Mode-Dependent Structural Damping of Suspension Bridge

*Shouqiang Wang1), Lin Zhao2) and Yaojun Ge3)

1), 2), 3)

Key Laboratory for Disaster Reduction in Civil Engineering，Tongji

University，Shanghai 200092，China 1)

shouqiangwang2007@gmail.com

ABSTRACT

A reliable structural modal damping is very essential to the wind-resistant design of suspension bridges, because it possess the key to obtain the accurate wind-induced dynamic response of the structure. Data-Driven Stochastic Subspace Identification Method was adopted to address the one year records from the Structural Health Monitoring (SHM) system of a long-span suspension bridge in China. And two major phenomena have been found, one is that the structural modal damping increases with the increasing of the acceleration or wind velocity when the modal order remain invariable and the other is that the identified modal damping decreases as the modal order or frequency increases, which is different from that of the building structure.

1. INTRODUCTION

The modal parameters are the accurate reflection of the structural properties such as stiffness and damage (H. Li, Li, Ou, & Li, 2010). The modal structural damping is one of the most important parameters, but comparitively difficult to measure (Yamaguchi & Ito, 1997).

Nowadays, more and more field dynamic experiments at the sites of long-span bridges (Hui Li et al., 2011; Wang, Li, Guo, & Xie, 2008; Zhu & Xu, 2005) are built after the completion of construction, in order to investigate the validity of assumed values of modal parameters and monitor the structural response under strong wind conditions. Although data of damping ratio for different types of bridges have been collected through such field vibration tests, there have not been many examples of newly constructed long-span bridges built in China. Since many numbers of long-span bridges will be constructed all over the world in the future, more accurate modal damping is needed to make the wind-resistant design of long-span bridges more rational.

One of the main objectives of the present study is to discuss the general characteristics of modal damping of long-span suspension bridges by using the field-

1)

Student 2), 3)

Professor

measured data. Another objective of this investigation is to find the relationship between between modal damping and other parameters, e.g. frequency, wind speed and RMS of acceleration.

2. SHM SYSTEM AND SENSORS

In respect of the identification of the on-site measured data, Data-Driven Stochastic Subspace Identification Method (Khan, Shan, & Li, 2015) was adopted to address the one year records from the Structural Health Monitoring (SHM) system of a long-span suspension bridge in China. And two major phenomena have been found, one is that the structural modal damping increases with the increasing of the acceleration or wind velocity when the modal order remain invariable, but due to the effect of the aerodynamic damping under strong wind, the growth rate of the modal damping slows down when the value of acceleration or wind velocity is relevantly high. The other phenomenon is that the identified modal damping decreases as the modal order or frequency increases, which is different from that of the building structure.

Using a long-span suspension bridge as an example, parameter identification was analysised on the basis of the health monitoring system. The bridge is two span steel box girder suspension bridge with a mid-span 1650m and total length 2588m. Figure 1 shows the layout of the configuration of the bridge.

Fig. 1 the layout of the configuration of the bridge.

Fig. 2 the layout of the anemometers and accelerometers.

The Structural Health Monitoring System (SHMS) of this Bridge has totally 6 anemometers which are installed to measure wind and provide wind characteristics to analysis bridge response and 12 accelerometers, as shown in figure 1. The sampling frequency of anemometer is 32Hz, with the north, the west and the vertically upward for its monitoring X, Y, Z, respectively. The sampling frequency of acceleration is 100Hz.

3. METHOD AND RESULTS

The bridge health monitoring system has been put on trial at the end of 2009, here the data of acceleration and anemometer is analyzed for one consecutive year, from 2009/12/1 to 2010/11/12. The bridge officially opened from December 25, 2009. As a result, the data of first month is not affected by the influence of vehicle load, which means that the bridge load of motivation is mainly from wind. To ensure more accurate, the sampling period is mainly from 23:00 to 04:00 and the data that the 10min mean wind speed is lower than 5m/s was not analysed.

S

bridge axis

225o

0o

30o

45o

60o

90o

120o

150o

180o

210o

240o

270o

300o

330o

5

15

25m/s

N

0 5 10 15 20 250

2

4

6

RMS fitted curve

RMS of acceleration(cm/s2)

mean wind speed(m/s)

Fig. 3 wind rose. Fig. 4 RMS of acceleration with wind speed.

Figure 3 and figure 4 are shown wind rose and RMS of acceleration at different wind speed, respectively. The direction of wind speed, mainly from northwest to southeast, is perpendicular to bridge axis while the RMS of acceleration, between 0 and 6cm/s2, increases with the increase of wind speed.

