MODAL CHARACTERISTICS OF TOWER-CABLE SYSTEM
IN CONSTRUCTION STATE OF SUSPENSION BRIDGE
S.D.Xue, Y.Q.Ni, a n d J.M.Ko
Department of Civil & Structural Engineering
The Hong Kong Polytechnic University
Hung Horn, Kowloon, Hong Kong
ABSTRACT
NOMENCLATURE
(0 : circular frequency
f
: natural frequency
[k] : stiffness matrix of element
[k,] : linear stiffness matrix of element
[k,] : geometric stiffness matrix of element
[k,] : large deflection stiffness matrix of element
[K] : stiffness matrix of structure
[KIT : tangent stiffness matrix
{U) : modal vector
[M] : mass matrix
1. INTRODUCTlON
The tower-cable system, two main cables being suspended
over bridge towers before the hoisting of bridge decks, is a
key structural state during the construction of long span
suspension bridges. The dynamic characteristics at this state
are quite different from those of completed bridge. With the
increase of the bridge span. the dynamic behaviors become
more complex and the bridge structure may encounter large
damage-causing vibrations during construction. In order to
keep the structure safe, a vibration monitoring system may
be needed during the construction stage to monitor the
performance and the condition of the bridge? and some
dynamic control measures may have to be employed to
suppress large vibrations during construction. At first. the
dynamic characteristics of the system have to be identified
before the implementation of the monitoring and the
employment of dynamic control measures. At present,
studies on dynamic behaviors of the tower-cable system in
the construction state of long span suspension bridges are
still not found within our knowledge. This paper presents
theoretical studies on the modal properties of the towercable system of suspension bridge. Some important modal
characteristics of the system are found, such as the
characteristic of cable local modes with closely-spaced
natural frequencies in low frequency range, the behavior of
locally combined modes in lateral direction or globally
combined modes in vertical and longitudinal direction, the
coupling relationship between the lateral, vertical and
longitudinal modes, the dynamic interaction between towers
and cables, and the stiffness contribution or influence of
suspension cable to the bridge tower, etc. The conclusions
obtained give a good understanding to the dynamic
behaviors of the tower-cable system, and they will provide
necessary information for further application in vibration
monitoring and control.
1911
The Tsing Ma Bridge, which is being constructed in Hong
Kong, is a long span cable suspension bridge with a main
span of 1377m and an overall length of 2 160111. This bridge,
when completed in the mid-1997, will have the world’s
longest span to carry both highway and railway traffic”.“. In
the present paper, the modal characteristics of the towercable system of the Tsing Ma Bridge are investigated as a
case study.
suspenders have been included. Under free cable condition,
the geometry of the cable is regarded as catenary. The static
horizontal tension has the same value of 122,642 KN for
both main span and side span cables. The sag of the cable is
ll2.483m for main span cable, 5.693m for Tsing Yi side
span cable and 7.953m for Ma Wan side span cable. The
Modulus of Elasticity for cables is taken as 200 kN/mm*.
2. DESCRIPTION OF THE SYSTEM
Figure I shows the schematic diagram of the tower-cable
system of the Tsing Ma Bridge, which has the main span of
1377m stretching from Tsing Yi Island to Ma Wan Island.
When completed, the 41 meter wide hybrid truss/box bridge
deck will be suspended on the two main cables to carry the
traffic. The two side spans of the bridge are of different
lengths. The eastern Tsing Yi side span measures 300
meters and will be supported on the ground by a series of
concrete piers. The western Ma Wan side span has the
length of 355.5 meters and will be suspended from the main
cables by hangers after completion.
The two bridge towers, Tsing Yi tower and Ma Wan tower,
are made of reinforced concrete. Each of the towers is
composed of two legs connected by 4 horizontal prestressed
concrete cross-beams at different levels, measuring 54.5m.
107.0m, 152.0m, a n d 19l.Sm f r o m t h e b a s e l e v e l
respectively. The four cross-beams have the depth of l2m,
IOm. 8m and 7m from the lowest one to the topmost one.
Bath towers are 206 m high from the base level. Each
tower leg is 6m wide and tapers from I8 to 9m in the
longitudinal direction. The center-lines between two tower
legs are 4Om apart at the base level reducing to 36m at the
top of the tower.
