ENGINEERING SERVICES FOR THE FINAL DESIGN OF A BRIDGE ACROSS THE CANAL AT THE ATLANTIC SIDE
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-1-
1 Introduction
The proposed Panama Canal Bridge crosses the canal at the Atlantic side near its outfall,
and will be the key section of the transportation between the two sides of the canal. As shown
in the annex figure of Fig. A-1, the bridge is a 1050m-long PC cable-stayed bridge with spans of
79+181+530+181+79m. A cross section of twin side boxes (see Fig. A-2), 23.6m in width and
2.834m in height, is designed for the PC bridge deck, which is 80m high above the water surface
at the mid-span. The figure on annex Fig. A-3 shows the general layout of the PC bridge towers.
The height of the tower above the bearing platform upper surface is 207.5m at the P022 axis,
and is 211.5m at the P023 axis. However, the altitudes of the two towers are same and are
equal to 212.5m.
The bridge is prone to be strongly affected by the trade wind and marine climate because it
is located in the west coastal-area of the Atlantic with low latitudes in the inner tropics. The
historical wind data recorded at the weather observatories near the bridge site show that the
wind in the canal region at the Atlantic side is very strong and will exert a large unfavorable
influence on the bridge safety. In this content, the State Key Laboratory for Disaster Reduction in
Civil Engineering (SLDRCE) at Tongji University of China and the CCCC Highway Consultants Co.,
LTD. has carried out the research on wind-resistant performances of the bridge. The content is
comprised of the following three parts: (1) Wind tunnel tests of rigid sectional model of bridge
deck; (2) Full bridge analyses on wind-induced static instability and buffeting responses; (3)
Wind tunnel test of aeroelastic model for free-standing full tower. This report mainly focuses on
the relevant results of the wind tunnel tests of rigid sectional model of the bridge deck, and
includes the following contents:
(1) Analysis of natural dynamic properties of the service state and the construction state with
the longest single cantilever (LSC) of deck.
(2) Testing the wind-resistant performances and aerodynamic parameters of the service state
and the construction state of LSC via a series of wind tunnel tests of rigid sectional model
in smooth flow. The details are as follows:
(2.1) Determining the lock-in ranges of wind speed and amplitudes of vortex-excited
resonances of SS under normal wind via the vibration tests of spring-suspended rigid
sectional model. If necessary, presenting the effective aerodynamic control
measures. The considered wind attack angles are -5, -3, 0, 3and 5.
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-2-
(2.2) Checking the flutter performances of the service state and the construction state of
LSC under normal wind via the vibration test of spring-suspended rigid sectional
model. If necessary, presenting the effective aerodynamic control measures. The
considered wind attack angles are -3, 0 and 3.
(2.3) Identifying the aerodynamic derivatives of the bridge deck of the service state and
the construction state of LSC under normal wind via the vibration tests of springsuspended rigid sectional model. The considered wind attack angles are -3, 0 and
3.
(2.4) Measuring the three-component aerodynamic coefficients of the bridge deck of the
service state and the construction state of LSC under normal wind via the force
balance tests of rigid sectional model. The considered wind attack angles are from 10 to 10 at an interval of 1.
2 Wind speed parameters at the bridge site
Because there isn’t any observed wind data at the bridge site，the long-term observed
data of monthly-maximal 1hr-average wind speed recorded between Jan. 1985 and Dec. 2005 at
the GATUN weather station near the bridge site, which were given in the Annex C Hydrometeorological Report (SE-09-16) of The Tender for Engineering Services for the Final
Design of a Bridge across the Canal at the Atlantic Side[1], were used to determine the relevant
wind speed parameters at the bridge site according to the design codes of AASHTO [2] and
ASCE/SEI 7-05[3] and some relevant references[4,5].
2.1 Basic wind speed
The basic wind speed (U10) is conventionally defined as the annually-maximal 10min-average
wind speed at a height of 10m above the ground of a standard “OPEN COUNTRY” terrain, and
varies with the return period. The weather stations, such as GATUN, are therefore required to
be built in a region with “OPEN COUNTRY” terrain. The statistic value of the annually-maximal
1hr-average wind speeds at a height of 10m above the ground at GATUN weather station
provided by the designer are listed in Table 1 for different return period.
Table 1 Annually-max. wind speeds at 10m height at GATUN weather station
Return period
(year)
1hr-average wind speed
(m/s)
10min-average wind speed
(m/s)
25
24.1
25.6
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-3-
100
28.8
30.6
1000
36.6
39.0
10000
44.4
57.2
Fig. A-4, excerpted from ASCE/SEI 7-05[3], gives the empirical curve of the ratio of VT/V3600
vs. the gust duration (or averaging duration) of time T in the unit of second, where, V3600
indicates the 1hr-average wind speed, and VT represents the average wind speed for an
duration of T seconds. One can then determine that the wind speed ratio is about 1.064 for
T=600s (i.e. 10min). The basic wind speed (U10) for different return period, which is
corresponding to an gust/averaging duration of 10min=600s, can then be converted from the
1hr-average wind speed, and is also listed in Table 1.
Because both the terrain conditions at the bridge site and at the GATUN weather station
can be classified into the terrain category of “OPEN COUNTRY”, the basic wind speed at the
bridge site (Us10), i.e., the 10min-average wind speed at a height of 10m above the ground at
the bridge site for a return period of 100 years, is as same as that at the GATUN weather station.
2.2 Design reference wind speeds at different levels
According to the design codes of AASHTO[2], the vertical profile of mean wind speed within
the atmospheric boundary layer can be supposed to comply with the logarithm law as follows:
U z  2.5u* ln( z z0 )
(1.)
Where, z indicates the height of the deck upper surface from the water surface; Uz represents
the mean wind speed at the height of z; z0 is the roughness length of the ground surface at the
bridge site, and is equal to 0.07m for the “OPEN COUNTRY” terrain in light of the code of
AASHTO; u* is the friction or shear velocity near the ground surface. One can then derive from
Eq.(1),
Uz  Ur
ln( z z0 )
ln( zr z0 )
(2.)
Where, zr is the reference height and U r is the reference mean wind speed at the reference
height of zr . If taking 10m as the reference height, i.e. zr＝10m , then the reference mean wind
speeds for an averaging duration of 10min are 25.6m/s, 30.6m/s, 38.9m/s and 47.2m/s for the
return periods of 25, 100, 1000 and 10000 years, respectively. The design reference wind
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-4-
speeds at different levels can then be calculated using Eq.(2), and are listed in the following
Table 2.
Table 2 Design reference wind speeds at different levels (10min duration)
Height above
water surface
(m)
10.000
20.000
30.000
40.000
50.000
60.000
70.000
80.000
83.374
90.000
100.000
110.000
120.000
130.000
138.175
140.000
150.000
160.000
170.000
180.000
190.000
200.000
212.500
Return period
25
100
1000
10000
25.6
29.2
31.3
32.8
34.0
34.9
35.7
36.4
36.6
37.0
37.5
38.0
38.5
38.9
39.2
39.3
39.6
40.0
40.3
40.6
40.9
41.1
41.4
30.6
34.9
37.4
39.2
40.6
41.7
42.7
43.5
43.7
44.2
44.9
45.5
46.0
46.5
46.9
46.9
47.4
47.8
48.1
48.5
48.8
49.1
49.5
38.9
44.4
47.6
49.8
51.6
53.0
54.2
55.3
55.6
56.2
57.0
57.8
58.4
59.1
59.6
59.7
60.2
60.7
61.2
61.6
62.1
62.5
62.9
47.2
53.8
57.7
60.4
62.6
64.3
65.8
67.0
67.4
68.2
69.2
70.1
70.9
71.7
72.2
72.4
73.0
73.6
74.2
74.8
75.3
75.8
76.3
According to the design codes of AASHTO[2], when determining the design wind speed of
bridges, the return period should be taken as 100 years for the service state, and 25 years for
the construction states. One can know from the design drawings, the height of the deck upper
surface at the mid-span is about 83.374m above the water surface. Therefore, the design
reference wind speeds at the deck level for the service state（ U d ）and for the construction
state ( U ds ) are as follows:
U d  43.7 m/s (100-year return period)
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
(3.)
-5-
U ds  36.6 m/s (25-year return period)
(4.)
2.3 Flutter checking wind speeds at deck level
According to the design codes of AASHTO[2], when determining the flutter checking wind
speeds of bridges, the return period should be taken as 10000 years for the service state, and
1000 years for the construction states. Therefore, the flutter checking wind speeds at the deck
level for the service state（ U cr  ）and for the construction states ( U crs  ) are as follows:
U cr   67.4 m/s
(10000-year return period)
U crs   55.6 m/s (1000-year return period)
(5.)
(6.)
3 Analyses of natural dynamic properties
The natural dynamic properties of the service state and the construction state of LSC of the
bridge were analyzed by using the FEM software of ANSYS based on the design materials
provided by the CCCC Highway Consultants Co., LTD. The geometric and material properties of
the bridge deck, cables, towers and piers are listed in Table A-1, Table A-2 and Table A-3
respectively, where the locations of the typical tower sections of A, B, C1-C5, E1-E3 and F1-F3
are shown in Fig. A-3. In the FEM model of the bridge, both the tower and the deck were
modeled with 3D beam element of 12 degrees of freedom (DOFs) in conjunction with the spine
girder model, whilst the truss element was used for modeling the cables, where the Ernst
equivalent elastic module was introduced to consider the influence of the initial tension and
gravity of the cables on their stiffness. The constraint conditions of the service state and the
construction state of LSC of the bridge are as follows:
For the service state: The deck vertical and lateral translation DOFs and the deck torsional
DOF around the longitudinal axis are rigidly constrained to the towers at the axes of P022 and
P023, whiles the deck longitudinal translation DOF and the deck rotation DOFs around the
vertical and lateral axes are free from the tower constraint. The deck vertical, lateral translation
DOFs and the deck torsional DOF around the longitudinal axis are also rigidly constrained to the
abutment piers at the axes of P020 and P025 and the auxiliary tie-down piers at the axes of
P201 and P024, whiles the deck longitudinal DOF and the rotation DOFs around the vertical and
lateral axes are free from the constraints of the piers. The towers and the piers are fixed on their
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-6-
own bearing platforms.
For construction state of LSC: The side spans of the deck are connected to the abutment
piers, the main span decks of two half bridges is approaching to each other, but not connected
yet. The length of the deck cantilever is set as 263m in this study. The constraint conditions of
the deck at the abutment piers and the auxiliary tie-down piers are as same as those for SS.
However, all the six DOFs of the deck are rigidly constrained to the tower. The towers and the
piers are also fixed on their own bearing platforms.
According to the literature [7], the equivalent mass and mass moment of inertia of deck,
considering the effect of vibration of whole bridge structure and the spatial behavior of
vibration, should be employed while determining the mass system of the sectional model. The
equivalent mass and mass moment of inertia of deck of the prototype bridge can be
determined as follows:
meqd  M
~
J mx eq  M

