EACWE 5
Florence, Italy
19th – 23rd July 2009
Flying Sphere image © Museo Ideale L. Da Vinci
Aerodynamic stabilization for box-girder suspension bridges with
super-long span
Y.J. Ge, H.F. Xiang
State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University –
[email protected] – 1239 Siping Road, Shanghai 200092, China
College of Civil Engineering, Tongji University – [email protected] – 1239 Siping Road,
Shanghai 200092, China
Keywords: aerodynamic stabilization, suspension bridge, box girder, central stabilizer, central slot.
ABSTRACT
As one of the most formidable challenges on long-span bridges, recent advances in aerodynamic
studies are presented in the aspects of flutter instability for ten longest suspension bridges.
Aerodynamic stabilization for two box-girder suspension bridges in China with super long span is
introduced, including Runyang Yangtze River Bridge with a central stabilizer and Xihoumen
Sea-Crossing Bridge adopted twin-box girder. The aerodynamic feasibility study of a box-girder
suspension bridge with a main span of 5000m is followed. It can be concluded that the intrinsic limit
of span length due to aerodynamic stability is about 1,500m for a traditional suspension bridge, but
either a widely slotted deck or a narrowly slotted deck with vertical and horizontal stabilizers could
provide a 5,000m suspension bridge with high enough critical flutter speed.
1. INTRODUCTION
Human beings have been building bridges in girder, arch, cable-stayed and suspension types for very
long history in order to cross streams and rivers. Even though those bridges in ancient time were very
Contact person: 1st Y. J. Ge, State Key Laboratory for Disaster Reduction in Civil Engineering, Tongji University, 1239
Siping Road, Shanghai 200092, China, Tel: +86-21-65983451 and Fax: +86-21-65984882.
E-mail [email protected]
small and primitive, they were bridges nevertheless. Our bridges today are certainly bigger and more
sophisticated, in particular, with greater bridging capacity in both longitudinal direction, span length,
and transversal aspect, deck width. Among these four types of bridges, suspension bridge has the
greatest longitudinal bridging capacity, and followed by cable-supported bridge.
The construction of long-span suspension bridges around the world has experienced a
considerable development for more than a century. It took about 54 years for the span length of
suspension bridges to grow from 483m of Brooklyn Bridge in 1883 to 1,280m of Golden Gate Bridge
in 1937, and had an increase by a great factor of about 2.7. Although the further increase in the next
44 years from Golden Gate Bridge to Verrazano Bridge and to Humber Bridge of 1410m in 1981 was
only 10% or by a factor of 1.1, another factor of about 1.4 was realized in Akashi Kaikyo Bridge with
a 1,991m main span within 17 years in 1998. With the ever-growing span-length of suspension
bridges, bridge structures are becoming lighter and more flexible, whose structural characteristics
result in great challenge on bridge wind resistance, in particular, aerodynamic problems including
flutter instability and torsional divergence.
Among ten longest-span suspension bridges in the world (Internet address A, 2007) listed in Table
1, the top four suspension bridges, including Akashi Kaikyo Bridge in Japan, Xihoumen Bridge in
China, Great Belt Bridge in Denmark and Runyang Bridge in China, as well as Tsing Ma Bridge in
Hong Kong China have been suffered in wind-induced problems in aerodynamic flutter or vortex
shedding, and some control measures have been adopted to improve aerodynamic performance, for
example, central stabilizer for Runyang Bridge, central slot for Xihoumen Bridge and Tsing Ma
Bridge, both slot and stabilizer for Akashi Kaikyo, and guide vane for Great Belt (Ge 2008).
Aerodynamic stabilization for two box-girder suspension bridges in China with super long span is
introduced hereafter, including Runyang Yangtze River Bridge with a central stabilizer and
Xihoumen Sea-Crossing Bridge adopted twin-box girder, and followed by the aerodynamic
feasibility study of a box-girder suspension bridge with a main span of 5000m.
Table 1: Ten longest span suspension bridges in the world.
