masses.
The masses of the cable bands and footbridge cross-walk
trusses used for constructing the main cables were taken
into account as additional distributed masses on the main
cables. After such adjustment, a good agreement between
the numerical and measured natural frequencies was achieved. The extensive numerical analysis was then carried
out and the results are presented in the following sections.
out-of-plane vibration is symmetric of a halt-wave (see
Figure 4). The computed mode shapes are in excellent
agreement with the theoretical mode shapes. Such a comparison indicates the accuracy of the cable element and that
enough cable elements are used in this study.
3.
Isolated cables and free-standing towers
The natural frequencies and modes of vibration of the isolated cables and free-standing towers were calculated first.
The isolated main span cable has a sag-to-span ratio of
about 1/12 and is suspended at its two fixed ends at the
same level. The isolated side span cables with two fixed
ends are inclined cables of a sag-to-span ratio less than
1/40. The catenary curves were used to calculate the static
equilibrium configuration of each cable. Because of small
sags in the cables, the linear theory (Irvine's theory) of free
vibration of cables can also be used to obtain the theoretical
natural frequencies and mode shapes. The same geometric
and material properties and the same number of elements
for each cable were used in the modal analysis of the
tower-cable system so that a reasonable comparison could
be made between the isolated cables and the cables in the
coupled tower-cable system.
3.1. Isolated main span cable
Table 1 lists only the first ten natural frequencies for inplane and out-of-plane vibration of the isolated main span
cable. The in-plane and out-of-plane motions of the cable
were uncoupled. The first natural frequency of in-plane
vibration of the cable is almost twice the first natural frequency of out-of-plane vibration. The natural frequencies
obtained from the linear vibration theory of cable (Irvine's
theory) are also displayed in Table 1. Table 1 only lists the
frequencies up to 0.53 Hz, but the two sets of results are
almost the same until 1.4 Hz. After that, relative differences, about 7%, appear between the two sets of results of
natural frequencies. The vertical component of the first
mode of in-plane vibration of the isolated main span cable
is antisymmetric of a single wave whilst the first mode of
3.2. Isolated side span cables
The natural frequencies for in-plane and out-of-plane
vibrations of both Ma Wan and Tsing Yi side span cables
are tabulated in Table 2 up to 1.4 Hz. These numerical
results are almost the same as those predicted by the linear
vibration theory of cable. The longer span of the Ma Wan
side cable leads to smaller natural frequencies than the
Tsing Yi side cable. This difference may affect the modal
properties of the coupled tower-cable system. The first outof-plane mode and the vertical component of the first inplane mode of the side span cable are both symmetric of
a half-wave, as shown in Figure 5. The numerical mode
shapes are again in good agreement with the theoretical
mode shapes.
The natural frequencies of both side span cables are,
however, much higher than those of the same order and in
the same plane of the main span cable. Some lower-order
natural frequencies of the two side span cables are close to
some higher-order natural frequencies of the main span
cable. One can expect some internal resonance between the
side span cables and the main span cable in the towercable system.
3.3. Free-standing towers
The first six natural frequencies, including the first two
bending modes, the first two torsional modes, and the first
two sway modes are tabulated in Table 3 for both the Ma
Wan and Tsing Yi towers. The mass matrix of each tower
included additional masses from the saddles, the cranes and
the working platforms at the top of the tower. For the Tsing
Yi tower, the additional masses from part of the deck on
the first cross-beam were considered. Therefore, there are
slight differences in natural frequencies between the two
towers. The first bending, sway, and torsional mode shapes
of the Ma Wan tower are shown in Figure 6.
Comparing the natural frequencies of the towers with
those of the cables, one may find that the natural frequencies of the first sway mode of the towers are close
to the ninth and tenth natural frequencies of out-of-plane
vibration of the main span cable. They are also close to the
Table I Natural frequencies of isolated main span cable
Mode no.
