Advances in Steel Structures - part 55 pot

520
Y.Q. Ni et al.
vertical dynamic response of the undamped main span cable at the cable midspan when a vertical
harmonic excitation is exerted at the same position. The harmonic excitation is
F(t)
= F0-cos2nfi with
F0 = 1000 kN and f= 0.05 Hz. Both the responses with and without consideration of bending stiffness
are given. The difference of response amplitudes between the two sequences is not significant, while
the transient responses at a same instant may be distinct from each other due to the phase shift.
CONCLUDING REMARKS
This paper reports on the development of a finite element formulation for free and forced vibration
analysis of structural cables taking into account both sag extensibility and bending stiffness. The
predicted results by the proposed formulation agree favorably with the analytical results available in
the literature and with the measurement results of real bridge cables. The numerical simulations show
that the cable bending stiffness contributes a considerable effect on the natural frequencies when the
tension force is relatively small, and affects higher modes more significantly than lower modes. The
proposed method will be used to provide the training data required for developing a multi-layer neural
network for identifying the cable tension from measured multi-mode frequencies. By interchanging the
input and output roles in the training of the network, a functional mapping for the inverse relation can
be directly established using the neural network which then serves as a tension force identifier.
ACKNOWLEDGEMENTS
This study was supported by The Hong Kong Polytechnic University under grants G-YW29 and G-
V785. These supports are gratefully acknowledged.
References
Casas J.R. (1994). A Combined Method for Measuring Cable Forces: The Cable-Stayed Alamillo Bridge, Spain.
Structural Engineering International
4:4, 235-240.
Ko J.M. and Ni Y.Q. (1998). Tsing Ma Suspension Bridge: Ambient Vibration Survey Campaigns 1994-1996.
Preprint, The
Hong Kong Polytechnic University.
Kroneberger-Stanton K.J. and Hartsough B.R. (1992). A Monitor for Indirect Measurement of Cable Vibration

Frequency and Tension.
Transactions of the ASAE
35:1, 341-346.
Mehrabi A.B. and Tabatabai H. (1998). Unified Finite Difference Formulation for Free Vibration of Cables.
ASCE Journal of Structural Engineering
124:11, 1313-1322.
Ni Y.Q., Lou W.J. and Ko J.M. (1999a). Nonlinear Transient Dynamic Response of a Suspended Cable.
Submitted to
Journal of Sound and Vibration.
Ni Y.Q., Zheng G. and Ko J.M. (1999b). Nonlinear Steady-State Dynamic Response of Three-Dimensional
Cables.
Intermediate Progress Report No. DG1999-03C, The
Hong Kong Polytechnic University.
Okamura H. (1986). Measuring Submarine Optical Cable Tension from Cable Vibration.
Bulletin of JSME
29:248, 548-555.
Russell J.C. and Lardner T.J. (1998). Experimental Determination of Frequencies and Tension for Elastic
Cables.
ASCE Journal of Engineering Mechanics
124:10, 1067-1072.
Takahashi M., Tabata S., Hara H., Shimada T. and Ohashi Y. (1983). Tension Measurement by Microtremor-
Induced Vibration Method and Development of Tension Meter.
IHI Engineering Review
16:1, 1-6.
Yen W H.P., Mehrabi A.B. and Tabatabai H. (1997). Evaluation of Stay Cable Tension Using a Non-
Destructive Vibration Technique.
Building to Last Structures Congress." Proceedings of the 15th Structures
Congress,
ASCE, Vol. I, 503-507.
Zui H., Shinke T. and Namita Y. (1996). Practical Formulas for Estimation of Cable Tension by Vibration