Span
Order
1
2
3
4
5
6
7
8
9
10
Bridge Name
Akashi Kaikyo
Xihoumen
Great Belt
Runyang
Humber
Jiangyin
Tsing Ma
Verrazano
Golden Gate
Hubei Yangluo
Main
Span
1991m
1650m
1624m
1490m
1410m
1385m
1377m
1298m
1280m
1280m
Girder
Type
Truss
Box
Box
Box
Box
Box
Box
Truss
Truss
Box
Wind-Induced
Problem
Flutter
Flutter
Vortex
Flutter
None
None
Flutter
None
None
None
Control
Measure
Slot/Stabilizer
Slot
Guide vane
Stabilizer
None
None
Slot
None
None
None
Country
Japan
China
Denmark
China
U.K.
China
H.K. China
U.S.A.
U.S.A.
China
Year
Built
1998
2009
1998
2005
1981
1999
1997
1964
1937
2007
Aerodynamic stabilization for two box-girder suspension bridges in China with super long span is
introduced hereafter, including the 1490m spanned Runyang Yangtze River Bridge with a central
stabilizer and the 1650m spanned Xihoumen Sea-Crossing Bridge adopted twin-box girder, and
followed by the aerodynamic feasibility study of a box-girder suspension bridge with a main span of
5000m.
2. CENTRAL STABILIZER MOUNTED ON SINGLE BOX GIRDER
2.1
General arrangement.
Among the top ten suspension bridges in Table 1, Runyang Bridge completed in 2005 is the second
longest suspension bridge in China and the fourth longest in the world. The bridge connects
Zhenjiang City and Yangzhou City over Yangtze River at Jiangsu Province in eastern China. The
main section of the bridge was designed as a typical three-span suspension bridge with span
arrangement of 510m + 1490m + 510m as shown in Figure 1. The deck cross section is a traditional
closed steel box, 36.3m wide and 3m deep, and carries three 3.75m wide traffic lanes in each direction
with 3.5m wide shoulders on both sides for emergency as shown in Figure 2. The box girder is
equipped with classical barriers and sharp fairings intended to improve the aerodynamic streamlining
as well as aesthetic quality (Chen et al., 2002).
Figure 1: Longitudinal arrangement of Runyang Bridge (Unit: m)
Figure 2: Deck cross-section of Runyang Bridge (Unit: m)
2.2
Dynamic characteristics.
With the structural properties provided in the reference (Chen et al., 2002), finite element analysis of
dynamic characteristics of the prototype bridge was performed, and the symmetrical and
antisymmetrical fundamental natural frequencies of lateral, vertical and torsional vibration modes
were numerically extracted and compared with those of the box-girder suspension bridges, including
Great Belt Bridge in Denmark and Xihoumen Bridge in China in Table 2. The fundamental vertical
and lateral vibration frequencies of Runyang Bridge are quite reasonable, but the torsional vibration
frequencies are relatively lower than those of the other two bridges mainly because of the small depth
of the box section.
Table 2: Fundamental natural frequencies of lateral, vertical and torsional vibration modes
Bridge
Name
Span
(m)
Runyang
Great Belt
Xihoumen
1490
1624
1650
Lateral Frequency (Hz)
Symmetric Antisymm.
0.0489
0.1229
0.0521
0.1180
0.0484
0.1086
Vertical Frequency (Hz)
Symmetric Antisymm.
0.1241
0.0884
0.0839
0.0998
0.1000
0.0791
Torsional Frequency (Hz)
Symmetric Antisymm.
0.2308
0.2698
0.2780
0.3830
0.2323
0.2380
2.3
Aerodynamic stability.
To study the aerodynamic stability, a wind tunnel experiment with a 1:70 sectional model was carried
out in the TJ (Tongji University) -1 Boundary Layer Wind Tunnel with the working section of 1.8m
width, 1.8m height and 15m length. It was found in the first phase of the testing that the original
structure could not meet the requirement of flutter speed of 54m/s. Some preventive means had to be
considered to stabilize the original structure. With a stabilizer in the center of the bridge deck as
shown in Figure 2, further sectional model testing was conducted, and the confirmation wind tunnel
tests with the full aeroelastic model were also performed in TJ-3 Wind Tunnel with the working
section of 15m width, 2m height, and 14m length. The critical flutter speeds obtained from the
sectional model (SM) and full model (FM) wind tunnel tests are collected and compared in Table 3.
