The Application Approach and Experimental Study on Spanwise Correlation of Vortex Induced Force of Box Girder

*Yanguo Sun1), Haili Liao2) and Mingshui Li3)

1), 2) ,3) Research Centre for Wind Eng., Southwest Jiaotong University, Chengdu, China

1) syg222@163.com

ABSTRACT

Section model wind tunnel testing is a general methodology to estimate vortex-induced vibration (VIV) performance of a long-span bridge. The test method and some existing semi-empirical models are two-dimensional (2-D) theories. As a matter of fact the VIV of actual bridge structure is a three-dimensional (3-D) problem. Incomplete spanwise correlation of vortex-induced aerodynamic forces along span leads to the inconsistency between section model tests and prototype structure response. A section model of box girder was selected to study the spanwise correlation of vortex-induced force by pressure measurement tests, and the correlation function was fitted. The VIV response of section model was interpreted into an aeroelastic model by applying a nonlinear VIV method in which the spanwise correlation of vortex-induced force was take into consideration, and the validity of the method was conformed.

1. INTRODUCTION

Long-span bridges often suffer from vortex-induced vibration due to its flexibility, light weight and low damping etc. At present, wind tunnel testing for section model is a general methodology to evaluate its VIV performance. However, wind tunnel tests method and the existing semi-empirical models are based on 2-D theory, the section model in wind tunnel tests is assumed to behave in a 2-D manner, for an actual structure, it is a 3-D problem. In addition to the differences in vibration mode, the aerodynamic forces are not perfectly correlated along its span. The consistence between the observation of prototype and predictions based on 2-D model testing is still uncertain due to the contribution of spanwise correlation of vortex-induced aerodynamic forces.

The correlation of vortex-induced force acting on square cylinders along the span was studied in detail by Wilkinson (1981), and then correlation of vortex-induced force varied with amplitude was found through direct pressure measurement on rigid cylinder section model, and after that a double exponential correlation semi-empirical formula

1)

Ph.D., Lecture 2),

3)

Ph.D., Professor

was fitted. On the basis of Scanlan’s semi-empirical nonlinear model and Wilkinson’s correlation function, Eshan (1990) discussed the vortex-induced force along the span roughly, and proposed a relatively simple method to estimate the response of VIV. A nonlinear method proposed by Sun (2014) based on Scanlan’s semi-empirical model was used to estimate prototype bridge by the result of section model wind tunnel tests. But their correlation function comes from the pressure tests of square cylinders (Wilkinson 1981), and it is not suitable for VIV analysis of different structure.

In this paper, a section model of box girder was selected to study the spanwise correlation of vortex-induced force by pressure measurement tests, and the correlation function was fitted. The VIV response of section model was interpreted into an aeroelastic model by applying a nonlinear VIV method proposed by Sun (2014).

2. TESTS DESCRIPTION

A trapezoidal section model and a simplely supported aeroelastic model were selected for pressure measurement tests. The length of section models is 2.095m, and the other dimensions are shown in Fig. 1. 5 rows measurement points with different row spacing were set in the model and 10 sets spanwise spacing were generated. There are 34 points in trapezoidal model. 5 groups pressure scanning valve (type: DSM-3400) were used for pressure tests. The valves were fixed in the internal model, and the data lines were led out from the end plates at both sides. The total length of the simple supported aeroelastic model is 3m, and the model was divided into 12 sections, and the spacing between two sections is 3mm. 6 sections were selected for pressure measurement tests, and the layout of pressure measurement points is the same with section model (Fig. 2). In order to be close to the actual structures, and observe the VIV phenomenon easily, railings were installed on the model.

146.3

166.7

397

110

19.7

Scanning valve1

18

Fig. 1 The section model

386

167

110

18

46

100

18

46 芯

梁

steelbeam

Fig. 2 The simply supported aeroelastic model

The wind tunnel (Type: XNJD-1) of Southwest Jiaotong University, a closed circuit wind tunnel with two tandem closed test sections, was used to carry out the investigation. The section model was tested in the second test section, and the dimension of the test section is 2.4m×2.0m (W×H). The simply supported aeroelastic model was tested in the first test section, and the dimension of this section is 3.6m×3.0m (W×H).

The VIV tests were conducted by free vibrating method in smooth flow. The vibration amplitude was recorded by laser displacement meters and the lock-in area was found, the acquisition of pressure data were preceded in the three stages (before the VIV region, in the VIV region and after the VIV region). In the VIV region, pressure data were acquired at different amplitudes which depend on three different damping levels, and the effects of VIV amplitude on the correlation of vortex-induced force were studied in detail. In addition, the pressure distribution characteristics of different cross sections were analyzed in detail, especially the points near the wake. This paper only pays attention to the correlation of the model in different amplitude.

