Dynamic Analysis of Self-Anchored Suspension Bridges
Chang-Huan Kou*, Jeng-Lin Tsai
Department of Civil Engineering and Engineering Informatics
Chung Hua University
707, Sec.2, WuFu Rd., Hsinchu, Taiwan 30012, R.O.C.
Tel.: +886-3-5186706
Fax.: +886-3-5372188
E-mail：chkou@chu.edu.tw
Abstract
The main cables of self-anchored suspension bridge are directly anchored at the two ends of the main girder
with the axial pressure transferred from the main cable to the main girder used as a source of pre-stressed force
that indirectly enhances the bending resistance capability of the reinforced concrete of the main girder, which
then enables the self-anchored suspension bridge to be an economically realistic long span structure. Aside from
developing the free vibration equations of self-anchored suspension bridges, the paper will analyze and explore
when a self-anchored suspension bridge’s main cables, main girders, towers, hangers, and other structural
elements’ material elastic modulus is individually altered, what the effect on the bridge’s vibration frequency,
would be in order to clearly understand the self-anchored suspension bridge’s dynamic characteristics. The
results of this research indicate that the vibration frequencies of self-anchored suspension bridges are less than
that of earth-anchored suspension bridges, furthermore, the effect of an alteration in the main girder elastic
modulus on the self-anchored suspension bridge’s vibration frequencies are the largest, followed by the effect of
an alteration in the elastic modulus of the towers, main cables, and hangers.
Keywords: Suspension Bridge, Self-anchored, Dynamic Analysis.
1. Preface
self-anchored and earth-anchored suspension bridge
Because the main cables of self-anchored
differ in vibration frequencies and in earthquake
suspension bridges are already directly anchored at
responses in order to clearly understand the dynamic
the two ends of the main girder, this structure saves
characteristics of the self-anchored suspension
the cost of installing other anchoring devices;
bridge in addition to the impact that different
furthermore, the axial pressure is transferred from
material characteristics may have on the earthquake
the main cable to the main girder and used as a
responses of this very structure.
source of pre-stressed force that indirectly enhances
2. Previous Research
the bending resistance capability of the main girder,
Research regarding the earthquake responses of
which then renders the self-anchored suspension
suspension bridges have continually progressed in
bridge worthy of further research and widespread
the U.S., Europe, Japan, and China. Much research
use as a water-spanning structure. Therefore, this
[2,3,4] using three-dimensional
paper, upon completing a stochastic static analysis of
models to simulate and analyze twin tower
the self-anchored suspension bridge [1], further
suspension
explores and compares the extent to which a
discovered that the vibration of vertical, and
bridges’
vibration
finite element
properties
has
longitudinal and twisted directions to some extent
previewed
encompasses a coupling relationship. In 1988, Wang
characteristics quite different from those in Taiwan,
Baoxi [5] created an initial shape analysis set of
besides
procedures for the suspension bridge, in effect
bridges’ free vibration formulas, this paper uses an
procuring the initial shape under statical loadings, in
example from past research [10], a three-span double
order to understand the initial geometric position and
suspension bridge with a long span length of 160
pre-stressed force of all the elements of bridge. This
meters and side span length of 76 meters, and
paper considered the nonlinear geometric effect of
applies the ANSYS program analysis to compare the
large structural displacements in the suspension
dynamic
bridge and used the virtual work method to create
earth-anchored suspension bridges. In addition, the
other nonlinear systematic equations. The statical
paper further explores when a self-anchored
analysis employed the Newton-Raphson iteration
suspension bridge’s main cable, main girder, tower,
method to search for displacement explanations;
hanger, and other structural elements’ material
then, dynamic analysis of the suspension bridges
elastic modulus is individually altered, the effect
used the numerical method to search for the
such change produces in the bridge’s vibration
vibration frequencies and mode shapes. In 1997,
frequency, in order to clearly understand the
Huang Chaoguang and Peng Dawen [6], according
self-anchored
to the 3-D finite element model, used a mixed frame
characteristics and thus provide research that may
form, with the single tower suspension bridge
benefit both the academic and engineering worlds.
serving as a comparison, to apply the subspace
3.Theoretical Analysis of the Dynamic
iteration method calculations in analyzing the
Characteristics
influence of the statical load, rise-span ratio,
stiffness of the main girder, and tower stiffness on
single
tower
suspension
bridges’
dynamic
characteristics and then subsequently compare the
differences between the single tower and twin tower
suspension bridges. In 2000, Tang Maolin, et al. [7]
pointed to the long span suspension bridge as a
typical flexible frame system with receiving force
characteristics
demonstrated
to
be
strongly
geometrically nonlinear. The paper introduces a
literature
introducing
all
consider
self-anchored
characteristics
of
suspension
of
earthquake
suspension
self-anchored
bridge’s
and
dynamic
Self-Anchored
Suspension Bridges
3.1 Large Displacement Generalized Potential
Energy Function
In considering the self-anchored suspension
bridge’s large displacement generalized potential
energy function, this paper will be based on the
following assumptions：
(a) All the materials must comply with Hook’s
law.
model for finite element analysis and applies such
(b) At the complete stage, the statical load is
model to one specific long span suspension bridge.
evenly distributed along the span, and the
In 2001, Gregor P. Wollmann [8, 9] researched and
cable forms a parabolic curve.
organized all basic formulas regarding the force
analysis aspect of suspension bridges and their
origin and then introduced a finite element model
concerning self-anchored suspension bridges.
Given
that
research
on
self-anchored
suspension bridges within Taiwan is lacking and the
(c) At its points of connection with the towers, the
main girder receives vertical support.
(d) Given that the hangers are distributed densely,
they are treated as even membranes with only
vertical resistance and malleability is not taken
into account.
(e) Without considering the bending resistance
rigidity and axial compression deformation in
the direction along the main girder of the
tower, the paper will assume that the rigidity
in the transverse direction is infinitely large.
(f) The paper considers only the warping
deformation of the twist in the main girder but
does not consider the cross-section’s distorted
deformation and will assume the diaphragm is
distributed densely and its shearing rigidity is
infinitely large.
Assume the displacements along the main
girder
are
u,
vertical
displacements
are
v,
2
  2v 
1 3 Li q s    2 w 
  I z 

