A comparison on nonlinear analysis of a highly redundant cable-stayed bridge is performed in the study. The initial shapes including geometry and prestress distribution of the bridge are determined by using a two-loop iteration method, i.e., an equilibrium iteration loop and a shape iteration loop. For the initial shape analysis a linear and a nonlinear computation procedure are set up. In the former all nonlinearities of cable-stayed bridges are disregarded, and the shape iteration is carried out without considering equilibrium. In the latter all nonlinearities of the bridges are taken into consideration and both the equilibrium and the shape iteration are carried out. Based on the convergent initial shapes determined by the different procedures, the natural frequencies and vibration modes are then examined in details. Numerical results show that a convergent initial shape can be found rapidly by the two-loop iteration method, a reasonable initial shape can be determined by using the linear computation procedure, and a lot of computation efforts can thus be saved. There are only small differences in geometry and prestress distribution between the results determined by linear and nonlinear computation procedures. However, for the analysis of natural frequency and vibration modes, significant differences in the fundamental frequencies and vibration modes will occur, and the nonlinearities of the cable-stayed bridge response appear only in the modes determined on basis of the initial shape found by the nonlinear computation.
2. Introduction
Rapid progress in the analysis and construction of cable-stayed bridges has been made over the last three decades. The progress is mainly due to developments in the fields of computer technology, high strength steel cables, orthotropic steel decks and construction technology. Since the first modern cable-stayed bridge was built in Sweden in 1955, their popularity has rapidly been increasing all over the world. Because of its aesthetic appeal, economic grounds and ease of erection, the cable-stayed bridge is considered as the most suitable construction type for spans ranging from 200 to about 1000 m. The world’s longest cable-stayed bridge today is the Tatara bridge across the Seto Inland Sea, linking the main islands Honshu and Shikoku in Japan. The Tatara cable-stayed bridge was opened in 1 May, 1999 and has a center span of 890m and a total length of 1480m. A cable-stayed bridge consists of three principal components, namely girders, towers and inclined cable stays. The girder is supported elastically at points along its length by inclined cable stays so that the girder can span a much longer distance without intermediate piers. The dead load and traffic load on the girders are transmitted to the towers by inclined cables. High tensile forces exist in cable-stays which induce high compression forces in towers and part of girders. The sources of nonlinearity in cable-stayed bridges mainly include the cable sag, beam-column and large deflection effects. Since high pretension force exists in inclined cables before live loads are applied, the initial geometry and the prestress of cable-stayed bridges depend on each other. They cannot be specified independently as for conventional steel or reinforced concrete bridges. Therefore the initial shape has to be determined correctly prior to analyzing the bridge. Only based on the correct initial shape a correct deflection and vibration analysis can be achieved. The purpose of this paper is to present a comparison on the nonlinear analysis of a highly redundant stiff cable-stayed bridge, in which the initial shape of the bridge will be determined iteratively by using both linear and nonlinear computation procedures. Based on the initial shapes evaluated, the vibration frequencies and modes of the bridge are examined.
3. System equations
3.1. General system equation
When only nonlinearities in stiffness are taken into account, and the system mass and damping matrices are considered as constant, the general system equation of a finite element model of structures in nonlinear dynamics can be derived from the Lagrange’s virtual work principle and written as follows:
Kjbαj-∑Sjajα= Mαβqβ”+ Dαβqβ’
3.2. Linearized system equation
In order to incrementally solve the large deflection problem, the linearized system equations has to be derived. By taking the first order terms of the Taylor’s expansion of the general system equation, the linearized equation for a small time (or load) interval is obtained as follows:
MαβΔqβ”+ΔDαβqβ’ +2KαβΔqβ=Δpα- upα
3.3. Linearized system equation in statics
In nonlinear statics, the linearized system equation becomes
2KαβΔqβ=Δpα- upα
4. Nonlinear analysis
4.1. Initial shape analysis