Lg

d2 ( x)dx (d  x, y, z)
Lg
2x ( x)dx
(7.)
(8.)
~
where, M is the generalized mass of the corresponding mode; d = x, y, z; x(x), y(x), z(x) and
x(x) are the mode function values of deck longitudinal, vertical and lateral displacement and
torsional angle at the coordinate of x, respectively; Lg is the total length of the bridge deck.
Fig. A-5 and Fig. A-6 show the schematic diagrams of FE mechanical models of the service
state and the construction state of LSC, respectively. The first 20 natural frequencies of the
service state, are listed in Table A-4 together with the descriptions of the corresponding major
mode features. The corresponding mode shapes are given in Fig.A-7. The equivalent mass and
mass moment of inertia of these modes computed with Eq.(7)and Eq.(8) are also presented in
Table A-5. It can be found that the fundamental symmetrical natural frequencies of deck vertical
bending and torsion of the service state are 0.2394Hz and 0.7013Hz, respectively, and the
corresponding mode shapes are symmetric. The frequency ratio of torsion to vertical bending is
2.929. The equivalent mass of deck corresponding to the fundamental vertical bending mode is
4.52104kg/m and the equivalent mass moment of inertia of deck corresponding to the
fundamental natural torsional mode is 2.28106kgm2/m.
The first 10 natural frequencies of the construction state of LSC, are listed in Table A-6
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-7-
together with the annotations of the corresponding major mode features.. The corresponding
mode shapes are given in Fig.A-8. The equivalent mass and mass moment of inertia of these
modes computed with Eq.(7)and Eq.(8) are also presented in Table A-7. The fundamental
symmetrical natural frequencies of deck vertical bending and torsion of the construction state
of LSC are 0.2464Hz and 0.7738Hz, respectively, and the corresponding mode shapes are
symmetric. The frequency ratio of torsion to vertical bending is 3.140. The equivalent mass of
deck corresponding to the fundamental vertical bending mode is 4.90104kg/m and the
equivalent mass moment of inertia of deck corresponding to the fundamental natural torsional
mode is 2.04106kgm2/m.
The above mentioned fundamental natural model will then be considered in the following
wind tunnel tests of sectional model for the service state and the construction state of LSC. The
modal parameters of these fundamental modes are listed in Table 3.
Table 3 Modal parameters of fundamental modes
Structural state
Vertical fundamental mode
(symmetrical)
Frequency
(Hz)
Torsional fundamental mode
(symmetrical)
Equivalent mass Frequency Equivalent mass moment
(Hz)
of inertia (kgm2/m)
(kg/m)
Service state
0.2394
4.52E+04
0.7013
2.28E+06
Construction state
of LSC
0.2464
4.90E+04
0.7738
2.04E+06
4 Wind tunnel test of sectional model for wind-induced vibration
4.1 Equipment and instrumentation
The sectional model wind tunnel tests for wind-induced vibration were carried out in the
TJ-2 Boundary Layer Wind Tunnel of the State Key Laboratory for Disaster Reduction in Civil
Engineering of Tongji University. The TJ-2 Wind Tunnel is a boundary layer tunnel of closedcircuit-type as shown in Fig. A-9. The working section of the tunnel is 3m wide, 2.5m high, and
15m long. The achievable mean wind speed ranges from 0.5m/s to 68.0m/s, adjustable
continuously. The non-uniformity index of smooth flow (U/U), excluding the boundary areas, is
less than 1.0% and the corresponding turbulence intensity (Iu) within the same cross section is
less than 0.5%. Both the horizontal yaw angle and vertical inclination angle of the mean wind in
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-8-
the tunnel are less than 0.5.
The signals of wind-induced vibration were acquired and processed with the a system
composed of three M353B15 miniature accelerometers made by PCB Piezotronics Inc. of USA, a
PCI-6052E data acquisition A/D board manufactured by NI Inc. of USA, and a PC and
corresponding computer programs for data acquisition and processing.
4.2 Basic similarity requirements and model parameters
The 2D spring-suspended rigid sectional model was employed in the tests. The geometric
length scale (L) was determined to be 1/65 according to the dimensions of the prototype deck
cross-section of the bridge, the mass and mass moment of inertia of the bridge, the testing
section size of the wind tunnel, the request of the direct testing approach for wind-induced
vibration. The width and length of the deck sectional model were 0.363m and 1.72m,
respectively. The ratio of length over width of the sectional model was thus about 4.7.
The rigid sectional models were suspended in the TJ-2 wind tunnel with 4 springs from the
ceiling and other 4 springs from the floor, as shown in Fig. A-10 for the service state, and in Fig.
A-11 for the construction state of LSC. The rigid sectional model comprised a steel frame and a
set of “coat”. The steel frame was 1.72m long and composed of two longitudinal bars of steel
box, two steel end plates and several steel transverse beams welded together. The coat was
made of wood three-ply and high-dense foam used to ensure the similarity of the deck shape.
For the service state, the handrails and crash barriers on the upper surface of the deck, and the
rails for the overhaul dolly on the lower surface of the deck were also simulated on the
sectional model, and were made of ABS plastic plates via computer-controlled sculpture.
Besides the geometric similarity, the following three groups of dimensionless parameters
should be kept in consistence between the model and the prototype in the wind tunnel test of
rigid spring-suspended sectional model:
Elastic parameters: U f v B , U f t B or f t f v （frequency ratio）
Inertia parameters: meq b 2 , J meq b 4 or re b （ratio of gyration radius）
Damping parameters: v, t（damping ratio）
where, U is mean wind speed, fv and ft are the natural frequencies of vertical and torsional
vibrations, respectively; B is deck width, b is half width of deck; meq and Jmeq are equivalent mass
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-9-
and mass moment of inertia of the bridge deck per unit length;  is air density; re is equivalent
gyration radius of the bridge deck; v and t are damping ratios of vertical and torsional
vibrations, respectively. According to the suggestions about the damping ratio presented in
AASHTO “Bridge Design Specifications” [1] and the Chinese “Wind-resistant Design Specification
for Highway Bridges” [7] , the modal damping ratios of the Panama Canal Bridge can be set to
2.0% because the PC structure is adopted for the bridge. However, in this study of windresistant performances of the bridge, a value of 1% was used as the target damping ratio for the
purpose of safety.
As a result, the designed and measured parameters of sectional model and the
corresponding parameters of prototype, obtained in the light of the similarity requirements
mentioned above, are presented in Table A-8 for the service state and in Table A-9 for the
construction state of LSC. The fundamental natural frequencies of vertical and torsional
vibrations of the both structural states were selected for the simulation of elastic parameters.
The mass and mass moment of inertia of the sectional model were designed according to the
equivalent mass and mass moment of inertia of the prototype bridge deck to consider the
spatial behavior of the vibration of the prototype bridge and the effects of the vibrations of
towers and cables [7].
The damping ratios of the original spring-suspended sectional model system were quite
low, and were increased in the tests by adding some deformable circles, made of steel wires
wound together, on the springs. The measured damping ratios of vertical and torsional
vibrations undulated a little bit for the various attitudes of the sectional model with different
attack angle of wind. The torsional damping ratios are, respectively, 0.94~0.97% and
0.79~0.88% for the flutter tests of the service state and the construction state of LSC. The
vertical damping ratios were, respectively, 0.92~1.05% and 0.86~0.93% for the vertical and
torsional vortex-excited resonances (VERs) of the service state. The major testing cases of the
wind tunnel test as well as the ranges of testing wind speed and the velocity scales are listed in
Table A-10. All the tests were conducted in smooth flow.
4.3 Testing on VER of the service state
Because evident vortex-excited resonances (VER) of the bridge were found in the
preliminary trial, the actual tests stared from the tests on VER so as to devise some proper
aerodynamic measures which might be necessary before the flutter tests and force
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-10-
measurement tests. Considering the VER is a kind of limited-amplitude vibration and the
construction stage of the bridge will last only a few years, the wind tunnel test on the VER was
carried out only for the service state.
4.3.1 Allowable amplitude of VER
In the light of the Chinese “Wind-resistant Design Specification for Highway Bridges”[6], the
allowable amplitudes of the vertical and torsional VERs are, respectively, as follows:
[ha] = 0.04/fb = 0.04/0.2394 = 0.167m
[a] = 4.56/Bft = 4.56/(23.60.7013) = 0.276o
(9.)
(10.)
4.3.2 Amplitude conversion between the sectional model system and the prototype
bridge
To take the 3D spatial effect of full bridge vibration, the mass and mass moment of inertia
of the sectional model for the VER test should also be simulated the equivalent mass and mass
moment of inertia of the prototype bridge, determined using Eq.(7) and Eq.(8)
[7,8].
Furthermore, while calculating the amplitudes of the VERs of the prototype bridge base on the
sectional model test results, the mode shape effect should be taken into account besides the
consideration of scale conversion. This may be attained approximately by using the following
equations [8]:
ymax  C RvCv max y0 m L
(11.)
 max  CRtC
(12.)
t
max
 0m
Where, ymax and max are, respectively, the maximal amplitudes of the vertical and torsional
resonances of the prototype bridge along the spans. y0m and 0m, respectively, the tested
amplitudes of the vertical and torsional VERs of the sectional model. L is the length scale. CRv
and CRt are the reduction coefficient with values less than 1.0, and are used to consider the
spanwise incomplete correlation of vortex-shedding forces due to turbulence and other factors.
Because of existence of the abduction action of the structural vibration on the vortex shedding
during the VER, the spanwise correlation of the vortex-shedding forces is better than that while
the absence of the resonance. Therefore, the values of CRv and CRt may safely set as 1.0 in the
practice. C v max and C t max are, respectively, the mode shape corrective coefficient for the
maximal amplitude of vertical and torsional VERs, and are defined as follows [8]:
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-11-
C v max  v max 
Ct max  t max 
Lg
0
Lg
0
 yv ( x) dx
t ( x) dx


