# Traveling Wave Resonance and Simplified Analysis Method for Long-Span Symmetrical Cable-Stayed Bridges under Seismic Traveling Wave Excitation

Authors: Zhong-ye Tian; Meng-lin Lou
Journal: Shock and Vibration
 (2014)

---

## Abstract

The seismic responses of a long-span cable-stayed bridge under uniform excitation and traveling wave excitation in the longitudinal direction are, respectively, computed. The numerical results show that the bridge’s peak seismic responses vary significantly as the apparent wave velocity decreases. Therefore, the traveling wave effect must be considered in the seismic design of long-span bridges. The bridge’s peak seismic responses do not vary monotonously with the apparent wave velocity due to the traveling wave resonance. A new traveling wave excitation method that can simplify the multisupport excitation process into a two-support excitation process is developed.


---

## Body

## 1. Introduction

Because of the ever-increasing economic and technological developments, long-span cable-stayed bridges have increased in both number and span lengths, examples include the Tatara Bridge, built in Japan in 1999 with a main span of 890 m, the Sutong Bridge, built in China in 2008 with a main span of 1088 m, and the Russky Island Bridge, built in Russia in 2012 with a main span of 1104 m. The evaluation of seismic performance becomes particularly important for these types of structures, as the distances between their multiple support points are great, sometimes even greater than the seismic wavelength. Therefore, a uniform excitation evaluation method is not suitable and the traveling wave effect must be considered [1]. Many scholars have conducted studies on the seismic performance of these structures. References [2–8] studied the features of seismic responses of long-span cable-stayed bridges under uniform excitation and multisupport excitations, respectively. Allam et al. made researches using random vibration method [9–12]. Researchers have also explored effective seismic control strategies for cable-stayed bridges [13–16] and experimental tests have been carried out [17, 18]. The methods for obtaining seismic sources play very important roles for investigating safety performance of cable-stayed bridges as well, and Dong and Li have conducted primitive work in this area [19–23]. They optimized and simplified the sensor location coordinates to find the analytical solution of the acoustic emission/microseismic source location coordinates. They also developed a new location method using P-wave and S-wave arrivals for unknown velocity system to eliminate the location error of monitoring system caused by the measurement deviations of the wave velocity. These research results can be used in the seismic response analysis of cable-stayed bridges for monitoring the seismic location in the future.But research on the relationship between the structures’ seismic response and the apparent wave velocity is insufficient. This paper, by comparing between the seismic responses of a highway bridge under both traveling wave and uniform excitations, demonstrates the seismic response characteristics of long-span cable-stayed bridges under traveling wave excitation and develops a new traveling wave excitation method that can simplify the multisupport excitation process into a two-support excitation process.


## 2. Finite Element Procedures

### 2.1. Equations of Motion

The cable-stayed bridge is assumed to be a linear and lumped-mass system withn unconstrained degrees of freedom and m support degrees of freedom. The damping force is assumed to be proportional to the relative velocity [24, 25]. The equations of motion for such a system subjected to m support motions along the longitudinal direction can be written in the following matrix form [24]:
(1)
M
0
0
M
g
x
¨
t
u
¨
t
+
C
C
c
C
c
T
C
g
x
˙
t
u
˙
t
+
K
K
c
K
c
T
K
g
x
t
u
t
=
0
f
(
t
)
,
where M, C, and K are the n
×
n mass, damping, and stiffness matrices associated with the unconstrained degrees of freedom, respectively; M
g, C
g, and K
g are the m
×
m matrices associated with the support degrees of freedom; M
c, C
c, and K
c are the n
×
m coupling matrices associated with both sets of degrees of freedom; x
¨
(
t
), x
˙
(
t
), and x
(
t
) refer to the n vectors of absolute accelerations, velocities, and displacements at the unconstrained degrees of freedom; u
¨
(
t
), u
˙
(
t
), and u
(
t
) refer to the m vectors of prescribed support accelerations, velocities, and displacements; and f
(
t
) is the m vector of the reacting forces at the support degrees of freedom.It is common to decompose the response into pseudostatic and dynamic components [24, 25]; that is,
(2)
x
t
=
x
s
t
+
x
d
t
,
where the pseudostatic component, x
s
(
t
), is the solution to (1) without the inertia and damping terms and is given by
(3)
x
s
t
=
-
K
-
1
K
c
u
t
=
R
u
t
in which R
=
-
K
-
1
K
c denotes the influence matrix. Substituting (2) and (3) into (1), while neglecting the damping forces multiplied by u
˙
(
t
) because they are typically much smaller than the inertia forces on the same side, the dynamic component of the response is obtained in the differential form
(4)
M
x
¨
d
t
+
C
x
˙
d
t
+
K
x
d
t
=
-
M
R
u
¨
t
.
The multisupport acceleration excitation u
¨
(
t
) in (1) is
(5)
u
¨
t
T
=
u
¨
1
t
,
u
¨
2
t
,
…
,
u
¨
m
t
,
where the seismic acceleration excitation at the ith support is u
¨
i
(
t
)
T
=
u
¨
i
x
(
t
)
,
u
¨
i
y
(
t
)
,
u
¨
i
z
(
t
). If only the seismic excitation in the longitudinal direction is considered, then u
¨
i
(
t
)
T
=
u
¨
g
(
t
-
Δ
i
/
v
a
)
,
0,0, where u
¨
g
(
t
) refers to the time history of the ground motion, Δ
i denotes the horizontal distance between the ith and the first station, v
a is the surface apparent wave velocity, and m becomes the number of supports.


### 2.2. Description of the Bridge and Support Excitations

We consider a highway bridge with an elevation as shown in Figure1. The bridge is a long-span symmetrical cable-stayed structure with double towers, double cable planes, and a total length of 2088 m. The distances between the adjacent piers and towers are 100 m, 100 m, 300 m, 1088 m, 300 m, 100 m, and 100 m from left to right. The inverted Y-shaped main tower is 300.4 meters high. Flat streamlined steel box girders with a width of 35.4 m are used for the deck beams. The longitudinal slope is 1.5% and the main span lies in a circular curve with R
=
36300 m. Deck beams are connected with the piers and towers by using sliding supports and dynamic dampers, which are used to confine the rated travel in a longitudinal direction, thus making the bridge a floating system. High strength parallel wire cable is adopted. The finite element method is employed and the beam element is used to simulate the piers, towers, and deck beams. The link element is used to simulate the cables. The beam element is linear, two-node, and three-dimensional. It has six degrees of freedom at each node and is based on Timoshenko beam theory. The link element is linear, two-node, and three-dimensional and has three translational degrees of freedom at each node. The bridge is discretized into 2050 elements with 10622 degrees of freedom.Figure 1
Elevation of the highway bridge.Nine sets of seismic acceleration data are used as excitation in the analysis, which are all obtained from the seismic safety evaluation of the specific engineering site. Three sets are marked asa
1, a
2, and a
3 for an exceeding probability of 2% in 50 years; three sets are marked as b
1, b
2, and b
3 for an exceeding probability of 2% in 100 years, and the other three sets are marked as c
1, c
2, and c
3 for an exceeding probability of 10% in 100 years. The time steps of the acceleration histories are all 0.02 s. The motion at each support and the propagation of the waves are all assumed to be in longitudinal direction.Figures2, 3, 4, 5, 6, and 7 show the Fourier amplitude spectra and the acceleration response spectra of a
1, b
1, and c
1.Figure 2
Fourier amplitude ofa
1.Figure 3
Acceleration response spectrum ofa
1.Figure 4
Fourier amplitude ofb
1.Figure 5
Acceleration response spectrum ofb
1.Figure 6
Fourier amplitude ofc
1.Figure 7
Acceleration response spectrum ofc
1.


