Dynamic Stress Estimation Method for Bridge Elements
Utilizing Linear Regression Algorithm
Y.H. Shao1, 2, L. Zhao1, Y.J. Ge1, M.L. Levitan2
1
State Key Laboratory for Disaster Reduction in Civil Engineering, Department of Bridge
Engineering, Tongji University, Shanghai, China, yahuishao@hotmail.com
2
LSU Hurricane Center, Louisiana State University, Baton Rouge, USA, levitan@hurricane.lsu.edu
ABSTRACT
A linear-regression-based framework for estimating bridge elements’ dynamic stress under both
normal and skew wind excitations is developed in this paper, utilizing both the wind tunnel test
results and flutter-buffeting numerical simulation results in time domain. Wind tunnel test results
come from whole bridge aeroelastic wind tunnel test in TJ-3 Boundary Layer wind tunnel and
sectional model wind tunnel tests for wind-induced vibration carried out in the TJ-1 Boundary
Layer Wind Tunnel; numerical simulation utilize the linear quasi-steady theory and the strip
theory of aerodynamics. The proposed method is finally employed to solve efficiently the hanger
dynamic stress time history prediction problems of Jiangdong long-span cable-supported bridges
under normal and skew winds, not only avoiding complicated calculation in numerical
simulation for skew winds excitation but also making up the shortcoming of wind tunnel test for
not being able to get structural inner forces. Furthermore, approach in this paper hoped to deal
with other elements’ dynamic stress estimation and problems in such bridges.
INTRODUCTION
With the rapid development of economy in China, a great number of modern long-span cablesupported bridges have been built or under construction, wind tunnel test of full bridge aeroelastic model will still be a most important and indispensable approach for evaluating the windresistant performance of long-span bridges (Zhu et al., 2007).
In whole bridge aero-elastic wind tunnel tests, displacements of the critical points on the
bridge (especially the points on the bridge deck and pylon) are easily gathered through laser
displacement sensors or accelerometers. However, dynamic stress and strain acquisition of
bridge elements such as cables and hangers is a difficult problem in bridge aero-elastic wind
tunnel tests. Pressure transducers must be attached to the object by a suitable adhesive in order
to measure the strain, but the transducers and associated power/data cables may have certain
influences on the surrounding wind field. The transducers and their cables will also alter the
mode shape and vibration frequency of the original model.
In pure finite element and numerical wind tunnel methods, it is easy to determine the
entire dynamic stress and strain; however, their accuracy and reliability are questionable and they
can only be considered as supplements to wind tunnel tests at present (Gu, 2005). Furthermore,
in the condition of skew wind excitation, six-component aerodynamic derivatives and
complicated coordinate transformations are absolutely necessary, and a new finite-element-based
framework is required (Zhu et al., 2002；Zhu et al., 2005).
Cable and hangers are the elements that bearing reiterate loads and repeated stresses
failure plays an important role in its failure mode. With regards to suspension bridge, dynamic
stress time history of hangers is of extreme importance, as is widely known, in its fatigue
performance evaluation, strength inspections, fatigue life prediction, etc. How to make up the
shortcomings of wind tunnel test in measuring dynamic stress time history of bridge element,
especially when under skew winds? The emphasis of this paper is on how to derive the cable and
hanger dynamic stress through wind tunnel tests indirectly, under both normal and skew wind
excitations based on bridge buffeting, correlation theory and linear regression method.
LINEAR REGRESSION THEORY
Linear regression is a form of regression analysis in which the relationship between one or more
independent variables and another variable, called the dependent variable, is modeled by a least
squares function, called a linear regression equation. This function is a linear combination of one
or more model parameters, called regression coefficients. A linear regression equation with one
independent variable represents a straight line when the predicted value (i.e. the dependant
variable from the regression equation) is plotted against the independent variable: this is called a
simple linear regression (Wikipedia).