Lg
0
Lg
0
 yv2 ( x)dx
(13.)
2t ( x)dx
(14.)
Where, v max and  t max are, respectively, spanwise maximal values of the functions of vertical
and torsional mode shapes.  yv (x) and t (x) are, respectively, the vertical component of the
vertical mode shape function or the torsional component of the torsional mode shape function
at the coordinate x. Lg is the length of the bridge deck.
For the Panama Canal Bridge, the values of C v max and C t max are, respectively, 1.6774 and
1.4456 according to the computed mode shape results. And the responses of VER provided in
this report have been corrected by using the above-mentioned approach.
4.3.3 Lock-in ranges of wind speed and amplitudes of the VER
The testing wind attack angle is +5, +3, 0, -3 and -5 for the VER test of the service
state. The scale of wind velocity is about 1:3.3（see TableA-10），and the testing wind speed
range is from 0 to 15m/s, which is corresponding to a prototype wind speed range of 049.5m/s
and exceeds the design reference wind speed of deck 43.7m/s.
The test was carried out at first on the original case of the bridge deck, in which the rails
for overhaul dolly are close to the outer edges of the bottom plates of the deck boxes with a
central distance only 20cm (see Fig. A-2). In this case of the overhaul dolly rail location, the
evident vertical VER was observed in the test when the wind attack angle was 3 and 5. The
torsional VER, however, was not found in the test. The corresponding curves of the standard
deviation values of the vertical VER responses vs. wind speed are plotted in Fig. A-12, and the
lock-in ranges of wind speed, the vibration frequencies, the maximal standard deviation and
amplitudes are listed in Table 4.
The test results show that, although the vertical VER was observed in the test in the case of
positive wind attack angle, the amplitudes didn’t exceed the allowable one determined
according to the Chinese code [6]. However, the lock-in ranges of wind speed of the vertical VER
are only between about 9m/s, and are quite low, and the maximal amplitude of VER reaches to
an evident level of 6.2cm in the case of +5 attack angle. In such a low range of wind speed, the
turbulent intensity is normally low and the +5 attack angle often occurs, therefore, the
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-12-
occurrence probability of VER may be rather high.
Table 4 Response data of vertical VER of the service state with the original locations of overhaul dolly rails
Wind Lock-in range
VER
attack of wind speed Frequency
angle
(m/s)
(Hz)
Parameters of VER Peak
Allowable
amplitude
Wind speed
Standard
Peak
（m）
(m/s)
deviation (m) amplitude (m)
+5 o
8.58～9.9
0.2394
8.91
0.0436
0.0616
o
8.58～9.9
0.2394
8.91
0.0185
0.0261
+3
0.167
0o
-3 o
—
-5 o
Note:
1. The values given in the table is corresponding to the prototype; 2. “—“ indicates there is no VER observed.
4.3.4 Aerodynamic measures for the VER mitigation
To mitigate the vertical VER and consequently to reduce the occurrence probability of
evident VER as possible, the location of the overhaul dolly rails was improved via further wind
tunnel tests. It was then found that moving the overhaul dolly rails to the deck central by 65cm
in prototype scale (totally 85cm between the rail center and the outer edge of the box bottom
plate, see Fig. A-2) could effectively reduce the vertical VER responses (see Table 5 and Fig. A13). The vertical VER was reduced by about 21% in the case of +5 attack angle, and was
thoroughly eliminated in the case of +3 attack angle. At the same time, the vertical VERs in the
cases of 0, -3, -5 attack angles and the torsional VERs in the cases of -5 to 5 attack angels
still remained dormant.
Table 5 Response data of vertical VER of the service state with the improved locations of overhaul dolly
rails
Wind Lock-in range
VER
attack of wind speed Frequency
angle
(m/s)
(Hz)
+5 o
8.58～10.23
0.2394
Parameters of VER Peak
Allowable
amplitude
Wind speed
Standard
Peak
（m）
(m/s)
deviation (m) amplitude (m)
9.24
0.0345
0.0487
+3 o
0o
-3 o
—
0.167
-5 o
Note:
1. The values given in the table is corresponding to the prototype; 2. “—“ indicates there is no VER observed.
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-13-
4.4 Checking of flutter performance
The flutter tests were conducted in smooth flow for both the service state and the
construction state of LSC and for the wind attack angles of -3, 0 and +3, and the
corresponding flutter critical wind speed were measured and checked for the two structural
states by means of direct testing approach. The test cases, the corresponding testing wind
speed range and the wind speed scale are shown in Table A-10.
4.4.1 Aeroelastic stability of the service state
The flutter test of the service state was carried out on the bridge deck with the improved
locations of the overhaul dolly rails. The variations of damping ratios (v, t) and frequencies (fv,
ft) of the vertical and torsional vibrations of the sectional model system with the testing wind
speed (Um) are shown in Fig. A-14 and Fig. A-15, respectively. From the curves of t－Um , one
can determine the flutter critical wind speeds by seeking the zero-damping points on the
curves. The test results of the flutter critical wind speed of the service state are listed in Table 6.
It can then be found that the flutter critical wind speed of the service state exceeds 104m/s for
all three wind attack angles of +3, 0and -3, and also significantly higher than the
corresponding flutter check wind speed 67.4m/s. The service state has thus the enough
aeroelastic stability.
Table 6 Test results of flutter critical wind speeds
Critical wind
speed (m/s)
Service state
Construction state of LSC
t  0.94~0.97%
t 0.79~0.88%
+3
104.5
113.8
0
117.3
>140
-3
>134
>140
Flutter checking wind speed (m/s)
[67.4]
[55.6]
Attack angle
Note:
1. The values given in the table is corresponding to the prototype.
4.4.2 Aeroelastic stability of the construction state of LSC
The variations of damping ratios (v, t) and frequencies (fv, ft) of the vertical and torsional
vibrations of the sectional model system of the construction state of LSC with the testing wind
speed (Um) are shown in Fig. A-16 and Fig. A-17, respectively. The flutter critical wind speeds
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-14-
can then be determined from the curves of t－Um by finding the zero-damping points on the
curves. The test results of the flutter critical wind speed of the construction state of LSC are also
listed in Table 6. It can be seen that all the flutter critical wind speeds of the construction state
of LSC for three wind attack angles of +3, 0and -3 are over 113m/s, and also notably higher
than the corresponding flutter check wind speed 55.6m/s. Thus, the construction state of LSC
also possesses the enough aeroelastic stability.
4.5 Identification of aerodynamic derivatives
The sectional model wind tunnel tests for the identification of aerodynamic derivatives of
the bridge deck of the service and construction states were carried out in the smooth flow with
the attack angles of 3, 0 and -3. The sectional models were the same as used in the tests for
checking the flutter stability of the bridge. The approach of 2DOF-coupled vertical bending and
torsional free decay vibration stimulated with an inertial displacement was employed in the
tests. The aerodynamic derivatives were then identified through the analysis of the recorded
acceleration signals of 2DOF-coupled vibration by using the modified least square method
introduced in the literature [9]. The identified data of the 8 aerodynamic derivatives Ai* and H i*
（i=1, …, 4） varying with reduced wind speed are listed in Table A-11 to Table A-13 for the
service state, and in the Table A-14 to Table A-16 for the construction state under the winds
with the attack angles of 3,0 and -3. The corresponding curves are plotted in Fig. A-18 and
Fig. A-19. The expressions of the aeroelastic self-excited forces employed in this report are as
follows:

h
B
h
Lse  U 2 B  KH 1* ( K )  KH 2* ( K )
 K 2 H 3* ( K )  K 2 H 4* ( K ) 
U
U
B

(15.)