## 2.1. Equations of Motion

The cable-stayed bridge is assumed to be a linear and lumped-mass system withn unconstrained degrees of freedom and m support degrees of freedom. The damping force is assumed to be proportional to the relative velocity [24, 25]. The equations of motion for such a system subjected to m support motions along the longitudinal direction can be written in the following matrix form [24]:
(1)
M
0
0
M
g
x
¨
t
u
¨
t
+
C
C
c
C
c
T
C
g
x
˙
t
u
˙
t
+
K
K
c
K
c
T
K
g
x
t
u
t
=
0
f
(
t
)
,
where M, C, and K are the n
×
n mass, damping, and stiffness matrices associated with the unconstrained degrees of freedom, respectively; M
g, C
g, and K
g are the m
×
m matrices associated with the support degrees of freedom; M
c, C
c, and K
c are the n
×
m coupling matrices associated with both sets of degrees of freedom; x
¨
(
t
), x
˙
(
t
), and x
(
t
) refer to the n vectors of absolute accelerations, velocities, and displacements at the unconstrained degrees of freedom; u
¨
(
t
), u
˙
(
t
), and u
(
t
) refer to the m vectors of prescribed support accelerations, velocities, and displacements; and f
(
t
) is the m vector of the reacting forces at the support degrees of freedom.It is common to decompose the response into pseudostatic and dynamic components [24, 25]; that is,
(2)
x
t
=
x
s
t
+
x
d
t
,
where the pseudostatic component, x
s
(
t
), is the solution to (1) without the inertia and damping terms and is given by
(3)
x
s
t
=
-
K
-
1
K
c
u
t
=
R
u
t
in which R
=
-
K
-
1
K
c denotes the influence matrix. Substituting (2) and (3) into (1), while neglecting the damping forces multiplied by u
˙
(
t
) because they are typically much smaller than the inertia forces on the same side, the dynamic component of the response is obtained in the differential form
(4)
M
x
¨
d
t
+
C
x
˙
d
t
+
K
x
d
t
=
-
M
R
u
¨
t
.
The multisupport acceleration excitation u
¨
(
t
) in (1) is
(5)
u
¨
t
T
=
u
¨
1
t
,
u
¨
2
t
,
…
,
u
¨
m
t
,
where the seismic acceleration excitation at the ith support is u
¨
i
(
t
)
T
=
u
¨
i
x
(
t
)
,
u
¨
i
y
(
t
)
,
u
¨
i
z
(
t
). If only the seismic excitation in the longitudinal direction is considered, then u
¨
i
(
t
)
T
=
u
¨
g
(
t
-
Δ
i
/
v
a
)
,
0,0, where u
¨
g
(
t
) refers to the time history of the ground motion, Δ
i denotes the horizontal distance between the ith and the first station, v
a is the surface apparent wave velocity, and m becomes the number of supports.


## 2.2. Description of the Bridge and Support Excitations

We consider a highway bridge with an elevation as shown in Figure1. The bridge is a long-span symmetrical cable-stayed structure with double towers, double cable planes, and a total length of 2088 m. The distances between the adjacent piers and towers are 100 m, 100 m, 300 m, 1088 m, 300 m, 100 m, and 100 m from left to right. The inverted Y-shaped main tower is 300.4 meters high. Flat streamlined steel box girders with a width of 35.4 m are used for the deck beams. The longitudinal slope is 1.5% and the main span lies in a circular curve with R
=
36300 m. Deck beams are connected with the piers and towers by using sliding supports and dynamic dampers, which are used to confine the rated travel in a longitudinal direction, thus making the bridge a floating system. High strength parallel wire cable is adopted. The finite element method is employed and the beam element is used to simulate the piers, towers, and deck beams. The link element is used to simulate the cables. The beam element is linear, two-node, and three-dimensional. It has six degrees of freedom at each node and is based on Timoshenko beam theory. The link element is linear, two-node, and three-dimensional and has three translational degrees of freedom at each node. The bridge is discretized into 2050 elements with 10622 degrees of freedom.Figure 1
Elevation of the highway bridge.Nine sets of seismic acceleration data are used as excitation in the analysis, which are all obtained from the seismic safety evaluation of the specific engineering site. Three sets are marked asa
1, a
2, and a
3 for an exceeding probability of 2% in 50 years; three sets are marked as b
1, b
2, and b
3 for an exceeding probability of 2% in 100 years, and the other three sets are marked as c
1, c
2, and c
3 for an exceeding probability of 10% in 100 years. The time steps of the acceleration histories are all 0.02 s. The motion at each support and the propagation of the waves are all assumed to be in longitudinal direction.Figures2, 3, 4, 5, 6, and 7 show the Fourier amplitude spectra and the acceleration response spectra of a
1, b
1, and c
1.Figure 2
Fourier amplitude ofa
1.Figure 3
Acceleration response spectrum ofa
1.Figure 4
Fourier amplitude ofb
1.Figure 5
Acceleration response spectrum ofb
1.Figure 6
Fourier amplitude ofc
1.Figure 7
Acceleration response spectrum ofc
1.


## 3. Natural Vibration Analysis of the Bridge Model

The free vibration analysis of the bridge is conducted and the first 14 natural frequencies are shown in Table1.Table 1
Frequencies (Hz) and characteristics of the first 14 modes of the cable-stayed bridge.
Order
Frequency
Characteristic
1
0.0656
Longitudinal floating
2
0.1004
Transverse symmetric bending of deck beam
3
0.1779
Vertical symmetric bending of deck beam
4
0.2195
Vertical antisymmetric bending of deck beam
5
0.2819
Transverse antisymmetric bending of deck beam
6
0.3083
Vertical symmetric bending of deck beam + longitudinal bending of towers in opposite direction
7
0.3718
Vertical antisymmetric bending of deck beam + longitudinal bending of towers in the same direction
8
0.4175
Vertical symmetric bending of deck beam + longitudinal bending of towers in opposite direction
9
0.4663
Transverse bending of towers in the same direction
10
0.4678
Transverse bending of towers in opposite direction
11
0.4692
Vertical antisymmetric bending of deck beam + longitudinal bending of towers in the same direction
12
0.5066
Vertical symmetric bending of deck beam + longitudinal bending of towers in opposite direction
13
0.5214
Torsion and transverse symmetric bending of deck beam
14
0.5534
Vertical antisymmetric bending of deck beam