The two main cables are 36m apart, each with 91 strands of
parallel steel wires in main span and 97 in side spans. Each
cable has an overall diameter of approximately I. I m and a
cross sectional area of 759,278mm’
in main span and
800,743mm’
in side spans. The cables are accommodated
with saddles located at the top of the tower legs and at the
main anchorages and adjacent piers at the height of deck
level. The height of the anchorages from the base level
measures 49.279m for Tsing Yi side cable and 64.866m for
Ma Wan side cable respectively. The loadings on the cables
will be transferred to the towers and anchorages.
The mass density of reinforced concrete is estimated to be
2.53 tonnes/m’. The Modulus of Elasticity for reinforced
concrete is taken as 30 kNlmm”. The mass density per unit
of cable length is respectively taken as 5.8320 tonnes/m for
main span cable and 6.1505 tonnes/m for side span cable, in
which the weights of cable bands used for mounting
Figure I Tower-Cable System of Tsing Ma Bridge
3. FINITE ELEMENT MODELLING
The bridge towers are represented by three-dimensional
multi-level portal frames in the finite element modelling. As
the tower legs and the cross-beams are relatively thick, the
shear deformation and rotational inertia of the members are
needed to be considered together with the bending
deformation. They are modelled as a number of threedimensional Timoshenko beam elements with six-degreeof-freedoms at each unconstrained joint, IX three
translational movements along local x, y and L axes of the
beam and three rotational movements about these axes.
Since the dimensions of the cross-section of the tower legs
and the cross-beams are of considerable size, the member
length for calculating the flexural rigidities of the beam
element is virtually shorter than the length taken center to
center between the connection points of the legs and the
cross-beams. Therefore three-dimensional Timoshenko
beam element with rigid ends is used to model these
members. The 12x12 stiffness and consistent mass matrices
were derived for these beam elements.
Cable is a flexible structural element which has hardly any
bending resistance. Cable structures are generally elastic in
nature but are nonlinear in the geometric sense. In the
present paper, each cable is modelled by a number of twonode cable elements with six-degree-of-freedoms, i.e. three
translational movements in horizontal, vertical and
transverse directions for each node. This kind of elements
takes account of the cable tension and the geometric
deflection. The 6x6 stiffness matrix, which includes linear
elastic stiffness, geometric stiffness and large deflection
1912
stiffness, and the consistent mass matrix were established by
The first 170
considering the static equilibrium state as an initial state.
the Tsing Ma Bridge are computed by the finite element
In the dynamic analysis of cable structures, the nonlinear
the main span cables and the side span cables. It is noticed
effects of the cables have to be taken into account in order
that none of the lateral modes of the system is coupled with
to
its
vibration modes of the tower-cable system of
program. Most of them are found to be the local modes of
get
accurate
structur.@‘,
estimation
of
dynamic
behaviors
of
the
The stiffness matrix of a cable element can be
expressed as:
vertical
and
longitudinal
modes.
However,
the
vertical
and longitudinal modes of the system cannot be decoupled.
The
modal
components
in
vertical
and
longitudinal
directions are always related to each other.
(1)
[kl = [kel + kl + WJ
where [k,,] is the linear stiffness matrix. which is the same as
4.1 The Modes in Lateral Direction
the stiffness matrix of a bar element; [k,] is the geometric
stiffness matrix, which denotes the effect of cable tensions:
[k,J
is the large deflection stiffness matrix. Both [k,] and
The lateral modes of the system do not couple with its
vertical and longitudinal modes. There are many cable local
[k,,] being matrices due to the effect of geometric
modes in lateral (sway) direction. These modes are
nonlinearity of the cable. They vary with the displacements
and internal forces and give a contribution to the stiffness of
dominated by the vibration of the cables whilst the towers
only contribute a little or no modal motion. As there are two
the structure.
cables, the south cable and the north cable, in each span, the
Additional stiffness induced by the masts of the cranes and
local modes occur in pairs. one having in phase modal
vectors between the two cables and the other having 180”
the hoists connected to the tower legs for construction might
out of phase modal vectors between the two
have
these
south cable and the north cable have identical geometrical
to
be
considered.