h
B
h
M se  U 2 B 2  KA1* ( K )  KA2* ( K )
 K 2 A3* ( K )  K 2 A4* ( K ) 
U
U
B

(16.)
where, Lse is the aeroelastic self-excited lift force; Mse is the aeroelastic self-excited pitching
moment;  =1.225kg/m3 is the air density; B is the deck width; U is the wind speed; K=B/U is
the reduced frequency; h and  are the vertical displacement and torsional angle of the deck

motion, respectively, ( ) represents the derivative with respect to time.
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-15-
5 Wind tunnel test of sectional model for aerodynamic coefficients
5.1 General situation of the test
The wind tunnel test of force balance-supported sectional model was also carried out in
the TJ-2 boundary wind tunnel for measuring the three-component aerodynamic coefficients of
the deck cross section for both the service state and the construction state of LSC. As shown in
Fig. A-20 and Fig. A-21, the sectional model used in the force measurement wind tunnel test
was comprised of a measured segment and a dummy segment for compensating the
surroundings and reducing the effect of 3D flow around the upper end of the measured
segment. The dummy segment was rigidly connected to a steel frame with its upper end, and
was adjusted properly to be in alignment with the measured segment, and to ensure its lower
end very close to the upper end of the measured segment with a narrow gap of about 1mm and
without any contact. The measured segment was vertically fixed on a five-component force
balance with its lower end through a horizontal circular Perspex plate, supported directly on the
turntable, without any touch to each other. The circular Perspex plate then serves as a 2D end
plate to eliminate the 3D flow around the lower end of the measured segment, and to reduce
the thickness of boundary layer. The measured segment was mainly made of wood three-ply
whilst its lower end plate was made of aluminum to raise the link stiffness, and some of its
middle transverse beam were made of ABS plates. The dummy segment was made of Perspex
and ABS plate. For the service state, the handrails and crash barriers on the upper surface of the
deck, and the rails for the overhaul dolly on the lower surface of the deck were also simulated
on the sectional model, and were made of ABS plastic plates via computer-controlled sculpture.
The length scale of this model was also selected as 1/65, and the major parameters of the
model are given in Table 7. The testing wind speed was 10m/s.
Table 7 Model parameters for force measurement test
Name of parameter
Symbol
unit
Model value (m)
Length scale
L
－
1/65
Length of measured model
L
m
0.458
Length of dummy model
L1
m
0.454
Width of model
B
m
0.363
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-16-
Height of model
H
m
0.0436
5.2 Three-component aerodynamic coefficients of the bridge deck
The definitions of the aerodynamic forces/moment acting on the bridge deck are shown in
Fig. A-22, where, FH and FV are the lateral and vertical aerodynamic forces associated with the
body coordinate system respectively; FL and FD are the aerodynamic lift and drag forces
associated with the wind coordinate system; M is the aerodynamic pitching moment, and is
same in the both coordinate systems;  is the attack angle of the incident wind, and is positive
when the mean wind is upward.
The three-component aerodynamic coefficients associated with the body coordinate
system are defined as follows:
Aerodynamic coefficient of lateral force:
CH 
FH
1 / 2 U 2 DL
(17.)
Aerodynamic coefficient of vertical force:
CV 
FV
1 / 2 U 2 BL
(18.)
Aerodynamic coefficient of pitching moment:
CM 
M
1 / 2 U 2 B 2 L
(19.)
where, U is testing wind speed;  is air density and is equal to 1.225kg/m3; L is the length of the
sectional model; D is the height of the deck and serves as the reference length of the
aerodynamic coefficients of lateral force and drag force; B is the width of the deck and used as
the reference length of the aerodynamic coefficients of vertical force, lift force and pitching
moment.
The three-component aerodynamic coefficients associated with the wind coordinate
system are defined and transformed from the body-axis aerodynamic coefficients as follows:
CD 
FD
 CH  cos   CV  B / D  sin 
1 / 2 U 2 DL
(20.)
CL 
FL
 CH  D / B  sin   CV  cos 
1 / 2 U 2 BL
(21.)
where, CD is the aerodynamic coefficient of drag; CL is the aerodynamic coefficient of lift; and
the aerodynamic coefficient of pitching moment CM is same in the both coordinate systems.
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-17-
5.3 Measurement results of aerodynamic coefficients
The measured results of the three-component aerodynamic coefficients of the deck crosssection of the bridge are shown in Fig. A-23 and Table A-17 for the service state, and in Fig. A-24
and Table A-18 for the construction state.
6 Summary
Based on the modal analyses of the Panama Canal Bridge and the sectional model testing
on the wind-resistant performance of the bridge, the following major conclusions can be
drawn:
(1) The design reference wind speed at deck level of the Panama Canal Bridge is 43.7m/s and
36.6m/s, respectively, for the service and construction states, and the corresponding
flutter checking wind speed is 67.4m/s and 55.6m/s, respectively.
(2) The results of modal analyses show that the natural frequencies of the fundamental
symmetrical vertical bending and torsional modes of the service state are 0.2394Hz and
0.7013Hz respectively, and the ratio of torsional frequency to vertical bending frequency is
2.93. The longest cantilever state is the most unfavorable construction state, and the
natural frequencies of the fundamental vertical bending and torsional modes are 0.2464Hz
and 0.7738Hz, the corresponding ratio of torsional frequency to vertical bending frequency
is 3.148.
(3) The results of vortex-excited resonance test of the service state with the overhaul dolly
rails near the edge of the deck bottom plate indicate that for the wind attack angles
ranging from -5 to 5, the torsional vortex-excited resonance will not occur, and the
vertical vortex-excited resonance may occur only at the wind attack angle from 3 to 5.
The maximal amplitude of the vertical vortex-excited resonance is about 6.2cm and occurs
at the wind attack angle of 5 with the value of damping ratio being about 1%. It is only
about 37% of the allowable value. Thus the performance of the vortex-excited resonance
of the bridge service state of agrees with the relevant code requirement.
(4) Moving the overhaul dolly rails to the deck central by 65cm in prototype scale (totally
85cm between the rail center and the outer edge of the box bottom plate) can further
improve the performance of the vortex-excited resonance of the bridge service state.
(5) The results of flutter test demonstrate that the flutter critical wind speeds of both the
service state and the construction state of the bridge exceed 104m/s within the wind
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-18-
attack angle range from -3 to 3, and remarkably higher than the corresponding flutter
checking wind speed. Thus the bridge possess enough aeroelastic stability.
(6) The aerodynamic coefficients of the bridge deck were measured for both the service and
construction states. Because of adoption of a narrow deck and the handrails and crash
barriers with relatively low ventilating ratios, the aerodynamic drag coefficients of the
bridge deck are relatively large for both the service and construction states, and reach to
1.8013 and 1.1244 at the wind attack angle of 0, respectively.
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-19-
References
AASHTO LRFD，Bridge Design Specifications, Customary U.S. Units, 4th Edition, 2007.
The American Society of Civil Engineers，ASCE STANDARD, ASCE/SEI 7-05, Minimum Design
Loads for Buildings and Other Structures, 2006.
John D. Holmes, Wind Loading of Structures, Spon Press (Taylor & Francis Group)，London and
New York, 2001.
Simiu, E., and Scanlan, R.H., Wind Effects on Structures: Fundamentals and Applications to
Design, 3rd Edition, John Wiley & Sons, INC., New York, USA，1996.
Panama Canal Authority, Tender for Engineering Services for the Final Design of a Bridge across
the Canal at the Atlantic Side: Annex C-Hydrometeorological Report (SE-09-16).
Ministry of Communications of P.R. China, Wind-resistant Design Specification for Highway
Bridges（JTG/T D60-01-2004）, China Communications Press, 2004.
Zhu, L.D., and Xiang, H.F., Mass-system simulation of sectional model for bridge flutter,
Structural Engineers, 1995, 11:4, 39-45+38. (in Chinese).
Zhu L.D., Mass simulation and amplitude conversion of bridge sectional model test for vortexexcited resonance, Engineering Mechanics, 2005, 22(5): 204-208+176.
[9]
Ding Q.S.; Chen A.R., and Xiang H.F., Modified least-square method for identification of
bridge deck aerodynamic derivatives，Journal of Tongji University, 2001, 29(1): 25-29.
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-20-
Table A-1
Geometric
properties
Geometric and material properties of typical deck cross-section
A (m2)
11.6
Material
properties
Distributed mass
and mass moment
of inertia
Jd (m4)
17.64
Iz (m4)
Iy (m4)
9.722
638.6
E (Pa)