3. SPANWISE CORRELATION ANALYSIS

The correlation of trapezoidal section model is affected at a large extent by the

change of attack angel. The correlation coefficient will increase with the increase of amplitude at 0º attack angel (Fig. 3 ).The change of attack angle of oncoming flow will affect the aerodynamic characteristics of section. With the increase of attack angle the points with weak correlation at the wake will reduce the correlation (Fig. 4).

0 1 2 3 4 5 6 7 8 9 10

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

3.97%

2.97%

2.32%

static

Co

rre

latio

n

Spacing (X/D) 0 1 2 3 4 5 6 7 8 9 10 11

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

0°

3°

5°

Co

rre

latio

n

Spacing (X/D) Fig. 3 The correlation coefficients of

section model at different amplitudes

Fig. 4 The correlation of section model at different attack angles

The correlation of vortex-induced force act on square cylinders along the span was

studied in detail by Wilkinson (1981). Also, he found the correlation of vortex-induced force varied with vibration amplitude, a double exponential correlation semi-empirical formula (Eq. 1) was fitted by Ehsan and Scanlan (1990) based on Wilkinson’s test results.

]))((exp[)()(

12 f

D

xfxR

(1)

in which, η is the ratio between vibration amplitude and cross wind size D; △x/D is spanwise spacing; and f1 and f2 were fitted as following form:

25.01298.0

052.0)(

f ，

042.0

065.0)(

2f (2)

For the test results of trapezoidal section model mentioned in this paper, the two

functions f1 and f2 can be fitted as Eq. 3, and the correlation coefficients of the test and fitted results can be seen in Fig. 5.

1 0.3441

2

0.1038( )

4.3259

0.1798( )

0.4734

f

f

(3)

0 1 2 3 4 5 6 7 8 9 10 11 12 13

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FittedCo

rre

latio

n

Spacing (X/D)

=3.97%

=2.97%

=2.32%

Tested

0 2 4 6 8 10 12 14 16 18 20 22

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

Co

rre

latio

n

Spacing (X/D)

Test results of aeroelastic model

Fitted correlation function(η=mid span)

Fitted correlation function(η=1/4 span)

Fig.5 The correlation function fitted by section model test result

Fig.6 The comparation between test results of aeroelastic model and fitted correlation function by section model

The correlation coefficients of simply supported aeroelastic model were shown in Fig. 6. The test results of aeroelastic model are closed to the correlation function fitted by section model when the span spacing is relatively small, and the difference increase with the spacing grows. Although the correlation function fitted by aeroelastic model is more reasonable than section model’s, it is difficult to conduct.

4. NONLINEAR VIV METHOD

Based on Scanlan’s semi-empirical nonlinear mathematical models, a method for interpreting the test results of section model into prototype bridge was proposed by the author (Sun 2014), and the uncompleted correlation of vortex-induced aerodynamic force along span was considered in the method. The amplitude in the form of generalized coordinates was defined as following form:

2/1

0

2

124

2

0)(

212

dxx

Ym

K L

r

(4)

The VIV amplitude for considering the partial correlation can be obtained from

Dxx )()(0y . The express of reduced factors can be written as Eq. 5 and Eq. 6, and

they can be calculated by self-convolution integration of mode shape.

L

xdxRxL

:

02

)(1

(5)

xdxRdxxxxL

l xL

L

)()()(2

1

0

2/

2/

22

22

(6)

xdxRdxxxxL

l xL

L

)()()(2

1

0

2/

2/

44

24

As the reduced factor is an amplitude dependent variable, the iterative process is needed. The VIV amplitude of prototype bridge under partial correlation can be estimated by following steps:

(1) Conduct section model wind tunnel tests, and obtain VIV amplitude and lock-in wind velocity;

(2) Identify VIV parameters 1

Y and by using experimental data, and /Y11

Y ,

; (3) Estimate VIV amplitude of the actual bridge by using Eq. 4 for full correlation

case (R = 1); (4) Calculate reduced factors with Eq. 5 and Eq. 6 by using the calculated

amplitude of the previous step; (5) Calculate the amplitude by using Eq. 4 with iterative method until convergence

or meaningful amplitude is obtained. The amplitude dependent reduced factors included in Eq. 4 should be calculated by Eq. 5 and Eq. 6 in every iterative step.

