Research Article  Open Access
Geometric Nonlinear Analysis of SelfAnchored CableStayed Suspension Bridges
Abstract
Geometric nonlinearity of selfanchored cablestayed suspension bridges is studied in this paper. The repercussion of shrinkage and creep of concrete, risetospan ratio, and girder camber on the system is discussed. A selfanchored cablestayed suspension bridge with a main span of 800 m is analyzed with linear theory, secondorder theory, and nonlinear theory, respectively. In the condition of various risetospan ratios and girder cambers, the moments and displacements of both the girder and the pylon under live load are acquired. Based on the results it is derived that the secondorder theory can be adopted to analyze a selfanchored cablestayed suspension bridge with a main span of 800 m, and the error is less than 6%. The shrinkage and creep of concrete impose a conspicuous impact on the structure. And it outmatches suspension bridges for system stiffness. As the risetospan ratio increases, the axial forces of the main cable and the girder decline. The system stiffness rises with the girder camber being employed.
1. Introduction
The selfanchored cablestayed suspension bridge dates back to the early 19th century [1, 2]. First a mixed structure of cablestayed and suspension bridge was built in France. After years of trial and effort, it has evolved to Roebling system, Dichinger system, and then the improved Dichinger system. The suspended part of the selfanchored cablestayed suspension bridge is much shorter than that of suspension bridge with the same overall span; therefore the main cable force can be reduced in a large extent. What is more, the cantilever length of cablestayed part can be decreased greatly during the construction period, thereby improving aerodynamic stability of the structure. This bridge system has been proposed for many design projects, such as the Strait of Gibraltar Bridge project and the Lingdingyang Bridge project [3].
Until now the cablestayed suspension bridge still remains in the design proposal phase. No largespan cablestayed suspension bridge has ever been built in the world. And all the design proposals are earthanchored systems. Both the anchorage and the construction period can be saved through adopting the selfanchored system. The selfanchored cablestayed suspension bridge suits the longspan needs well. Longspan cablestayed bridges, suspension bridges, and cablestayed suspension bridges have been discussed a lot, but there are few papers relating to selfanchored cablestayed suspension bridges [4]. Based on a selfanchored cablestayed suspension bridge with an 800 m main span, its geometric nonlinearity under live load is studied.
2. The Geometric Nonlinear Characteristics of the Cooperation System
Three main factors cause the geometric nonlinearity of the cooperation system, including the cable sag, large displacements, and the initial internal force.
2.1. The Cable Sag Effect
The cable would sag in the free suspension state. The sag and the chord length of the cable change as the internal force alters. A nonlinear relationship exists between the chord length and the cable force. The method of equivalent elasticity modulus can be adopted to simulate the sag effect. The wellknown Ernst formula for equivalent elasticity modulus is shown as follows: where indicates the elasticity modulus of the cable, the horizontal projection length of the cable element, the cable weight per unit length, the crosssectional area of the cable, and the axial force of the cable element. The cable element can be set up as a straight bar with the elastic modulus correction.
2.2. The Effect of the Initial Internal Force and Large Displacements
The girder and main pylon of the cooperation system bear tremendous pressure. The axial force causes additional bending moments, which affects the bending stiffness of components; meanwhile the bending moment alters the length of structure components, which furthermore affects the axial stiffness of components. By introducing the initial stress stiffness matrix to simulate the effect of the initial internal force and stiffness matrix of displacement to model the effect of large displacements, the initial tangent stiffness matrix can be attained [5]: where represents linear stiffness matrix.
2.3. SecondOrder Analysis Theory
Geometric nonlinearity leads to the failure in the application of superposition principles. The moment and displacement caused by live load cannot be calculated with influence lines. The response of the cooperation system under all kinds of loads can be attained with the method described above. However, it is timeconsuming, especially for the live load which requires repeated iterations. Secondorder theory, a simplified method of approximate calculation, will be discussed as follows.
The girder and pylon of the cooperation system belong to compressionbending members. A simply planar differential equation for compression bending beams is shown below [6]:
The axial force of the cooperation system consists of two parts. The one caused by the dead load is recorded as ; the other one caused by live load is denoted by . Therefore, (3) is a nonlinear differential equation.