## 4. Numerical Analysis for the Seismic Responses of the Bridge

To study the influence of the traveling wave effect on the seismic response of the bridge, five values of apparent wave velocityc are considered: 500 m/s, 1000 m/s, 2000 m/s, 3000 m/s, and 5000 m/s; in addition, uniform excitation can also be observed as a case, where c
=
∞. Reference [26] indicated that the traveling wave resonance effect existed in the seismic response of long-span arch bridges when the apparent wave velocity obtained specific values. To see if this phenomenon exists in the seismic response of long-span double-tower cable-stayed bridges, three additional values of apparent wave velocity are adopted: 1712 m/s, 1100 m/s, and 900 m/s; the theoretical foundation is provided below.If we assume the longitudinal support excitation to be a sine wave with the same frequency as theith natural frequency of the structure, then the wavelength of the sine wave would be λ
i
=
c
/
f
i, with c and f
i denoting the surface apparent wave velocity and the ith natural frequency of the structure, respectively. In the case of uniform longitudinal excitation, the wavelength is equivalent to ∞, which means that the support motions of the two main towers are antisymmetric; therefore, the seismic response of the symmetric modes of the structure will not be excited. But, in the case of longitudinal traveling wave excitation, if a specific surface apparent wave velocity can make the main span L
=
0.5
λ
i, then the support motions of the two main towers would be equal and opposite, thus forming longitudinal symmetric excitation for the two towers. If the ith mode shape is symmetric, then it will be excited along with other symmetric modes and will remain in a resonance state, thus maximizing its modal response.Based on a free vibration analysis of the bridge, we know that the 8th, 12th, and 26th modes are symmetric. Figures8, 9, and 10 show the mode shapes and Table 2 shows the relationships between the main span, the structural natural frequencies, the apparent wave velocity, and the wavelength.Table 2
Relationships between the main span, structural natural frequencies, apparent wave velocity, and wavelength.
Mode
Surface apparent wave velocity
c = 1712 m/s
c = 1100 m/s
c = 1000 m/s
c = 900 m/s
Order
Characteristic
f
i (Hz)
λ
i (m)
L/λ
i
λ
i (m)
L/λ
i
λ
i (m)
L/λ
i
λ
i (m)
L/λ
i
8
Symmetric in longitudinal direction
0.41747
2395
0.45
2156
0.50
12
Symmetric in longitudinal direction
0.50662
2171
0.50
1974
0.55
26
Symmetric in longitudinal direction
0.78681
2176
0.50Figure 8
The 8th mode shape of the bridge.Figure 9
The 12th mode shape of the bridge.Figure 10
The 26th mode shape of the bridge.We call this phenomenon as the traveling wave resonance. At a specific apparent wave velocity, the span of a symmetric structure attains sufficiently close to an integral multiple of the half-wavelength of the excitation wave, and the frequency of the excitation becames the frequency of one of the structural modes. When the excitation and the mode are both symmetric or antisymmetric, the mode reaches a resonance state and its contribution to the response of the structure is maximized.When the frequency of the resonant mode is within the predominant frequencies of the ground motion, the contribution maximization phenomenon will be more significant. It may increase or decrease the structural response depending on the orientation combination of the modal superposition, and the results will vary with different ground motions.The structural seismic response may reach an extreme value, or even maxima and minima, at specific apparent wave velocities because of the traveling wave resonance. Therefore, significant importance should be attached to this phenomenon.


