3D NONLINEAR AEROSTATIC STABILITY ANALYSIS FOR SUSPENSION BRIDGES WITH ULTRA-LONG SPAN

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1 The Eighth Asia-Pacific Conference on Wind Engineering, December 1 14, 13, Chennai, India 3D NONLINEAR AEROSTATIC STABILITY ANALYSIS FOR SUSPENSION BRIDGES WITH ULTRA-LONG SPAN Yaojun Ge 1, Yahui Shao 1 Professor of Civil Engineering, SLDRCE, Tongji University, Shanghai, China, yaojunge@tongji.edu.cn Lecturer of Civil Engineering, Hefei University of Technology, Hefei, Anhui, China, yahuishao@16.com ABSTRACT With the rapid increase of span length, suspension bridges are becoming lighter, more flexible, and lower damping, which result in more and more sensitive to wind actions, in particular related to aerostatic and aerodynamic instability. The intrinsic limit of span length due to aerodynamic flutter stability seems to be about 1,5m for a traditional suspension bridge, but both twin and triple box stiffening girders could provide a 5,m suspension bridge with high enough critical flutter speed. Three-dimensional nonlinear aerostatic stability analysis has been performed on the suspension bridges with single and multi-box girders having ultralong spans from 1,5m to 5,m. It can be concluded that the single and the multi-box girders can support a 1,5m and 5,m spanned suspension bridge with high enough aerostatic critical speed, which can meet with the requirement of strong wind. Keywords: Suspension bridge, Ultra-long span, Aerostatic stability, Nonlinear analysis, Multi-box girder Introduction Bridges have been generally built in four types, including girder bridge, arch bridge, cable-stayed bridge and suspension bridge, and suspension bridge has the greatest longitudinal bridging capacity. The evolution and achievements of suspension bridges greatly promote the development of modern bridge engineering as well as advanced bridge aerodynamics. Ancient suspension bridges were firstly built in China long before the history of the Anno Domini. The construction of modern suspension bridges around the world has experienced a considerable development since 1883, when the first modern suspension bridge, Brooklyn Bridge, was built. It took about 54 years for the span length of suspension bridges to grow from 483m of Brooklyn Bridge to 1,67m of George Washington Bridge as the first bridge with a span over 1 km, and to 1,8m of Golden Gate Bridge in 1937, and had a great span increase factor of.7. Although the further span increase in the next 44 years to Verrazano Bridge and to Humber Bridge of 1,41m in 1981 was only 1%, another factor of about 1.4 was realized in Akashi Kaikyo Bridge with a 1,991m span within 17 years in 1998 (Ge, 11a). It seems that, except Tacoma Narrows Bridge failure in 194, we didn t come across serious wind-induced problem until Great Belt Bridge and Akashi Kaikyo Bridge built in This is not true if we take a look of Table 1, which gives main information about ten longest span suspension bridges completed in the world (Ge, 11b). Among these ten longest bridges, six of them have encountered serious wind-induced problems, and dominant one is aerodynamic flutter instability. The intrinsic limit of span length due to aerodynamic stability seems to be about 1,4m for a traditional suspension bridge with either a box or truss girder. In order to improve flutter stability, three kinds of control measures have been adopted in these six bridges, including twin box or slotted box, central stabilizer and sharping box (Ge, 11a). Proc. of the 8th Asia-Pacific Conference on Wind Engineering Nagesh R. Iyer, Prem Krishna, S. Selvi Rajan and P. Harikrishna (eds) Copyright c 13 APCWE-VIII. All rights reserved. Published by Research Publishing, Singapore. ISBN: doi:1.385/