Both experimental results show good agreement with each other and the central stabilizer of 0.88 m
height as shown in Figure 3 can raise the critical flutter speed over the required value (Chen et al.,
2002).
Table 3: Critical flutter speeds of Runyang Bridge
Deck Box Girder
Configuration
Original box girder
With a 0.65m stabilizer
With a 0.88m stabilizer
With a 1.1m stabilizer
SM at 0°
64.4
Critical flutter speed (m/s)
FM at 0°
SM at +3°
64.3
50.8
69.5
58.1
72.1
64.9
>75
67.4
FM at +3°
52.5
53.8
55.1
56.4
Required
(m/s)
54
54
54
54
Figure 3: Central stabilizer mounted on Runyang Bridge
3. CENTRAL SLOT IN TWIN BOX GIRDER
3.1
Main span selection.
Xihoumen Sea-Crossing Bridge is part of the Zhoushan Island-Mainland Connection Project linking
Zhoushan Archipelago and Ningbo City in Zhejiang Province around China East Sea. It crosses the
Xihoumen channel, one of the most important national deep waterways. A very long span is required
in order to minimize technical complexity and unpredictable costs in constructing deep-water
foundation. The bridge route is selected at the shortest distance of the Xihoumen Strait between
Jintang Island and Cezi Island, about 2200 m far away. Between these two islands and near Cezi,
there is a small island, called Tiger Island, which can be used to hold on a pylon for a cable-supported
bridge. If one pylon of a traditional three-span suspension bridge sets on Tiger Island, the other one
may be placed at the inclined reef of Jintang Island. The location of the pylon foundation on Jintang
was compared with different span lengths, for example, above the water level with a minimum span
of 1650 m, 20 m under the water surface with a 1520 m span, 35 m under the water with a 1310 m
span, and so on. In order to avoid from constructing deep-water foundation, Xihoumen Bridge is
finally designed as a two-continuous-span suspended bridge with the span arrangement of 578m +
1650m + 485m shown in Figure 4, and is the longest suspension bridge in China and to create a world
record for box-girder suspension bridges in 2009.
Figure 4: Longitudinal arrangement of Xihoumen Bridge (Unit: m)
3.2
Box girder innovation.
Based on the experience gained from the 1490 m Runyang Bridge with flutter speed of 51 m/s and the
1624 m Great Belt Bridge with 65 m/s flutter speed, the span length of 1650 m may cause problems of
aerodynamic instability for suspension bridges, even with the stricter stability requirement of 78.4
m/s in Xihoumen Bridge. In other words, some countermeasures should be adopted to increase the
aerodynamic stability for box-girder suspension bridges with the span like Xihoumen Bridge. Apart
from a traditional single box deck and the box deck with a central stabilizer in Figure 5a, two more
innovative box decks called twin box girder with a central slot of 6m in Figure 5b and 10.6m in Figure
5c, were proposed and were investigated through wind tunnel testing.
The wind tunnel testing with these four sectional models in the scale of 1:80 was performed in the
TJ-1 Wind Tunnel. The experimental results of critical flutter speeds for these four prototypes are
summarized in Table 4. Apart from the traditional single box, the rest three deck cross sections can
meet with the flutter stability requirement under the attack angles from −3° to +3°, and the twin box
girder with 6m slot (Figure 5b) was selected as the proposed scheme, which was further modified to
the final configuration as shown in Figure 5d (Ge et al. 2003).