### 4.1. Seismic Responses of the Bridge

We examine the peak values of the following structural responses: the cross-sectional axial forces, bending moments, and shear forces at the bottoms of piersP
1, P
2, P
3, and main tower T
1 (see Figure 1), denoted by F
P
1, M
P
1, S
P
1, F
P
2, M
P
2, S
P
2, F
P
3, M
P
3, S
P
3, F
T
1, M
T
1, and S
T
1; the relative displacement between the connection node of tower T
1 and the beam and the node at the bottom of tower T
1, denoted by U
c; the relative displacement between the top and bottom nodes of the finite element, which is at the bottom of tower T
1, referred to as U
b; the relative displacement between the top and bottom nodes on tower T
1 and the absolute acceleration of the top node of tower T
1, referred to as U
t and A
t.To clearly compare the peak values of structural seismic responses under uniform excitation and traveling wave excitation, we define the traveling wave effect influence coefficiente
t
=
m
t
-
m
u
/
m
u
×
100
%, where m
t denotes the peak response under traveling wave excitation, and m
u refers to the peak response under uniform excitation. If e
t is a positive number, then the structural response increases when considering the traveling wave effect; if not, then it decreases.The results show that structural seismic responses vary in a similar pattern under the excitations of the nine different sets of seismic acceleration data. Due to space limitations in this paper, only the results under the excitations ofa
1, b
1, and c
1 are listed (See Tables 3, 4, and 5); however, the conclusions are based on all computing results.Table 3
Traveling wave effect influence coefficient (%) for seismic responses under excitation ofa
1.
Apparent wave velocity (m/s)
5000
3000
2000
1712
1100
1000
900
500
U
t
−2.7
−3.8
−2.8
−2.4
−3.9
−4.6
−6.0
−8.5
A
t
−19.2
−20.8
−15.0
−12.1
−14.1
−15.9
−16.8
−17.4
U
b
−6.5
−12.2
−17.9
−19.5
−14.5
−11.1
−7.3
−18.3
U
c
−4.6
−9.4
−14.5
−16.1
−12.9
−10.3
−7.0
−18.8
F
P
1
−3.4
5.1
4.6
−1.2
−12.8
−3.6
5.5
−13.4
M
P
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
S
P
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
F
P
2
−12.4
−23.9
−26.5
−22.9
−17.3
−17.5
−17.7
−18.2
M
P
2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
S
P
2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
F
P
3
4.9
7.8
9.4
10.7
17.9
19.1
18.9
1.4
M
P
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
S
P
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
F
T
1
5.4
12.5
21.2
26.3
38.8
40.0
39.8
−8.9
M
T
1
−6.9
−12.8
−18.6
−19.9
−14.7
−11.3
−7.1
−18.4
S
T
1
3.3
5.1
4.1
2.3
−5.6
−7.4
−8.5
6.7Table 4
Traveling wave effect influence coefficient (%) for seismic responses under excitation ofb
1.
Apparent wave velocity (m/s)
5000
3000
2000
1712
1100
1000
900
500
U
t
0.1
−0.1
−1.4
−2.5
−5.9
−7.2
−8.7
−11.4
A
t
−11.5
−15.9
−10.0
−3.9
−4.2
−12.7
−18.2
0.4
U
b
−8.8
−15.6
−21.2
−21.8
−19.1
−18.2
−17.9
−17.9
U
c
−6.4
−12.3
−18.1
−19.6
−18.4
−17.7
−17.2
−17.1
F
P
1
−1.9
−12.2
−29.3
−21.5
−6.6
−8.2
0.4
−25.2
M
P
1
0.0
0.0
0.0
0.0
−0.1
−0.1
−0.1
−0.1
S
P
1
0.0
0.0
−1.0
0.0
0.0
0.0
0.1
0.1
F
P
2
8.9
13.9
15.1
9.3
10.2
10.2
15.3
−14.8
M
P
2
0.0
0.1
0.1
0.0
0.0
0.0
0.0
0.0
S
P
2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
F
P
3
−4.8
−5.5
0.8
1.5
−8.2
−13.2
−20.0
−7.5
M
P
3
0.0
0.1
0.1
0.0
0.0
0.0
0.0
−0.1
S
P
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
−0.1
F
T
1
9.2
26.3
48.1
55.2
51.2
42.8
29.4
23.6
M
T
1
−9.4
−16.2
−21.3
−22.0
−19.0
−18.2
−18.2
−18.0
S
T
1
0.1
1.7
−3.7
−10.7
−12.5
−13.7
−15.5
−14.7Table 5
Traveling wave effect influence coefficient (%) for seismic responses under excitation ofc
1.
Apparent wave velocity (m/s)
5000
3000
2000
1712
1100
1000
900
500
U
t
−1.3
−1.3
−1.3
−1.5
−2.8
−3.2
−3.9
−13.4
A
t
1.3
3.4
14.3
17.1
15.9
13.5
12.1
9.8
U
b
2.3
−0.9
−5.1
−7.5
7.5
9.3
8.9
7.5
U
c
−0.4
−2.4
−5.5
−6.3
2.0
3.9
4.4
6.6
F
P
1
−10.9
−16.0
−11.1
−5.2
2.2
7.0
7.8
−13.9
M
P
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
−0.1
S
P
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
F
P
2
−15.6
−12.1
18.0
29.5
22.0
9.6
−5.4
−9.0
M
P
2
0.0
0.0
0.0
−0.1
−0.2
−0.2
−0.2
−0.3
S
P
2
0.0
0.0
0.0
0.0
−0.1
−0.1
−0.1
−0.2
F
P
3
4.3
9.2
11.1
9.7
10.7
11.6
14.1
6.3
M
P
3
0.0
0.0
0.0
−0.1
−0.1
−0.1
−0.2
−0.3
S
P
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
F
T
1
21.0
33.6
42.9
44.3
28.2
18.4
15.6
13.2
M
T
1
3.7
0.2
−7.2
−4.8
9.4
11.1
10.8
7.7
S
T
1
3.1
3.9
5.0
1.6
7.9
8.7
8.7
8.7The results show that responses considering the traveling wave effect significantly differ from those under uniform excitation. For example, let us compare the results in the case wherec
=
1000 m/s with those in the case of uniform excitation: under the excitation of a
1, F
P
3 increases by 19.1%, M
T
1 decreases by 11.3%, and F
T
1 increases by 40%. And under the excitation of b
1, F
T
1, M
Y
1, and S
T
1 are amplified by 18.4%, 11.1%, and 8.7%, respectively. These responses threaten structural safety and should be given sufficient attention in the seismic design of long-span cable-stayed bridges.However, the seismic responses of the bridge do not always follow the same changing rules with the decrease of the surface apparent wave velocity. The responses may be amplified under the excitation of one set of ground motions but may be reduced under the excitation of another set. Even when the input is certain, some responses may remain the same, some may be amplified, and others may be reduced. For instance, the cross-sectional bending moment and shear force responses at the bottom of the piers are barely influenced by the wave passage effect because the bridge is a floating system (see Section2.2). The longitudinal displacement responses of the nodes on each pier almost fully depend on the pier support motion itself; if only the ground motion remains constant, then the longitudinal displacements of the nodes will not change and neither will the bending moments nor the shear forces. The time histories of the responses will be slightly translated, because the starting time of the support motion varies with the apparent wave velocity (see Figure 11). As another example, U
t is reduced under the excitation of a
1 and b
1 when the wave passage effect is considered, but it is amplified under the excitation of c
1 when c
=
1000 m/s and 500 m/s. Reference [27] drew similar conclusions when analyzing another long-span cable-stayed bridge model.Figure 11
Time history ofM
P
3 under the excitation of c
3.As for the traveling wave resonance, we can see from Tables3 through 5 that numerous structural peak responses reach their extreme values, either local maxima or minima, when c
=
1712 m/s, 1100 m/s, 1000 m/s, or 900 m/s. To make the expression clear and distinct, we graph the response data for which the resonance phenomena are obvious. Figures 12, 13, and 14 are based on the results under the excitation of a
1, b
1, and c
1, respectively. The dotted lines in the figures reflect the results under uniform excitation, which shows the influence trend of different apparent wave velocities.Variation trend of peak structural seismic responses with apparent wave velocity under the excitation ofa
1.
(a)
Variation trend ofU
b with c
(b)
Variation trend ofU
c with c
(c)
Variation trend ofF
P
2 with c
(d)
Variation trend ofF
P
3 with c
(e)
Variation trend ofF
T
1 with c
(f)
Variation trend ofM
T
1 with cVariation trend of peak structural seismic responses with apparent wave velocity under the excitation ofb
1.
(a)
Variation trend ofU
c with c
(b)
Variation trend ofF
P
3 with c
(c)
Variation trend ofF
T
1 with c
(d)
Variation trend ofM
T
1 with cVariation trend of peak structural seismic responses with apparent wave velocity under the excitation ofc
1.
(a)
Variation trend ofU
b with c
(b)
Variation trend ofF
P
2 with c
(c)
Variation trend ofF
T
1 with c
(d)
Variation trend ofS
T
1 with cBased on Figures2 through 7, we can see that 0.50662 Hz (the 12th natural frequency of the bridge) is within the predominant frequency range of a
1 and 0.41747 Hz (the 8th natural frequency of the bridge) is close to the range; therefore, the resonance phenomenon is evident when c
=
1100 m/s, 1000 m/s, or 900 m/s under the excitation of a
1. For b
1 and c
1, the 8th and 12th natural frequencies are beyond the predominant frequency ranges, but the 26th natural frequency (0.78681 Hz) is within range; therefore, the resonance phenomenon when c
=
1712 m/s is more remarkable than when c is near 1000 m/s.