Natural frequency (Hz)
In-plane vibration
1
2
3
4
5
6
7
8
9
10
Out-of-plane vibration
FEM
Irvine
Difference (%)
FEM
Irvine
Difference (%)
0.1028
0.1489
0.2095
0.2553
0.3157
0.3454
0.3872
0.4223
0.4789
0.5296
0.1058
0.1499
0.2115
0.2563
0.3173
0.3486
0.3931
0.4231
0.4796
0.5289
-2.84
-0.67
-0.95
-0.39
-0.50
-0.92
-1.50
-0.19
-0.15
0.13
0.0529
0.1054
0.1581
0.2107
0.2634
0.3162
0.3691
0.4220
0.4751
0.5284
0.0529
0.1058
0.1587
0.2115
0.2644
0.3173
0.3702
0.4231
0.4760
0.5289
0.0
-0.38
-0.38
-0.38
-0.38
-0.35
-0.30
-0.26
-0.19
-0.09
861
Modal analysis of Tsing Ma bridge: Y. L. Xu et al.
1
0.8
--TheoreUc~
0.8
0.6
0.4
0.2
0
0.4
-0.2
-0.4
-0.6
-0.8
-1
0
200
400
600
0
8(]0 1000 1200 1400
200
400
600
x(m)
ta)~ v ~
Figure 4
Table 2
~
dn= F-m ~o=m Mode
fo)~ r-rs o u t ~
~x~
First mode shape.,; of isolated main span cable
Natural frequencies of isolated side span cable
Mode no.
Natural frequency (Hz)
Ma Wan side span
1
2
3
4
5
6
7
800 1000 1200 1400
x(m)
Tsing Yi side span
In-plane
Out-of-plane
In-plane
Out-of-plane
0.3493
0.3931
0.6022
0.7914
0.9982
1.2077
1.4288
0.1965
0.3931
0.5905
0.7897
0.9926
1.2014
1.4174
0.3560
0.4691
0.7120
0.9434
1.1878
1.4370
0.2346
0.4692
0.7047
0.9422
1.1839
1.4325
0
o! ....iNuii......
0.8
0.6
O.
4, Numerk~l
0.4
0.2
0
0.
. . . . . . .
(]
50
i
. . . .
J
.
.
.
.
.
.
100 150 200 250 300 350 400
0-
0
x(m)
50
100 150 200 250 300 350 400
x(m)
(a)the v~oal Coml)on~ofb Fr~ mn..p~neMode
Figure 5
First mode shapes of isolated Ma Wan side span cable
third natural frequency of the Ma Wan side span cable and
the second natural frequency of the Tsing Yi side cable in
out-of-plane vibration. The first sway mode of the tower
combining with high-order out-of-plane vibration modes of
the cables are anticipated in the tower-cable system. For
in-plane vibration, both the main span and side span cables
provide additional restraints to the bending and torsional
vibrations of the towers, significant changes of the corresponding modal properties of the tower-cable system are
expected.
4.
Tower-cable system
Three-dimensional finite element modal analysis of the
tower-cable system shows that at lower frequencies, the
modes of vibration of the system can be reasonably separated into in-plane modes of vibration and out-of-plane
modes of vibration. There are no significantly coupled
mode shapes between in-plane vibration and out-of-plane
vibration. The in-plane modes of vibration of the system
can be further separated into local in-plane vibration modes
862
Modal analysis of Tsing Ma bridge: Y. L. Xu et al.
Table 3
Natural frequencies of free-standing towers
Mode no.
Ma Wan tower
1
2
3
4
5
6
Tsing Yi tower
Frequency (Hz)
Mode type
Frequency (Hz)
Mode type
0.1819
0.5154
0.6328
0.9785
1.4587
1.6330
Bending
Sway
Torsion
Bending
Sway
Torsion
0.1819
0.5149
0.6326
0.9757
1.4446
1.6291
Bending
Sway
Torsion
Bending
Sway
Torsion
I"'
I
P
t
.
(a) The First Bending Mode (b) The First Sway Mode (c) The First Torsional Mode
Figure 6 First mode shapes of free-standing Ma Wan tower
modal motions in the same direction (in-phase modal
motion) and in the other mode two cables have opposite
modal motions (out-of-phase modal motion). Figure 7a
shows the first pair of the in-plane local vibration modes of
the main span cables. Compared with the in-plane natural
frequencies of the isolated cables (see Tables 1 and 2), each
pair of natural frequencies of the cables in the coupled system are almost the same as those of the corresponding isolated cables. This indicates that the tower-cable system
retains all the natural frequencies and mode shapes of the
isolated cables as local modal properties.