(a) Single box
(b) Twin box with a 6m slot
(c) Twin box with a 10.6m slot
(d) Final scheme
Figure 5: Proposed box girder sections for Xihoumen Bridge (Unit: m)
Table 4: Critical flutter speeds of Xihoumen Bridge
Deck Box Girder
Configuration
Single box girder
Single box with 1.2m stabilizer
Single box with 1.7m stabilizer
Single box with 2.2m stabilizer
Twin boxes with 6m slot
Twin boxes with 10.6m slot
3.3
−3°
50.7
>89.3
88.0
>89.3
88.4
>89.3
Critical flutter speed (m/s)
0°
+3°
46.2
48.7
>89.3
37.7
>89.3
43.4
>89.3
88.0
>89.3
>89.3
>89.3
>89.3
Minimum
46.2
37.7
43.4
88.0
88.4
>89.3
Required
(m/s)
78.4
78.4
78.4
78.4
78.4
78.4
Final experimental confirmation.
The final experimental investigation of aerodynamic instability of Xihoumen Bridge through a full
aeroelastic model was carried out in the TJ-3 Wind Tunnel. Based on the prototype structure and the
overall principals of the aeroelastic model design, the 1:208 full aeroelastic model was designed and
constructed based on the bridge with the final scheme. The full bridge model testing was respectively
conducted in smooth flow at the attack angles of −3°, 0° and +3° without yaw angle and at the yaw
angles of 5° and 15° without attack angle, and turbulent flow without yaw and attack angles (Figure
6). Under smooth flow, the flutter stability limits were not experimentally found up to the wind
speeds from 95m/s to 115m/s, under which torsional displacements increased rapidly in the tested full
aeroelastic model, so that the critical wind speeds were related to another kind of aerodynamic
instability, torsional divergence. Flutter oscillation did not also happen in turbulent flow up to the
wind speed of 85m/s, beyond which the aeroelastic model stochastically vibrated with large
amplitude but not related to any aerodynamic instability (Ge et al. 2005).
The steady aerodynamic effect on bridge structures is usually treated as an action of three steady
aerodynamic components of wind forces, represented by drag force coefficient CD, lift force
coefficient CL and pitching moment coefficient CM, which were experimentally identified at the
attack angles from −12° to +12° shown in Figure 7 (Ge et al. 2005). With the consideration of
nonlinear effects in aerostatic performance, the foremost problem is to determine nonlinear
deformation and deformed states of the bridge structure induced by steady aerodynamic force, the
static equilibrium status of a bridge structure with nonlinear effect can be determined by the
following iteration approach (Xiang and Ge 2002).
([K ]
e j −1
)
+ [K σ ] j −1 {Δδ j } = {F (α j ) − F (α j −1 )}
(1)
Values of coefficients
2.4
CD
2.0
CL
1.6
10CM
1.2
0.8
0.4
0.0
-0.4
-0.8
-1.2
-15
-10
-5
0
5
10
15
Angle of attach (degree)
Figure 6: Full bridge aeroelastic model
Figure 7: Steady force coefficients
in which [Ke]j-1 and [Kσ]j-1 are the linear elasticity stiffness matrix and the nonlinear geometry
stiffness matrix due to the displacement {Δδj}, and {F(αj-1)} and {F(αj)} are the aerostatic force
vectors corresponding to the effective attacked angles of αj-1 in step j-1 and αj in step j under the wind
speed of U, respectively. The Euclidean Norm of aerostatic force coefficients of lift, drag and
pitching moment is taken as convergence criterion, which can be expressed as (Xiang and Ge 2002)
∑ [C (α ) − C (α )]
N
2
k
j
j −1
k
n
∑ [C (α )]
N
n
≤ ε k2
(k = D, L, M )
(2)
2
k
j −1
n
n
in which εk is the convergence accuracy, and N is the total number of the structural nodes applied with
aerostatic force.
Having performed the full aeroelastic model testing and the iteration approach defined in Equation
(2), the torsional angles at the mid-span of the suspension bridge can be determined and the
experimental results are shown in Figure 8. The critical wind speeds due to torsional divergence are
given in Table 5. It can be concluded that Xihoumen Bridge with the 6m slotted deck section shown
in Figure 5d can meet with the aerodynamic stability requirement of critical speed of 78.7 m/s for
flutter instability and torsional divergence.
