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 . 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. Structural Engineering 122:6, 65 1-6 56. 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. (1999), Wind Effects Improve Tower Shape, Structural Engineering International, IABSE, Vol.8, No.4, pp.25 6-2 57. Yang-Cheng Wang (1999a), Kao-Pin Hsi Cable-Stayed Bridge, Taiwan, China, Structural