It is important to note that though these local natural
frequencies of the system are listed separately for the main
and side span cables, the interactions between the main
span cables and side span cables are observed at some local
natural frequencies. For instance, the first pair of in-plane
of cables in which the towers have no modal motion or
have relatively small modal motion, and the global in-plane
modes of vibration where the one or two towers vibrate
significantly together with one or more cables. The out-ofplane modes of vibration of the system are classified in the
same way.
4.1.
Local in-plane vibration
The natural frequencies corresponding to in-plane local
modes of vibration are listed in Table 4 for the main span
cables and in Table 5 for the side span cables. Since the
two identical main cables are connected to the bridge towers, there is some connection between the two main cables.
The local vibration modes of the main cables, therefore,
appear in pairs. Each pair of the local modes have very
close natural frequencies, but in one mode two cables have
Table 4
Natural frequencies of local in-plane modes of vibration of main span cables
Mode no.
Natural frequency (Hz)
Mode property
of two cables
Mode no.
Natural frequency (Hz)
Mode property
of two cables
1
3
5
7
9
11
13
15
17
19
0.1027
0.1425
0.2091
0.2610
0.3158
0.3468
0.3798
0.4222
0.4777
0.5295
In
In
In
In
In
In
In
In
In
In
2
4
6
8
10
12
14
16
18
20
0.1027
0.1462
0.2096
0.2639
0.3158
0.3469
0.3800
0.4222
0.4778
0.5295
Out
Out
Out
Out
Out
Out
Out
Out
Out
Out
phase
phase
phase
phase
phase
phase
phase
phase
phase
phase
of
of
of
of
of
of
of
of
of
of
)hase
)hase
)hase
)hase
)hase
)hase
)hase
)hase
)hase
)hase
863
Modal analysis of Tsing Ma bridge: Y. L. Xu et al.
Table 5
Natural frequencies of local in-plane modes of vibration of side span cables
Tsing Yi side span cable
Ma Wan side span cable
Mode no.
Natural frequency (Hz)
Mode property
of two cables
Mode no.
Natural frequency (Hz)
Mode property
of two cables
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.3468
0.3469
0.3927
0.3927
0.5910
0.5943
0,7910
0.7912
0.9967
0.9983
1.2075
1.2079
1.4286
1.4289
In phase
Out of phase
In phase
Out of phase
In phase
Out of phase
Out of phase
In phase
In phase
Out of phase
Out of phase
In phase
In phase
Out of phase
1
2
3
4
5
6
7
8
9
10
11
12
0.3468
0.3469
0.4687
0.4687
0.7071
0.7086
0.9427
0.9454
1.1877
1.1878
1.4362
1.4372
In phase
Out of phase
In phase
Out of phase
Out of phase
In phase
In phase
Out of phase
Out of phase
In phase
In phase
Out of phase
Out of phase
In phase
(a) Main span cables
//
//
Out of phase
In phase
(b) Side span cables
Figure 7 First pair of in-plane local mode shapes of cables
local vibrations of the Ma Wan side span cables cause the
main span cable to vibrate at the same natural frequency
as the Ma Wan side span cables but in a mode shape similar
to its sixth local mode shape (see Figure 7b). Correspondingly, the power spectrum of the main span cable in the
tower-cable system obtained from the ambient measurement exhibits an additional peak between its fifth and sixth
local natural frequencies. On the contrary, the first pair of
in-plane local vibrations of the main span cables do not
excite the side span cables (see Figure 7a), as the antisymmetric mode shapes of the main span cables cause no
additional cable tension in the system. It is also observed
that when one of the local natural frequencies of the cables
is near one of the global natural frequencies of the system,
the tower(s) may have a relatively small modal motion.
4.2. Global in-plane vibration
Table 6 lists the first six global natural frequencies of the
tower-cable system with the towers in bending or torsional
vibration. As the two towers have similar dynamic characteristics and are connected by two sets of identical cables,
the modes of vibration of the two towers appear in pairs:
in one mode the two towers have in-phase modal motion
but in the other mode the two towers have out-of-phase
modal motion. Figure 8 shows the first pair of the global
modes of the system with the towers in bending. The two
towers have an out-of-phase modal motion at the natural
frequency of 0.1788 Hz, which induces the second mode
of vibration in the main span cables and the first mode of
vibration in each side span cable. At the natural frequency
of 0.2225 Hz, the two towers have an in-phase modal
864
Table 6
Modal analysis of Tsing Ma bridge: Y. L. Xu et al.