The VIV amplitude of prototype bridge under partial correlation can be obtained after several iteration. More details about this method can be seen in (Sun 2014).

5. APPLICATION OF FITTED CORRELATION FUNTION

To verify the validity of the fitted correlation function, the VIV response of section model was interpreted into an aeroelastic model by applying a nonlinear VIV method in which the spanwise correlation of vortex-induced force was taken into consideration, and compared with the test results of simply supported aeroelastic model.

The test results of section model and simply supported aeroelastic model were listed in Table 1 and Table 2. The test results of two models cannot be compared with each other directly, as their damping and mass are quite different. The VIV amplitude of section model could be converted to any damping and mass system by using the logarithmic equation between VIV amplitude and Scruton number (Eq. 7).

bLogScaLog A (7)

in which, A is VIV amplitude；Sc is Scruton number; 24 Dm Sc ; a and b can be

fitted by several tests. The converted results were listed in Table 3, the analytical results of other methods were listed as well.

Table 1 Test results of section model

Attack angle(°)

M (kg/m)

ζ

(%) Sc

A (%)

Log(A) Log(Sc)

0

11.12 0.53 49.973 3.971 -1.401 1.699

11.12 0.73 68.831 2.97 -1.527 1.838

11.12 1.2 113.147 2.32 -1.634 2.054

8.03 0.29 19.669 7.007 -1.154 1.294

3

11.12 0.53 49.973 6.459 -1.190 1.699

11.12 0.73 68.831 4.987 -1.302 1.838

11.12 1.2 113.147 3.633 -1.440 2.054

8.03 0.29 19.669 12.22 -0.913 1.294

Table 2 Test results of simply supported aeroelatic model

Attack angle(°)

m (kg/m)

ζ (%)

A (%)

Sc Log(Sc)

0 8.03 0.29 5.881 19.67 1.294

3 8.03 0.29 9.098 19.67 1.294

Table 3 The analytical results of different method

Method 0º 3º

Section model 7.007 12.22

4mA

(Zhu 2005) 8.922 15.564

2 33 mA (Zhang 2011) 8.091 14.115

Nonlinear VIV method 6.13 8.05

Aeroelastic model 5.88 9.098

It shows from Table 3 that if the section model test result is directly interpreted

into prototype bridge, very conservative results may be yield. By using the traditional interpreting methods which only consider the effects of mode shape without taking the spanwise correlation of vortex-induced force into account may yield similar conclusion. It’s obvious that a good consistency between section model and aeroelastic model is only found by considering the contribution of spanwise correlation of vortex-induced

force. And the fitted correlation function by section model tests can be adopted by the nonlinear VIV method to evaluate the amplitude of aeroelastic model.

5. CONCLUSIONS

If the section model test result is directly interpreted into prototype bridge, or only the effects of mode shape is considered without taking the spanwise correlation of vortex-induced force, conservative results may be yield. By considering the contribution of spanwise correlation of vortex-induced force, the test results of section model can be used to estimate the VIV performance of the prototype bridge.

It is more reasonable to get the correlation function by aeroelastic model, but it is difficult to conduct the test. For long-span bridge, it is acceptable to use the correlation function fitted from section model tests.

ACKNOWLEDGMENTS

Financial supported by "the National Natural Science Foundation of China"

(No.51408505)

REFERENCES Ehsan, F., Scanlan, R. H. (1990), “Vortex-Induced Vibrations of Flexible Bridges”.

Journal of Engineering Mechanics, 116: 1392-1410. Simiu, E., Scanlan, R. H., (1986). Wind effects on structures, (2nd Edn). John Wiley &

Sons, New York. Sun, Y. G., Li, M. S., Liao, H. L. (2014), “Nonlinear Approach of Vortex-induced

Vibration for Line-like Structures”. Journal of Wind Engineering & Industrial Aerodynamics, 124: 1–6.

Wilkinson, R. H. (1981), “Fluctuating pressures on an oscillating square prism. Part II. Spanwise correlation and loading”, Aero. Quarterly, 32(2): 111-125.

Zhang, Z. T., Chen, Z. Q., Cai, Y. Y. (2011), “Vortex-induced oscillations of bridges: linkages between sectional model tests and full bridge responses”. 13th International Conference on Wind Engineering, Netherlands, Amsterdam.

Zhu, L. D. (2005), “Mass simulation and amplitude conversion of bridge sectional model test for vortex excited resonance”. Engineering mechanics, 22(5): 204-208, in Chinese.