T
A linear regression model assumes, given a random sample Yi , X i1 ,..., X ip  , i  1, 2,..., n ,
a possibly imperfect relationship between Yi , the regressand, and regressors X i1 , X i 2,... , X ip , a
T
disturbance term     1 ,  2 ,...,  n  which is a random variable too, is added to this assumed
relationship to capture the influence of everything else on Yi other than X i1 , X i 2,... , X ip . Hence, the
multiple linear regression models take the following form in equation 1, 2 and 3:
Yi   0  1 X i1   2 X i 2  ...   p X ip   i
i  1, 2,..., n
(1)
Y    X      
(2)
 Y1  1 x11
  1 x
21
 Y2   
    
  
 Yn  1 xn1
(3)
 x1 p   1   1 

 x2 p    2    2 
  
       
   
 xnp    n    n 
Regressors are also called independent variables; similarly, regressands are also called
dependent variables. Models which do not conform to this specification may be treated by
nonlinear regression.
The first objective of regression analysis is to best-fit the data by estimating the
parameters of the model. Of the different criteria that can be used to define what constitutes a
best fit, the least squares criterion is a very powerful one. This estimation is given by equation 4,
ˆ   X T X  X T Y
1
(4)
For the univariate linear case, we consider here the case of the simplest regression model
in equation 5,
y  ax  b  e
(5)
In order to estimate a and b ，we have a sample  yi , xi  ， i  1, 2,..., n of observations
which are, here, not seen as random variables and denoted by lower case letters. According to the
idea of least square estimation, the estimated value of a and b are as follows in equation 6,
(Wikipedia),
n
a
 x y  nxy
i
i
y
2
i 1
n
i 1
i
 ny
2
;
b  x  ay
(6)
In which x and y means the mean value of the samples of  x1 , x2 ,..., xn 
and  y1 , y2 ,..., yn  , respectively.
All the above is the basic theory. To illustrate the various goals of regression, procedures
for using univariate linear regression method should be illustrated as follows:
a) Data collection and the determination of regressand, and regressors.
b) Make scattering points diagram between regressand, and regressors, that is, plotting the
regressand as a function of regressors. Obtaining its linear trend. The points should lie along
a straight line.
c) Do mathematical statistics analysis and calculate correlation coefficients in order to
assessing the goodness of linearity.
d) Solve the linear regression equation to get regression coefficients a and b .
e) Analysis of the coefficient of determination. It gives what fraction of the observed variance
of the response variable can be explained by the given variables. In linear regression, the
coefficient of determination is simply the square of the sample correlation coefficient
between the outcomes and their predicted values.
f) Examine the observational and prediction confidence intervals. In most contexts, the smaller
they are the better.
g) Problem solved.
BRIEF INTRODUCTION TO PROJECT
Jiangdong Bridge is a two-span suspension bridge with two towers supporting center span of 260
meters and two identical side spans of 83 meters each. The bridge towers are made of reinforced
concrete, and the heights of the left and the right side tower above pylon base are 97.765 meters
and 99.915 meters, respectively. The prestressed concrete bridge deck has a twin-box cross
section in a width of 47 meters, in a height of 3.5 meters and in a central slotting width of 6
meters. It is supported by two specially inclined hanger planes emanating from deck anchorages
to main cable, each plane comprising 26 cables. The horizontal distance between two adjacent
cables is 9 meters (Zhao et al., 2008).
Figure 1: General configuration of the whole bridge
(units: m)
Figure 2: Whole bridge aero-elastic model
in the wind tunnel TJ-3
Whole bridge aero-elastic wind tunnel tests for the long-span, cable-supported Jiangdong
Bridge were carried out in the TJ-3 Boundary Layer Wind Tunnel of the State Key Laboratory
for Disaster Reduction in Civil Engineering of Tongji University (see Figures 1 and 2 for the
general test configuration and a picture of the model in the wind tunnel).
The main static and dynamic parameters (see Table 1); such as flutter derivatives, threecomponent coefficients has been determined by wind tunnel tests of a scaled section model of
the bridge deck.