Natural frequencies of global modes of vibration of tower-cable system
Mode no.
1
2
3
4
5
6
Natural frequencies (Hz)
Bending mode
Torsional mode
Sway mode
0.1788
0.2225
0.6797
0.7199
1.1102
1.5831
0.2075
0.2504
0.9241
1.0489
1.3675
1.9291
0.5244
0.5408
1.4664
1.4720
(out of phase)
(in phase)
(in phase)
(out of phase)
(in phase)
(out of phase)
(out of phase)
(in phase)
(in phase)
(out of phase)
(in phase)
(out of phase)
(b) In pha*e
(a) Out of phue
Figure 8
(in phase)
(out of phase)
(out of phase)
(in phase)
First pair of in-plane global mode shapes of system with towers in bending
motion, resulting in the third mode of vibration in the main
span cables and the first mode of vibration in the side span
cable. These vibration modes of the cables and the corresponding natural frequencies are in addition to the local
vibrations of the cables. The cables in the coupled system
therefore have more natural frequencies and mode shapes
than the isolated cables. The two natural frequencies in
each pair of the global modes of vibration are relatively
separated compared with those in each pair of the local
modes of cable vibration. This may be attributed to the
different geometry between the Tsing Yi and Ma Wan
side spans.
The natural frequencies of the towers with the cables in
bending mode are considerably different from those of the
free-standing towers in the same mode. There are two natural frequencies in the tower-cable system corresponding to
one natural frequency of the free-standing tower. For
instance, the first pair of natural frequencies in the towercable system correspond to the first natural frequency of
the free-standing tower while the third pair of natural frequencies in the system conform to the second natural fi'equency of the free-standing tower. Apart from the first and
third pairs of natural frequencies, the existence of the cables
creates the second pair of natural frequencies in the towercable system. The bending vibration mode shapes of the
towers in the coupled system at the second pair of natural
frequencies appear between the first and second bending
mode shapes of the free-standing tower.
The natural frequencies and the associated torsional
modes of the tower-cable system are very different from
those of the free-standing towers. The first pair of natural
frequencies of the tower-cable system is significantly
lower than the first natural frequency of the free-standing
tower. The second natural frequency of the free-standing
tower is, however, between the third pair of natural frequencies of the tower-cable system. Again, between the
first and third pairs of natural frequencies in torsional
modes there are the second pair of natural frequencies
which do not appear in the free-standing towers. Figure 9
shows the first pair of the global torsional mode of the system. The two towers have an out-of-phase modal motion at
the natural frequency of 0.2075 Hz and an in-phase modal
motion at the frequency of 0.2504 Hz. Again, the torsional
vibrations of the towers are accompanied by the vibrations
of both main and side span cables.
From the above modal analysis, the important conclusion
may be drawn that the in-plane dynamic response of the
tower-cable system to wind excitation should be computed
using the coupled system model rather than the separated
free-standing tower model and isolated cable model. As the
global natural frequencies of the tower-cable system are
found to be much more closely spaced and lower (in the
torsional mode case) than those of the free-standing towers,
the computation of the dynamic response of the system to
wind excitation may also have to consider contributions
from both higher modes of vibration and cross-correlation
items between the participating modes. The dynamic
response of the tower with cables, therefore, may not be
less than that of the free-standing tower.
4.3.
Local out-of-plane vibration
The natural frequencies corresponding to the local out-ofplane modes of vibration are listed in Table 7 for the main
span cables and in Table 8 for the side span cables. Again,
as two identical cables in each span are connected to the
flexible towers, the local modes of vibration appear in pairs.
Each pair of the local modes consists of one in-phase mode
and one out-of-phase mode, and have very similar natural
frequencies. The local natural frequencies of the cables in
the tower-cable system are also very close to those of the
corresponding isolated cables. The out-of-plane interaction
between the main and side span cables are relatively weak,
compared with the in-plane interaction between the cables.
865
Modal analysis of Tsing Ma bridge: Y. L. X u et al.
(a) Out of phase
Figure 9
(b) I* pha=e
First pair of in-plane global mode shapes of system with towers in torsion
Table 7 Natural frequencies of local out-of-plane modes of vibration of main span cables
Mode no.
Natural frequency (Hz)
Mode property
of t w o cables
Mode no.