3.47381010
0.167
M (kg/m)
J m (kg·m2/m)
m I meqI (kg/m)
J mI (kg·m2/m)
35166
1.96E+06
29017.64
1.62E+06
Note：1. The subscripts y and z indicate the vertical and horizontal axis of the elemental
coordinates of deck, respectively;
2. A, Iz, Iy and Jd represent the area, the second moments of area about axis z and axis y,
and the second pole moment of area (pure torsional constant) of bridge deck crosssection, respectively;
3 m, Jm represent the total mass and mass moment of inertia per unit length of bridge
deck in the service stage, respectively;
4 mI, J mI represent the total mass and mass moment of inertia per unit length of bridge
deck in the construction stage; respectively;
5 E、 are the module of elasticity and Poisson ratio;
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-21-
Table A-2
Geometric and material properties of bridge cables
No.
Tension of
one cable
(kN)
Cable length
(m)
Equivalent
Module (Pa)
S32
4974
283.5
1.90081011
S31
4805
276.2
1.89851011
S30
4614
268.9
1.89521011
S29
4455
261.6
1.89271011
S28
3695
254.4
1.92081011
S27
3560
247.1
1.91941011
S26
3424
239.8
1.91781011
S25
3292
232.5
1.91621011
S24
2871
225.3
1.92201011
S23
2906
217.2
1.92511011
S22
2882
209.0
1.92661011
S21
2894
201.0
1.92891011
S20
2711
192.9
1.92901011
S19
2680
184.8
1.92821011
S18
2714
176.8
1.93111011
S17
2734
168.9
1.93341011
S16
2737
160.9
1.93531011
S15
2685
153.0
1.93631011
S14
2637
145.2
1.93731011
S13
2582
137.4
1.93831011
S12
2505
129.7
1.93901011
S11
2431
122.1
1.93991011
S10
2345
114.5
1.94061011
S9
2245
107.2
1.94121011
S8
2119
100.0
1.94701011
S7
2003
93.0
1.94731011
S6
1960
86.0
1.94781011
S5
1897
79.3
1.94891011
S4
1775
72.7
1.94911011
S3
1890
66.0
1.94961011
S2
1901
58.5
1.94981011
S1
2700
48.8
1.94991011
Area
(mm2)
Equivalent
density
(kg/m3)
7645
7645
7645
7645
5143
5143
5143
5143
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
3058
3058
3058
2641
2641
2641
2641
4309
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
8325
-22-
Note: 1. The module of elasticity of straight cable is 1.951011Pa, and the Poisson ratio is 0.3;
2. Cable No. S1~S14 are of side-span cables numbered from tower to abutment pier; M15~M28
are of main-span cables numbered from tower to mid-span.
Table A-2 (cont.)
Geometric and material properties of bridge cables
No.
Tension of
one cable
(kN)
Cable length
(m)
Equivalent
Module (Pa)
M1
2690
49.0
1.94991011
M2
1862
58.0
1.94981011
M3
1841
64.9
1.94961011
M4
1720
71.2
1.94921011
M5
1827
77.4
1.94901011
M6
1877
83.8
1.94791011
M7
1911
90.5
1.94731011
M8
2017
97.3
1.94701011
M9
2135
104.2
1.94681011
M10
2239
111.3
1.94031011
M11
2337
118.7
1.93971011
M12
2432
126.1
1.93911011
M13
2546
133.6
1.93891011
M14
2634
141.2
1.93831011
M15
2718
148.9
1.93781011
M16
2864
156.6
1.93821011
M17
2928
164.4
1.93751011
M18
2980
172.2
1.93671011
M19
3016
180.1
1.93571011
M20
3130
187.9
1.93581011
M21
3414
195.9
1.92961011
M22
3461
203.8
1.92851011
M23
3509
211.8
1.92751011
M24
3589
219.8
1.92721011
M25
3522
227.8
1.92371011
M26
3553
235.8
1.92231011
M27
3574
243.9
1.92061011
M28
3588
251.9
1.91881011
M29
3633
260.0
1.91781011
Area
(mm2)
Equivalent
density
(kg/m3)
4309
2641
2641
2641
2641
3058
3058
3058
3058
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
4309
5143
5143
5143
5143
5143
5143
5143
5143
5143
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
8325
-23-
No.
Tension of
one cable
(kN)
Cable length
(m)
Equivalent
Module (Pa)
M30
3608
268.1
1.91491011
M31
3577
276.2
1.91151011
M32
3517
284.3
1.90701011
Area
(mm2)
Equivalent
density
(kg/m3)
5143
5143
5143
Note: 1. The module of elasticity of straight cable is 1.951011Pa, and the Poisson ratio is 0.3
2. Cable No. S1~S14 are of side-span cables numbered from tower to abutment pier; M15~M28 are
of main-span cables numbered from tower to mid-span.
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-24-
Table A-3
Typical crosssection
Tower
Area (m2)
Geometric and material properties of towers and piers
Second moment of area of longitudinal Second moment of area of
lateral bending (m4)
/ vertical bending (m4)
Second pole moment of
area (m4)
A
22.002
150.99
125.75
217.34
B
56.230
298.48
546.02
616.71
C1
20.576
114.73
47.43
114.13
C2
20.863
121.10
48.33
117.55
C3
21.207
129.04
49.42
121.65
C4
21.608
138.70
50.68
126.45
C5
22.239
154.75
52.67
134.02
E1
115.942
951.91
17117.1
2578.85
E2
116.115
1043.69
10273.8
2761.44
E3
54.526
559.56
1193.73
1294.69
F1
54.246
556.18
1196.25
1292.08
F2
57.272
662.24
1389.64
1522.52
F3
61.31
823.46
1678.38
1869.30
23.47
132.57
202.74
271.44
23.47
132.57
202.74
271.44
Transition
pier
Pier auxiliary
pier
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-25-
E(Pa)
v
 ( kg/m3)
3.271010
0.167
2600
2.841010
0.2
2600
Table A-4
Mode No.
Natural frequencies of the service state
Frequency（Hz）
Major feature of mode
1
0.0759
Tower longitudinal bending + deck
longitudinal floating and asymmetric
vertical bending
2
0.1972
Deck symmetric lateral bending
3
0.2394
Deck symmetric vertical bending
4
0.2625
Lateral bending of north tower
5
0.2770
Lateral bending of south tower
6
0.2957
Deck asymmetric vertical bending
7
0.4226
Deck symmetric vertical bending
8
0.5087
Deck asymmetric vertical bending
9
0.5096
Deck asymmetric lateral bending
10
0.5812
Deck symmetric vertical bending
11
0.6248
Deck symmetric lateral bending + torsion
12
0.6440
Deck asymmetric vertical bending
13
0.6979
Deck symmetric vertical bending
14
0.7014
Deck symmetric torsion
15
0.7648
Deck asymmetric lateral bending
16
0.7739
Deck asymmetric vertical bending
17
0.8274
Deck symmetric vertical bending
18
0.8770
Deck asymmetric vertical bending
19
0.8936
Deck symmetric vertical bending
20
0.9315
Deck asymmetric vertical bending
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-26-
Table A-5
Equivalent mass and mass moment of inertia of the service state
No.
meqx (t/m)
meqy (t/m)
meqz (t/m)
x
(tm2/m)
J meq
1
5.55E+01



2


4.55E+01

3

4.52E+01


4


4.66E+02

5


2.78E+02

6

4.75E+01


7

4.75E+01


8

4.79E+01


9


4.17E+01

10

4.33E+01


11


4.53E+01
1.24E+05
12

4.05E+01


13

4.42E+01


14



2.28E+03
15


4.54E+01

16

4.77E+01


17

5.15E+01


18

4.31E+01


19

7.92E+01


20

1.13E+02


WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-27-
Table A-6
Natural frequencies of the construction state of LSC
Mode No.
Frequency (Hz)
Major features of mode
1
0.1638
Deck lateral bending
2
0.2464
Deck vertical bending
3
0.2647
Tower lateral bending
4
0.3663
Deck vertical bending
5
0.4954
Deck vertical bending
6
0.6611
Deck vertical bending
7
0.685
Deck lateral bending + deck torsion
8
0.7487
Deck vertical bending
9
0.7738
Deck torsion
10
0.9349
Deck vertical bending
Table A-7
Equivalent mass and mass moment of inertia of the construction state of LSC
Table A-8
No.
meqx (t/m)
meqy (t/m)
meqz (t/m)
x
(tm2/m)
J meq
1


3.58E+01

2
7.62E+02
4.90E+01


3


4.21E+02

4
1.66E+02
7.84E+01


5
3.80E+02
4.15E+01


6
7.89E+02
3.68E+01


7


4.03E+01
3.06E+04
8

3.49E+01


9


4.98E+02
2.04E+03
10
1.89E+03
3.93E+01


WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-28-
Table A-9
Parameters of prototype and sectional model for the service state
Scale
Parameter Symb
name
ol
Unit
Value of model
VortexValue of
prototype Flutter excited
resonanc
e
Vortexexcited
resonance
Flutter
Deck length
L
m
111.8
1/65
1.72
Deck width
B
m
23.6
1/65
0.363
Deck height
H
m
2.83
1/65
0.0436
Equivalent
mass
meq
kg/m 4.52510 4
1/65 2
10.71
Equivalent
kgm 2 /
mass
J meq
2.27710 6
moment of
m
inertia
1/65 4
0.1276
Equivalent
gyration
radius
re
m
7.09
1/65
0.109
Basic vertical
frequency
fv
Hz
0.2394
9.7
19.7
2.32
4.72
Basic
torsional
frequency
ft
Hz
0.7013
9.7
19.7
6.80
13.81
Frequency
ratio


2.93


2.93
2.93
Vertical
damping
ratio
v
%
1


0.86~1.12 0.92~1.05
Torsional
damping
ratio
t
%
1


0.94~0.97 0.86~0.93
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-29-
Table A-10 Parameters of prototype and sectional model for the construction state of LSC
Parameter name
Symbol
Unit
Value of
prototype
Scale
Value of
model
Flutter
Flutter
Deck length
L
m
111.8
1/65
1.72
Deck width
B
m
23.6
1/65
0.363
Deck height
H
m
2.83
1/65
0.0436
Equivalent mass
meq
kg/m
4.904104
1/65 2
11.607
Equivalent mass
moment of
inertia
Jmeq
kgm2/m
2.036106
1/65 4
0.1141
Equivalent
gyration radius
re
m
6.443
1/65
0.0991
Basic vertical
frequency
fv
Hz
0.2464
9.3
2.29
Basic torsional
frequency
ft
Hz
0.7738
9.3
7.19
Frequency ratio