## 4.1. Seismic Responses of the Bridge

We examine the peak values of the following structural responses: the cross-sectional axial forces, bending moments, and shear forces at the bottoms of piersP
1, P
2, P
3, and main tower T
1 (see Figure 1), denoted by F
P
1, M
P
1, S
P
1, F
P
2, M
P
2, S
P
2, F
P
3, M
P
3, S
P
3, F
T
1, M
T
1, and S
T
1; the relative displacement between the connection node of tower T
1 and the beam and the node at the bottom of tower T
1, denoted by U
c; the relative displacement between the top and bottom nodes of the finite element, which is at the bottom of tower T
1, referred to as U
b; the relative displacement between the top and bottom nodes on tower T
1 and the absolute acceleration of the top node of tower T
1, referred to as U
t and A
t.To clearly compare the peak values of structural seismic responses under uniform excitation and traveling wave excitation, we define the traveling wave effect influence coefficiente
t
=
m
t
-
m
u
/
m
u
×
100
%, where m
t denotes the peak response under traveling wave excitation, and m
u refers to the peak response under uniform excitation. If e
t is a positive number, then the structural response increases when considering the traveling wave effect; if not, then it decreases.The results show that structural seismic responses vary in a similar pattern under the excitations of the nine different sets of seismic acceleration data. Due to space limitations in this paper, only the results under the excitations ofa
1, b
1, and c
1 are listed (See Tables 3, 4, and 5); however, the conclusions are based on all computing results.Table 3
Traveling wave effect influence coefficient (%) for seismic responses under excitation ofa
1.
Apparent wave velocity (m/s)
5000
3000
2000
1712
1100
1000
900
500
U
t
−2.7
−3.8
−2.8
−2.4
−3.9
−4.6
−6.0
−8.5
A
t
−19.2
−20.8
−15.0
−12.1
−14.1
−15.9
−16.8
−17.4
U
b
−6.5
−12.2
−17.9
−19.5
−14.5
−11.1
−7.3
−18.3
U
c
−4.6
−9.4
−14.5
−16.1
−12.9
−10.3
−7.0
−18.8
F
P
1
−3.4
5.1
4.6
−1.2
−12.8
−3.6
5.5
−13.4
M
P
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
S
P
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
F
P
2
−12.4
−23.9
−26.5
−22.9
−17.3
−17.5
−17.7
−18.2
M
P
2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
S
P
2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
F
P
3
4.9
7.8
9.4
10.7
17.9
19.1
18.9
1.4
M
P
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
S
P
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
F
T
1
5.4
12.5
21.2
26.3
38.8
40.0
39.8
−8.9
M
T
1
−6.9
−12.8
−18.6
−19.9
−14.7
−11.3
−7.1
−18.4
S
T
1
3.3
5.1
4.1
2.3
−5.6
−7.4
−8.5
6.7Table 4
Traveling wave effect influence coefficient (%) for seismic responses under excitation ofb
1.
Apparent wave velocity (m/s)
5000
3000
2000
1712
1100
1000
900
500
U
t
0.1
−0.1
−1.4
−2.5
−5.9
−7.2
−8.7
−11.4
A
t
−11.5
−15.9
−10.0
−3.9
−4.2
−12.7
−18.2
0.4
U
b
−8.8
−15.6
−21.2
−21.8
−19.1
−18.2
−17.9
−17.9
U
c
−6.4
−12.3
−18.1
−19.6
−18.4
−17.7
−17.2
−17.1
F
P
1
−1.9
−12.2
−29.3
−21.5
−6.6
−8.2
0.4
−25.2
M
P
1
0.0
0.0
0.0
0.0
−0.1
−0.1
−0.1
−0.1
S
P
1
0.0
0.0
−1.0
0.0
0.0
0.0
0.1
0.1
F
P
2
8.9
13.9
15.1
9.3
10.2
10.2
15.3
−14.8
M
P
2
0.0
0.1
0.1
0.0
0.0
0.0
0.0
0.0
S
P
2
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0
F
P
3
−4.8
−5.5
0.8
1.5
−8.2
−13.2
−20.0
−7.5
M
P
3
0.0
0.1
0.1
0.0
0.0
0.0
0.0
−0.1
S
P
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
−0.1
F
T
1
9.2
26.3
48.1
55.2
51.2
42.8
29.4
23.6
M
T
1
−9.4
−16.2
−21.3
−22.0
−19.0
−18.2
−18.2
−18.0
S
T
1
0.1
1.7
−3.7
−10.7
−12.5
−13.7
−15.5
−14.7Table 5
Traveling wave effect influence coefficient (%) for seismic responses under excitation ofc
1.
Apparent wave velocity (m/s)
5000
3000
2000
1712
1100
1000
900
500
U
t
−1.3
−1.3
−1.3
−1.5
−2.8
−3.2
−3.9
−13.4
A
t
1.3
3.4
14.3
17.1
15.9
13.5
12.1
9.8
U
b
2.3
−0.9
−5.1
−7.5
7.5
9.3
8.9
7.5
U
c
−0.4
−2.4
−5.5
−6.3
2.0
3.9
4.4
6.6
F
P
1
−10.9
−16.0
−11.1
−5.2
2.2
7.0
7.8
−13.9
M
P
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
−0.1
S
P
1
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
F
P
2
−15.6
−12.1
18.0
29.5
22.0
9.6
−5.4
−9.0
M
P
2
0.0
0.0
0.0
−0.1
−0.2
−0.2
−0.2
−0.3
S
P
2
0.0
0.0
0.0
0.0
−0.1
−0.1
−0.1
−0.2
F
P
3
4.3
9.2
11.1
9.7
10.7
11.6
14.1
6.3
M
P
3
0.0
0.0
0.0
−0.1
−0.1
−0.1
−0.2
−0.3
S
P
3
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.1
F
T
1
21.0
33.6
42.9
44.3
28.2
18.4
15.6
13.2
M
T
1
3.7
0.2
−7.2
−4.8
9.4
11.1
10.8
7.7
S
T
1
3.1
3.9
5.0
1.6
7.9
8.7
8.7
8.7The results show that responses considering the traveling wave effect significantly differ from those under uniform excitation. For example, let us compare the results in the case wherec
=
1000 m/s with those in the case of uniform excitation: under the excitation of a
1, F
P
3 increases by 19.1%, M
T
1 decreases by 11.3%, and F
T
1 increases by 40%. And under the excitation of b
1, F
T
1, M
Y
1, and S
T
1 are amplified by 18.4%, 11.1%, and 8.7%, respectively. These responses threaten structural safety and should be given sufficient attention in the seismic design of long-span cable-stayed bridges.However, the seismic responses of the bridge do not always follow the same changing rules with the decrease of the surface apparent wave velocity. The responses may be amplified under the excitation of one set of ground motions but may be reduced under the excitation of another set. Even when the input is certain, some responses may remain the same, some may be amplified, and others may be reduced. For instance, the cross-sectional bending moment and shear force responses at the bottom of the piers are barely influenced by the wave passage effect because the bridge is a floating system (see Section2.2). The longitudinal displacement responses of the nodes on each pier almost fully depend on the pier support motion itself; if only the ground motion remains constant, then the longitudinal displacements of the nodes will not change and neither will the bending moments nor the shear forces. The time histories of the responses will be slightly translated, because the starting time of the support motion varies with the apparent wave velocity (see Figure 11). As another example, U
t is reduced under the excitation of a
1 and b
1 when the wave passage effect is considered, but it is amplified under the excitation of c
1 when c
=
1000 m/s and 500 m/s. Reference [27] drew similar conclusions when analyzing another long-span cable-stayed bridge model.Figure 11
Time history ofM
P
3 under the excitation of c
3.As for the traveling wave resonance, we can see from Tables3 through 5 that numerous structural peak responses reach their extreme values, either local maxima or minima, when c
=
1712 m/s, 1100 m/s, 1000 m/s, or 900 m/s. To make the expression clear and distinct, we graph the response data for which the resonance phenomena are obvious. Figures 12, 13, and 14 are based on the results under the excitation of a
1, b
1, and c
1, respectively. The dotted lines in the figures reflect the results under uniform excitation, which shows the influence trend of different apparent wave velocities.Variation trend of peak structural seismic responses with apparent wave velocity under the excitation ofa
1.
(a)
Variation trend ofU
b with c
(b)
Variation trend ofU
c with c
(c)
Variation trend ofF
P
2 with c
(d)
Variation trend ofF
P
3 with c
(e)
Variation trend ofF
T
1 with c
(f)
Variation trend ofM
T
1 with cVariation trend of peak structural seismic responses with apparent wave velocity under the excitation ofb
1.
(a)
Variation trend ofU
c with c
(b)
Variation trend ofF
P
3 with c
(c)
Variation trend ofF
T
1 with c
(d)
Variation trend ofM
T
1 with cVariation trend of peak structural seismic responses with apparent wave velocity under the excitation ofc
1.
(a)
Variation trend ofU
b with c
(b)
Variation trend ofF
P
2 with c
(c)
Variation trend ofF
T
1 with c
(d)
Variation trend ofS
T
1 with cBased on Figures2 through 7, we can see that 0.50662 Hz (the 12th natural frequency of the bridge) is within the predominant frequency range of a
1 and 0.41747 Hz (the 8th natural frequency of the bridge) is close to the range; therefore, the resonance phenomenon is evident when c
=
1100 m/s, 1000 m/s, or 900 m/s under the excitation of a
1. For b
1 and c
1, the 8th and 12th natural frequencies are beyond the predominant frequency ranges, but the 26th natural frequency (0.78681 Hz) is within range; therefore, the resonance phenomenon when c
=
1712 m/s is more remarkable than when c is near 1000 m/s.