2 Span Order Table 1: Ten longest span suspension bridges completed in the world Main Girder Wind-Induced Control Bridge Name Country Span Type Problem Measure Year Built 1 Akashi Kaikyo 1991m Truss Flutter Slot+Stabilizer Japan 1998 Xihoumen 165m Box Flutter Twin box China 9 3 Great Belt 164m Box Vortex Guide vane Denmark Yi Sun-Sin 1545m Box Flutter Twin box Korea 1 5 Runyang 149m Box Flutter Stabilizer China 5 6 Nanjing 4th 1418m Box Flutter Sharping China 1 7 Humber 141m Box None None UK Jiangyin 1385m Box None None China Tsing Ma 1377m B/T Flutter Slot H.K. China Verrazano 198m Truss None None USA 1964 As a human dream and an engineering challenge, the structural engineering of bridging larger obstacles has entered into a new era of crossing wide rivers and sea straits. Table collects some proposed suspension bridges with ultra-long span, including,1m for Bali Strait in Indonesia (Wangsadinata et al., 199),,3m for Tokyo Bay in Japan (Ge, 11a),,8m for Qiongzhou Strait in China (Ge, 11a), 3,m for Sunda Strait in Indonesia (Wangsadinata et al., 199), 3,3m for Messina Strait in Italy (Castellani, 1994), 3,5m for Gibraltar Strait I linking Spain and Morocco (Lin and Chow, 1991), and 5,m as Gibraltar Strait II and the limit span scheme in China (Xiang and Ge, 3). With the rapid increase of span length, suspension bridges are becoming lighter, more flexible, and lower damping, which result in more and more sensitive to wind actions, in particular related to aerostatic and aerodynamic instability. As a result, besides single box girder, twin and triple box girders have been considered to improve aerodynamic performance against strong winds (Ge and Shao, 13). Span Order Table : Super long span suspension bridges proposed in the world Main Girder Wind-Induced Control Bridge Name Country Span Type Problem Measure Year Prop. 1 Bali Strait 1m Box Instability Single box Indonesia 199 Tokyo Bay 3m Box Instability Single box Japan Qiongzhou Strait 8m Box Instability Twin box China 4 Sunda Strait 3m Box Instability Triple box Indonesia Messina Strait 33m Box Instability Triple box Italy Gibraltar Strait I 35m Box Instability Twin box Spain/Morocco Gibraltar Strait II 5m Box Instability Triple box Spain/Morocco Limit span scheme 5m Box Instability Twin box China 3 Aerostatic stability of ultra-long span suspension bridges is a much concerned issue during the design stage. Typical aerostatic instability is the so-called torsional divergence which may lead to abrupt structural failure. The traditional way to analyze torsional divergence is two-dimensional linear method, which is quite simple but may lead to unsafe result. The 3D static iterative finite element method, which generally entails the linear assumption, has been now used to predict aerostatic instability of the bridge concerned. For ultra-long bridge span, nonlinearities of wind loading and structural performance must be taken into account, and the 3D nonlinear aerostatic stability analysis method should be adopted in practice. 41

3 Ultra-long Span Suspension Bridge Models and Properties The conceptual design of traditional three-span suspension bridges with ultra-long span has to be made before aerostatic stability analysis can be performed. The span arrangement of an ultra-long span suspension bridge is set as Fig. 1, in which the main span L is from 15 to 5m and both side spans are.3l (Shao, 1)..3L L=15-5.3L Fig.1 Span arrangement of a three-span suspension bridge The main cables are designed with the minimum cable area as (Xiang and Ge, 3) 1+ 16n ( wc + wd ) A (1) 8nσ / L a in which n is the ratio of sag to span of main cables, and n = f/l; σ a is the allowable stress of steel wires, and σ a = 1, MPa for simplification; w c is the cable weight per unit length; and w d is the deck load per unit length including dead weight of stiffening girder and live loads of traffic, and w d = w dd + w dl = = 8 kn/m. The basic parameters of main cables are listed in Table 3 for two common ratios of sag to span, n = 1/8 and 1/1 (Shao, 1). Main span L (m) Ratio n = f/l Table 3: Basic parameters of main cables Cable area Cable weight Deck load Total load A (m ) w c (N/m) w d (N/m) w c +w d (N/m) Load ratio w c /(w c +w d ) 1/ / / / / / / / / / In order to make systematic comparison of aerostatic stability performance, two groups of box girder cross sections have been considered. The first group of box girders includes three types of single box, the deep single box (4m in depth) corresponding to Great Belt Bridge in Fig. a, the shallow single box (3m in depth) of Jiangying Bridge in Fig. b and an ideal thin plate with the same stiffness and mass of Jiangying Bridge. The other group of box girders involves three types of multi-box cross sections, the narrowly slotted twin box of Xihoumen Bridge in Fig. 3a, the widely slotted twin box proposed for a 5,m spanned suspension bridge (Xiang and Ge, 3) in Fig. 3b and the triple box girder designed for 411