Method.
ASCE Journal of Structural Engineering
122:6, 651-656.
STABILITY ANALYSIS OF CURVED CABLE-STAYED BRIDGES
Yang-Cheng Wang I , Hung-Shan Shu
!
and John Ermopoulos 2
l Department of Civil Engineering, Chinese Military Academy, Taiwan, ROC
P.O. Box 90602-6, Feng-Shan, 83000, Taiwan, ROC
2 Department of Civil Engineering, National Technical University of Athens
42 Patission Street, 10682 Athens, Greece
ABSTRACT
The objective of this study is to investigate the stability behaviour of curved cable-stayed bridges.
In recent days, cable-stayed bridges become more popular due to their pleasant aesthetic and their
long span length. When the span length increases, cable-stayed bridges become more flexible than
the conventional continuous bridges and therefore, their stability analysis is essential. In this study,
a curved cable-stayed bridge with a variety of geometric parameters including the radius of the
curved bridge deck is investigated. In order to study the stability effects of the curved cable-stayed
bridges, a three-dimensional finite element model is used in which the eigen-buckling analysis is
applied to find the minimum critical loads. The numerical results first indicate that as the radius of
the bridge deck increases the fundamental critical load decreases. Furthermore, as the radius of the
curved bridge deck becomes greater than 500m, the fundamental critical loads are not significantly
decreased and they are approaching to those of the bridge with straight bridge deck. The
comparison of the results between the curved bridges with various radiuses and that of a straight
bridge deck determines the curvature effects on stability analysis. In order to make the results useful,
they are non-dimensionalized and presented in graphical form, for various values of the parameters
that are interested in the problem.
KEYWORDS
Stability Analysis, Curved, Cable-Stayed Bridges, Bridges, Buckling
INTRODUCTION

Cable-stayed bridges have been known since the beginning of the 18th Century (Leonhardt, 1982
and Chang et al., 1981), but they have been widely used only in the last 50 years (O'Connor 1971,
Troitsky 1988). The span length of cable-stayed bridges increases (Ito 1998 and Wang 1999a) due
to the use of computer technology and the high strength material; some of them have curved decks
due to the pleasant aesthetic and the functional reasons (Menn 1998, and Ito 1998). These structures
521
522 Y C. Wang et al.
utilize their material well since all of their components are mainly axially loaded (Wang 1999b).
The geometric nonlinearity induced by the pylon, the deck and the cables' arrangement influences
the analysis results (Ermopoulos et al. 1992, Troitsky 1988, and Xanthakos 1994), especially for the
curved cable-stayed bridges. Generally, this influence is small, but if the pylons and the deck are
flexible, and cables' slope is small, then this influence becomes significant and stability analysis
may be necessary.
In this paper an elastic stability analysis of a cable-stayed bridge with two pylons and curved deck
is performed. The considered loads include a uniform load along the entire span and a concentrated
moving load. A nonlinear finite element program and the Jocobi eigen-solver technique are used to
determine the critical loads and their corresponding buckling mode shapes. The results are
presented in graphical form for a wide range of the parameters of the problem.
GEOMETRY AND LOADING
The geometry, the notation, and the loading of the curved cable-stayed bridges structural model are
presented in Figure 1. The bridge is symmetric and is composed of three major elements: (a) the
bridge deck with various radiuses ranging from 250m to infinity, i.e., straight roadway, (b) the two
pylons and (c) the cables. Two cases of the bridge span lengths are considered. In Case I (Figure 1)
the projective length of the bridge remains constant no matter what is the radius, and in Case II
(Figure 2) the total curved bridge length remains 460m; the bridge deck has a constant cross-section
along the whole span. It is supported at the ends of the both side spans by rollers while at the
intersection points with the pylons is attached with a pinned connection. The pylons are fixed at
their bases; they have a constant cross-section and their intersection with the deck lies on the one
third of their total height from the supports. The projective distance between the two pylons is
Ll=220m; the projective distance of the side spans is L 2 =120m each, for both cases. The height H