Table 1: Static and dynamic parameters for Jiangdong Bridge
Vibration mode
Dynamic characteristics
In service stage
Vertical bending Lateral bending
Frequency(Hz)
0.495
0.858
Damping ratio
4.7‰~5.0‰
Torsional
moment
1.037
0.013
-0.663
0.601
Torsional moment
Lift
Static coefficients under
0°attack angle excitation
Mean value
Slope
Flutter derivatives under design wind
speed and 0°attack angle excitation
Torsion
1.320
Drag
-0.088
1.457
Lift
H1*
H4*
-0.571
-0.116
A2*
A3*
-0.018
0.017
FLUTTER AND BUFFETING
Utilizing APDL language programming techniques, a finite-element-based framework for flutterbuffeting analysis of long-span cable-supported bridges is developed in the time domain utilizing
the linear quasi-steady theory and the strip theory of aerodynamics and considering admittance
function modification. Traditional Liepmann’s approximation to Sears function derived from
thin plate and streamline aerofoil, or so-called maximal value 1.0 approximated as the possible
upper limitation are usually adopted in realistic engineering application (Zhao, 2008).
Simiu and Panofsky functions are selected as horizontal and vertical wind spectrum
function, which can be referred to Chinese Codes (Communication Ministry of PRC 1989, 2004).
Spatial correlation coefficient is defined as 10.0. Harmonic wave combination technique was
used in wind filed simulation (Deodatis, 1996).
Table 2: Wind excited vibration displacement mean square deviation
under design wind speed for Jiangdong Bridge
Climate mode
Normal
climate
Typhoon
climate
Analysis
methods
Aerodynamic
admittance
Wind tunnel test result
Numerical
result
Wind tunnel test result
Numerical
result
1.0
Sears
1.0
Sears
Mid-main span of
deck
Vert.disp (mm)
21.40
87.20
26.43
33.08
135.59
40.09
1/4-main span of
deck
Vert.disp (mm)
14.80
45.97
14.15
23.24
71.86
21.17
Root variance of displacem ent: m m
A user-defined element in ANSYS, namely Matrix27 (SASI, 2004), is adapted to model
aeroelastic forces acting on the deck of long-span bridges. The aeroelastic stiffness and damping
matrices in Matrix27 elements are derived and expressed in terms of the flutter derivatives.
Table 2 and Figure 3 show both the wind tunnel tests and the numerical simulation results
(mean square derivation of displacements) for the girder’s vertical bending in the test cases of 0
attack angle and 0 yaw angles. Wind tunnel results come from whole bridge aeroelastic wind
tunnel test (Zhao et al., 2008). It is discovered that the numerical results coincide with the wind
tunnel test results when the aerodynamic admittance equals to Sears function.
-250 -200 -150 -100
100
-50
0
50
100
90
150
200
1
Sears
0 attack angle
in wind tunnel
80
70
250
100
90
80
70
60
60
50
50
40
40
30
30
20
20
10
10
0
0
-10
-250 -200 -150 -100
-50
0
50
100
150
200
-10
250
Bridge layout: m
Figure 3: Wind tunnel and numerical results for displacements of bridge deck
at different span-wise locations
CORRELATION AND LINEAR REGRESSION
In probability theory and statistics, correlation (often measured as a correlation coefficient)
indicates the strength and direction of a linear relationship between two random variables.
Motivation for correlation comes from inspecting the method of simple linear regression. The
correlation coefficient  X ,Y between two random variables X and Y with expected values  X and
Y and standard deviations  X and  Y is defined as (in equation 7):
 X ,Y 
Cov  X , Y 
 XY

E   X   X Y  Y  
 XY
(7
)
It describes the dependencies between two variables. With regard to interpret the size of a
correlation, several authors have offered guidelines for the interpretation of a correlation
coefficient. Cohen (1988) has observed, however, that all such criteria are in some ways arbitrary
and should not be observed too strictly. This is because the interpretation of a correlation
coefficient depends on the context and purposes. A correlation of 0.9 may be very low if one is
verifying a physical law using high-quality instruments, but may be regarded as very high in the
social sciences where there may be a greater contribution from complicating factors (see Table
3).
