Unstrained Element Length-Based Methods for Determining One Optimized Initial State of Cable-Stayed Bridges
Abstract
Two robust procedures evaluating all unstrained element lengths are presented to find one practically optimized initial shape of cable-stayed bridges under dead loads. An analytical method based on the continuous girder model accounting for P-Δeffects due to stay-cable tensions is first proposed to calculate optimized cable tensions and unstrained element lengths without recourse to refined nonlinear FE analysis method. And then it is addressed how the G.TCUD method [10] developed for suspension bridges should be applied to determine an optimized initial state of cable-stayed bridges. For this, the extended nonlinear formulations of the co-rotational frame element as well as the elastic catenary cable element are briefly summarized by adding unstrained lengths of all finite elements to the unknown. Finally, based on the unstrained lengths determined from two methods, the unstrained length methods are presented to effectively perform nonlinear FE analysis of cable stayed bridges subjected to various load combinations. Consequently accuracy and effectiveness of the proposed schemes are demonstrated by showing that not only the unstrained lengths of a long-span cable-stayed bridge model by the analytical method are nearly same as those by the G.TCUD method but also these two methods lead to essentially one optimized initial configuration which is in suit with the target geometry.
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Introduction
Generally one initial configuration satisfying the equilibrium condition between external dead loads and internal member forces including cable tensions should be predetermined in the preliminary design stage of cable-supported bridges because cable members cannot be defined in the stress-free state. Moreover it is of extreme importance to obtain the minimized bending moment distributions by determining optimized cable tensioning forces because the internal forces due to dead loads can be significantly large as the span length of cable bridges is increased. This analysis process finding one initial equilibrium state close to the target configuration of cable structures under full dead loads is referred to as shape finding, form finding or initial shape analysis.
With relation to shape finding problems of cable-stayed bridges, a set of optimized tensioning forces for stay-cables should be found such that the vertical displacements of the main girder vanish except for the fabrication camber and the horizontal displacements of the pylon are minimized within the allowable limit. Otherwise, huge bending moments in the deck and pylons of cable-stayed bridges under dead loads can be induced due to the P - ∆ effect by horizontal or vertical components of the cable tension. Furthermore, in case of fan- and harp-typed cable-stayed bridges, one practically optimized initial state should be searched because there can exist several initial configurations. Particularly as the span length of cable-supported bridges is greatly increased, the maximum bending moment occurring in the main girder and the pylon can become rapidly outsized depending on the fabrication camber and the balanced condition with respect to self-weights.
Until now, to find the initial state solution of cable-stayed bridges, various analysis methods have been developed such as the zero displacement method [1], the force equilibrium method [2], the optimization method [3, 4, 5], the initial force method [6], the TCUD (Target Configuration Under Dead loads) method [7], and the combination method of initial force method and TCUD method [8, 9].
Conclusion
Two unstrained-length calculation procedures for determining one optimized initial state solution of cable-stayed bridges, the analytical method and the G.TCUD method, have been presented, in which the former method is based on the continuous beam analysis and the nonlinear algebraic equations but the latter method adopts the FE Newton iteration method using the elastic catenary cable element and the consistent frame element based on the co-rotational formulation. Moreover, the unstrained length method strongly depending on the unstrained-length calculation schemes are presented to effectively perform nonlinear FE analysis of cable-stayed bridges subjected to various load combinations. Finally initial shaping analysis of a cable-stayed bridge having one intermediate pier is performed and numerical results are analyzed. The important concluding remarks can be made as follows:
1. The initial state solutions by AM1 look reasonable at first glance but ULM1 based on unstrained lengths by AM1 leads to explosively large bending moments in both the main girder and the pylon for the bridge model having the initial camber.
2. The G.TCUD introduces the corresponding additional boundary constraints instead of adding all the unstrained element lengths to the nodal unknown while the ULM adopts Newton iteration method with keeping the predetermined unstrained lengths constant. And G.TCUD provides the optimized initial solution converging nearly to the target configuration in case of balanced cable-stayed bridges under dead loads.
3. Interestingly, even though any additional constraints in the ULM method are not enforced except for the essential boundary condition, the initial state solutions by ULM3 and ULM4 are nearly identical to those by G.TCUD irrespective of the weight-balanced condition and the fabrication camber.
4. Initial state solutions by AM3 and ULM3 are in excellently good agreement with those by G.TCUD and ULM3 while the results by AM1and ULM1 display large difference. Furthermore, the initial solution by AM3 shows little difference with that by ULM3, which means that one optimized initial state of balanced cable-stayed bridges can be easily found by adopting AM3 without recourse to relatively complicated G.TCUD.
5. Practically AM2 and ULM2 can be applied to the initial shaping and the construction stage analysis of cable-stayed bridges having moderate span lengths.
6. Bending moments in the main girder can be always localized by applying the G.TCUD method but the maximum moments in pylons in case of the unbalanced cable-stayed bridge can be extremely huge than those in the balanced bridge which means that the weight balancing between the center span and side spans should be carefully taken into account in the preliminary design.
7. Finally, it is judged that ULM3 and ULM4 based on the unstrained-lengths by AM3 and G.TCUD, respectively, can be the most powerful tool for not only the initial shaping analysis but also the subsequent construction stage analysis and structural nonlinear analysis under various load combinations.