T  
 I y 
2 i1 0 gAs   xt 
 xt 
  
 u 
 v 
 I y  I z    As    As  
 t 
 t 
 t 
2
1 3 Li q c
 w  
dx

1  h 2
 As    i


2 i 1 0 2 g
 t  
2
cross-sectional area, lateral, and vertical
direction of the main girder, in the vertical direction,
flexural inertial moments of the main girder,
and transverse direction of the left side cables of the
respectively.
main girder are ul、vl、wl，displacements along the
the main girder are ur、vr、wr.
If considering the effects of axial bending
pressure coupling, bending in the vertical and
transverse
direction,
warping,
energy as Ug，dynamic energy as T，then the large
  2
1 3  Li
    E l J w  2
2 i 1  0
 x

displacement generalized potential energy function
Li
are designated as Uce and Use，gravitational potential
is of the suspension bridge[1]：
3
 
i 1
t2
t1
Li
0
T  U
3
ce
and
shearing
deformation, the strain energy of the main girder is：
If the strain energy of the cable and main girder
II   
2
length along the x-axis； As , I y , I z are the
twisted angles are θ, displacements along the
and transverse direction of the right side cables of
2
 ul  2  vl  2  wl  2  u r  2
 
 
 


 t   t   t   t 
2
2
 vr   wr  
（2）

 
 
 t   t  
where： g is gravity acceleration； h  the cable
displacements in the transverse direction are w,
direction of the main girder, in the vertical direction,
2
 U se  U g   
i 1
Li
0
 r f 1 dx i

 r f r dx i  dt


（1）
U se

0
2
Li
 2w 
  
GJ t 
 dx i   EI y  2  dx i
0
 x 
 x 
2
2


where： l , r are the Lagrange Multipliers；
Li
0
Li
0
Li
2

 dx i

Li
  2v 
 u 
EI z  2  dx i   EAs 
 dx i
0
 x 
 x 
2
2
Li
 u v 
 v 
N x   dx i   GAs 
  dx i
0
 x 
 y x 
2
2

 u w 
GAs  
 dx i 
 z x 

f l , f r are the forces constraining the

extension of the left and right hangers.
where：E is the elastic modulus of the main
Organizing terms, vibration dynamic energy of
0
girder； 
（3）
is the shearing influence
the suspension bridge is：
coefficient.
The first term in the equation（3） is the
warping strain energy， El J w is the main girder’s
B 
warping rigidity；the second term is the twisting
strain energy ； the third term and fourth terms
1
4 f  h1  ；
4 Li
where： h1 is the hanger’s length；F is the rise
height；Li is the length of each span.
encompass the bending strain energy；the fifth term
is the axial compression strain energy；the sixth term
is the bending pressure coupling strain energy；the
seventh and eight terms are the shearing strain
energy.
Where : 