The initial shape of a cable-stayed bridge provides the geometric configuration as well as the prestress distribution of the bridge under action of dead loads of girders and towers and under pretension force in inclined cable stays. The relations for the equilibrium conditions, the specified boundary conditions, and the requirements of architectural design should be satisfied. For shape finding computations, only the dead load of girders and towers is taken into account, and the dead load of cables is neglected, but cable sag nonlinearity is included. The computation for shape finding is performed by using the two-loop iteration method, i.e., equilibrium iteration and shape iteration loop. This can start with an arbitrary small tension force in inclined cables. Based on a reference configuration (the architectural designed form), having no deflection and zero prestress in girders and towers, the equilibrium position of the cable-stayed bridges under dead load is first determined iteratively (equilibrium iteration). Although this first determined configuration satisfies the equilibrium conditions and the boundary conditions, the requirements of architectural design are, in general, not fulfilled. Since the bridge span is large and no pretension forces exist in inclined cables, quite large deflections and very large bending moments may appear in the girders and towers. Another iteration then has to be carried out in order to reduce the deflection and to smooth the bending moments in the girder and finally to find the correct initial shape. Such an iteration procedure is named here the ‘shape iteration’. For shape iteration, the element axial forces determined in the previous step will be taken as initial element forces for the next iteration, and a new equilibrium configuration under the action of dead load and such initial forces will be determined again. During shape iteration, several control points (nodes intersected by the girder and the cable) will be chosen for checking the convergence tolerance. In each shape iteration the ratio of the vertical displacement at control points to the main span length will be checked, i.e.,
The shape iteration will be repeated until the convergence toleranceε, say 10-4, is achieved. When the convergence tolerance is reached, the computation will stop and the initial shape of the cable-stayed bridges is found. Numerical experiments show that the iteration converges monotonously and that all three nonlinearities have less influence on the final geometry of the initial shape. Only the cable sag effect is significant for cable forces determined in the initial shape analysis, and the beam-column and large deflection effects become insignificant.
The initial analysis can be performed in two different ways: a linear and a nonlinear computation procedure. 1. Linear computation procedure: To find the equilibrium configuration of the bridge, all nonlinearities of cable stayed bridges are neglected and only the linear elastic cable, beam-column elements and linear constant coordinate transformation coefficients are used. The shape iteration is carried out without considering the equilibrium iteration. A reasonable convergent initial shape is found, and a lot of computation efforts can be saved.
2. Nonlinear computation procedure: All nonlinearities of cable-stayed bridges are taken into consideration during the whole computation process. The nonlinear cable element with sag effect and the beam-column element including stability coefficients and nonlinear coordinate transformation coefficients are used. Both the shape iteration and the equilibrium iteration are carried out in the nonlinear computation. Newton–Raphson method is utilized here for equilibrium iteration.
4.2. Static deflection analysis
Based on the determined initial shape, the nonlinear static deflection analysis of cable-stayed bridges under live load can be performed incrementwise or iterationwise. It is well known that the load increment method leads to large numerical errors. The iteration method would be preferred for the nonlinear computation and a desired convergence tolerance can be achieved. Newton– Raphson iteration procedure is employed. For nonlinear analysis of large or complex structural systems, a ‘full’iteration procedure (iteration performed for a single full load step) will often fail. An increment–iteration procedure is highly recommended, in which the load will be incremented, and the iteration will be carried out in each load step. The static deflection analysis of the cable stayed bridge will start from the initial shape determined by the shape finding procedure using a linear or nonlinear computation. The algorithm of the static deflection analysis of cable-stayed bridges is summarized in Section 4.4.2.
4.3. Linearized vibration analysis
When a structural system is stiff enough and the external excitation is not too intensive, the system may vibrate with small amplitude around a certain nonlinear static state, where the change of the nonlinear static state induced by the vibration is very small and negligible. Such vibration with small amplitude around a certain nonlinear static state is termed linearized vibration. The linearized vibration is different from the linear vibration, where the system vibrates with small amplitude around a linear static state. The nonlinear static state qαa can be statically determined by nonlinear deflection analysis. After determining qαa , the system matrices may be established with respect to such a nonlinear static state, and the linearized system equation has the form as follows:
MαβAqβ”+ DαβAqβ’+ 2KαβAqβ=pα(t)- TαA
where the superscript ‘A’ denotes the quantity calculated at the nonlinear static state qαa . This equation represents a set of linear ordinary differential equations of second order with constant coefficient matrices MαβA, DαβA and 2KαβA. The equation can be solved by the modal superposition method, the integral transformation methods or the direct integration methods.