To illustrate the identification process of modal parameters, one hour acceleration data is used as an example. Figure 5 shows the time history of acceleration for an hour, with the sampling length T=1h=3600s, frequency f=100Hz and the total point in the figure N=100*3600=360000. It can be seen from the diagram, the change range of the bridge vertical acceleration is between -20cm/s2 to 20cm/s2 in this hour.

0 600 1200 1800 2400 3000 3600-20

-10

0

10

20

Vert

ical accele

ration(c

m/s

2)

Time(s)

Fig. 5 time history of vertical acceleration for an hour.

Using the time history of acceleration shown in figure 5, Figure 6 shows the measured spectra of deck vertical acceleration by the method of SSI-DATA. Table 1 presents the natural frequencies and modal damping ratios identified from the measured vertical vibration of the deck, the numerical results from finite element analysis, and the results from Qw Zhang and Hui Li (Hui Li, Laima, Zhang, Li, & Liu, 2014).

Fig. 6 Measured spectra of deck vertical acceleration.

Table 1 The measured natural frequencies and modal damping ratios from vertical vibration of the deck.

Order of

modes

Calculated value

Zhang SSI-COV Li SSI-DATA Here SSI-

DATA

20/7/2009~28/7/2009 29/09/2009~30/11/

2009 01/12/2009~12/11/20

10

U=2~8m/s U≤4m/s U=15~22m/s

Natural frequenc

y (Hz)

Natural frequenc

y (Hz)

Damping ratio (%)

Natural freque

ncy (Hz)

Damping ratio (%)

Natural frequency

(Hz)

Damping ratio (%)

1 0.094 0.095 1.80~2.18 0.0953 0.57 0.0947 1.7703

2 0.102 0.103 1.02~1.62 -- -- 0.1031 1.1973

3 0.132 0.133 0.90~1.46 0.1328 0.52 0.1322 0.9829

4 0.178 0.183 0.37~0.61 0.1825 0.5 0.1826 0.8802

5 0.229 0.229 0.23~0.62 0.2301 0.51 0.2303 0.7794

6 0.273 0.276 0.43~0.83 0.2757 0.39 0.2786 0.5138

7 0.323 0.327 0.36~0.68 0.3237 0.42 0.3297 0.2110

8 0.372 0.379 0.44~0.72 -- -- 0.3818 0.0010

9 0.426 0.435 0.21~0.37 0.4351 0.37 0.4379 0.3044

10 0.479 0.491 0.10~0.50 0.4918 0.32 0.4956 0.5019

It can be seen from table 1 that the deviations of natural frenquencies recognized from different time and under the inflence of diffferent wind speed are relative small. But for modal damping ratios, the deviation of the results for different conditions is more than 10%. Compared with the results of other two scholars, the results of this paper is relatively close to Zhang, especially in the first three order modal damping ratio. In addition, it can be found that with the increase of order of mode, natural frequency increases meanwhile the modal damping ratio decreases gradually. The accuracy of identification may be lower for the higher modal order and the decreasing trend of damping ratio is significant for the previous order.

The natural frequencies and damping ratio calculated by the above-mentioned method are shown in figure 7 and 8, respectively. It can be seen from figure 7 that the natural frequencies of this suspension bridge change slightly for all these four orders. From figure 8, the damping ratio has nonlinear characteristics changing with time and decreases with the increase of order.

Table 2 shows statistic characteristics of natural frequency and damping ratio. It can be seen from the table that the natural frequency fluctuates slightly with the gap between minimum and maximum small, while the damping ratio swings widly with a big gap between minimum and maximum.

0.092

0.094

0.096

0.098

0.100 natrual frequency

fitting curve y=a+b*x

Na

tru

al fr

eq

ue

ncy

natrual frequency

fitting curve y=a+b*x

0 1000 2000 3000 4000 5000 6000 7000 80000.130

0.132

0.134

0.136

natrual frequency

fitting curve y=a+b*x

Na

tru

al fr

eq

ue

ncy

time (h)

0 1000 2000 3000 4000 5000 6000 7000 8000

natrual frequency

fitting curve y=a+b*x

time (h)

0.100

0.102

0.104

0.106

0.108

0.180

0.182

0.184

0.186

Fig. 7 Natrual frequency along with time.

