Nonstationarity analysis in wind-rain-induced vibration of stay cables.
He, Xu-hui ; Yu, Xiang-dong ; Chen, Zheng-qing 等
1. Introduction
Owing to the structural elegance and relative economy, more and
more large-span cable-stayed bridges have been built in recent years
throughout the world. However, cables in cable-stayed bridges are prone
to vibration induced by weather conditions and the structure (girders
and/or towers) to which they are connected due to their large
flexibility, relatively small mass and low inherent damping. For
example, large amplitude oscillations of stay cables under the
simultaneous occurrence of moderate wind and rain conditions have been
reported in a number of cable-stayed bridges worldwide (Bosdogianni,
Olivari 1996; Chen et al. 2004; Main et al. 2001; Hikami, Shiraishi
1988). This vibration can cause reduced cable and connection life due to
fatigue or rapid deterioration of the corrosion protection system and
may result in the loss of public confidence in the bridge (Hwang et al.
2009). Intensive studies have thus been carried out to explore the
mechanism and explain the complex phenomenon of wind-rain-induced
excessive vibration of stay cables. Research on wind-rain-induced cable
vibration includes theoretical analyses (Cao et al. 2003; Gu, Lu 2001;
Xu, Wang 2003; Yamaguchi 1990), wind tunnel simulation tests
(Bosdogianni, Olivari 1996; Gu et al. 2009; Matsumoto et al. 1992) and
field observation (Hikami, Shiraishi 1988; Ni et al. 2007; Zuo et al.
2008). Some main features for wind-rain-induced vibration such as
occurrence conditions of moderate wind and rain combination, water
rivulet formation on upper surface, axial flow in a near-wake of cable,
low-frequency vortex shedding along the cable axis, and vortex-induced
vibration at high reduced wind speed have been captured (Main et al.
2001). Almost all previous research studies were based on the assumption
of considering wind and cable vibration as a stationary random process.
In fact, the wind speed and the cable large-amplitude oscillation cannot
maintain a stationary level for a long time (Ni et al. 2007).
This paper aims to develop a wavelet-based method to investigate
wind and wind-rain-induced response on the basis of field measured wind
data from the DLB in China. Combining the field measurements of wind and
cable vibration and wavelet multiscale analysis, the time-varying mean
wind speed is extracted and a nonstationary wind speed model is proposed
based on the typical wind samples. The wind parameters in
rain-wind-induced vibration are obtained by using the proposed
nonstationary model and compared with those based on EMD and design
values. The correlation between cable acceleration response peak factors
and mean wind velocity are also discussed. It is concluded that the
multiscale-based approach is more appropriate for investigating
nonstationary wind and wind-rain-induced vibration of stayed cables.
2. Description of bridge and field measurements
The Dongting Lake Bridge (DLB), as shown in Fig. 1, is the first
three-tower prestressed concrete cable-stayed bridge located in the
influx of Dongting Lake in to the Yangtze River, China. The bridge
consists of two main spans of 310 m each and two side spans of 130 m
each with 25.0 m clearance height above water level. The deck is 23.4 m
wide with four lanes of traffic. The central tower is 125.7 m high and
side towers are 99.3 m high each. There are a total of 222 cables with
size ranging from 28 to 201 m in length and 99 to 159 mm in diameter
with polyethylene (PE) pipes. Shortly after it opened to traffic in
2000, excessive and unanticipated wind-rain-induced cable vibrations
were observed every April, July. The large-amplitude cable vibration
cause concerns of the bridge administrative authority and engineers.
Unlike the mass dampers used in buliding structures and other type
bridges, the MR dampers were finally installed on the cables to mitigate
cable vibration after a series of field observation and measurements
were conducted (Chen et al. 2004).
The field tests included measurements of wind and rain
characteristics, cable vibration and its mitigation by using MR dampers.