 J p J p  J t , El  E 1  v 2

（9）
Gravitational potential energy is：
3 

Li
Li 1
U g      q s vdxi  
q c  v l  v r dx i 
0 2
i 1  0


（10）
The forces constraining the extension of the left and
；J p is the pole moment inertia of the
cross section；Jt is St. Venant’s torsional
constant；E is the elastic modulus of the
right hangers are：
 b w

2
fl  
 ul   w  d h  wl 
 2 x

2
b


 h  v 
 vl   h 2  x   0
2


2
main girder； v is the Poison ratio.
The strain energy of cables the main is：
（11）
 b w

2
fr  
 u r   w  d h  wr 
 2 x

2
Lc  1
 H q H l  H r 
i 1 Ec Acl  2
1
2
2 
 Hl  Hr 
2

3
U ce  


3
Where： Lc
 
i 1
Li
0
1  y  
2
b


 h  v 
 vr   h 2  x   0
2


2
（4）
where：h is the hanger’s length；b is the width of
3
2
the bridge；dh is the vertical height of the
dx ，which can
distance from the main girder’s twisting
also be written as：


Li
0
center to the point of hanger’s position.
1
(1  y  2 ) dx  Li   A  B 
2
3
2
h1
 A  B 
8f

Organizing terms will give the large displacement
generalized potential energy function：
（5）
Where：

15
2
2
A    16 A  1  16 A
4 2

1


2
ln 4 A  1  16 A 2  ；


15
2
2
B    16 B  1  16 B
4 2

1


2
ln 4 B  1  16 B 2  ；


1
4 f  h1  ；
A 
4 Li


1
2


（6）

1
2


 

t2
t1
2
2
 1 3 Li q s    w 

 I y 
 
 2 i 1 0 gAs   xt 
2
3
32 A



（12）
3
32 B
  2v 
  
 u 
  I y  I z    As  
 I z 
 t 
 t 
 xt 
2
2
Li q
1 3
 v 
 w  
c
 As    As 
  dx i   
2 i 1 0 2 g
 t 
 t  
2
2
2
2

 v1   w1   u r 
2  u1 
1  h' 
 
 
 

 t   t   t   t 
2
（7）
（8）
2
2
3 L 1
 v   w  
  r    r   dxi   c  H q
 t   t  
i 1 Ec Ac1  2
2
3 
L
 H l  H r   1 H 12  H r2   1    i E1 J w
2
 2 i 1  0

  2
 2
 x

2
2
2
2
Li
Li

  
 dx  GJ t 
 dx i   EI y
0
0
 x 

2
Li
 w
 v
 2  dx i   EI z  2  dx i
0
 x 
 x 
2
Li
0
Li
  2u 
 v 
EAs  2  dx i   N x   dx i
0
 x 
 x 
2
2

Li
0
Li
 u v 
  dx i   GAs
0
 y x 
GAs 
2
 3  Li
 u v 
  dx i      q s vdxi

 z x 
 i 1  0
u  2 v
 2v


GA
0
s
y x 2
x 2
（15）
the equation for vibration in the transverse direction
qs
q
4w
4w
  s I y 2 2  EI y 4
w
g
gAs x t
x
Li
w  d h  wl   qc
b  1