0.01

0.02

0.03

0.04

0.05 modal damping

fitting curve

mo

da

l d

am

pin

g

modal damping

fitting curve

0 1000 2000 3000 4000 5000 6000 7000 80000.00

0.01

0.02

0.03

0.04

0.05 modal damping

fitting curve

mo

da

l d

am

pin

g

time (h)

0 1000 2000 3000 4000 5000 6000 7000 8000

modal damping

fitting curve

time (h)

0.01

0.02

0.03

0.04

0.05

0.00

0.01

0.02

0.03

0.04

0.05

Fig. 8 Structural modal damping along with time.

Table 2 Statistic characteristics of natural frequency and damping ratio.

Order of modes Natural frequency (Hz) Damping ratio (%)

Mean SD(10-4) Min. Max. Mean SD(10-3) Min. Max.

1 0.095 4.404 0.093 0.097 0.017 6.28 0.0003 0.045 2 0.103 5.82 0.102 0.106 0.015 6.05 0.0011 0.039 3 0.133 5.29 0.131 0.136 0.010 4.84 0.0002 0.029 4 0.183 4.51 0.181 0.186 0.007 4.85 0.0001 0.024

SD for Standard Deviation.

0.01

0.02

0.03

0.04

0.05 modal damping

fitting curve

mo

da

l d

am

pin

g

modal damping

fitting curve

0 1 2 3 4 50.00

0.01

0.02

0.03

0.04

0.05

modal damping

fitting curve

mo

da

l d

am

pin

g

RMS of acceleration (cn/s2)

1 2 3 4 5

modal damping

fitting curve

RMS of acceleration (cn/s2)

0.01

0.02

0.03

0.04

0.05

0.00

0.01

0.02

0.03

0.04

0.05

Fig. 9 Structural modal damping along with acceleration.

Figure 9 shows the measured modal damping ratio changes of wind speed with RMS of acceleration. The modal damping ratio increases with the increase of acceleration, but the growth rate of damping ratio under the high RMS of acceleration is smaller than the low RMS.

4. CONCLUSIONS

The mechanism of environmental effects on damping ratio is much more complicated and is not completely understood at present. With the development of the structural system identification technique and the popularization of the bridge Structural Health Monitoring system, the research of the modal damping has stepped into a new era, and the following conclusions can be obtained.

(i) Wind speed has no clear impact on the natural frequencies of bridges. For all four orders of modes, the deviations of natural frenquencies recognized under the inflence of diffferent wind speed are relative small.

(ii) The deviation of the results for modal damping ratios is more than 10% and the data is discrete.

(iii) Under the same wind speed, different modal order has the corresponding damping ratio. For suspension bridge, modal damping ratio decreases with the increase of natural frequency, while it changes in contrast for structure.

The application of SSI-DATA method to this suspension bridge leads that abundance of modal damping data has been collected. However, to reveal the true nature of the structural modal damping still has a long way to go.

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CABLE STAYED BRIDGE BASED ON EXPLORATORY DATA ANALYSIS. Archives of Civil Engineering, 61(2), 3-22. doi:10.1515/ace-2015-0011

Li, H., Laima, S., Ou, J., Zhao, X., Zhou, W., Yu, Y., . . . Liu, Z. (2011). Investigation of vortex-induced vibration of a suspension bridge with two separated steel box girders based on field measurements. Engineering Structures, 33(6), 1894-1907. doi:10.1016/j.engstruct.2011.02.017

Li, H., Laima, S., Zhang, Q., Li, N., & Liu, Z. (2014). Field monitoring and validation of vortex-induced vibrations of a long-span suspension bridge. Journal of Wind Engineering and Industrial Aerodynamics, 124, 54-67. doi:10.1016/j.jweia.2013.11.006

Li, H., Li, S. L., Ou, J. P., & Li, H. W. (2010). Modal identification of bridges under varying environmental conditions: Temperature and wind effects. Structural Control & Health Monitoring, 17(5), 495-512. doi:10.1002/stc.319

Wang, H., Li, A., Guo, T., & Xie, J. (2008). Field measurement on wind characteristic and buffeting response of the Runyang Suspension Bridge during typhoon Matsa. Science in China Series E: Technological Sciences, 52(5), 1354-1362. doi:10.1007/s11431-008-0238-y

Yamaguchi, H., & Ito, M. (1997). Mode-dependence of structural damping in cable-stayed bridges. Journal of Wind Engineering and Industrial Aerodynamics, 72(1-3), 289-300. doi:10.1016/s0167-6105(97)00249-3

Zhu, L. D., & Xu, Y. L. (2005). Buffeting response of long-span cable-supported bridges under skew winds. Part 1: theory. Journal of Sound and Vibration, 281(3-5), 647-673. doi:10.1016/j.jsv.2004.01.026