However,
when
comparing
cables. The
stiffnesses with those of the towers, they are too small to
and sectional properties, therefore each pair of the modes
affect the global modal behavior of the towers and the
possess the same or very close frequency values. This also
tower-cable system. Therefore
the finite element modelling.
indicates that the relatively rigid bridge towers participate
they
are
not
considered
in
very slightly in the lateral local modes of the cables. Tables
1 to 3 show respectively the first eight lateral local modes of
A total of four saddles, weighted 500 tonnes each; at the top
of the tower legs of both Ma Wan tower and Tsing Yi tower
the main span cables, Ma Wan side span cables and Tsing
Yi side span cables. The mode shape for the first lateral
are treated as additional lumped masses in the modelling.
local mode of the cables behaves a single-half-wave for
both main span and side span cables.
4.
DYNAMIC
ANALYSIS
No global mode of the system is found in lateral direction.
There are only the locally combined modes of the Ma Wan
Based on the proposed finite element modelling, a three-
tower or Tsing Yi tower with its side span cables. For each
dimensional finite element program has been developed to
locally combined mode in lateral direction, the combined
perform the modal analysis of the structure. ‘This program
takes into account the nonlinear influence of the cables.
modal motions occur only between the tower and its side
span cables. The main span cables just swing accompanying
Assuming
around
stiffness
that
the
structure
its static equilibrium
matrix [K] can be
vibrate
amplitude
the tower at their local connecting ends. There is no
configuration; the structural
approximately regarded as
with
small
vibrational transmission or interaction between the Ma Wan
and Tsing Yi towers through the main span cables. Table 4
constant during vibration. When the damping is neglected,
shows the first four locally combined modes in Ma Wan
the free vibrational equation of the structure can be written
side and Tsing Yi side respectively. It is noticed that the
a*:
modal motion of the tower in each locally combined mode
is similar to the corresponding mode shape of the free[KIT ( U } a’ [ M ] {U) = 0
standing tower. Therefore, the locally combined modes can
(2
be regarded as the tower-dominated modes. It is seen from
where [M] = mass matrix; {U} = modal vector; w = circular
Table 4 that, the natural frequencies of the locally combined
frequency of free vibration. and [KIT = tangent stiffness
matrix in initial state of vibration, which are obtained by
modes in Tsing Yi side are greater than the corresponding
considering
cables.
Yi side span cables contribute more additional stiffness to
iteration
the bridge tower.
The
above
the
geometric
equation
are
nonlinear
solved
effect
by
of
the
sub-space
method in the computer program.
1913
values in Ma Wan side. It can be explained that the Tsing
Table 5 illustrates the comparison of the natural frequencies
between the locally combined modes of the tower-cable
system and the sway (lateral) modes of the corresponding
free-standing tower. It is seen that there is a correspondence
between the hvo groups of modes, and the difference
between the natural frequencies of the cable-connected and
free-standing towers is small. It indicates that the main span
cables and the side span cables only make a little
contribution to the lateral stiffness ofthe towers.
Table 4 Locally Combined Modes in Lateral Direction
of the Tower-Cable System
Table I Lateral Local Modes of Main Span Cables
Table 2 Lateral Local Modes of Ma Wan Side Span
Cables
Table 5 Comparison between the Natural Frequencies
of Lateral Modes of the Cable-Connected and FreeStanding Bridge Tower
Table 3 Lateral Local Modes of Tsing Yi Side Span
Cables
4.2 The Modes in Vertical and Longitudinal Directions
The vibration modes of the tower-cable system are always
coupled together in vertical and longitudinal directions, i.e.
each mode has both the vertical and longitudinal
components which are related to each other. Similar to the
vibrations in lateral direction, in vertical and longitudinal
directions there are also many cable local modes with
closely spaced natural frequencies. These local modes also
appear in pairs, one having in phase modal vectors between
the hvo cables and the other having 180” out of phase modal
vectors between the two cables. Each pair of the modes
have the same or very close frequency values. This implies
that the bridge towers participate very slightly in the local
1914
modes of the cables. Tables 6 to 8 show respectively the
first eight local modes of the main span cables, Ma Wan
side span cables and Tsing Yi side span cables.