Dr. Li Guohao solved the secondorder nonlinear catenaries theory with linear method [7]. By drawing on this idea, the nonlinear analysis of the cooperation system can be simplified. The proportion of live load to dead load for longspan bridges usually ranges from 10% to 20% [8]. Therefore the effect of is negligible. Then (3) can be simplified into
Equation (4) is a linear differential equation, which satisfies the condition of linear superposition. This simplified method can be called secondorder theory [9].
3. Case Study
The Dalian Gulf Bridge with a main span of 800 m, is in the form of a selfanchored cablestayed suspension bridge. Its total length is 1326 m. The main girder section adopts streamlined flat box girder, which is 3.5 m high and 34 m wide. There are two kinds of main girder for the whole bridge including steel girder and prestressed concrete girder. The steel part is in the middle suspension segment and the prestressed concrete part in the cablestayed area. The Hshaped pylon is 127 m above the main girder. The elevation layout of the bridge is shown in Figure 1.
(a) Elevation of bridge/m
(b) Standard section of steel box beam
(c) Standard section of concrete box beam
3.1. Study on the Nonlinearity of Live Load
Here the nonlinearity of live load is analyzed with nonlinear theory, secondorder theory, and linear theory, respectively. Since the superposition principle becomes inappropriate in the nonlinear analysis, the influence zone method is adopted. Under the automobile load of grade I (China), the moment and displacement envelope diagram of main girder and main pylon are as shown in Figures 2, 3, 4, and 5. The results of these calculations are summarized in Table 1.(1)The result of linear theory radically differs from that of nonlinear theory, while the result of secondorder theory is close to that of nonlinear theory. The maximum relative difference between the results of secondorder theory and nonlinear theory is less than 6%, which is acceptable for the actual project. The calculation of secondorder theory is simpler than that of nonlinear theory; therefore for the cooperation system bridge with a main span of 800 m, the secondorder theory is feasible to calculate live load response.(2)The maximum positive moment of the main girder occurs at the middle, while the maximum negative moment occurs near the junction. Due to the great change of stiffness and the enormous negative moment, the junction between steel girder and prestressed concrete girder has to be strengthened specifically.(3)The maximum deflection at the middle of the girder is 1.18 m, which is about 1/650 of the middle span length. Therefore, the integral stiffness of the cooperation system bridge is higher than that of normal suspension bridges with the same length, due to the fact that the cablestayed part enhances the global stiffness.

3.2. Study on Concrete Shrinkage and Creep
Shrinkage and creep of concrete can make girder and pylon shorter, causing the main cable and stayedcable sag, so the bending moment and deformation of the girder increase. Based on the Bridge Criterion (China), the concrete shrinkage and creep effects within 15 years are analyzed. The results are summarized in Table 2.
 
The moment of pylon takes place at the root. The displacement of pylon takes place at top and the positive direction points to midspan. The positive direction of girder displacement points to upward side. 
The results show that concrete shrinkage and creep have an effect on internal forces and the shape of cooperation system. Effective measures should be taken to reduce the influence, such as extending the load age of concrete, using microexpansion concrete and so on.
3.3. Analysis on the Impact of RisetoSpan Ratio
The risetospan ratio of the main cable is an important parameter for the cooperation system bridge, which affects both structural stiffness and internal forces. The impact of risetospan ratio under live load is given in Figure 6. For the sake of convenience, dimensionless forms are adopted. It is convenient to make the parameters dimensionless using results of risetospan ratio 1/10 as a reference.
Figure 6 shows that global stiffness increases as the risetospan ratio rises, which is the same as the cooperation system bridge but opposite to earthanchored suspension bridges [10]. The axial forces of the girder and the main cable descend as the risetospan ratio increases. The axial force of the girder is extremely sensitive to the risetospan ratio. The axial forces of the girder and the main cable are enormous, which affects the crosssectional areas of the girder and the main cable. So the smaller risetospan ratio is not recommended for the cooperation system bridge.