3.14

3.14
Vertical damping
ratio
v
%
1

0.86~0.91
Torsional
damping ratio
t
%
1

0.79~0.88
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-30-
Table A-11 Cases of wind tunnel test (Smooth flow)
Structural
Attack
angle
Testing wind
speed (m/s)
1
+5 o
0.015.0
2
+3o
0.015.0
0o
0.015.0
4
-3o
0.015.0
5
-5 o
0.015.0
6
+5 o
0.015.0
7
+3o
0.015.0
0o
0.015.0
9
-3o
0.015.0
10
-5 o
0.015.0
+3 o
0.020.0
0o
0.020.0
-3 o
0.020.0
+3 o
0.020.0
0o
0.020.0
-3 o
0.020.0
-10o+10o
10
No.
3
8
state
Service state
(original)
Service state
(improved)
11
13
Service state
(improved)
15
16
18
Construction
state of LSC
20
21
22
Testing contents
Vortex-excited
resonance
Vortex-excited
resonance
Flutter instability;
Aerodynamic
derivatives
Flutter instability;
Aerodynamic
derivatives
Service state
(improved)
Aerodynamic
coefficients of three
components
Construction
state
Aerodynamic
coefficients of three
components
Scale of wind
speed
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
1/3.3
1/3.3
1/6.7
1/7.0
——
-31-
Table A-12 Aerodynamic derivatives of deck cross-section of the service state (=+3)
U/fB
H 2*
H 3*
A2*
A3*
0.0000
0.8179
1.6289
2.4601
3.2863
4.1276
4.5592
4.9537
5.3656
5.7566
6.1623
6.5012
0.0000
-0.0190
-0.2408
-0.0749
0.3285
0.4143
0.5174
-0.2429
-0.4796
-0.8248
-1.3772
-0.7844
0.0000
0.1934
-0.1429
-0.0397
-0.2742
-1.3011
-1.4178
-2.0109
-2.4586
-2.8620
-2.9548
-2.8616
0.0000
0.0166
-0.0340
-0.0437
-0.0140
0.0319
0.0072
0.0343
0.0451
0.0280
0.0108
0.2600
0.0000
0.0651
0.0097
0.0959
0.1190
0.1755
0.2274
0.1746
0.1693
0.1141
0.1105
0.2775
U/fB
H 1*
H 4*
A1*
A4*
0.0000
-0.5971
-0.2798
0.8626
1.6801
2.3559
2.3633
2.3694
3.0573
3.4315
3.7926
1.1966
0.0000
0.1890
0.2033
0.1995
0.1171
0.1783
0.1997
0.2504
-0.0326
0.1820
1.4269
-6.4034
0.0000
0.0223
-0.0894
-0.2364
-0.2844
-0.2221
-0.0443
-0.1424
0.0368
0.9509
1.1576
-3.0682
0.0000
2.4561
4.9202
7.4451
9.9768
12.4878
13.7249
14.9508
16.2423
17.4381
18.7150
19.4641
0.0000
0.0442
-2.0457
-2.2301
-1.8403
-1.5935
-1.5663
-1.7792
-1.8207
-1.9223
-0.9876
-6.8832
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-32-
Table A-13 Aerodynamic derivatives of deck cross-section of the service state (＝0)
U/fB
H 2*
H 3*
A2*
A3*
0.0000
0.8124
1.6198
2.4382
3.2571
4.0871
4.9418
5.3811
5.7890
6.2274
6.6509
7.0206
7.2833
0.0000
-0.0419
-0.1021
0.0487
0.0760
-0.2122
-0.1927
-0.1713
-0.8304
-0.8725
-1.1791
-1.5921
-1.8581
0.0000
-0.0374
-0.1709
-0.1843
-0.7947
-1.1707
-1.8448
-2.1604
-2.3613
-2.8239
-3.0457
-3.5681
-3.5950
0.0000
-0.0172
-0.0379
-0.0366
-0.0054
0.0070
0.0138
0.0192
0.0725
0.0623
0.0930
0.0645
0.1164
0.0000
0.0838
0.0451
0.0891
0.1096
0.1577
0.2520
0.3165
0.3023
0.3535
0.3688
0.2733
0.3906
U/fB
H 1*
H 4*
A1*
A4*
0.0000
0.0008
-0.0992
1.3146
2.9037
4.0149
5.0508
4.9814
5.4546
5.4024
5.7698
5.4130
5.1699
0.0000
0.2004
0.2398
0.2355
0.0873
0.1942
0.1879
0.2400
0.1594
0.1851
0.2264
0.3818
0.6563
0.0000
-0.2267
-0.4937
-0.4981
-0.4630
-0.7092
-0.6903
-0.6021
-0.6862
-0.8382
-0.7298
0.3549
-1.1918
0.0000
2.4499
4.8970
7.4211
9.9818
12.5610
15.1412
16.3683
17.6786
18.9164
20.2016
21.2534
22.0327
0.0000
0.1366
-1.7952
-2.5096
-2.5641
-1.5706
-1.0283
-0.7680
-0.7129
-0.4683
-0.4669
-0.2301
-0.4358
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-33-
Table A-14 Aerodynamic derivatives of deck cross-section of the service state (＝-3)
U/fB
H 2*
H 3*
A2*
A3*
0.0000
0.8186
1.6347
2.4563
3.2851
4.1182
4.9805
5.8220
6.6658
7.5492
8.4262
0.0000
-0.1674
-0.1702
-0.2686
-0.1365
-0.6609
-0.8598
-1.2934
-1.4789
-2.0103
-2.4892
0.0000
0.0317
-0.1909
-0.3826
-0.7551
-0.6382
-0.9551
-1.0888
-1.4339
-1.8025
-2.2212
0.0000
-0.0140
-0.0035
-0.0229
-0.0311
-0.0438
-0.0504
-0.0188
-0.0242
-0.0073
0.0235
0.0000
0.0322
0.0130
0.0337
0.0702
0.1082
0.2067
0.2334
0.2595
0.3463
0.4067
U/fB
H 1*
H 4*
A1*
A4*
0.0000
-0.4017
0.1203
0.5380
0.7592
1.0879
1.4340
1.5993
1.3785
1.6825
1.6265
0.0000
0.1344
0.2259
0.2754
0.2566
0.3695
0.5438
0.5557
0.8044
0.6167
0.5744
0.0000
0.0330
-0.0689
-0.1707
-0.1681
-0.2147
0.0347
0.0530
-0.2839
-0.4236
-0.5685
0.0000
2.4628
4.9419
7.4323
9.9135
12.4273
14.9277
17.4296
19.8912
22.4107
24.8796
0.0000
-0.2281
-1.1907
-1.5642
-1.6019
-1.7590
-1.9518
-1.9706
-2.1854
-1.8910
-2.1809
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-34-
Table A-15 Aerodynamic derivatives of deck cross-section of the construction state (＝+3)
U/fB
H 2*
H 3*
A2*
A3*
0.0000
0.8062
1.6096
2.4260
3.2467
4.0852
4.5055
4.9444
5.3590
5.7980
6.2221
6.7141
6.8377
0.0000
-0.2083
-0.2333
-0.2689
-0.0498
0.3231
0.5299
0.5246
0.5693
0.7622
0.7126
0.6600
0.5201
0.0000
0.0854
-0.0343
-0.0749
-0.3294
-0.9962
-1.4973
-1.9335
-2.2072
-2.6018
-3.1218
-3.5548
-3.6922
0.0000
-0.0119
-0.0475
-0.0685
-0.0806
-0.0803
-0.0847
-0.0677
-0.0505
-0.0359
-0.0016
0.1129
0.1087
0.0000
0.0481
0.0267
0.0816
0.1242
0.1990
0.2302
0.3016
0.3129
0.3749
0.4043
0.5597
0.5651
U/fB
H 1*
H 4*
A1*
A4*
0.0000
2.4731
4.9458
7.4154
9.8967
12.4407
13.7226
15.0164
16.2612
17.6617
18.8975
20.2091
20.4886
0.0000
-0.1489
-1.5522
-2.6868
-4.2490
-5.4673
-6.2244
-6.4113
-6.9534
-7.2952
-7.9272
-7.4866
-4.0037
0.0000
-1.1948
-1.1347
-1.2187
-1.1772
-0.2321
0.1565
0.5529
0.0406
0.4651
-0.6919
-1.5377
1.1762
0.0000
0.0385
0.2588
0.3588
0.4638
0.2093
-0.1447
-0.1762
-0.6704
-0.5445
-0.7678
-1.2666
-0.3464
0.0000
-0.3459
-0.2210
-0.3379
-0.9964
-1.3491
-1.7077
-2.0579
-2.7600
-3.8983
-4.9200
-6.1335
-3.2158
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-35-
Table A-16 Aerodynamic derivatives of deck cross-section of the construction state (＝0)
U/fB
H 2*
H 3*
A2*
A3*
0.0000
0.8039
1.6096
2.4224
3.2437
4.0785
4.9319
5.7967
6.6514
7.5459
8.4961
0.0000
-0.1852
-0.2806
-0.3882
-0.3881
-0.6070
-0.6360
-0.6596
-0.7266
-0.8446
-1.0915
0.0000
-0.1014
-0.1256
-0.3340
-0.4414
-1.0091
-1.0193
-1.6682
-2.6710
-3.4113
-4.3228
0.0000
-0.0043
-0.0356
-0.0560
-0.0672
-0.0948
-0.1391
-0.1518
-0.2497
-0.2354
-0.2526
0.0000
0.0124
0.0255
0.0623
0.1118
0.1784
0.2694
0.3570
0.4012
0.4752
0.6535
U/fB
H 1*
H 4*
A1*
A4*
0.0000
2.4869
4.9648
7.4616
9.9589
12.4873
14.9822
17.4564
19.9011
22.3185
24.8433
0.0000
-0.1311
-1.3974
-2.1811
-3.0580
-3.5910
-4.2719
-4.5480
-2.9925
-5.4047
-4.4553
0.0000
-0.2452
-0.5140
-0.1399
-0.0198
0.7050
0.6319
0.6967
1.5144
3.1801
3.1071
0.0000
0.1160
0.3619
0.3631
0.4451
0.6262
0.9198
1.2388
3.7421
1.9173
2.6757
0.0000
-0.0232
-0.0606
-0.1389
-0.3671
-0.4750
-0.5264
-0.7376
0.1880
1.8220