## 5. Simplified Time History Analysis Method

When only considering the seismic excitation in the longitudinal direction for the long-span floating system cable-stayed bridge shown in Figure1, that is, the longitudinal displacement responses of nodes on each pier, the cross-sectional bending moment and shear force responses of each pier are only affected by the pier support motion and are irrelevant to the traveling time of the seismic wave. However, other structural seismic responses are the results of both the input seismic wave and the traveling wave from one main tower to the other. Therefore, we can ignore the traveling time from the far left pier to the left tower and from the right tower to the far right pier and only consider the wave passage time from the left tower to the right tower. Two different traveling wave excitation methods are used below to discuss the influence of the passage time between the piers and towers on the structural seismic responses.


### 5.1. Excitation Method 1:Full Traveling Time Excitation

This method is commonly adopted in current practice when the traveling times of the seismic waves between each pair of adjacent supports are considered. The support motions are denoted asu
1
(
t
)
~
u
8
(
t
), as shown in Figure 15: u
1
(
t
)
=
u
g
(
t
), u
i
(
t
)
=
u
g
(
t
-
x
i
-
x
1
/
c
), where u
g
(
t
) denotes the ground motion and x
i is the longitudinal coordinate of the ith support.Figure 15
Full traveling time excitation method.


### 5.2. Excitation Method 2:Grouping Uniform Excitation

The traveling time from the left-most pier to the left tower and that from the right-most tower to the far right pier are ignored. We assign the three left piers and the left tower to the left group and utilize the uniform excitation method. The three right piers and the right tower are assigned to the right group and are also exposed to uniform excitation. Only the traveling times between the two towers are considered, as shown in Figure16: u
l
(
t
)
=
u
g
(
t
) and u
r
(
t
)
=
u
g
(
t
-
1088
/
c
).Figure 16
Grouping uniform excitation method.