of the pylon above the deck varies between 0.165 x L and 0.542x L. These limiting values
correspond to the top cable's slope of 20 ~ and 50 ~ , respectively. The ratio Ip/I b (where Iv is the
moment of inertia of the pylon and I b is the moment of inertia of the bridge deck) varies between
0.25 and 4. In order to take the cables' arrangement into account in buckling analysis, the distance d
is introduced as shown in Figure 1. The ratio d/H varies between 0.2 (harp-system) and 0.95 (fan-
system). The cables are of constant cross-section, they support the deck every 20m and are attached
to the pylons by hinges.
Figure 1 (a) Side view and (b) Plan View of the Curved Cable-Stayed Bridge
Case I: with variable total curved length
Stability Analysis of Curved Cable-Stayed Bridges
523
Figure (2) Plane View of the Curved Cable-Stayed Bridge
Case II" with constant total curved length
Element
Deck
Pylon
Cable
TABLE 1
ELEMENT'S PROPERTIES
Area (m 2 )
0.300
0.100
Moment of Inertia (m
4 )
0.200
0.050
0.300 0.200
0.500
0.005
0.800

Table 1 shows the area and the moment of inertia used in different elements. The Young's modulus
(E) is taken to be 21 x 10
6
t/m 2 for the deck and the pylons, and 17 x 10
6
t/m 2 for the cables. The
applied loading is consisted of a uniformly distributed load (q) along the deck, and a moving
concentrated load (P) at a distance (e) from the left deck's support. Two values of the q/P ratio are
considered, i.e. 0 and 0.07(m -! ). During the critical load search this ratio remains constant for a
given set of geometric parameters. The total number of finite elements used in the whole structure
was 96.
FINITE ELEMENT MODEL AND IDEALIZATION
Numerical methods such as finite difference and finite element methods are powerful tools in recent
days (Bathes 1982). In this study finite element method is used.
Finite Element Model
Two different types of three-dimensional element such as beam and spar have modeled the curved
cable-stayed bridge. Forty-six beam elements model the bridge deck; fifteen beam elements model
each pylon; and twenty spar elements which can only resist tension forces, model the stayed cables.
Each beam element consists of six degrees of freedom, i.e. translation in x-, y- and z-direction and
rotation about x-, y- and z-axis. Each spar element consists of three degrees of freedom, i.e.
translation in the three directions.
Boundary condition is one of the most important factors in buckling analysis. The bases of pylons
are considered as fixed; the end of side span is simply supported; and the connection between the
pylon and the bridge deck is coupled in both vertical and lateral directions.
524 Y C. Wang et al.
Idealization
An exact formulation and finite element analysis were made within the limitations of the following
assumptions:
1 .Members are initially straight and piecewise prismatic.

2.The material behavior is linearly elastic and the moduli of elasticity E in tension and compression
are equal.
3.Statically concentrated and uniformly distributed loads only apply on the structure. The loading is
proportional to each other thus the load state increases in a manner such that the ratios of the
forces to one another remain constant.
4.No local buckling is considered.
5.The effect of residual stress is assumed negligibly small.
NUMERICAL RESULTS AND DISCUSSION
Based on the finite element model and the eigen-buckling analysis procedure (Ermopoulos et al.
1992, Vlahinos et al. 1993), the critical loads for various sets of geometric parameters are calculated.
The fundamental critical load and its corresponding mode shape are found. In all cases the anti-
symmetric modes' critical load was the lowest while the second mode is always symmetric. Figure
3 shows the undeformed and the first three buckling mode shapes for a set of geometric parameters
(for radius R=300m, Ip / I b = 4, H/L=0.262, d/H=0.6 and the deck's dead load only).
Figure 3. Buckling Mode Shapes of the Curved Cable-Stayed Bridges
Figure 4 shows several curves of critical loads Pcr versus the distance e from the left deck support
(load eccentricity), for H/L=0.262, d/H=0.20 (harp-type) and d/H=0.95 (fan-type) represented in (a)
and (b), respectively with the uniform load q=0. The solid lines correspond to Ip /I b = 4 and the
dashed lines correspond to I p /I b = 1.
Stability Analysis of Curved Cable-Stayed Bridges
525
Figure 4 indicates that the ratio of Ip/I
b
is one of the most important factors for the minimum
critical loads of this type of structure. The harp-type bridge (d/H=0.2) represented in (a) has the
ratio of H/L=0.262 and the fan-type bridge (d/H=0.95) represented in (b) has H/L=0.126. Figure
4(a) shows that the minimum critical loads occurs around the mid-span and are almost the same for
the curved-deck bridges with Ip/I b = 1.0. When the ratio becomes Ip/Ib= 4, the fundamental
critical loads increase for the curved-deck bridge with radius less than 500m. Figure 4(b) first
shows that fan-type bridge has lower fundamental critical loads than harp-type, and if the radius