Natural frequency (Hz)
Mode property
of t w o cables
1
3
5
7
9
11
13
15
17
19
0.0528
0.1052
0.1577
0.2102
0.2627
0.3152
0.3675
0.4197
0.4722
0.5104
In
In
In
In
In
In
In
In
In
In
2
4
6
8
10
12
14
16
18
20
0.0529
0.1054
0.1581
0.2107
0.2634
0.3162
0.3691
0.4220
0.4751
0.5284
Out
Out
Out
Out
Out
Out
Out
Out
Out
Out
Table 8
)hase
)hase
)hase
)hase
)hase
)hase
)hase
)hase
)hase
)hase
of
of
of
of
of
of
of
of
of
of
3hase
3hase
3hase
3hase
3hase
3hase
)hase
)hase
)hase
~hase
Natural frequencies of local out-of-plane modes of vibration of side span cables
Ma Wan side span cable
Tsing Yi side span cable
Mode no.
Natural frequency (Hz)
Mode property
of t w o cables
Mode no.
Natural frequency (Hz)
Mode property
of two cables
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.1955
0.1965
0.3897
0.3931
0.5905
0.5979
0.7897
0.7914
0.9926
0.9931
1.2005
1.2014
1.4085
1.4173
In phase
Out of phase
In phase
Out of phase
Out of phase
In phase
Out of phase
In phase
Out of phase
In phase
In phase
Out of phase
In phase
Out of phase
1
2
3
4
5
6
7
8
9
10
11
12
0.2332
0.2346
0.4590
0.4692
0.7046
0.7078
0.9421
0.9430
1.1831
1.1838
1.4230
1.4324
In phase
Out of phase
In phase
Out of phase
Out of phase
in phase
Out of phase
In phase
In phase
Out of phase
In phase
Out of phase
4.4. Global out-of-plane vibration
The first four global natural frequencies of the tower-cable
system with the towers being in the sway mode of vibration
are listed in Table 6. Again, the modes of vibration of the
two towers appear in pairs. Figure 10 shows the first pair
of the global modes of the system with the towers in sway
motion. The two towers have an in-phase modal motion at
the natural frequency of 0.5244 Hz and an out-of-phase
modal motion at 0.5408 Hz. The vibrations of the towers
cause both main and side span cables to vibrate in higherorder mode shapes. There are nearly five 'sinusoidal' waves
in each main span cable, and each wave is approximately
plotted by ten points because the three-node curved cable
elements were used to model the main cables. However,
referring to Table 3, one can see that the first pair of natural
frequencies in the global sway mode of the system is close
to the first natural frequency of the free-standing tower in
the sway mode, and the second pair of natural frequencies
in the tower-cable system is close to the second natural
frequency of the free-standing tower. This indicates that the
restraints from the cables to the sway motion of the towers
are very weak compared with the in-plane modal motion
of the towers.
Based on the aforementioned dynamic characteristics of
the system in out-of-plane vibration, the dynamic response
of cables in the tower-cable system may be computed as
the sum of the local cable response and the global cable
response, while the dynamic response of the towers in the
coupled system is simply the global tower response.
5.
Comparison with measured results
With the permission of the Highways Department of Hong
Kong Government, a brief comparison with the measured
Modal analysis of Tsing Ma bridge: Y. L. Xu et al.
866
(a) I . ph~c
(b) Out of phase
Figure 10 First pair of out-of-plane global mode shapes of system with towers in sway
results is presented in this section. The details of the
arrangement of field measurements, instrumentation,
environment, signal processing and data analysis for the
tower-cable system of the Tsing Ma Bridge can be found
in the literature 9. The finest frequency resolution available
in the spectral analysis of measured data was 0.002 Hz.
Therefore, very closely spaced natural frequencies in each
pair of local cable modes could not be identified, but the
natural frequencies in each pair of global modes were identified.
Table 9 lists the numerical and experimental natural frequencies of local in-plane modes of vibration of both main
span and side span cables. For the main span cable, the
numerical results are very close to the measured results.
For the side span cables, the first two natural frequencies
are in good agreement, and only small differences of less
than 7% exist in the third and fourth natural frequencies.
Table 10 displays the natural frequencies of local out-ofplane modes of vibration of both main and side span cables.