Two ultrasonic anemometers were installed on the top of the south side
tower and deck level near cable A12, respectively. Tower one is situated
at an elevation of 102 m, 2 m high above the tower top. Deck level one
is situated at an elevation of 26 m, 4 m stretching out from the deck
edge with a horizontal cantilever, as shown in Fig. 2. Four uniaxial
accelerometers were installed on the locations of L/6 and L/20 from the
lower anchorage for cable A12 in-plane and out-of-plane acceleration
measurement. One rain gauge was installed at the deck level near by
cable A12. One data acquisition and processing system in the bridge site
can record the data while wind-rain-induced vibration occurs. The sample
frequency of wind speed and acceleration response are 4 Hz and 100 Hz,
respectively. Continuous field measurements were conducted for 47 days
from 24 March to 11 May 2003.
[FIGURE 1 OMITTED]
[FIGURE 2 OMITTED]
3. Nonstationary wind speed and time-varying mean wind speed
extraction
Some research studies (Li et al. 2000; Ni et al. 2007) have shown
that based on field measurements, wind speed usually has obvious
nonstationary characteristics. The characteristics of nonstationary
random process represent that the information of time-domain,
frequency-domain and so on are related with time and are not ergodic.
Thus, it will be unreasonable to assume wind speed or wind pressure as a
Gaussian stationary process supposition of in the study of civil
engineering. Based on the Gramer theorem, the nonstationary wind speed
can be modeled as a deterministic time-varying mean wind speed plus a
zero-mean stationary random process for fluctuating wind speed (Xu, Chen
2004):
U(t) = [bar.U](t) + u(t), (1)
where: [bar.U](t) is a deterministic time-varying mean wind speed
reflecting the temporal trend of wind speed; and u(t) is a fluctuating
wind speed of a zero-mean stationary process. The above nonstationary
model can be expanded to lateral and vertical wind speed. In fact, the
stationary wind speed model Eq. (1) can be looked at as an especial case
of the nonstationary model.
The key issue in using nonstationary wind speed model is how to
extract the trend in nonstationary signals. The wavelet transform (WT)
overcomes the limitations of traditional Fourier transform (FT) and
short-time Fourier transform (STFT). It can be thought of as a
generalized STFT, with a frequency-dependent window size (Bienkiewicz,
Ham 1997). Through dilation of a mother function (wavelet) [PSI](u),
adjustment in the window can be accomplished, and localization of
frequency resolution is achieved by translation of this function. The
resulting wavelet function [[PSI].sub.[lambda]t](u) is thus obtained:
[[PSI].sub.a,b](t) = [1/[square root of ([lambda])]] [PSI]([t -
b]/a), (2)
where b is an instant at which the wavelet function is centered and
a [not equal to] 0 is a scale parameter controlling the function spread.
The signal is then decomposed into a series of basis functions of length
consisting of dilated (stretched) and translated (shifted) versions of
the mother function, i.e., wavelets of different scales and positions in
time or space (Gurley, Kareem 1999).
Based on the wavelet theory, the wavelet has the character of
conservation of energy when the wavelet function is a series of
orthogonal basis functions. The wavelet energy of signal scale can be
defined as the sum of squares of wavelet coefficients as follows (Zunino
et al. 2007):
[E.sub.j] = [[parallel][r.sub.j][parallel].sup.2] = [summation]
[[absolute vallue of [W.sub.j,k]].sup.2] (j = 1,...,N). (3)
For a complex nonstationary signal, the longest period component
obtained by WT decomposing in maximum layers, is not always the optimal
component reflecting the local information of nonstationary time-varying
mean. Therefore, the key issue in using WT to extract the trend in
nonstationary signals is how to make sure the most reasonable number of
decomposition layers is considered. For a discrete WT, the energy of
each layer detail coefficient can be obtained using Eq. (3). The
appropriate levels for decomposing the time-varying mean wind speed were
quantitatively determined by the sudden change of simple scale wavelet
energy.
[FIGURE 3 OMITTED]
4. Nonstationary wind characteristics
4.1. Wind speed and direction
The two typical measured wind speed data segments (1 h) from
anemometers installed on the top of the south side tower and bridge deck
level during wind-rain-induced vibration duration on 1 April 2003 are
considered here. Shown in Fig. 3 (a-b) are the 1 h duration wind speed
and wind direction samples from bridge deck level anemometer between
16:51 to 17:51, 1 April 2003, respectively. Shown in Fig. 3 (c-d) are
the 1h duration wind speed and wind direction samples from the tower top
anemometer between 16:51 to 17:51, 1 April 2003. It is seen that the
mean wind speed and wind direction in 1 h is time-varying, and it is not
appropriate to adopt the constant mean wind speed assumption. The
time-varying mean wind speeds and directions obtained by WT are a
continuous function of time with a designated frequency level, which is
more natural than the traditional time-averaged mean wind speed with the
certain time interval.