Table 3: Interpretation of the size of a correlation
Correlation coefficients
Small
Medium
Large
Negative
-0.3 to -0.1
-0.5 to-0.3
-1.0 to -0.5
Positive
0.1 to 0.3
0.3 to 0.5
0.5 to 1.0
In both whole bridge aeroelastic wind tunnel test and numerical simulation, it is observed
that the first order vertical bending vibration is the main vibration modes for Jiangdong bridge
girder under design wind speed excitation; its vibration mode keeps simple, without complicated
modes coupled. The above characteristics make the suspension hangers keeping synchronized
with the bridge girder in vertical bending more easily. In order to check the hypothesis, the
author relooked into the numerical simulation of Jiangdong Bridge flutter-buffeting results, and
calculated the correlation coefficient between axial force time history of every hanger and the
vertical displacements time histories of mid-span of girder (See Figure 3 and Table 4).
Table 4: Correlations between hanger axial force and mid-span of girder’s vertical displacement
Hanger No. Correlation coefficients Size of correlation
Hanger1
0.9128
Large
Hanger2
0.9765
Large
Hanger3
0.9846
Large
Hanger4
0.9857
Large
Hanger5
0.9862
Large
Hanger6
0.9862
Large
Hanger7
0.9859
Large
Hanger8
0.9874
Large
Hanger9
0.9877
Large
Hanger10
0.9873
Large
Hanger11
0.9868
Large
Hanger12
0.9866
Large
Hanger13
0.9867
Large
It was discovered that the mid-span vertical displacement time history kept a high
correlation with the hanger axial forces, meaning these two variables maintain a nearly linear
relationship (Figure 4). Based on this relationship, it is not difficult to draw a conclusion that
linear regression theory can be used to establish a linear regression equation between mid-span
vertical displacement time history and hanger axial force time history. Here go back the linear
regression theory, assume a linear regression model exists, given a sample in equation 8, a
possibly imperfect relationship between S girder , the regressand, and regressor F hanger . A
disturbance term  which is a random variable too, is added to this assumed relationship to
capture the influence of everything else on S girder other than F hanger . Hence, the linear regression
model takes the following form in equation 9,
F
hanger
, S girder 
T
(8
)
F hanger  aS girder  b  
(9
)
Correlation coefficient
-150 -120 -90
-60
-30
0
30
60
90
120
150
0.990
0.990
0.975
0.975
0.960
0.960
0.945
0.945
0.930
0.930
0.915
0.915
0.900
0.900
0.885
0.885
0.870
-210-180-150-120 -90 -60 -30 0
0.870
30 60 90 120 150 180 210
Bridge layout: m
Figure 4: Correlation coefficient of mid-span vertical displacement and hanger axial force
According to the idea of least square estimation, the estimated value of a and b are as
follows,
n
a
xF
i 1
i
n
hanger
F
i 1
i
 nS girder F hanger
hanger 2
i
 nF
hanger
2
;
b  S girder  aF hanger
(10
)
Here we get the complete regression equation for one sample. There are totally 26 pairs
of hangers in Jiangdong Bridge, so we can finally get 26 different such functions. In order to
make their relationship clear, both the regressand and the regressors are non-dimensional zed.
That is let F hanger / F maincable substitute F hanger and S girder substitute S girder / Lspan . Since F maincable and
Lspan are constants, the substitution could not alter the original correlation relationship in the
sample. And, with the help of the final regression equation and wind tunnel test results of midspan vertical displacement time history, we finally got the elements’ stress in wind tunnel tests.
Figure 5 shows an example regression analysis for hanger 13, and Figure 6 shows the
corresponding stress time history.