10
Table 5: Critical wind speeds due to torsional divergence
9
Torsional displacement (deg.)
o
α=-3
8
o
α=0
7
o
α=+3
6
5
Flow
Field
4
3
2
Smooth
1
0
0
10 20 30 40 50 60 70 80 90 100 110 120
Wind speed (m/s)
Figure 8: Torsional displacement
Turbulence
Attack
Angle
Yaw
Angle
−3°
0°
+3°
0°
0°
0°
0°
0°
0°
5°
15°
0°
Speed (m/s)
Exp.
Cal.
115
120
105
108
95
96
100
100
>85
-
4. AERODYNAMIC METHODS FOR SUPER-LONG SPAN
4.1
Design scheme.
f
As a long-time dream and an engineering challenge, the technology of bridging larger obstacles has
entered into a new era of crossing wider sea straits, for example, Messina Strait in Italy, Qiongzhou
Strait in China, Tsugaru Strait in Japan, and Gibraltar Strait linking the European and African
Continents. One of the most interesting challenges has been identified as bridge span length
limitation, in particular the span limits of suspension bridges as a bridge type with potential longest
span. The dominant concerns of super long-span bridges to bridge designers are basically
technological feasibility and aerodynamic considerations. With the emphasis on aerodynamic
stabilization for longer span length, a typical three-span suspension bridge with a 5,000m central span
and two 1,600m side spans is considered as the limitation of span length as shown in Figure 9.
1600
5000
1600
Figure 9: Elevation of the 5,000m long suspension bridge (Unit: m)
In order to push up the aerodynamic stability limit, two kinds of generic deck sections, namely a
widely slotted deck (WS) without any stabilizers (Figure 10) and a narrowly slotted deck with vertical
and horizontal stabilizers (NS) (Figure 11), were investigated. The WS cross section has a total deck
width of 80m and four main cables for a 5,000m-span suspension bridge while the NS provides a
narrower deck solution of 50m and two main cables (Xiang & Ge 2003, Ge & Xiang 2006).
Figure 10: Geometry of WS Cross section (Unit: m)
4.2
Figure 11: Geometry of NS (Unit: m)
Fundamental natural frequencies.
Having performed a dynamic finite-element analysis based on the structural parameters listed in
Table 6, the fundamental natural frequencies of the structures have been calculated for all four ratios
n of cable sag to span and the two deck configurations in Table 7.
Table 6: Parameters of stiffness and mass of the 5,000m suspension bridge
Section
WS
NS
EA (Nm2)
0.61~1.12×106
0.61~1.12×106
Main Cables
m (kg/m)
2.62~4.82×104
2.62~4.82×104
Stiffening Girder
Im (kgm2/m) EIy (Nm2) GId (Nm2) m (kg/m) Im(kgm2/m)
24000
2.8×1011
2.16×107
2.36~4.33×107 4.7×1011
11
11
7
24000
8.1×10
4.1×10
5.40×106
1.27~2.33×10
The fundamental lateral bending frequencies vary about 16% for the WS section and 17% for the
NS section from n =1/8 to n =1/11, but almost remain the same between the WS and NS deck
configurations. The fundamental vertical bending frequencies are not influenced significantly by both
deck configurations and the sag-span ratios. The fundamental torsional frequencies vary differently
with the ratio n in the two deck configurations, in which the frequency values go up in the WS section
and go down in the NS section with the decrease of the ratio n, but it is interesting to see that the
frequency ratio of torsion to vertical bending monotonically decreases with reduction of the ratio n.
Table 7: Fundamental natural frequencies of the 5,000m suspension bridge
Lateral (Hz)
WS
NS
0.02199
0.02156
0.02322
0.02285
0.02438
0.02406
0.02548
0.02520
Ratio
n = 1/8
n = 1/9
n = 1/10
n = 1/11
4.3
Vertical (Hz)
WS
NS
0.05955
0.05936
0.06126
0.06115
0.06219
0.06204
0.06237
0.06219
Torsional (Hz)
WS
NS
0.07090
0.09073
0.07207
0.08928
0.07268
0.08653
0.07269
0.08403
Frequency Ratio
WS
NS
1.191
1.528
1.176
1.460
1.168
1.395
1.165
1.351
Critical flutter speeds.