 v     H q

h
2  2
 2g 
2
q 1
b 
 
 Hl 
  Hq  Hl  2 v 

Hq  2
2 
 x 


 3 Li
1
q c  v l  v r dx i      l f l dx i
0 2
 i 1 0
3

Li

    r f r dx i  dt
0
i 1


（13）
Where H l t  , H r t  are the horizontal

 GAs
of the main girder：
2

qc
 2 vl
 2 vr
 2v
 Hr 
 Nx 2  Hl

H
r
Hq
x
x 2
x 2
qc  w  d h  wr   qc 
b 

 v  


2
h
2 
 2g 
2
1
 q 1

  Hq  Hr 
  Hq  Hr  2
2
 Hq  2
 r
b  qc 
u  2 w

v




GA


s
2  2 
z x 2

 2u
 GAs 2  0
x
（16）
the equation for vibration in the direction along the
increced forces caused by inertial force； Ec is the
main girder：
elastic modulus of main cable ； Acl is the
cross-section of main cable； q is the total loading；
  2u v  2u
qs
 2u
u  EAs 2  GAs  2 
g
x y 2
x
 y
H q is the sum of horizontal forces triggered by the

statical load of the two main cables.
the equation for twisted vibration of the main girder：
3.2
The
free
vibration
equations
of
the
self-anchored suspension bridge
Based on the following variation operation
 * 
 *
i  0 ψi= u，v，w，θ…（14）
i
one can get the vertical vibration equation of the
main girder：
qs
q
q
 4v
v  c vl  vr   s I z 2 2
g
2g
gAs x t
 EI z
 4v H q

x 4
2
  2 vl  2 vr
 2  2
x
 x

  H r

 2u w  2u 
0

z 2 x z 2 
（17）
qs
 4
( I y  I z )  El J w 4
gAs
x

  2 q  b 
b2
  GJ t  H q  2  c   
4
g 2

 x
q
b
q
l  w
r )
 H l  H r 
 c d h (w
2
H q 2g
2
H q   2 wl  2 wr  b  2 vl

  Hl 2

2  x 2
x 2  2
x
2
2
 w 
 v
b
 H r 2r  d h H l  2 l   d h H r
2
x
 x 
 dh
 qc b  2  w  d h θ  wl 2
 
2 
2h
 2 g 2 t 
w  d h θ  wr 2   b H q  2


2
2h
 2 2 x
 w  d θ  w 2 w  d θ  w 2 
h
l
h
r


0
2h
2h


（18）
with a standard cross-section of 5 single box cells,
the lateral vibration equation for the left side of the
considers the effect of initial stress rigidity on the
main cable：
main cables and hangers and partitions the structure
 Hq
  2 w w  d h  wl 
 
 H l  2 l 
h
 2
 x
 q  Hq
q 
b   Hq
[ c  v    
 H l 
 
2g 
2   2
 Hq  2
into 306 spatial elements. Among them, the towers,
  2 wr
 2
 x
qc
l
w
2g
 H1 
 
b  q
v    c ]  0
2 
x 
2  2
geometric center of 2.5 meters, and main girder
height-span ratio of 1:64, all of which are displayed
in Figure 1.
The analysis model (as demonstrated in Figure
2) uses a three-dimensional girder module that
establishes a dynamic finite element analysis, which
foundation, main girder, and transverse girder use
girder elements while the main cables and hangers
use cable elements. The main girder and towers
encompass 255 nodes and connect by using hinge
2
（19）
main cable：
 Hq
  2 wr w  d h  wr 
qc
r  
w
 H r  2 
2g
h
 2
 x
 q  Hq
q 
b   Hq
[ c  v    
 H r 
 