4 Messina Strait Bridge in Fig. 3c. The main parameters of two sets of box girders are listed in Table 4 (Shao, 1). a) Deep single box girder (Great Belt Bridge) b) Shallow single box girder (Jiangying Bridge) Fig. Cross sections of single box girders a) Narrowly slotted twin box girder (Xihoumen Bridge) ,5,5 % % % %,5 1,75 4 1,75 8 b) Widely slotted twin box girder (Limit span scheme) c) Triple box girder (Messina Strait Bridge) Fig. 3 Cross sections of multi-box girders 41

5 Cross Sections Module E (MPa) Table 4: Main parameters of box girders Width Depth Mass B (m) H (m) m (kg/m) Mass moment I m (kg m /m) Stiffness K α (MN m ) Torsion f t (Hz) Deep box Shallow box Thin plate N.S. twin W.S. twin Triple box Aerostatic Stability Analysis Approaches Aerostatic instability of long-span bridges relates generally to lateral buckling or torsional divergence of bridge girder under extreme wind loads. Torsional divergence is, relative to the lateral buckling, more likely to be encountered by ultra-long span suspension bridges. Torsional divergence is an aeroelastic phenomenon that can be described by using aerodynamic properties measured on a girder section at rest. Referring to a linear aerostatic instability deck model of Fig. 4, the equation of motion of the girder section can be written as (Simiu and Scanlan, 1996) Fig. 4 Deck aerostatic instability model 1 1 I α α + C αα + Kαα = ρu B CM ( α) ρu B [ CM + C M α ] () in which I is the mass moment of inertia; C is the torsional damping; K is the torsional stiffness; is the density of the air; U is the horizontal wind speed; B is the reference width of a bridge girder; is the angle of rotation about the girder s elastic center; and C M ( ) is the aerodynamic pitching moment coefficient, and C M and C M are the coefficient value at the initial stage or the angle of attack and its derivative with the respect to rotational angle, respectively. Torsional divergence is defined as aerostatic instability usually without considering girder vibration, by which both and are equal to zero. The wind loads on girder in Eq. () only depend on the structural deformation and the wind speed U, at which the elastic rotation increases rapidly to the point of failure, is called divergence wind speed (Dowell et al., 4). Simiu and Scanlan (1996) presented a simple equation derived from Eq. () to predict the divergence wind speed, as Kα U cr = (3) ' ρb C Eq. (3) is based upon the assumptions, the linear theory, the small deformation and the uniform distribution of aerostatic forces along the longitudinal axes of the concerned bridge, M 413

6 and generally gives rise to an over-estimation of aerostatic instability of the concerned bridge (Ge, 11b). The refined analysis of aerostatic stability of ultra-long span bridges can be conducted by three dimensional calculation model with considering both structural nonlinearities and the nonlinear dependence of the wind loads to structural deformation. This is generally described with the incremental static equilibrium equation (Simiu and Scanlan, 1996) K δ Δ δ = ΔP δ (4) [ ( )]{ } { ( )} in which [K( )] is the tangent stiffness matrix of the bridge in consideration; { } is the displacement increment vector with nonlinear geometric properties; and { P( )} is the nodal force increment vector induced by nonlinear aerostatic loads. The solution of Eq. (4) can be accomplished by a nonlinear static iterative finite element method (Boonyapinyo et al., 1994, Cheng et al., and Boonyapinyo et al., 6). Aerostatic Force Coefficients The aerostatic force of a bridge deck is commonly consisted of lift component L as, drag component D as and pitching moment component M as. These three components of the aerostatic force can be expressed with three aerostatic force coefficients,, and C M, as follows: 1 Las = U BCL( ) (5) 1 Das = U BCD( ) (6) 1 M as = U B CM( ) (7) in which is the aerostatic lift force coefficient; is the aerostatic drag force coefficient; and C M is the aerostatic pitching moment coefficient. These three aerostatic force coefficients for the single and multi-box girder groups have been identified through the sectional wind tunnel tests. The aerostatic force coefficients tested are plotted in Fig. 5 for the single box girder group and Fig. 6 for the multi-box girder group, respectively (Shao, 1) C M C -.5 M a) Deep single box girder a) Narrowly slotted twin box girder 414