With the dynamic characteristics given above and the numerically identified flutter derivatives shown
in Figure 12, the critical flutter speeds of the suspension bridges were calculated by multi-mode
flutter analysis assuming a structural damping ratio of 0.5%.
0. 6
0. 5
0. 4
0. 3
0. 2
0. 1
0
- 0. 1
- 0. 2
- 0. 3
- 0. 4
- 0. 5
- 0. 6
0. 5
A1
0
A2
- 0. 5
A3
-1
A4
- 1. 5
-2
- 2. 5
H1
-3
H2
- 3. 5
H3
-4
H4
- 4. 5
0
1
2
3
4
5
6
Vr
7
8
1
2
3
4
5
6
7
8
9 10 11 12 13
Vr
(a) Flutter derivatives Ai of WS section
0. 6
0. 5
0. 4
0. 3
0. 2
0. 1
0
- 0. 1
- 0. 2
- 0. 3
- 0. 4
- 0. 5
- 0. 6
0
9 10 11 12 13
(b) Flutter derivatives Hi of WS section
A1
0
A2
-1
A3
-2
A4
-3
-4
-5
H1
-6
H2
-7
H3
H4
-8
0
1
2
3
4
5
6 7
Vr
8
9 10 11 12 13
0
1
2
3
4
5
6
Vr
7
8
9 10 11 12 13
(c) Flutter derivatives Ai of NS section
(d) Flutter derivatives Hi of NS section
Figure 12: Flutter derivatives at the 0° angle of attack
The results of critical flutter speeds Ucr together with the generalized mass m and mass moment of
inertia Im are summarized in Table 8. For both deck sections the critical wind speed increases with
decrease of the ratio n, although the frequency ratio of torsion to vertical bending slightly decreases.
The most important reason is the considerable increase of the generalized properties in the
aerodynamic stability analysis. The minimum critical wind speeds for the WS and NS sections are
82.9 m/s and 74.7 m/s, respectively (Ge & Xiang 2006, Ge & Xiang 2007).
Table 8: Critical flutter wind speeds of the 5,000m suspension bridge
Ratio
n = 1/8
n = 1/9
n = 1/10
n = 1/11
m (×104kg/m)
WS
NS
6.01
6.79
6.27
7.43
6.73
8.33
7.66
9.52
Im (×107kgm2/m)
WS
NS
5.28
2.37
5.36
3.22
5.92
3.29
6.77
3.62
fh (Hz)
WS
0.05955
0.06126
0.06219
0.06237
NS
0.05936
0.06115
0.06204
0.06219
fα (Hz)
WS
0.07090
0.07207
0.07268
0.07269
NS
0.09073
0.08928
0.08653
0.08403
Ucr (m/s)
WS
NS
82.9 74.7
88.8 77.4
90.9 78.9
98.9 82.7
5. CONCLUSIONS
With the experience gained from the recently built suspension bridges, such as Akashi Kaikyo
Bridge, Xihoumen Bridge, Great Belt Bridge, Runyang Bridge and Hong Kong Tsing Ma Bridge, the
intrinsic limit of span length due to aerodynamic stability is about 1,500m for a traditional suspension
bridge with either a streamlined box deck or a ventilative truss girder. Beyond or even approaching
this limit, designers should be prepared to improve aerodynamic stability of a bridge by adopting
some countermeasures in girder, for example, central stabilizer in Runyang Bridge and twin box deck
in Xihoumen Bridge. Based on a preliminary study, either a widely slotted deck or a narrowly slotted
deck with vertical and horizontal stabilizers could provide a 5,000m span-length suspension bridge
with high enough critical wind speed, which can meet aerodynamic requirement in most
typhoon-prone areas in the world.
The work described in this paper is partially supported by the Natural Science Foundation of China under the Grant
50538050, and the Ministry of Science and Technology under the Grants 2006AA11Z108 and 2008BAG07B02.
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