0.6842
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-36-
Table A-17 Aerodynamic derivatives of deck cross-section of the construction state (＝-3)
U/fB
H 2*
H 3*
A2*
A3*
0.0000
0.8032
1.6109
2.4201
3.2354
4.0703
4.9128
5.7778
6.6546
7.5469
8.4802
0.0000
-0.1286
-0.1548
-0.3361
-0.5327
-0.7151
-1.1337
-1.2151
-1.4073
-2.0123
-2.4180
0.0000
-0.2814
-0.4289
-0.5845
-0.8703
-0.9314
-1.2378
-1.6803
-2.2053
-2.9609
-4.0871
0.0000
0.0027
-0.0140
-0.0413
-0.0646
-0.0924
-0.1061
-0.0764
-0.0910
-0.0843
-0.1107
0.0000
0.0121
0.0440
0.0611
0.0915
0.1663
0.2350
0.3292
0.4259
0.5274
0.6649
U/fB
H 1*
H 4*
A1*
A4*
0.0000
2.4876
4.9788
7.4823
9.9874
12.5498
15.0731
17.6649
20.2894
22.9455
25.6312
0.0000
-0.1715
-1.2785
-1.8749
-2.5642
-2.8845
-3.5398
-3.5586
-3.9277
-3.6361
-3.9238
0.0000
0.2252
0.3903
0.7314
1.0094
1.7636
2.0034
2.9281
3.5475
4.5018
5.8064
0.0000
0.0917
0.3089
0.3563
0.5768
0.6468
0.7085
0.6740
0.8948
0.9956
0.9434
0.0000
0.3508
0.2568
0.2070
0.0853
0.0200
-0.0859
-0.1274
-0.7758
-1.0024
-1.0370
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-37-
Table A-18 Three-component aerodynamic coefficients of the deck cross-section (service
state)
Attack angle ()
CH
CV
CM
CD
CL
10
2.2307
0.1592
0.1170
2.4270
0.1103
9
2.1867
0.1251
0.1095
2.3226
0.0824
8
2.1773
0.0865
0.1052
2.2563
0.0492
7
2.1483
0.0549
0.1000
2.1880
0.0230
6
2.1027
0.0213
0.0943
2.1097
-0.0052
5
2.0389
-0.0108
0.0889
2.0233
-0.0321
4
1.9825
-0.0356
0.0850
1.9570
-0.0521
3
1.9224
-0.0579
0.0813
1.8945
-0.0699
2
1.8667
-0.0838
0.0769
1.8412
-0.0916
1
1.8230
-0.1093
0.0719
1.8069
-0.1131
0
1.8013
-0.1270
0.0657
1.8013
-0.1270
-1
1.7974
-0.1364
0.0585
1.8170
-0.1326
-2
1.8224
-0.1377
0.0513
1.8613
-0.1300
-3
1.8845
-0.1500
0.0420
1.9473
-0.1380
-4
1.9348
-0.1686
0.0338
2.0280
-0.1520
-5
1.9947
-0.2062
0.0209
2.1367
-0.1845
-6
2.0180
-0.2407
0.0092
2.2164
-0.2141
-7
2.0258
-0.2846
-0.0026
2.2995
-0.2528
-8
2.0115
-0.3376
-0.0153
2.3830
-0.3007
-9
1.9781
-0.3881
-0.0285
2.4592
-0.3461
-10
1.9223
-0.4487
-0.0430
2.5418
-0.4018
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-38-
Table A-19 Three-component aerodynamic coefficients of the deck cross-section
(construction state)
Attack angle ()
CH
CV
CM
CD
CL
10
1.6122
0.5018
0.1248
2.3132
0.4606
9
1.4763
0.5714
0.1481
2.2024
0.5366
8
1.3623
0.5905
0.1728
2.0332
0.5620
7
1.3057
0.5634
0.1824
1.8676
0.5401
6
1.2610
0.4941
0.1875
1.6841
0.4756
5
1.2278
0.4028
0.1857
1.5154
0.3884
4
1.2087
0.3116
0.1785
1.3868
0.3008
3
1.1966
0.2153
0.1669
1.2888
0.2075
2
1.1748
0.1164
0.1517
1.2079
0.1114
1
1.1516
0.0230
0.1346
1.1548
0.0206
0
1.1244
-0.0660
0.1157
1.1244
-0.0660
-1
1.0960
-0.1406
0.0980
1.1162
-0.1382
-2
1.0725
-0.2005
0.0826
1.1301
-0.1958
-3
1.0594
-0.2469
0.0688
1.1655
-0.2399
-4
1.0688
-0.2797
0.0584
1.2287
-0.2701
-5
1.0755
-0.3175
0.0470
1.3019
-0.3051
-6
1.0661
-0.3643
0.0343
1.3773
-0.3489
-7
1.0460
-0.4131
0.0212
1.4574
-0.3947
-8
1.0175
-0.4589
0.0086
1.5393
-0.4374
-9
0.9717
-0.5050
-0.0052
1.6175
-0.4805
-10
0.9110
-0.5545
-0.0187
1.6989
-0.5271
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-39-
General layout of the bridge (mm)
Fig. A-1
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-40-
Typical cross section of the bridge deck (m\cm)
Fig. A-2
20
original location for
overhaul dolly rail
improved location for
overhaul dolly rail
140
140
12
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-41-
Fig. A-3
General lay out of the bridge towers (mm)
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-42-
Convertion relation between mean wind speed with different average -duration and 1hr-mean wind
speed ( from ASCE7-05 )
Fig. A-4
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-43-
Fig. A-5
Shematic diagram of FE mechanical model of the service state
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-44-
Fig. A-6
Shematic diagram of FE mechanical model of the longest single canitilever state
1
2
3
4
Mode No.1
Freq=0.0759Hz
1
2
3
4
Mode No.2
Freq=0.1972Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-45-
Fig. A-7
Structural natural mode shapes of the service state
1
2
3
4
Mode No.3
Freq=0.2394Hz
1
2
3
4
Mode No.4
Freq=0.2624Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-46-
Fig. A-7 (cont.)
Structural natural mode shapes of the service state
1
2
3
4
Mode No.5
Freq=0.2769Hz
1
2
3
4
Mode No.6
Freq=0.2956Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-47-
Fig. A-7 (cont.)
Structural natural mode shapes of the service state
1
2
3
4
Mode No.7
Freq=0.4225Hz
1
2
3
4
Mode No.8
Freq=0.5086Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-48-
Fig. A-7 (cont.)
Structural natural mode shapes of the service state
1
2
3
4
Mode No.9
Freq=0.5096Hz
1
2
3
4
Mode No.10
Freq=0.5811Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-49-
Fig. A-7 (cont.)
Structural natural mode shapes of the service state
1
2
3
4
Mode No.11
Freq=0.6248Hz
1
2
3
4
Mode No.12
Freq=0.6439Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-50-
Fig. A-7 (cont.)
Structural natural mode shapes of the service state
1
2
3
4
Mode No.13
Freq=0.6978Hz
1
2
3
4
Mode No.14
Freq=0.7013Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-51-
Fig. A-7 (cont.)
Structural natural mode shapes of the service state
1
2
3
4
Mode No.16
Freq=0.7739Hz
1
2
3
4
Mode No.18
Freq=0.8770Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-52-
Fig. A-7 (cont.)
Structural natural mode shapes of the service state
1
2
3
4
Mode No.19
Freq=0.8936Hz
1
2
3
4
Mode No.20
Freq=0.9314Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-53-
Fig. A-7 (cont.)
Structural natural mode shapes of the service state
1
2
3
4
Mode No.1
Freq=0.1638Hz
1
2
3
4
Mode No.2
Freq=0.2464Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-54-
Fig. A-8
Structural natural mode shapes of the construction state of the longest single
cantilever
1
2
3
4
Mode No.3
Freq=0.2647Hz
1
2
3
4
Mode No.4
Freq=0.3663Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-55-
Fig. A-8 (cont.)
Structural natural mode shapes of the construction state of the
longest single cantilever
1
2
3
4
Mode No.5
Freq=0.4953Hz
1
2
3
4
Mode No.6
Freq=0.6611Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-56-
Fig. A-8 (cont.)
Structural natural mode shapes of the construction state of the
longest single cantilever
1
2
3
4
Mode No.7
Freq=0.6852Hz
1
2
3
4
Mode No.8
Freq=0.7486Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-57-