7 C M b) Shallow single box girder b) Widely slotted twin box girder C M C M c) Ideal thin plate girder c) Triple box girder Fig. 5 Aerostatic force coefficients of single box girder group Fig. 6 Aerostatic force coefficients of multi-box girder group 1C M Critical Speed and State of Aerostatic Instability The finite element models of suspension bridges with ultra-long span are firstly set up with the main parameters of the main cables and decks for five span lengths, including 1,5,,, 3,, 4, and 5,m, and each span length scheme covers with the main parameters of six stiffening girders, including the deep single box, the shallow single box, the ideal thin plate, the narrowly slotted twin box, the widely slotted twin box and the triple box. With the experimentally identified aerostatic force coefficients of these six cross sections in Figs. 5 and 6, the 3D nonlinear aerostatic stability analysis is similarly performed on the total 3 finite element bridge models, and critical speeds for aerostatic instability based on D linear and 3D nonlinear approaches are predicted and compared in Table 5 for the single box girder group and Table 6 for the multi-box girder group, respectively. Based on the results in Tables 5 and 6, the D linear analysis approach cannot be used for predicting divergence wind speeds for suspension bridges with ultra-long spans since the linear results are in bad precision and on unsafe side. All three single box girders can provide a 1,5m spanned suspension bridge with high enough aerostatic critical speed, but for longer span the deep box girder can only be tried. All three multi-box girders can provide a 5,m spanned suspension bridges with high enough aerostatic critical speed, which can meet with most strong wind prone areas over the world. 415

8 Table 5: Critical speed of aerostatic instability for the single box decks Deep single box Shallow single box Ideal thin plate Main span L (m) D Linear 3D Nonlinear D Linear 3D Nonlinear D Linear 3D Nonlinear Table 6: Critical speed of aerostatic instability for multi-box decks Narrowly slotted twin Widely slotted twin Triple box Main span L (m) D Linear 3D Nonlinear D Linear 3D Nonlinear D Linear 3D Nonlinear > 157 > > 148 > deep single box L=15m wind speed: 11m/s shallow single box L=15m wind speed: 9m/s a) Deep box girder thin plate L=15m wind speed: 91m/s b) Shallow box girder c) Thin plate girder Fig. 7 Aerostatic critical state of 1,5m spanned suspension bridges In order to further compare aerostatic instability performance, the structural deformations of main cables and stiffening girder at the critical state of aerostatic instability are described in Figs. 7 and 8. Fig. 7 shows the vertical and lateral displacements of main cables and stiffening girder and the torsional displacement of stiffening girder at aerostatic 416

9 critical state for 1,5m spanned single box girders. The deep box girder has the smallest torsional displacement at the critical state so that the corresponding bridge can provide with the highest aerostatic critical speed, 11m/s, and followed by the shallow box girder. Fig. 8 describes the same displacements at critical state for the suspension bridges with a 5,m span. It is interesting to see that the torsional displacements of three-type multi-box girders are almost same, and the widely slotted twin box has the smallest translational displacements narrowly slotted box L=5m wind speed: 9m/s widely slotted box L=5m wind speed: 1m/s a) Narrowly slotted twin box girder triple box L=5m wind speed: 17m/s b) Widely slotted twin box girder c) Triple box girder Fig. 8 Aerostatic critical state of 5,m spanned suspension bridges Conclusions Suspension bridges with ultra-long span may have some aerostatic and aerodynamic instability problems due to extreme wind actions. Referring to aerodynamic flutter stability, twin and triple box girders have been suggested, instead of traditional single box girder, to improve aerodynamic performance against strong winds. With the experimentally identified aerostatic force coefficients of three single box girders and three multi-box girders, D linear and 3D nonlinear aerostatic stability analysis has been conducted for five span lengths, including 1,5,,, 3,, 4, and 5,m, and the aerostatic critical speeds of the total 3 suspension bridge models have been obtained and compared. It is necessary to adopt 3D static iterative finite element approach to find out aerostatic critical speed. For ultra-long bridge span, in which the nonlinearities of wind loading and structural performance must be taken into account, the 3D nonlinear aerostatic stability analysis approach is recommended in practice. In the case of the single box girders, aerostatic stability limit of the ideal thin plate girder is slightly greater than that of the shallow single box. The deep single box has the highest aerostatic critical speed among three single box girders. The deep and the shallow 417