2g 
2   2
 Hq  2
2 
b  q
v    c ]  0
2 
x 
2  2
supports. The main structural constants of the main
girder, towers, main cables, and hangers are
the lateral vibration equation for the right side of the
 Hr 
girder width of 41 meters, height of girder’s
（20）
displayed in Table 1.
5.Results and Discussion
5.1 The Vibration Frequencies and Mode Shapes
of the Self-Anchored and Earth-Anchored
Suspension Bridge
This paper uses the subspace iteration method
to
retrieve
self-anchored
and
earth-anchored
suspension bridges’ various mode shapes and
Based on the above vibration equations, the
vibration frequencies and displays the first 20 mode
self-anchored suspension bridge’s upper structural
shapes, vibration frequencies, and mode shape
vibration can be treated as a combination of the
characteristics in Table 2. The table demonstrates
flexural vibration of the cables and hangers and twist,
that the vibration frequency of the self-anchored
horizontal and flexural displacements of main girder.
suspension
The vibration equations include nonlinear and
earth-anchored suspension bridges. This is caused by
coupling terms.
the bending pressure coupling effect that renders the
4. Basic Data Analysis
main girder bending resistance stiffness of the
bridges
is
less
than
that
of
In this paper’s analysis, the choice of bridge
self-anchored suspension bridge to be less than that
corresponds to that in reference [1], namely a double
of an earth-anchored suspension bridge. Furthermore,
tower, suspension bridge with a main span of 160
Figure 3 displays the self-anchored suspension
meters, two side spans of 76 meters, an entire length
bridge’s horizontal, vertical, transverse, and twisted
of 312 meters, tower height of 42.4 meters, rise-span
mode shapes with vibration frequencies of 0.18842
ratio of 1/6, hanger to hanger distance of 5 meters, a
HZ, 2.4788 HZ, 2.7930 HZ, and 3.7852 HZ
reinforced concrete box girder as the main girder
respectively.
those
shape’s vibration frequency the most and the
corresponding vibration frequency calculations for
horizontal mode shape’s vibration frequency the
earth-anchored suspension bridges.
least. This demonstrates that a reduction in the main
Remark ： Figures
in
parentheses
are
girder’s elastic modulus will have a larger effect on
5.2 The Effect of a Change in the Material Elastic
Modulus on Vibration Frequency
the entire suspension bridge’s vertical stiffness and a
smaller effect on its horizontal stiffness.
When the elastic modulus of the main cables,
main girder, towers, and hangers each changes
5.2.3The Effect of a Change in the Towers’
separately, the self-anchored suspension bridge’s
Elastic Modulus on Vibration Frequency
horizontal,
Table 3 indicates that reducing the towers’
vertical, transverse, and twisted mode shapes are
elastic modulus will cause the self-anchored
displayed in Table3.
suspension bridge’s horizontal, vertical, transverse,
vibration
frequencies
corresponding
and twisted mode shapes’ vibration frequencies to
5.2.1 The Effect of a Change in the Main Cables’
Elastic Modulus on Vibration Frequency
all decrease. This is because a reduction in the
towers’
Table 3 indicates that reducing the main cables’
elastic modulus will cause the self-anchored
suspension bridge’s horizontal, vertical, transverse,
and twisted mode shapes’ vibration frequencies to
all decrease. This is because a reduction in the main
cables’ elastic modulus renders the entire suspension
bridge’s stiffness to decrease relatively as well.
Furthermore, a reduction in the main cables’ elastic
modulus affects the horizontal mode shape’s
elastic
modulus
renders
the
entire
suspension bridge’s stiffness to decrease relatively
as well. Furthermore, a reduction in the towers’
elastic modulus affects the vertical mode shape’s
vibration frequency the most and the horizontal
mode shape’s vibration frequency the least. This
demonstrates that a reduction in the towers’ elastic
modulus will have a larger effect on the entire
suspension bridge’s vertical stiffness and a smaller
effect on its horizontal stiffness.
vibration frequency the most and the twisted mode
shape’s