Fig. A-8 (cont.)
Structural natural mode shapes of the construction state of the
longest single cantilever
1
2
3
4
Mode No.9
Freq=0.7738Hz
1
2
3
4
Mode No.10
Freq=0.9349Hz
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-58-
Fig. A-9
Layout of TJ-2 Boundary Layer Wind Tunnel
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-59-
Fig. A-10
Sectional model of the service state suspended in TJ-2 Wind Tunnel
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-60-
Fig. A-11
Sectional model of the construction state suspended in TJ-2 Wind Tunnel
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-61-
0.20
o
+5
allowable value
y (m)
0.16
0.12
0.08
0.04
0.00
4
5
6
7
8
9
10
11
12
13
14
15
11
12
13
14
15
11
12
13
14
15
11
12
13
14
15
11
12
13
14
15
U (m/s)
0.20
o
y (m)
0.16
+3
allowable value
0.12
0.08
0.04
0.00
4
5
6
7
8
9
10
U (m/s)
0.20
o
y (m)
0.16
+0
allowable value
0.12
0.08
0.04
0.00
4
5
6
7
8
9
10
U (m/s)
0.20
o
y (m)
0.16
-3
allowable value
0.12
0.08
0.04
0.00
4
5
6
7
8
9
10
U (m/s)
0.20
o
y (m)
0.16
-5
allowable value
0.12
0.08
0.04
0.00
4
5
6
7
8
9
10
U (m/s)
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-62-
Fig. A-12
Standard deviation of vortex-excited vibration of service state vs. wind
speed (Original locations of overhaul-dolly rails)
0.20
o
y (m)
0.16
+5
allowable value
0.12
0.08
0.04
0.00
4
5
6
7
8
9
10
11
12
13
14
15
U (m/s)
No vortex-excited resonances observed in cases of wind attack angle being +3， 0，-3 and 5
Fig. A-13
Standard deviation of vortex-excited vibration of service state vs. wind
speed (Improved locations of overhaul-dolly rails)
0.10
o
 = +3
0.09
= 0
0.08
o
 = -3
o
0.07
0.06
v
0.05
0.04
0.03
0.02
0.01
0.00
0
2
4
6
8
10
12
14
16
18
20
22
Um (m/s)
(a) Vertical vibration
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-63-
0.020
 = +3
0.016
= 0
o
o
 = -3
o
0.012
t
0.008
0.004
0.000
-0.004
-0.008
0
2
4
6
8
10
12
14
16
18
20
22
Um (m/s)
(b) Torsional vibration
Fig. A-14
Damping ratios of sectional model system vs. testing wind speed (service
state)
2.28
2.26
 = +3
2.24
 = -3
= 0
o
o
o
2.22
fv,m (Hz)
2.20
2.18
2.16
2.14
2.12
2.10
2.08
0
2
4
6
8
10
12
14
16
18
20
22
Um (m/s)
(a) Vertical vibration
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-64-
6.88
o
6.84
 = +3
6.80
 = -3
= 0
o
o
ft,m (Hz)
6.76
6.72
6.68
6.64
6.60
6.56
6.52
6.48
0
2
4
6
8
10
12
14
16
18
20
22
Um (m/s)
(b) Torsional vibration
Fig. A-15
Frequencies of sectional model system vs. testing wind speed (service
state)
0.14
o
 = +3
0.12
= 0
o
 = -3
o
0.10
v
0.08
0.06
0.04
0.02
0.00
0
2
4
6
8
10
12
14
16
18
20
22
Um (m/s)
(a) Vertical vibration
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-65-
0.040
0.036
 = +3
0.032
= 0
0.028
 = -3
o
o
o
0.024
0.020
t
0.016
0.012
0.008
0.004
0.000
-0.004
-0.008
0
2
4
6
8
10
12
14
16
18
20
22
Um (m/s)
(b) Torsional vibration
Fig. A-16
Damping ratios of sectional model system vs. testing wind speed
(construction state of the longest cantilever)
2.30
 = +3
2.28
= 0
2.26
o
o
 = -3
o
2.24
fv,m (Hz)
2.22
2.20
2.18
2.16
2.14
2.12
2.10
0
2
4
6
8
10
12
14
16
18
20
22
Um (m/s)
(a) Vertical vibration
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-66-
 = +3
7.2
= 0
o
o
 = -3
o
ft,m (Hz)
7.0
6.8
6.6
6.4
0
2
4
6
8
10
12
14
16
18
20
22
Um (m/s)
(b) Torsional vibration
Fig. A-17
Frequencies of sectional model system vs. testing wind speed
(construction state of the longest cantilever)
2
1
0
-1
*
-3
A1
H1
*
-2
-4
-5
 = +3
-6
= 0
-7
 = -3
-8
0
2
4
6
o
o
o
8 10 12 14 16 18 20 22 24 26
U/fB
3
2
1
0
-1
-2
-3
-4
-5
-6
-7
-8
 = +3
= 0
 = -3
0
2
4
6
o
o
o
8 10 12 14 16 18 20 22 24 26
U/fB
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-67-
0.30
0.25
 = +3
0.20
= 0
o
o
 = -3
0.15
o
*
0.10
A2
H2
*
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
 = +3
= 0
0
1
0.00
o
o
 = -3
0.05
-0.05
o
-0.10
2
3
4
5
6
7
8
9
0
1
2
3
U/fB
0.0
-0.5
-1.0
*
-2.0
A3
H3
*
-1.5
-2.5
 = +3
= 0
-3.5
-4.0
0
1
o
o
 = -3
o
2
3
4
5
6
7
8
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
9
 = +3
= 0
 = +3
= 0
4
 = -3
o
o
o
*
2
A4
H4
*
3
1
0
-1
-2
0
2
4
6
8 10 12 14 16 18 20 22 24 26
U/fB
Fig. A-18
7
8
9
 = -3
0
1
6
7
8
9
o
o
2
3
4
5
U/fB
7
5
6
o
U/fB
6
5
U/fB
0.5
-3.0
4
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
 = +3
= 0
 = -3
0
2
4
6
o
o
o
8 10 12 14 16 18 20 22 24 26
U/fB
Aerodynamic derivatives of deck cross-section vs. reduced wind
speed (service state)
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-68-
1
0
 = +3
-1
= 0
-2
o
 = -3
-3
o
o
*
A1
H1
*
-4
-5
-6
-7
-8
-9
0
2
4
6
8 10 12 14 16 18 20 22 24 26
4.0
3.5
3.0
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
 = +3
= 0
 = -3
0
2
4
o
o
o
6
8 10 12 14 16 18 20 22 24 26
U/fB
U/fB
1.5
0.20
1.0
0.15
0.5
0.10
0.0
0.05
0.00
*
-1.0
-1.5
 = +3
-2.0
= 0
-2.5
-3.0
-0.05
A2
H2
*
-0.5
o
 = -3
0
1
o
o
2
3
4
5
6
7
8
-0.10
-0.15
 = +3
-0.20
= 0
o
-0.25
 = -3
-0.30
9
0
1
o
o
2
3
4
 = +3
0.6
= 0
0.5
 = +3
= 0
9
o
 = -3
o
0.3
0
1
0.2
o
0.1
o
 = -3
0.0
o
2
3
4
5
6
7
8
-0.1
9
0
1
2
3
4
5
6
7
8
9
U/fB
2
6
 = +3
1
o
o
5
= 0
4
 = -3
0
o
-1
3
-2
*
2
A4
*
8
0.4
7
H4
7
o
U/fB
-3
1
-4
0
-5
-1
-6
-2
6
0.7
*
1.0
0.5
0.0
-0.5
-1.0
-1.5
-2.0
-2.5
-3.0
-3.5
-4.0
-4.5
-5.0
5
U/fB
A3
H3
*
U/fB
0
2
4
6
8 10 12 14 16 18 20 22 24 26
U/fB
Fig. A-19
-7
 = +3
= 0
 = -3
0
2
4
6
o
o
o
8 10 12 14 16 18 20 22 24 26
U/fB
Aerodynamic derivatives of deck cross-section vs. reduced wind
speed (construction state)
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-69-
Fig. A-20
Sectional model installed in TJ-2 Wind Tunnel for force measurement
(service state)
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-70-
Fig. A-21
Sectional model installed in TJ-2 Wind Tunnel for force measurement
(construction state)
Fig. A-22
Coordinates of aerodynamic forces on deck cross-section
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-71-
3.5
CD；
Aerodynamic Coeficient
3.0
CL；
CH；
CM;
CV
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-10 -8
-6
-4
-2
0
2
4
6
8
10
o
Wind attck angle  ( )
Fig. A-23
Three-component aerodynamic coefficients of deck vs. wind attack angle
(service state)
3.5
CD；
aerodynamic coefficient
3.0
CL；
CM;
CH；
CV
2.5
2.0
1.5
1.0
0.5
0.0
-0.5
-1.0
-10 -8
-6
-4
-2
0
2
4
6
8
10
o
Wind attck angle  ( )
WIND TUNNEL TESTS OF RIGID SECTIONAL MODEL OF BRIDGE DECK
Contract Number 247815: Engineering Services for the Final Design of a Bridge Across the Canal at the Atlantic Side
-72-