Again, there are very close results between computation
and field measurement for the main span cable and only
small disparities for the side span cables. The natural frequencies of global modes of vibration of the tower-cable
system for bending, torsion and sway are given in Table 11.
The numerical results are in good agreement with the measured results.
Owing to the restriction of access to the lower crossbeam levels of the towers during the measurement, the sensors were positioned only at the topmost cross-beam level.
Also due to the limitation on the number of the accelerometers and the time for access to the site, only two
accelerometers were arranged on each main span cable and
each side span cable, respectively. Therefore, a completed
picture of mode shapes was not available from the field
measurement. However, the trends of the first two meas-
ured mode shapes of vibration of cables are similar to the
corresponding numerical mode shapes.
6.
Conclusions
A three-dimensional dynamic finite element model has
been established for the tower-cable system of the Tsing
Ma long suspension bridge under construction. After the
model was validated by the measured dynamic properties,
modal analyses were carried out to investigate dynamic
interaction between the towers and cables. The results show
that at low natural frequencies, there are no significantly
coupled mode shapes between in-plane vibration and outof-plane vibration. Modes of vibration of the system can
be reasonably separated into in-plane modes and out-ofplane modes. In-plane dynamic characteristics of the bridge
towers with cables are significantly different from those of
the free-standing bridge towers. Out-of-plane dynamic
characteristics of the towers with cables, however, almost
remain the same as those of the free-standing towers.
Dynamic interactions between the towers and cables are
significant at global natural frequencies in either in-plane
or out-of-plane vibration. There are many local natural frequencies at which only the cables vibrate and the towers
remain still or have relatively small modal motion. The
interactions between the main and side span cables are also
observed at some local natural frequencies. The understanding of dynamic interaction between the towers and cables
for the Tsing Ma suspension bridge sheds light on both
the determination of dynamic response of the tower-cable
system to wind excitation, and the finite element analysis
of dynamic characteristics of the completed bridge which
are under way.
Table9 Comparison of natural frequencies of local in-plane modes of vibration of cables
Natural frequency (Hz)
Mode no
Main span cable
1
2
3
4
5
6
7
8
9
10
Ma Wan side span cable
Tsing Yi side span cable
Numerical
Measured
Numerical
Measured
Numerical
Measured
0.103
0.143
0.209
0.261
0.316
0.347
0.380
0.422
0.478
0.530
0.102
0.143
0.207
0.260
0.315
0.344
0.379
0.424
0.480
0.533
0.347
0.393
0.591
0.791
0.997
0.344
0.395
0.604
0.817
1.047
0.347
0.469
0.707
0.943
1.188
0.343
0.478
0.731
0.995
1.277
867
Modal analysis of Tsing Ma bridge: Y. L. Xu et al.
Table 10 Comparison of natural frequencies of local out-of-plane modes of vibration of cables
Mode no
Natural frequency (Hz)
Main span cable
1
2
3
4
5
6
7
8
9
10
Ma Wan side span cable
Tsing Yi side span cable
Numerical
Measured
Numerical
Measured
Numerical
Measured
0.053
0.105
0.158
0.210
0.263
0.315
0.368
0.420
0.472
0.510
0.053
0.105
0.156
0.208
0.264
0.316
0.369
0.423
0.477
0.537
0.196
0.390
0.591
0.790
0.993
0.195
0.395
0.613
0.826
1.057
0.233
0.459
0.705
0.942
1.183
0.236
0.477
0.740
1.006
1.285
Table 11 Comparison of natural frequencies of global modes of vibration of tower-cable system
Mode no
Natural frequency (Hz)
Bending mode
1
2
3
4
5
6
Torsional mode
Sway mode
Numerical
Measured
Numerical
Measured
Numerical
Measured
O.179
0.223
0.680
0.720
1.110
1.583
O.176
0.221
0,629
0.672
1.063
1.453
0.208
0.250
0.924
1.049
1.368
1.929
0.203
0.248
0.914
1.023
1.348
1.875
0.524
0.541
1.466
1.472
0.498
0.527
1,414
1.436
Acknowledgments
The writers are grateful for the financial support from the
Hong Kong Polytechnic University through a HKPU studentship to the third writer. The support from the Lantau
Fixed Crossing Project Management Office, Highways
Department of Hong Kong allowing the writers access to
the design details of the Tsing Ma Bridge and to use the
field measurement data9 is particularly appreciated.
6
7
8
9
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