Figure 6 also shows that horizontal displacements of the main girder descend as the risetospan ratio increases because the axial force of the girder descends as the risetospan ratio increases. As the risetospan ratio increases, the moment at the pylon root rises a little.
3.4. Analysis on Camber of the Main Girder
Because of the high level of component force of the main cable at both ends of the main girder, the camber of the main girder will induce an additional negative bending moment for the main girder and further increase the main girder’s moment due to PΔ effect. The moment under dead load can be adjusted through the cable tension adjustment; therefore it is only necessary to analyze the main girder camber’s effect under live load. Table 3 presents the main girder’s bending moments and deflections under live load with both 0 m and 2.9 m cambers at the midspan of the main girder. The results indicate that setting up a girder camber can effectively reduce the bending moment of the main girder and improve the structural stiffness.

4. Conclusions
Geometric nonlinear factors of the cooperation system bridge are discussed in this paper. Based on this, a cooperation system bridge with an 800 m main span is analyzed. And the following conclusions are reached.(1)The error is less than 6% using a simple secondorder approximation theory to calculate live load response of a cooperation system bridge with an 800 m main span.(2)The stiffness of the junction between the cablestayed segment and the suspension area varies. Great internal forces occur easily under live load, so it is necessary to strengthen the junction.(3)Concrete shrinkage and creep have a conspicuous impact on the internal force and deformation of the structure. It is necessary to take measures to alleviate the influence.(4)Global stiffness increases with the risetospan ratio ascending and the axial forces of the girder and the main cable descend as the risetospan ratio rises. Therefore the small risetospan ratios are not recommended for the cooperation system bridge.(5)Setting a girder camber can improve the integral stiffness of the cooperation system bridge.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work is funded by National Natural Science Foundation (51008047, 51108052) and West Transportation Construction Projects Foundation of Ministry of Communications, China (2006 318 823 50).
References
 N. J. Gimsing, Cable Supported Bridges: Analysis and Design Concept, John Wiley & Sons, Chichester, UK, 1997.
 B. H. Wang, “Cablestayed suspension bridges,” Journal of Liaoning Provincial College of Communications, vol. 2, no. 3, pp. 1–6, 2000. View at: Google Scholar
 R. C. Xiao, L. J. Jia, and E. L. Xue, “Research on the design of cablestayed suspension bridges,” China Civil Engineering Journal, vol. 33, no. 5, pp. 46–51, 2000. View at: Google Scholar
 H. L. Wang, Structure Properties Analysis and Experiment Study of SelfAnchored CableStayed Suspension Bridge [Doctoral dissertation], Dalian University of Technology, Dalian, China, 2007.
 C. X. Li and G. Y. Xia, “Geometric nonlinear analysis of long span bridge,” in The Calculation Theory of Long Span Bridge Structure, pp. 88–113, China Communications Press, Beijing, China, 2002. View at: Google Scholar
 X. F. Sun, X. S. Fang, and T. L. Guan, “The beam deflections,” in Mechanics of Materials, pp. 269–309, China Higher Education Press, Beijing, China, 1994. View at: Google Scholar
 G. H. Li, “Utility calculation of suspension bridges with secondorder theory,” in Study of Bridge and Structure Theory, Shanghai Science and Technology Publishing House, Shanghai, China, 1983. View at: Google Scholar
 H. F. Xiang, “The calculation theory of cablestayed bridges,” in Higher Theory of Bridge Structure, pp. 282–301, China Communications Press, Beijing, China, 2002. View at: Google Scholar
 Y. R. Pan, “Suspension bridge analysis under external loads,” in Nonlinear Analysis Theory and Method of Suspension Bridge Structure, pp. 69–89, China Communications Press, Beijing, China, 2004. View at: Google Scholar
 W. L. Qiu Zhang Z and C. L. Hang, “Mechanical properties of selfanchored concrete suspension bridge,” Journal of Harbin University of Civil Engineering and Architecture, vol. 35, no. 11, pp. 1388–1391, 2003. View at: Google Scholar
Copyright
Copyright © 2013 Wang HuiLi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.