10 single box can support a 1,5m spanned suspension bridge with about 11 and 9m/s aerostatic critical speed, but for longer span the deep box girder can only be tried. Among three kinds of multi-box girders, the widely slotted twin box girder has greater aerostatic critical speed than the triple box girder with the span length between, and 4,m, and slightly smaller aerostatic critical speed under the spans of 1,5 and 5,m, while the narrowly slotted twin box girder gets the smallest aerostatic critical speeds. According to aerostatic stability limits, all three-kind multi-box girders can support a 5,m span suspension bridge with high enough aerostatic critical speeds to meet with most typhoon prone areas all over the world. Acknowledgements The work described in this paper is partially supported by the NSFC under the Grant and by the MOST under the 973 Program Grant 13CB3631. References Boonyapinyo, V., Yamada, H. and Miyata, T. (1994), Wind-induced nonlinear lateral-torsional buckling of cable-stayed bridges, Journal of Structural Engineering, ASCE, 1(), Boonyapinyo, V., Lauhatanon, Y. and Lukkunaprasit, P. (6), Nonlinear aerostatic stability analysis of suspension bridges, Engineering Structures, 8(3), Castellani, A. (1994), The project for the bridge over the Messina Straits, Proc. AIPC-FIP Int. Conf. on Cablestayed and Suspension Bridges, Deauville, France, Oct. 1-15, 1994, Vol. 1. Cheng, J., Jiang, J.J., Xiao, R.C. and Xiang, H.F. (), Nonlinear aerostatic stability analysis of Jiang Yin suspension bridge, Engineering Structures, 4(6), Dowell, E.H., et al. (4), A Modern Course in Aeroelasticity, 4 th Edition, Kluwer Academic, New York, USA. Ge, Y.J. (11a), Wind Resistance of Long Span Suspension Bridges (in Chinese), China Communications Press, Beijing, China. Ge, Y.J. (11b), Challenging aerodynamic concerns of cable-supported bridges with super long span, Proceedings of the 13 th International Conference on Wind Engineering, Amsterdam, The Netherland, July 11-15, 11. Ge, Y.J. and Shao, Y.H. (13), Aerodynamic flutter prediction for super long span suspension bridges, Proceedings of the 1th Americas Conference on Wind Engineering, Seattle, Washington, USA, June 16-, 13. Lin, T.Y. and Chow, P. (1991), Gibraltar Strait Crossing, a challenge to bridge and structural engineering, J. Structural Engineering International, Vol. 1, No.. Shao, Y.H. (1), Refinements on aerodynamic and aerostatic stability of super long span suspension bridges with box girder (in Chinese), PhD thesis supervised by Y.J. Ge, Tongji University, China. Simiu, E and Scanlan, R.H. (1996), Wind Effects on Structures, 3 rd Edition, John Wiley & Sons, New York, USA. Wangsadinata, W., Chow, P., Lin, T.Y. (199), A prospective bridge crossing using ultra-long bridge spans between Java and Sumatera, Proceedings of the Seminar on Underground, Urban and Commuter Transportation Technology, BPPT-JICA, Jakarta, Indonesia, Sept , 199. Xiang, H.F. and Ge, Y.J. (3), On aerodynamic limit to suspension bridges, Proceedings of the 11 th International Conference on Wind Engineering. Texas, USA, June -5,