vibration
frequency
the
least.
This
demonstrates that a reduction in the main cables’
5.2.4 The Effect of a Change in the Hangers’
Elastic Modulus on Vibration Frequency
elastic modulus will have a larger effect on the entire
Table 3 indicates that reducing the hangers’
suspension bridge’s horizontal stiffness and a
elastic modulus will cause the self-anchored
smaller effect on its twisted stiffness.
suspension bridge’s horizontal, vertical, transverse,
and twisted mode shapes’ vibration frequencies to
5.2.2 The Effect of a Change in the Main Girder’s
Elastic Modulus on Vibration Frequency
hangers’
Table 3 indicates that reducing the main
girder’s elastic modulus will cause the self-anchored
suspension bridge’s horizontal, vertical, transverse,
and twisted mode shapes’ vibration frequencies to
all decrease. This is because a reduction in the main
girder’s
elastic
modulus
renders
the
all decrease. This is because a reduction in the
entire
suspension bridge’s stiffness to decrease relatively
as well. Furthermore, a reduction in the main
girder’s elastic modulus affects the vertical mode
elastic
modulus
renders
the
entire
suspension bridge’s stiffness to decrease relatively
as well. Furthermore, a reduction in the hangers’
elastic modulus affects the twisted mode shape’s
vibration frequency the most and the transverse
mode shape’s vibration frequency the least. This
demonstrates that a reduction in the hangers’ elastic
modulus will have a larger effect on the entire
suspension bridge’s twisted stiffness and a smaller
effect on its transverse stiffness.
the smallest effect on the horizontal mode
shape’s vibration frequency; then lastly, a
5.2.5 Overall Comparison
change in the hangers’ elastic modulus will
Table 3 indicates that a change in the main
have the greatest effect on the twisted mode
girder’s elastic modulus will affect the vibration
shape’s vibration frequency and the smallest
frequency the most followed by a the elastic
effect on the transverse mode shape’s vibration
modulus in the towers, main cables, and hangers.
This demonstrates that a change in the main girder’s
elastic modulus will have a larger effect on the entire
suspension bridge’s stiffness followed by a change
the elastic modulus in the towers, main cables, and
hangers.
frequency.
(d)
A change in the main girder’s elastic modulus
has the largest effect on the self-anchored
suspension
bridge’s
vibration
frequencies
followed by a change the elastic modulus in
the towers, main cables, and hangers.
6. Conclusion
7. References
In comparing a double tower self-anchored
versus earth-anchored suspension bridge with bridge
length 312 m, main span 160 m, two side spans of
[1] Chin-Sheng Kao, Chang-Huan Kou, Jeng-Lin
Tsai, Yu-Min Shao, 「Nonlinear Stochastic
76 m, tower height 42.4 m, and hanger to hanger
Static Analysis of Self-Anchored Suspension
distance of 5 m, this paper’s analysis yields the
Bridge,」Asia Pacific Review of Engineering
following conclusions：
Science
(a)
pp.653~668, 2006.
Vibration frequency of the self-anchored
suspension bridges is smaller than that of the
(b)
Technology,
Vol.4,
No.1,
[2] Ahmed M Abded – Ghaffar, 「Free Lateral
earth-anchored suspension bridges.
Vibration of Suspension Bridges,」Journal of
A reduction the elastic modulus in the main
the Structural Division ASCE, 104 (3) : 503～
cables, main girder, towers, and hangers will
525, 1978.
all cause the self-anchored suspension bridge’s
horizontal, vertical, transverse, and twisted
mode
shapes’
vibration
frequencies
to
decrease.
(c)
and
A change in the main cables’ elastic modulus
[3] Ahmed M Abded – Ghaffar, 「Free Torsional
Vabration of Suspension Bridges,」Journal of
the Structural Division ASCE, 105 (4) : 767～
787, 1979.
will have the greatest effect on the horizontal
[4] Ahmed M Abded – Ghaffar, 「 Vertical
mode shape’s vibration frequency and the
Vabration Analysis of Suspension Bridges,」
smallest effect on the twisted mode shape’s
Journal of the Structural Division, ASCE, 106
vibration frequency; a change in the main
(10) : 2053～2075, 1980.
girder’s elastic modulus will have the greatest
effect on the vertical mode shape’s vibration