Table 6 Local Modes in Vertical and Longitudinal
Directions of the Main Span Cables
It is found that the side span cables sometimes participate in
the modal motions of the local modes of the main span
cables, and vice versa. It should be specially noted that the
first local mode of the main span cables in vertical and
longitudinal directions behaves as an antisymmetric doublehalf-wave mode. This is different from that in the lateral
direction. For the Ma Wan and Tsing Yi side span cables,
the first local mode in vertical and longitudinal directions is
still a single-half-wave mode. As illustrated in Ref. [4], the
above difference for the mode shape is resulted from the
different structural parameters of the cables, such as the
cable tensions. the sag of cables, and the span of cables etc.
Some global modes of the tower-cable system are found in
vertical and longitudinal directions, in which the modal
motions of the two towers behave interactively through the
connecting main span cables. The global modes of the
system in vertical and longitudinal directions can be
classified as tower-bending globally combined modes and
tower-torsion globally combined modes. These global
modes appear in pairs, one having in phase modal vectors
between the two towers and the other having 180” out of
phase modal vectors between the two towers. Table 9 shows
the first six tower-bending globally combined modes of the
system. The first six tower-torsion globally combined
modes of the system are listed in Table IO. Different from
the local modes of the cables, in the present circumstances,
the frequency values between each pair of the globally
combined modes are considerably different from each other
due to the different effects of the two side span cables on
the towers.
Mode
I st
2nd
3rd
4th
5th
6th
7th
8th
Frequency (Hz)
0.1007
0.1007
0.1398
0.1429
0.201 I
0.201 I
0.2443
0.2479
Remarks on Two Cables
in phase
180” out of phase
in phase
180” out of phase
in phase
180” out of phase
180” out of phase
in phase
Table 7 Local Modes in Vertical and Longitudinal
Directions of the Ma Wan Side Span Cables
Table 8 Local Modes in Vertical and Longitudinal
Directions of the Tsing Yi Side Span Cables
In order to make comparison, the theoretical natural
frequencies of bending and torsional modes of the freestanding Tsing Ma bridge towers are listed in Table I I. It is
observed that the natural frequencies of the tower-bending
(or tower-torsion) global modes of the system are greatly
distinct from the corresponding values of the bending (or
torsion) modes of the free-standing towers, and there is no
correspondence between them. Although the natural
frequency of the first tower-bending global mode of the
system is close to that of the first bending mode of the freestanding towers, an additional tower-bending global mode
appears with the frequency between the first and second
modes of the free-standing towers. The natural frequency of
the first tower-torsion global mode of the system is much
lower than that of the first torsion mode of the free-standing
towers. All these phenomena indicate that the main and the
side span cables have a considerable influence on the
bending and torsional stiffness of the towers.
Mode 1Frequency (Hz) 1 Remarks on Two Cables
1st
I
0.362X
I
in
~~~ nhase
r~~~~-~
2nd
0.3637
180” out of mase
1~‘~
3rd
0.4514
in phase
4th
0.4514
180” out of phase
5th
0.6504
in phase
6th
0.6578
180” out of phase
7th
0.8346
180” out of phase
Xth
0~8147
in nhaw
1915
Table 9 Tower-Bending Globally Combined Modes in
Vertical and Longitudinal Directions of the System
Frequency (Hz:
T
Table 11 Natural Frequencies of Bending and Torsional
Modes of the Free-standing Tsing Ma Bridge Towers
Remarks on the Towers
‘F ‘base
relatior
Modal shape
180” out
of phase
0.2189
in phase
0.6730
in ohase
180” out
of phase
in phase
180” out
of phase
Table 10 Tower-Torsion Globallv Combined Modes in
Vertical and Longitudinal Directions of the System
Aode
-
Remarks on the Towers
Frequency
(HZ
Phaskrelation Modal sha~oe
1st
2nd
3rd
5. COMPARING WITH EXPERIMENTS
The ambient vibration measurements on both free-standing
fower~ and the tower-cable system of the Tsing Ma Bridge
were respectively carried out during the construction stage.