Using the same method of time-varying mean wind speed extraction,
the time-varying mean wind deviation angle and yaw angle are also
obtained. The results show that the wind deviation angles of the bridge
are nonstationary, the deck level wind deviation angle varied from 30
degree to 60 degree, and the tower top wind deviation angle among 60
degree to 70 degree; but wind yaw angles varied slightly, the deck level
wind yaw angle is almost 10 degree, and the tower top windyaw angle is
about 20 degree, respectively.
4.2. Turbulence intensity
For stationary wind speed, the ratio of the standard deviation of
fluctuating wind to mean wind speed is traditionally defined as the
turbulence intensity. For nonstationary wind speed, however, the mean
wind speed is time varying, and turbulence intensity is also time
dependent over time interval. Table 1 lists the measured turbulence
intensities at tower top and deck level during the wind-rain excitation
event on April 1-2 2003, where [I.sub.u] indicates the turbulence
intensity in the longitudinal direction, Iv the turbulence intensity in
the lateral direction, and [I.sub.w] the turbulence intensity in the
vertical direction. For comparison, the design values based on
stationary method (Ni et al. 2007) and corresponding design values are
also listed. It is found that the mean values of turbulence intensities
computed by nonstationary model are smaller than those obtained by the
traditional stationary model and design values. Among the two
nonstationary approaches based on WT and EMD, the results by WT are
slightly smaller than those by EMD.
[FIGURE 4 OMITTED]
4.3. Probability distribution
Displayed in Fig. 4 are the probability distributions of the
typical fluctuating wind speeds recorded from the deck level and south
tower anemometers together with Gaussian density functions,
respectively. The probability distributions based on WT are calculated
from fluctuating wind speed obtained by subtracting the time-varying
mean wind speed at a frequency level of 1/3600 Hz from original wind
sample. The probability distributions based on EMD nonstationary and
traditional stationary wind speed models are also computed and compared
in Fig. 4. It is seen that the probability densities obtained by the
nonstationary model based on WT comply with the Gaussian distribution
better than those calculated by nonstationary model based on EMD and
traditional stationary model. Especially, as shown in Fig. 4d, the
probability distribution of the fluctuating wind speed obtained based WT
from tower original wind sample 2 (22:10-02:48, 1-2 April 2003) complies
with Gaussian distribution well, but the results obtained based EMD and
traditional stationary model deviate from the Gaussian distributions
significantly. Thus, it may be concluded that wavelet-based stationary
model is more reasonable and reliable for characterizing field measured
nonstationary wind speeds.
5. Correlation between wind and wind-rain-induced response
5.1. Correlation between wind and response
Fig. 5 shows the correlations of wind-rain-induced cable A12
in-plane acceleration and wind speed of deck level and tower top. It is
seen that the root mean square (RMS) of in-plane accelerations show an
increasing tendency with the increase in the RMS of the wind speed of
deck level, whereas the relation between the wind speed and out-of-plane
acceleration is not very clear. Similar tendencies can be found in
correlations of tower top wind fluctuations and RMS of A12 cable
out-of-plane accelerations. It is evident from the Fig. 5 that the
critical RMS of wind velocity of deck level for wind-rain-induced cable
vibration is 7-9.5 m/s.
5.2. Relation between response peak factors and mean wind speed
The response peak factor is defined as the ratio of the maximum
value of acceleration to its RMS values and is taken as a parameter to
represent the fluctuations in the structural response (Adhikari,
Yamaguchi 1997). The relation between the mean wind velocity and
response peak factors is shown in Fig. 5. For comparison, the
theoretical average peak factors for resonant response based on a narrow
band stationary assumption made both on the wind turbulence and
structure response are also plotted in Fig. 6. The theoretical values of
average peak factors F were calculated by the following equation:
F = [square root of (2ln(nT))] + [[gamma]/[square root of
(2ln(nT))]], (4)
where n is the first natural frequency, T is the averaging time and
[gamma] is the Euler's constant equal to 0.5772.