decreases the fundamental critical load decreases for all ratios of Ip/I b . Based on Figure 4, the
ratio of Ip/I b , the cable arrangement, and the radius of the curved bridge deck play the most
important role for buckling analysis of this type of structures.
Figure 4 Minimum Critical Loads versus Eccentricity of the Concentrated Load for Various
Radiuses
Figure 5 represents the fundamental critical load for the same bridge but subjected only to its dead
load. It becomes obviously that the fundamental critical loads are almost the same when the radius
is greater than 500 m for the harp-type bridge (d/H=0.2) represented in (a). For fan-type bridge
(d/H=0.95) represented in Figure (b), if the radius decreases, the fundamental critical loads
decrease.
Figure 6 represents several curves of fundamental critical loads. It is for dead and the moving
concentrated load with ratio q/P=0.07 applied at the midpoint of the middle span versus the Ip /I b
ratio. Two cases are shown in Figure 6; the bridge with H/L = 0.262 is represented in (a), and with
H/L=0.165 is represented in (b) for various values of radius. It can be seem that the minimum
critical load of the straight-deck bridges decreases when the ratio of Ip /I b increases, which means
that the flexural interaction between the pylons and the bridge deck decrease. If the radius of the
curved bridge deck is less than 300m, the minimum critical load increases when the ratio of Ip /I b
increases, which is different from those bridges having large radius.
526
Y C. Wang et al.
Figure 5 Minimum Critical Loads versus Eccentricity of Concentrated and dead Loads for Radiuses
Figure 6 Fundamental Critical Loads versus Ratio Ip /I
b
for Various Radiuses
Figure 7 shows the radius effects on minimum critical load for various ratios of d/H and Ip /I b .
Coupling parameters of d/H and radius of curved-deck, the minimum critical loads of curved-deck
bridges are significantly different from those of straight-deck bridges.
Figure 7(a) shows that a curved-deck bridge with H/L=0.262 having the ratios of d/H=0.4 and
Ip/Ib=4.0 has the optimum critical load when the radius is less than 500 m. If the radius is.
greater than 500 m, the bridge with d/H=0.2 gives the optimum critical load. For H/L=0.165

(Figure 7b), there are different sets of geometric parameters and different radius for this optimum.
Stability Analysis of Curved Cable-Stayed Bridges
527
Figure 7 Fundamental Critical Loads versus the Ratio of d/H for Radiuses
Regarding Case II (as represented in Figure 2), the stability behavior of both cases is similar but the
minimum critical loads are greater than those of Case I. Figure 8 shows four curves to compare
the minimum critical loads for the optimum design parameters represented in Figure 7(a).
Figure 8 Comparison of the Fundamental Critical Load of Case I and II
CONCCLUDING REMARKS
In a common sense if a bridge's span length increases, the bridge becomes more flexible and then
the critical load decreases but this study shows that this kind of sense is not suitable to apply to the.
curved cable-stayed bridges. Due to the axial components of cable reactions (Wang 1999), the
curved bridge deck has less axial forces acting on the bridge deck than the straight bridge deck has.
For the geometric parameters considered, the minimum critical load significantly increases when
the radius of the curved-deck bridges less than 300 m. On the other hand, if the radius is greater
528
Y C. Wang et al.
than 500 m, he characteristics of the minimum critical loads are similar to those of straight-deck
bridge even though the curved-deck bridges have higher minimum critical loads.
Reference
F. Leonhardt (1982), Briiken/Bridges. Architectural Press, London.
Fu-Kuei Chang and E. Cohen (1981), Long-span bridges: state of art, Journal of structural Division,
ASCE.
C. O'Connor (1971), Design of Bridge Superstructures, John Wiley, New York.
M.S. Troitsky (1988), Cable-Stayed Bridges: An Approach to Modem Bridge Design, 2 nd Edition,
Van Nostrand Reinhold, New York.
Manabu Ito (1999), The Cable-Stayed Meiko Grand Bridges, Nagoya, Structural Engineering
Intemational (SEI), IABSE, Vol. 8, No.3, pp.168-171.
Christian Menn (1999), Functional Shaping of Piers and Pylons, Structural Engineering
International, IABSE, Vol.8, No.4, pp.249-251.