When damping effect and load terms are neglected, the system equation becomes
MαβAqβ” + 2KαβAqβ=0
This equation represents the natural vibrations of an undamped system based on the nonlinear static state qαa The natural vibration frequencies and modes can be obtained from the above equation by using eigensolution procedures, e.g., subspace iteration methods. For the cable-stayed bridge, its initial shape is the nonlinear static state qαa . When the cable-stayed bridge vibrates with small amplitude based on the initial shape, the natural frequencies and modes can be found by solving the above equation.
4.4. Computation algorithms of cable-stayed bridge analysis
The algorithms for shape finding computation, static deflection analysis and vibration analysis of cable-stayed bridges are briefly summarized in the following.
4.4.1. Initial shape analysis
1. Input of the geometric and physical data of the bridge.
2. Input of the dead load of girders and towers and suitably estimated initial forces in cable stays.
3. Find equilibrium position
(i) Linear procedure
• Linear cable and beam-column stiffness elements are used.
• Linear constant coordinate transformation coefficients ajαare used.
• Establish the linear system stiffness matrix Kαβ by assembling element stiffness matrices.
• Solve the linear system equation for qα (equilibrium position).
• No equilibrium iteration is carried out.
(ii) Nonlinear procedure
• Nonlinear cables with sag effect and beam-column elements are used.
• Nonlinear coordinate transformation coeffi- cients ajα; ajα，β are used.
• Establish the tangent system stiffness matrix 2Kαβ.
• Solve the incremental system equation for △qα.
• Equilibrium iteration is performed by using the Newton–Raphson method.
4. Shape iteration
5. Output of the initial shape including geometric shape and element forces.
6. For linear static deflection analysis, only linear stiff-ness elements and transformation coefficients are used and no equilibrium iteration is carried out.
4.4.3. Vibration analysis
1. Input of the geometric and physical data of the bridge. 2. Input of the initial shape data including initial geometry and initial element forces.
3. Set up the linearized system equation of free vibrations based on the initial shape.
4. Find vibration frequencies and modes by sub-space iteration methods, such as the Rutishauser Method.
5. Estimation of the trial initial cable forces
In the recent study of Wang and Lin, the shape finding of small cable-stayed bridges has been performed by using arbitrary small or large trial initial cable forces. There the iteration converges monotonously, and the convergent solutions have similar results, if different trial values of initial cable forces are used. However for large cable-stayed bridges, shape finding computations become more difficult to converge. In nonlinear analysis, the Newton-type iterative computation can converge, only when the estimated values of the solution is locate in the neighborhood of the true values. Difficulties in convergence may appear, when the shape finding analysis of cable-stayed bridges is started by use of arbitrary small initial cable forces suggested in the papers of Wang et al. Therefore, to estimate a suitable trial initial cable forces in order to get a convergent solution becomes important for the shape finding analysis. In the following, several methods to estimate trial initial cable forces will be discussed.
5.1. Balance of vertical loads
5.2. Zero moment control
5.3. Zero displacement control
5.4. Concept of cable equivalent modulus ratio
5.5. Consideration of the unsymmetry
If the estimated initial cable forces are determined independently for each cable stay by the methods mentioned above, there may exist unbalanced horizontal forces on the tower in unsymmetric cable-stayed bridges. Forsymmetric arrangements of the cable-stays on the central (main) span and the side span with respect to the tower, the resultant of the horizontal components of the cable-stays acting on the tower is zero, i.e., no unbalanced horizontal forces exist on the tower. For unsymmetric cable-stayed bridges, in which the arrangement of cable-stays on the central (main) span and the side span is unsymmetric, and if the forces of cable stays on the central span and the side span are determined independently, evidently unbalanced horizontal forces will exist on the tower and will induce large bending moments and deflections therein. Therefore, for unsymmetric cable-stayed bridges, this problem can be overcome as follows. The force of cable stays on the central (main) span Tim can be determined by the methods mentioned above independently, where the superscript m denotes the main span, the subscript I denotes the ith cable stay. Then the force of cable stays on the side span is found by taking the equilibrium of horizontal force components at the node on the tower attached with the cable stays, i.e., Tim cosαi= Tis cosβi, and Tis = Tim cosαi/ cosβi, where αi is the angle between the ith cable stay and the girder on the main span, andβi, angle between the ith cable stay and the girder on the side span.