### 5.3. Comparison of the Results

We define the relative errore
=
m
2
-
m
1
/
m
1
×
100
% to compare the peak seismic responses computed with the two different excitation methods, where m
2 and m
1 refer to the peak responses calculated using excitation method 2 and method 1, respectively.Figure17 shows the time histories of F
P
2 under the excitation of c
3 when c
=
5000 m/s and 1000 m/s. We can see that the time histories resulting from the two excitation methods are nearly identical except for a slight translation due to the starting times of the support motions.Time histories ofF
P
2 under the excitation of c
3.
(a)
c
=
5000 m/s
(b)
c
=
1000 m/sTables6, 7, and 8 list the peak seismic responses under the two excitation methods. The results indicate that the two excitation methods lead to nearly identical responses, with the maximum absolute value of the relative error at 0.08%. It is therefore acceptable to use the grouping uniform excitation in practice instead of the full traveling time excitation method.Table 6
Comparisons of the structural peak seismic responses under different excitation methods under the excitation ofa
1.
Responses
Cases
c = 5000 m/s
c = 3000 m/s
c = 1000 m/s
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
U
t (m)
0.620
0.620
0.00
0.613
0.613
0.00
0.608
0.608
0.00
A
t (m/s2)
1.835
1.835
0.00
1.798
1.798
0.00
1.910
1.909
−0.05
U
b (mm)
2.450
2.450
0.00
2.300
2.300
0.00
2.330
2.330
0.00
U
c (cm)
4.759
4.759
0.00
4.522
4.522
0.00
4.476
4.477
0.02
F
P
1 (105 N)
2.451
2.451
0.00
2.666
2.666
0.00
2.446
2.444
−0.08
M
P
1 (104 kN⋅m)
10.669
10.668
−0.01
10.670
10.667
−0.03
10.670
10.672
0.02
S
P
1 (106 N)
2.275
2.274
−0.04
2.275
2.274
−0.04
2.275
2.275
0.00
F
P
2 (105 N)
5.267
5.265
−0.04
4.573
4.571
−0.04
4.959
4.958
−0.02
M
P
2 (104 kN⋅m)
11.153
11.153
0.00
11.153
11.153
0.00
11.151
11.152
0.01
S
P
2 (106 N)
2.378
2.378
0.00
2.378
2.378
0.00
2.378
2.378
0.00
F
P
3 (106 N)
1.767
1.767
0.00
1.816
1.816
0.00
2.005
2.005
0.00
M
P
3 (104 kN⋅m)
11.804
11.804
0.00
11.804
11.804
0.00
11.802
11.803
0.01
S
P
3 (106 N)
2.466
2.466
0.00
2.466
2.466
0.00
2.465
2.466
0.04
F
T
1 (106 N)
3.096
3.096
0.00
3.303
3.303
0.00
4.113
4.114
0.02
M
T
1 (106 kN⋅m)
1.149
1.149
0.00
1.076
1.076
0.00
1.095
1.095
0.00
S
T
1 (107 N)
0.994
0.994
0.00
1.011
1.011
0.00
0.891
0.891
0.00Table 7
Comparisons of the structural peak seismic responses under different excitation methods under the excitation ofb
1.
Responses
Cases
c = 5000 m/s
c = 3000 m/s
c = 1000 m/s
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
U
t (m)
0.727
0.727
0.00
0.725
0.725
0.00
0.674
0.674
0.00
A
t (m/s2)
2.734
2.734
0.00
2.599
2.599
0.00
2.698
2.697
−0.04
U
b (mm)
3.100
3.100
0.00
2.870
2.870
0.00
2.780
2.780
0.00
U
c (cm)
5.874
5.873
−0.02
5.506
5.506
0.00
5.165
5.165
0.00
F
P
1 (105 N)
3.945
3.945
0.00
3.531
3.532
0.03
3.692
3.693
0.03
M
P
1 (105 kN⋅m)
1.243
1.243
0.00
1.243
1.243
0.00
1.242
1.243
0.08
S
P
1 (106 N)
2.698
2.698
0.00
2.699
2.698
−0.04
2.699
2.699
0.00
F
P
2 (105 N)
6.751
6.750
−0.01
7.061
7.061
0.00
6.832
6.831
−0.01
M
P
2 (105 kN⋅m)
1.280
1.280
0.00
1.281
1.281
0.00
1.280
1.280
0.00
S
P
2 (106 N)
2.856
2.856
0.00
2.856
2.856
0.00
2.854
2.855
0.04
F
P
3 (106 N)
2.071
2.071
0.00
2.056
2.056
0.00
1.889
1.889
0.00
M
P
3 (105 kN⋅m)
1.354
1.354
0.00
1.355
1.354
−0.07
1.354
1.354
0.00
S
P
3 (106 N)
2.965
2.965
0.00
2.966
2.966
0.00
2.964
2.965
0.03
F
T
1 (106 N)
3.885
3.885
0.00
4.493
4.493
0.00
5.079
5.079
0.00
M
T
1 (106 kN⋅m)
1.461
1.461
0.00
1.352
1.352
0.00
1.319
1.319
0.00
S
T
1 (107 N)
1.501
1.501
0.00
1.526
1.526
0.00
1.294
1.294
0.00Table 8
Comparisons of the structural peak seismic responses under different excitation methods under the excitation ofc
1.
Responses
Cases
c = 5000 m/s
c = 3000 m/s
c = 1000 m/s
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
U
t (m)
0.676
0.676
0.00
0.676
0.676
0.00
0.663
0.663
0.00
A
t (m/s2)
1.881
1.881
0.00
1.919
1.919
0.00
2.106
2.105
−0.05
U
b (mm)
2.190
2.190
0.00
2.120
2.120
0.00
2.340
2.340
0.00
U
c (cm)
4.182
4.182
0.00
4.098
4.099
0.02
4.361
4.361
0.00
F
P
1 (105 N)
2.628
2.628
0.00
2.476
2.478
0.08
3.155
3.154
−0.03
M
P
1 (105 kN⋅m)
1.060
1.060
0.00
1.060
1.060
0.00
1.060
1.060
0.00
S
P
1 (106 N)
2.361
2.361
0.00
2.361
2.361
0.00
2.361
2.362
0.04
F
P
2
(105 N)
4.878
4.879
0.02
5.081
5.081
0.00
6.336
6.332
−0.06
M
P
2 (105 kN⋅m)
1.003
1.003
0.00
1.003
1.003
0.00
1.001
1.001
0.00
S
P
2 (106 N)
2.283
2.283
0.00
2.283
2.283
0.00
2.281
2.283
0.09
F
P
3 (106 N)
2.099
2.099
0.00
2.198
2.198
0.00
2.246
2.246
0.00
M
P
3 (105 kN⋅m)
1.054
1.054
0.00
1.054
1.054
0.00
1.053
1.053
0.00
S
P
3 (106 N)
2.379
2.379
0.00
2.379
2.379
0.00
2.380
2.380
0.00
F
T
1 (106 N)
4.369
4.369
0.00
4.827
4.827
0.00
4.275
4.275
0.00
M
T
1 (106 kN⋅m)
1.036
1.036
0.00
1.001
1.001
0.00
1.110
1.110
0.00
S
T
1 (106 N)
9.952
9.952
0.00
10.025
10.025
0.00
10.489
10.487
−0.02The results demonstrate that only the wave passage time from one main tower to the other needs to be considered for long-span floating system cable-stayed bridges under longitudinal traveling wave excitation. The grouping uniform excitation method can significantly reduce work load with an acceptable error.


## 5.1. Excitation Method 1:Full Traveling Time Excitation

This method is commonly adopted in current practice when the traveling times of the seismic waves between each pair of adjacent supports are considered. The support motions are denoted asu
1
(
t
)
~
u
8
(
t
), as shown in Figure 15: u
1
(
t
)
=
u
g
(
t
), u
i
(
t
)
=
u
g
(
t
-
x
i
-
x
1
/
c
), where u
g
(
t
) denotes the ground motion and x
i is the longitudinal coordinate of the ith support.Figure 15
Full traveling time excitation method.


## 5.2. Excitation Method 2:Grouping Uniform Excitation

The traveling time from the left-most pier to the left tower and that from the right-most tower to the far right pier are ignored. We assign the three left piers and the left tower to the left group and utilize the uniform excitation method. The three right piers and the right tower are assigned to the right group and are also exposed to uniform excitation. Only the traveling times between the two towers are considered, as shown in Figure16: u
l
(
t
)
=
u
g
(
t
) and u
r
(
t
)
=
u
g
(
t
-
1088
/
c
).Figure 16
Grouping uniform excitation method.