Manabu Ito (1999), Wind Effects Improve Tower Shape, Structural Engineering International,
IABSE, Vol.8, No.4, pp.256-257.
Yang-Cheng Wang (1999a), Kao-Pin Hsi Cable-Stayed Bridge, Taiwan, China, Structural
Engineering International, Journal of IABSE, Vol.9, No.2, pp.94-95.
Yang-Cheng Wang (1999b), Number Effects of Cable-Stayed-Bridges on Buckling Analysis,
Journal of Bridge Engineering, ASCE, Vol.4, No.4.
Yang-Cheng Wang (1999c), Effects of Cable Stiffness on a Cable-Stayed Bridge, Structural
Engineering and Mechanics, Vol.8, No.l, pp.27-38.
John CH. Ermopoulos, Andreas S. Vlahinos and Yang-Cheng Wang (1992), Stability Analysis of
Cable-Stayed Bridges, Computers and Structures, Vol.44, No.5, pp. 1083-1089.
Andreas S. Vlahinos, John CH. Ermopoulos and Yang-Cheng Wang (1993), Buckling Analysis of
Steel Arch Bridges, Journal of Constructional Steel Research, Vol. 33, No.2, pp.100-108.
Klaus-Jurgen Bathe (1982), Finite Element Procedures in Engineering Analysis, Prentice-Hall, Inc.
Petros P. Xanthakos (1994), Theory and Design of Bridges, John Wiley & Sons, Inc. New York,
USA.
Anthony N. Kounadis (1989), AYNAMIKH Tf2N ZYNEXf2N EAAZTIK~N ZYZTHMAT~N,
EKDOZEIZ ZYMEQN, (in Greek).
EXPERT SYSTEM OF FLEXIBLE
PARAMETRIC STUDY ON CABLE-STAYED
BRIDGES WITH MACHINE LEARNING
Bi Zhou ~ and Masaaki Hoshino 2
1, 2
Dept. of Transportation Engineering
College of Science and Technology, Nihon University
(24-1, Narashinodai 7, Funabashi, Chiba 274-8501, Japan)
ABSTRACT
The development of practical expert systems is mostly concentrated on how to acquire experiential
knowledge from domain experts successfully. However, frequently, the acquiring progress is difficult
and the representation is incomplete. Furthermore, the experiential knowledge may be entirely lacking
when the design situation changes or technology comes new. The present study is to develop a cable-

stayed bridge expert system of how knowledge in the cable-stayed bridges may be generated from
hypothetical designs with machine learning for the parametric study processed as flexible as possible.
KEYWORDS
cable-stayed bridge, structural design, multiple regression analysis, expert system, machine learning,
object-oriented method
INTRODUCTION
Modern structures such as cable-stayed bridges involve a relatively new knowledge that may be
entirely lacking when the design situation changes or technology comes new. Formalised knowledge
and knowledge evolving procedures are difficult to acquire, store and represent. In view of expert
systems, the knowledge obtained from experts or documentary materials (such as guidelines, books or
papers) usually only contains general explanations about possible configurations with few
recommendations which play a conceptual control or value-restricted role in selecting candidate
designs.
By introducing the concepts of
static knowledge
and
dynamic knowledge,
this paper presents an
exploration for the expert systems of how to generate the domain knowledge from hypothetical designs
with the change of the design situation and apply it to the knowledge evolution with the ability of
learning. The
candidate related knowledge (CRK),
that is regarded as having influence on the design
situation, is used to supplement the relative knowledge constantly and is concentrated on hypothetical
529