The measurement results on the free-standing Tsing Ma
bridge towers are given in Ref. [5]. The detailed
measurement information and experimental results on the
tower-cable system of the Tsing Ma Bridge are presented in
the field measurement report “ 1 These field measurements
revealed the same dynamic characteristics as the theoretical
analysis on either the free-standing towers or the towercable system of the Tsing Ma bridge. Both theoretical and
experimental results yive a good agreement on the modal
properties (natural frequencies and mode shapes) of the
structure. The comparison with the field measurement
results proves that the modal characteristics predicted by the
theoretical analysis are correct. It indicates that the finite
element modelling established in the theoretical analysis is
reasonable and acceptable for predicting the modal
properties of the tower-cable system. It also implies that
there is a good agreement between the measured structural
stiffness and that used in the finite element modelling.
in phase
6. CONCLUDING REMARKS
4th
5th
!
1.0281
1~3486
180” out
of phase
Based on the above analysis, some conclusions on the
modal characteristics of the tower-cable system can be
drawn:
I” phase
(i)
6th
1916
In the low frequency range there are many local
modes of the cables with closely-spaced natural
frequencies, in which almost no modal motions of the
towers participate.
(ii)
(iii)
None of lateral modes of the system is coupled with
the vertical and longitudinal modes. But the modes in
vertical and longitudinal directions are always
coupled with each other.
In lateral (sway) direction, no global mode of the
system is found. There are only the cable local modes
and the locally combined modes of a tower with its
side span cables.
In lateral direction, there is no vibrational
transmission or interaction between the two towers
through the main span cables.
(VI
(4
Some global modes of the tower-cable system appear
in vertical and longitudinal directions, where the
modal motions
of the two towers behave
interactively through the connecting main span
cables.
The globally combined modes include both towerbending global modes and tower-torsion global
modes. Each type of the global modes appear in
pairs, one having in phase modal vectors between the
two towers and the other having 180” cut of phase
modal vectors between the two towers.
(vii) The tower-bendin g global modes and the towertorsion global modes of the tower-cable system are
significantly distinct from the bending or torsional
modes of the free-standing towers. There is no
correlation behveen them. However, in the lateral
direction, the locally combined modes of the towercable system seem to be dominated by tower
vibration.
(viii) The suspension
on the bending
but they give
stiffness of the
7.
cables have a considerable influence
and torsional stiffness of the towers,
a little contribution to the lateral
towers.
ACKNOWLEDGEMENTS
The authors wish to thank the Lantau Fixed Crossing
Project Management Office, Highways Department of the
Hong Kong Government, for providing the access to the
design and measurement information. Thanks also go to the
Hong Kong Polytechnic University for the financial support
of the research project.
1917
8.
REFERENCES
[I] Beard, A. S. and Youn’& J. S.
Aspects of the Design of the Tsing Ma Bridge,
Proc. of BRIDGES INTO THE ZIST CENTURY, Hong
Kong, 1995.
[Z] Hunter, 1. E.
Tsing Ma Bridge - Superstructure Construction and
Engineering
Aspects,
Proc. of BRIDGES INTO THE 2lST CENTURY, Hong
Kong, 1995.
131 Xne, S. D. and Cm, Z.
The Dynamic Characteristic and Response of Cable
Structures and a General Program on Microcomputer,
Proc. of International Colloquium on Space Structures for
Sports Buildings; Beijing, 1987.
141 Xue, S. D., Cm, Z., and Zhang, S. Y.
The Practical Analysis of Dynamic Characteristics for Cable
Roof Structures,
Proc. of the International Congress on Innovative Long
Span Structures. Toronto. 1992.
[5] Law, s. S., Ko, J. M., Lau, C. K. and Wang, K. Y.
Ambient Vibration Measurement of the Tsing Ma Bridge
Towers,
Proc. of BRIDGES INTO THE ZIST CENTURY, Hong
Kong, 1995.
161 The Hong Kong Polytechnic University
Lantau Fixed Crossing: Vibration Monitoring of Bridge
Towers-Field Measurement Report on the Tower-Cable
System of Tsing Ma Bridge, submitted to the Highways
Department of the Hong Kong Govemment~ Report No. 7;
1996.