Hence, bridge hanger dynamic stress estimation method on the basis of linear regression
theory is built up. Flow chart of the whole research approach shows in Figure 7. It is a half test
and half theory method. Cable supported bridges comprise a lot of hanger and cables, and it is
difficult to measure the displacements for hangers in whole bridge wind tunnel test, not
mentioning to get the stress of hangers during the test. Research in this paper helps to solve this
problem.
Ratio of hanger axial force
to main cable horizontal force
(N / N)
0.070
0.068
0.066
0.064
Y =0.06597-0.01126 X
0.062
-0.30 -0.25 -0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25 0.30
Ratio of mid-span vertical displacement to span (10-3 )
Figure 5: Linear regression equation for hanger 13
3.7x108
3.7x108
Stress (Pa)
3.7x108
3.7x108
3.7x108
3.7x108
3.7x108
3.7x108
3.7x108
0
50
100 150 200 250 300 350 400 450 500 550 600
Time (s)
Figure 6: Predicated stress time history for hanger 13 in service state
Figure 7: Research flow chart in this paper
APPLICATION IN SKEW WIND EXCITATION
A new finite element based method for buffeting analysis of long-span cable-supported bridges
under skew winds has been developed in the frequency domain utilizing the quasi-steady linear
theory and the oblique strip theory in conjunction with the pseudo-excitation method (Zhu,
2004). In this method, the results is consistent with those of wind tunnel tests in general, and
show that the variations of buffeting responses are not monotonous with wind yaw angle and
inclination, and the normal wind case may not correspond with the worst case scenario. In this
method, a set of universal expressions for six components of buffeting forces and coordinates
transformation are needed (Kimura 1999, Xie 1991, Kimura 1992, Kimura 1994, Scanlan 1993,
Zhu 2003, 2005).
The development of the proposed framework dictates that it would solve the problem of
hanger stress measurements under skew wind excitations. Take again hanger number 13 as
example, by using the regression equation and take into consideration the whole bridge wind
tunnel test results in the yaw angles of 0°, 22.5°, 45°, 67.5°, 90°for girder, dynamic stress for
hanger number 13 can thus be easily deduced without skew winds numerical simulation (See
table 5).
Table 5: Stress for hanger number 13 in various yaw angles
Yaw angle
(°)
0
22.5
45
67.5
90
Mean value
(×108Pa)
3.70705
3.70705
3.70704
3.70704
3.70704
Mean Square Derivation
(Pa)
4160.29
4256.90
3032.16
2286.90
1653.23
4500
Mean square deviation (Pa)
4000
°
3500
3000
2500
2000
1500
0.0
22.5
45.0
67.5
90.0
Yaw angle ( °)
Figure 8: Mean square derivation of stress for hanger 13
CONCLUDING REMARKS
(1) A linear regression method for predicting dynamic stress of bridge elements was proposed in
this paper. This method is the combination of whole bridge wind tunnel test and flutterbuffeting numerical simulation. It helps dealing with the problem of inner force
measurement of bridge elements.
(2) The developed linear regression method is a half test half theory method, and it is not only
the extension of wind tunnel but also the supplementation to numerical simulation. The
method was applied to Jiangdong Bridge to investigate the hanger dynamic stress under both
skew and non-skew winds excitation. The results are consistent with those of related
reference in general, and show that the hanger stress variations are not monotonous with
wind yaw angle and inclination, and the normal wind case may not correspond with the
worst case scenario.
(3) Furthermore, the proposed framework could be used to evaluate dynamic stress of other
bridge elements when proper correlation exists. More field-measured results are needed for
the verification of the proposed method in this paper.
ACKNOWLEDGEMENTS
The authors would like to gratefully acknowledge the supports of the Natural Science
Foundation of China under the grants 90715039 and 50408035. The writers also want to thank
Teng Wu, Yufen Zhou and Zhenghua Wang for their hardworking in the wind tunnel tests.