As seen in Fig. 6, the peak factors for wind-rain excitation event
at deck level during 16:51 to 21:36 of 1 April 2003 show no clearly
increasing tendency with increase in the mean wind speed when the
averaging time is 10 min or 1 min, the peak factors are not affected by
the increase in mean wind velocity. It is observed that when the
averaging time is set to be 10 min, large values of peak factors, as
compared to the theoretical values of the peak factors, can occur in the
real case, thus highlighting the nonstationary characteristics of the
wind turbulence of the cable. Similar results can be found in relation
between out-of-plane acceleration peak factors and the mean wind
velocity.
[FIGURE 5 OMITTED]
6. Conclusions
In this paper, the nonstationarity in the wind is discussed from
the point of view of developing a wavelet-based method. Based on the
analysis of the field measurements of the wind and wind-rain-induced
acceleration response of a stayed cable of DLB during wind-rain-induced
vibration in April 2003, some conclusions could be drawn as follows:
1. The measured wind speeds as well as the cable response
accelerations show remarkable nonstationarity. The nonstationarity in
wind may be one of the main factors for the nonstationarity in cable
acceleration response.
2. The mean values of turbulence intensities calculated by
wavelet-based nonstationary model are smaller than those obtained by the
traditional stationary model and design values. Therefore, traditional
stationary approach may lead to the overestimation of turbulence
intensity, and the turbulence intensity estimated by nonstationary model
seems to be more reasonable for wind-rain-induced vibration analyses.
3. The cable acceleration response peak factors obtained based on
nonstatioanry method are generally larger than the theoretical results
obtained by stationary assumption. In general, due to wind-rain-induced
vibration of stay cables in cable-stayed bridge and wind have obvious
nonstionarity, considering wavelet-based analyses is more reasonable for
investigating large-amplitude wind-rain-induced vibration of stayed
cables.
[FIGURE 6 OMITTED]
doi: 10.3846/13923730.2012.720933
Acknowledgements
The work described in this paper was supported by the China
National Natural Science Foundations (Grant no. 50808175, 51178471) to
which the authors gratefully appreciate.
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Xu-hui He (1), Xiang-dong Yu (2), Zheng-qing Chen (3)
(1,2) School of Civil Engineering, Central South University,
Changsha, Hunan, China
(3) Wind Engineering Research Center, Hunan University, Changsha,
Hunan, China
E-mails: (1) xuhuihe@csu.edu.cn (corresponding author); (2)
yxd77@139.com; (3) zqchen@hnu.edu.cn
Received 28 Feb. 2011; accepted 09 Apr. 2011
Xu-hui HE. Is a professor of Civil Engineering in the School of
Civil Engineering at Central South University, China. He is a member of
IABSE (International Association of Bridge and Structural Engineering),
and a member of IABMAS (International Association for Bridge Maintenance
and Safety). His interests include bridge wind engineering, bridge
structure health monitoring, parameter identification and condition
assessment of bridge.
Xiang-dong YU. Is an associate professor of Civil Engineering in
the School of Civil Engineering at Central South University, China. His
interests include bridge wind engineering, structural design theory and
condition assessment of bridge.
Zheng-qing CHEN. Is a professor and director of Wind Engineering
Research Center at Hunan University, China. He is a member of ASCE
(American Society of Civil Engineers). His interests include bridge wind
engineering, nonlinear analysis of large flexible bridge.
Table 1. Comparison of average values of turbulence intensity
16:51-21:36 22:10-02:48
Deck Tower Deck Tower
level top level top
Based on WT [I.sub.u] 0.084 0.0587 0.0888 0.063
[I.sub.v] 0.0815 0.0499 0.0963 0.0677
[I.sub.w] 0.0757 0.0684 0.0802 0.0687
Based on EMD [I.sub.u] 0.1147 0.0689 0.1014 0.0759
[I.sub.v] 0.0985 0.0631 0.1102 0.0789
[I.sub.w] 0.0871 0.0797 0.0889 0.099
Design value [I.sub.u] 0.13 0.1 0.13 0.1
[I.sub.v] 0.11 0.09 0.11 0.09
[I.sub.w] 0.07 0.05 0.07 0.05