[5] Gi-Chuan Yan,「Structure Analysis Suspension
Bridge,」Master Thesis, Department of Civil
frequency and the smallest effect on the
horizontal mode shape’s vibration frequency.
Moreover, a change in the towers’ elastic
modulus will have the greatest effect on the
vertical mode shape’s vibration frequency and
Engineering, Chung Yuan Christian University,
Taoyuan County, Taiwan, 1988.
[6] Daw-en Peng, Chaog-Uang Huang, Zhong
Wang, 「Analysis of Earthquake Respon for
Single-Tower Suspension Bridge, 」 China
Journal of Highway and Transport, Vol.10,
No.4, 1997.
[7] Mau-Lin Tang, Ruei-Li Shen, Shr-Jung
Chiang, 「 Analytic Theories and Software
Development of Spatial Non-linearity Static
and
Dynamic
of
Long-span
Suspension
Bridge ,」Bridge Railroading, No.1, 2000.
[8] Gregor P. Wollmann, M.ASCE , 「Preliminary
Analysis of Suspension Bridges,」Journal of
Bridge
Engineering,
ASCE,
Vol.
6,
M.ASCE
,
Figure 2：Suspension bridge finite element
Module
pp.227-233, 2001.
[9] Gregor
P.
Wollmann,
「Self-Anchored Suspension Bridges,」Journal
of Bridge Engineering, ASCE, Vol. 4,
pp.156-158, 2001.
[10] Hong-jin Zhang, 「Mechanical Performance
Analysis and Experimental Research on
Concrete Self-Anchored Suspension Bridge,」
Mode1：horizontal mode shape（0.18842 HZ）
Master
Thesis,
Department
of
Bridge
Engineering, School of Civil ＆ Hydraulic
Engineering, Dalian University of Technology,
Dalian, 2004.
Side diagram
Mode9：vertical mode shape（2.4788 HZ）
Main girder cross-section
Mode11：transverse mode shape（2.7930 HZ）
Tower structure
Figure 1：Suspension bridge measurements
（2.96058）
2.7943
（2.9620）
2.8888
（3.0621）
3.0513
（3.2344）
3.1183
（3.3054）
3.1419
（3.3304）
3.4016
（3.6057）
3.5359
（3.7481）
3.7788
（4.0055）
3.7852
（4.0123）
12
13
14
15
16
Mode20：twisted mode shape（3.7852 HZ）
17
Figure 3：T he fo ur main mode shapes of
self-anchored suspension bridges
18
Table 1：Suspension bridge main constants
Main
constants
20
Main
girder
Tower
Main
cable
Hanger
A (m2)
16.41
13.67
0.12
0.0047
Ix (m4)
50.40
28.03
─
─
Iy (m4)
15.00
19.38
─
─
Iz (m4)
896.18
14.47
─
E（N/m2）
3.5*1010
3.5*1010
1.0*1011
Structural
element
19
the tower
Longitudinal vibration
of the tower
Twist of the side span
main girder and tower
Twist of the side span
main girder
Third order symmetric
vertical vibration
Longitudinal vibration
of the tower
Fourth order symmetric
vertical vibration
Twist of the side span
main girder and tower
Twist of the main span
main girder
Longitudinal vibration
of the main girder
Table 3: The Effects of a Change in the Material
Elastic Modulus on Vibration Frequency
Horizontal Vertical Transverse Twisted
Mode
Mode
Mode
Mode
Shape
Shape
Shape
Shape
─
（HZ）
（HZ）
（HZ）
（HZ）
1.2*1011
Mode1
Mode9
Mode11
Mode20
-1.57％
-0.30％
-0.06％
-0.07％
+0.88％
+0.30％
+0.01％
+0.07％
-0.86％
-15.92％ -11.67％ -15.41％
Item
Reduce to 0.7
Table 2：The first 20 mode shapes, vibration frequencies
and characteristics of the self-anchored
suspension bridge
Order
1
2
3
Frequency
（HZ）
0.18842
（0.19407）
0.39521
（0.40706）
0.67485
（0.69510）
4
0.96317
（0.99207）
5
1.0467
（1.07810）
6
7
8
9
10
11
1.4694
（1.5135）
1.7216
（1.7732）
1.9485
（2.0070）
2.4788
（2.5532）
2.6467
（2.7261）
2.7930
Mode Shape
Characteristics
Longitudinal vibration
of the main girder
First order symmetric
vertical vibration
First order asymmetric
vertical vibration
Second order
symmetric vertical
vibration
Second order
asymmetric vertical
vibration
Hanger vibration and
main girder twist
Third order symmetric
vertical vibration
Transverse vibration of
the main girder
Hanger vibration and
main girder twist
Third order asymmetric
vertical vibration
Transverse vibration of
times the
Main
original value
Cable E
value
Increase to
1.3 times the
original value
Reduce to 0.7
times the
Main
original value
Girder E
value
Increase to
1.3 times the
+0.45％ +14.55％ +10.77％ +14.23％
original value
Reduce to 0.7
times the
-14.81％
-0.04％
-10.18％
-0.12％
1.3 times the +13.67％ +0.02％
+8.22％
+0.08％
Tower E original value
value
Increase to
original value
Reduce to 0.7
times the
-0.10％
-0.02％
-0.01％
-1.89％
+0.05％
+0.01％
+0.01％
+0.72％
Hanger original value
E value
Increase to
1.3 times the
original value