REFERENCES
[1] L.D. Zhu, M. Wang, D.L. Wang, Z.S. Guo, F.C. Cao, Flutter and buffeting performances of Third
Nanjing Bridge over Yangtze River under yaw wind via aeroelastic model test, J. Wind Eng. Ind.
Aerodyn. 95 (2007) 1579–1606.
[2] M. Gu, Civil structure wind engineering research progress and fundamental problems, Strategy
progress report of National Science Foundation of China, Shanghai, 2005.
[3] L.D. Zhu, Buffeting Response of Long Span Cable-supported Bridges under Skew winds: Field
Measurement and Analysis. Hong Kong: The Hong Kong Polytechnic University; 2002.
[4] L.D. Zhu, Y.L. Xu, H.F. Xiang, Tsing Ma Bridge deck under skew winds-Part I: aerodynamic
coefficients. J. Wind Eng. Ind. Aerodyn. 90(7) (2002) 781-805.
[5] L.D. Zhu, Y.L. Xu. Tsing Ma Bridge deck under skew winds--part II: flutter derivatives. J. Wind Eng.
Ind. Aerodyn. 90(7) (2002) 807-837.
[6] http://en.wikipedia.org/wiki/Linear_regression#Theoretical_model.
[7] http://en.wikipedia.org/wiki/Correlation.
[8] L. Zhao, Y.J. Ge, L.D. Zhu, Wind-excited Vibration of Long-span Steel-lattice Arch Bridge under
Typhoon Climate, The 4th International Conference on Advances in Wind and Structures. 2008. p.
1448-1462.
[9] Communication Ministry of PRC, Wind Resistant Design Code for Highway Bridges (JTG/T D60-012004), People’s Communication Press, 2004.
[10] G. Deodatis, Simulation of ergodic multivariate stochastic processes. J. Engrg. Mech. ASCE. 122(8)
(1996) 778-787.
[11] L. Zhao, Y.J. Ge, Wind Tunnel Study on Wind-Resistant Performance of Jiangdong Bridge in
Hangzhou, China-Part1,2,3. State Key Laboratory for Disaster Reduction in Civil Engineering,
Tongji University. 2008.
[12] J. Cohen, Statistical power analysis for the behavioral sciences (2nd edition.), 1988.
[13] L. Zhao, Y.J. Ge, Stochastic parameter sensitivity analysis of typhoon wind field, J. Tongji University
(Natural Science). 33(6) (2005) 727-311.
[14] Swanson Analysis Systems Inc. (SASI), ANSYS User’s Manual, Version 8.0, Houston, Pennsylvania.
2004.
[15] K. Kimura, T. Ohara, Lateral sway buffeting of bridge decks due to yawed wind. In: Proc of the
Tenth International Conference on Wind Engineering: Wind Engineering into 21st Century; 1999;
Copenhagen, Denmark; 1999.
[16] J. Xie, H. Tanaka, Buffeting analysis of long span bridges to turbulent wind with yaw angle. J. Wind
Eng. Ind. Aerodyn. 37(1) (1995) 65-77.
[17] K. Kimura, H. Tanaka, Bridge buffeting due to wind with yaw angles. J. Wind Eng. Ind. Aerodyn.
41~44(1992) 1390~1320.
[18] K. Kimura, S. Nakamura, H. Tanaka, Buffeting analysis for cable-stayed bridges during construction
in yawed wind. In: Proceedings of Symposium on Cable-stayed and Suspension Bridges; 1994;
Deauville, France; 1994. p. 109-116.
[19] R.H. Scanlan, Bridge buffeting by skew winds in erection stages. J .Engineering Mechanics, ASCE.
119(2) (1993) 251-269.
[20] L.D. Zhu, Y.L. Xu, Buffeting response of long span cable-supported bridges under skew winds-part I:
theory. J. Sound and Vibration. 281(3~5) (2005) 674-673.
[21] Y.L. Xu, L.D. Zhu, Buffeting response of long suspension bridges to skew winds. J. Wind &
Structures. 6(3) (2003)179-196.