## 5.3. Comparison of the Results

We define the relative errore
=
m
2
-
m
1
/
m
1
×
100
% to compare the peak seismic responses computed with the two different excitation methods, where m
2 and m
1 refer to the peak responses calculated using excitation method 2 and method 1, respectively.Figure17 shows the time histories of F
P
2 under the excitation of c
3 when c
=
5000 m/s and 1000 m/s. We can see that the time histories resulting from the two excitation methods are nearly identical except for a slight translation due to the starting times of the support motions.Time histories ofF
P
2 under the excitation of c
3.
(a)
c
=
5000 m/s
(b)
c
=
1000 m/sTables6, 7, and 8 list the peak seismic responses under the two excitation methods. The results indicate that the two excitation methods lead to nearly identical responses, with the maximum absolute value of the relative error at 0.08%. It is therefore acceptable to use the grouping uniform excitation in practice instead of the full traveling time excitation method.Table 6
Comparisons of the structural peak seismic responses under different excitation methods under the excitation ofa
1.
Responses
Cases
c = 5000 m/s
c = 3000 m/s
c = 1000 m/s
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
U
t (m)
0.620
0.620
0.00
0.613
0.613
0.00
0.608
0.608
0.00
A
t (m/s2)
1.835
1.835
0.00
1.798
1.798
0.00
1.910
1.909
−0.05
U
b (mm)
2.450
2.450
0.00
2.300
2.300
0.00
2.330
2.330
0.00
U
c (cm)
4.759
4.759
0.00
4.522
4.522
0.00
4.476
4.477
0.02
F
P
1 (105 N)
2.451
2.451
0.00
2.666
2.666
0.00
2.446
2.444
−0.08
M
P
1 (104 kN⋅m)
10.669
10.668
−0.01
10.670
10.667
−0.03
10.670
10.672
0.02
S
P
1 (106 N)
2.275
2.274
−0.04
2.275
2.274
−0.04
2.275
2.275
0.00
F
P
2 (105 N)
5.267
5.265
−0.04
4.573
4.571
−0.04
4.959
4.958
−0.02
M
P
2 (104 kN⋅m)
11.153
11.153
0.00
11.153
11.153
0.00
11.151
11.152
0.01
S
P
2 (106 N)
2.378
2.378
0.00
2.378
2.378
0.00
2.378
2.378
0.00
F
P
3 (106 N)
1.767
1.767
0.00
1.816
1.816
0.00
2.005
2.005
0.00
M
P
3 (104 kN⋅m)
11.804
11.804
0.00
11.804
11.804
0.00
11.802
11.803
0.01
S
P
3 (106 N)
2.466
2.466
0.00
2.466
2.466
0.00
2.465
2.466
0.04
F
T
1 (106 N)
3.096
3.096
0.00
3.303
3.303
0.00
4.113
4.114
0.02
M
T
1 (106 kN⋅m)
1.149
1.149
0.00
1.076
1.076
0.00
1.095
1.095
0.00
S
T
1 (107 N)
0.994
0.994
0.00
1.011
1.011
0.00
0.891
0.891
0.00Table 7
Comparisons of the structural peak seismic responses under different excitation methods under the excitation ofb
1.
Responses
Cases
c = 5000 m/s
c = 3000 m/s
c = 1000 m/s
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
U
t (m)
0.727
0.727
0.00
0.725
0.725
0.00
0.674
0.674
0.00
A
t (m/s2)
2.734
2.734
0.00
2.599
2.599
0.00
2.698
2.697
−0.04
U
b (mm)
3.100
3.100
0.00
2.870
2.870
0.00
2.780
2.780
0.00
U
c (cm)
5.874
5.873
−0.02
5.506
5.506
0.00
5.165
5.165
0.00
F
P
1 (105 N)
3.945
3.945
0.00
3.531
3.532
0.03
3.692
3.693
0.03
M
P
1 (105 kN⋅m)
1.243
1.243
0.00
1.243
1.243
0.00
1.242
1.243
0.08
S
P
1 (106 N)
2.698
2.698
0.00
2.699
2.698
−0.04
2.699
2.699
0.00
F
P
2 (105 N)
6.751
6.750
−0.01
7.061
7.061
0.00
6.832
6.831
−0.01
M
P
2 (105 kN⋅m)
1.280
1.280
0.00
1.281
1.281
0.00
1.280
1.280
0.00
S
P
2 (106 N)
2.856
2.856
0.00
2.856
2.856
0.00
2.854
2.855
0.04
F
P
3 (106 N)
2.071
2.071
0.00
2.056
2.056
0.00
1.889
1.889
0.00
M
P
3 (105 kN⋅m)
1.354
1.354
0.00
1.355
1.354
−0.07
1.354
1.354
0.00
S
P
3 (106 N)
2.965
2.965
0.00
2.966
2.966
0.00
2.964
2.965
0.03
F
T
1 (106 N)
3.885
3.885
0.00
4.493
4.493
0.00
5.079
5.079
0.00
M
T
1 (106 kN⋅m)
1.461
1.461
0.00
1.352
1.352
0.00
1.319
1.319
0.00
S
T
1 (107 N)
1.501
1.501
0.00
1.526
1.526
0.00
1.294
1.294
0.00Table 8
Comparisons of the structural peak seismic responses under different excitation methods under the excitation ofc
1.
Responses
Cases
c = 5000 m/s
c = 3000 m/s
c = 1000 m/s
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
Excitation method 1
Excitation method 2
e (%)
U
t (m)
0.676
0.676
0.00
0.676
0.676
0.00
0.663
0.663
0.00
A
t (m/s2)
1.881
1.881
0.00
1.919
1.919
0.00
2.106
2.105
−0.05
U
b (mm)
2.190
2.190
0.00
2.120
2.120
0.00
2.340
2.340
0.00
U
c (cm)
4.182
4.182
0.00
4.098
4.099
0.02
4.361
4.361
0.00
F
P
1 (105 N)
2.628
2.628
0.00
2.476
2.478
0.08
3.155
3.154
−0.03
M
P
1 (105 kN⋅m)
1.060
1.060
0.00
1.060
1.060
0.00
1.060
1.060
0.00
S
P
1 (106 N)
2.361
2.361
0.00
2.361
2.361
0.00
2.361
2.362
0.04
F
P
2
(105 N)
4.878
4.879
0.02
5.081
5.081
0.00
6.336
6.332
−0.06
M
P
2 (105 kN⋅m)
1.003
1.003
0.00
1.003
1.003
0.00
1.001
1.001
0.00
S
P
2 (106 N)
2.283
2.283
0.00
2.283
2.283
0.00
2.281
2.283
0.09
F
P
3 (106 N)
2.099
2.099
0.00
2.198
2.198
0.00
2.246
2.246
0.00
M
P
3 (105 kN⋅m)
1.054
1.054
0.00
1.054
1.054
0.00
1.053
1.053
0.00
S
P
3 (106 N)
2.379
2.379
0.00
2.379
2.379
0.00
2.380
2.380
0.00
F
T
1 (106 N)
4.369
4.369
0.00
4.827
4.827
0.00
4.275
4.275
0.00
M
T
1 (106 kN⋅m)
1.036
1.036
0.00
1.001
1.001
0.00
1.110
1.110
0.00
S
T
1 (106 N)
9.952
9.952
0.00
10.025
10.025
0.00
10.489
10.487
−0.02The results demonstrate that only the wave passage time from one main tower to the other needs to be considered for long-span floating system cable-stayed bridges under longitudinal traveling wave excitation. The grouping uniform excitation method can significantly reduce work load with an acceptable error.


## 6. Conclusions

The seismic responses of a long-span floating system double-tower symmetrical cable-stayed bridge under longitudinal uniform excitation and traveling wave excitation are calculated in this paper, and three main conclusions are obtained through a comparison of the results.(i)
Responses that consider the traveling wave effect significantly differ from those under uniform support motions. The responses may be amplified under the excitation of one set of ground motions but reduced under the excitation of another set. Even when the input is certain, some responses may remain the same, some may be amplified, and others are reduced. To guarantee the safety of the bridge, the traveling wave effect should be given sufficient attention in the seismic design of long-span cable-stayed bridges.
(ii)
The bridge’s seismic responses do not vary monotonously with the apparent wave velocity due to the traveling wave resonance. The responses may obtain an extreme value, or even a maxima and minima, at specific apparent wave velocities. This phenomenon is more remarkable when the frequency of the modes at the resonance state is within the predominant frequency range of the seismic wave.
(iii)
When only considering the seismic excitation in the longitudinal direction for a long-span floating system cable-stayed bridge, we can ignore the traveling time from the far left pier to the left tower and from the right tower to the far right pier and only consider the wave passage time from the left tower to the right tower.



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*Source: 602825-2014-12-31.xml*