6. Examples
In this study, two different types of small cable-stayed bridges are taken from literature, and their initial shapes will be determined by the previously described shape finding method using linear and nonlinear procedures. Finally, a highly redundant stiff cable-stayed bridge will be examined. A convergence tolerance e =10-4 is used for both the equilibrium iteration and the shape iteration. The maximum number of iteration cycles is set as 20. The computation is considered as not convergent, if the number of the iteration cycles exceeds 20.
The initial shapes of the following two small cable stayed bridges in Sections 6.1 and 6.2 are first determined by using arbitrary trial initial cable forces. The iteration converges monotonously in these two examples. Their convergent initial shapes can be obtained easily without difficulties. There are only small differences between the initial shapes determined by the linear and the nonlinear computation. Convergent solutions offer similar results, and they are independent of the trial initial cable forces.
7. Conclusion
The two-loop iteration with linear and nonlinear computation is established for finding the initial shapes of cable-stayed bridges. This method can achieve the architecturally designed form having uniform prestress distribution, and satisfies all equilibrium and boundary conditions. The determination of the initial shape is the most important work in the analysis of cable-stayed bridges. Only with a correct initial shape, a meaningful and accurate deflection and/or vibration analysis can be achieved. Based on numerical experiments in the study, some conclusions are summarized as follows:
(1). No great difficulties appear in convergence of the shape finding of small cable-stayed bridges, where arbitrary initial trial cable forces can be used to start the computation. However for large scale cable-stayed bridges, serious difficulties occurred in convergence of iterations.
(2). Difficulties often occur in convergence of the shape finding computation of large cable-stayed bridge, when trial initial cable forces are given by the methods of balance of vertical loads, zero moment control and zero displacement control.
(3). A converged initial shape can be found rapidly by the two-loop iteration method, if the cable stress corresponding to about 80% of Eeq=E value is used for the trial initial force of each cable stay in the main span, and the trial force of the cables in side spans is determined by taking horizontal equilibrium of the cable forces acting on the tower.
(4). There are only small differences in geometry and prestress distributionforces. The iteration converges monotonously in these two examples. Their convergent initial shapes can be obtained easily without difficulties. There are only small differences between the initial shapes determined by the linear and the nonlinear computation. Convergent solutions offer similar results, and they are independent of the trial initial cable forces.
7. Conclusion
The two-loop iteration with linear and nonlinear computation is established for finding the initial shapes of cable-stayed bridges. This method can achieve the architecturally designed form having uniform prestress distribution, and satisfies all equilibrium and boundary conditions. The determination of the initial shape is the most important work in the analysis of cable-stayed bridges. Only with a correct initial shape, a meaningful and accurate deflection and/or vibration analysis can be achieved. Based on numerical experiments in the study, some conclusions are summarized as follows:
(1). No great difficulties appear in convergence of the shape finding of small cable-stayed bridges, where arbitrary initial trial cable forces can be used to start the computation. However for large scale cable-stayed bridges, serious difficulties occurred in convergence of iterations.
(2). Difficulties often occur in convergence of the shape finding computation of large cable-stayed bridge, when trial initial cable forces are given by the methods of balance of vertical loads, zero moment control and zero displacement control.
(3). A converged initial shape can be found rapidly by the two-loop iteration method, if the cable stress corresponding to about 80% of Eeq=E value is used for the trial initial force of each cable stay in the main span, and the trial force of the cables in side spans is determined by taking horizontal equilibrium of the cable forces acting on the tower.
(4). There are only small differences in geometry and prestress distribution between the results of initial shapes determined by linear and nonlinear procedures.
(5). The shape finding using linear computation offers a reasonable initial shape and saves a lot of computation efforts, so that it is highly recommended from the point of view of engineering practices.
(6). In small cable-stayed bridges, there are only small difference in the natural frequencies based on initial shapes determined by linear and nonlinear computation procedures, and the mode shapes are the same in both cases.
(7). Significant differences in the fundamental frequency and in the mode shapes of highly redundant stiff cable stayed bridges is shown in the study. Only the vibration modes determined by the initial shape based on nonlinear procedures exhibit the nonlinear cable sag and beam-column effects of cable-stayed bridges, e.g., the first and third modes of the bridge are dominated by the transversal motion of the tower, not of the girder. The difference of the fundamental frequency in both cases is about 12%. Hence a correct analysis of vibration frequencies and modes of cable-stayed bridges can be obtained only when the ‘correct’ initial shape is determined by nonlinear computation, not by the linear computation.