UI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE

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1 UI FORMULATION FOR CABLE STATE OF EXISTING CABLE-STAYED BRIDGE Juan Huang, Ronghui Wang and Tao Tang Coege of Traffic and Communications, South China University of Technoogy, Guangzhou, Guangdong 51641, P.R. China ABSTRACT As incined cabes are primary oad-bearing members of cabe-stayed bridges, the accuracy of the method of anaysing cabe state is a key issue in keeping existing bridges safe. Typica cabe characteristics are that they are susceptibe to corrosion and tend to sag during their ong service ife, so it is essentia to take these characteristics into account in the structura anaysis to determine the actua behaviour of a cabe in service. However, most of the recent methods of cabe structure anaysis are done on the materia in a perfect state. The deterioration characteristics, such as cabe corrosion and initia sag caused by the cabe weight, are disregarded, which makes it difficut to appy the current methods when deaing with cabe structures that are in service. By soving the boundary probem of incined cabe using the governing differentia equation for the UL (Updated Lagrangian) formuation, this paper introduces a convergence iterative soution method for the anaysis of cabe structures of existing cabe-stayed bridges. When using the iterative soution, it is convenient to determine the reationship between the co-ordinate difference of cabe-end position, cabe tension and cabe weight. With the approach described in this paper, the effect of cabe sag can be incuded without any approximations. Moreover, cabe corrosion described by the method eads to good accuracy of resuts. The method meets the engineering requirements for the anaysis of existing ong-span cabe structures. The resuts obtained from the method show that it is efficient and reiabe. It can be convenienty appied in the anaysis of arge-dispacement cabe structures that are in service, which provides a new approach to structura heath monitoring of ong-span cabe-stayed bridges. 1. INTRODUCTION Incined cabes are primary oading-bearing members of cabe-stayed bridges. The accuracy of the method to determine cabe state is a key issue in keeping existing bridges safe. In view of the fact that extreme fexibiity and susceptibiity to corrosion are typica characteristics of stee cabe, this eads to high stress eves during its service ife. The geometric non-inearity and corrosion effects shoud therefore be taken into account to describe the actua behaviour of cabes. The geometric non-inearity of cabe structures has been studied extensivey, and a considerabe number of anaytica techniques and computer modes have been deveoped and are avaiabe. Ernst (1965), Kwan (1998), Mitsugi (1994), and Leonard (1988) modeed cabe as a series of straight inear eements. A arge number of eements and specific formuations have been proposed to improve the performance of the straight eements taking into account the non-inear behaviour of the cabe. Ahmadi-Kashani (1983), Jayaraman and Knudson (1981), and Gambhir and Batcheor (1986) used curved eements to describe the highy non-inear nature of the cabe probem. However, most avaiabe Proceedings of the 6 th Southern African Transport Conference (SATC 7) 9-1 Juy 7 ISBN Number: X Pretoria, South Africa Produced by: Document Transformation Technoogies cc Conference organised by: Conference Panners 745

2 methods are given with the materia in a perfect state, and disregard cabe deterioration characteristics, such as cabe corrosion and initia sag causing by cabe weight, which makes the appication of the current methods difficut when deaing with cabe structures that are in service. By soving the boundary probem of incined cabe with the governing differentia equation of the Updated Lagrangian formuation, a convergence iterative soution method is introduced for the anaysis of cabe state of existing cabe-stayed bridges. Using the approach described in this paper, the effect of cabe sag can be incuded without any approximations. Moreover, cabe corrosion characteristic are described, and the method presented makes it possibe to sove the divergence probem of stress reaxation of cabes that occur when using other techniques, which eads to good accuracy of resuts. The method meets the engineering requirements of the anaysis of existing ong-span cabe structures.. BEHAVIOUR OF INCLINED CABLE SEGMENT Basic assumptions As incined cabes of cabe-stayed bridges are highy fexibe, they undergo arge dispacements and shoud therefore be anaysed taking into account geometric non-inearity. In order to better describe the behaviour of incined cabes, some assumptions shoud be made as foows: 1. Incined cabes have no bending and bucking resistance and can sustain ony tensie forces.. Strains wi be sma, athough arge dispacements take pace when incined cabe is in service. 3. Cabe weight oading is uniformy distributed aong the ength of the cabe and cabe tension is directed aong the tangent to the cabe segment. Figure 1. Cabe equiibrium 746

3 Equiibrium equations of incined cabe The incined cabe shown in Figure 1 (a) is suspended between two fixed points A and B. Let q denote the intensity of cabe weight oading per unit ength of the segment. The horizonta coordinate x is the independent variabe. The dependent variabes wi be the components of tension H and V, the vertica defection z, and the deformed and undeformed engths of the cabe S and S. A free-body diagram of a differentia ength of the cabe segment is shown in Figure 1 (b). The equiibrium of forces on the differentia ength are given as: where dh ΣX =, dx = dx d dz ΣZ =, ( H ) dx + qds = dx dx (1) H is the horizonta component of cabe tension which is constant throughout the span since no ongitudina oads are acting. Because the foowing geometric constraint must be satisfied: dx dz + = 1 ds ds we can rewrite equation (1) as: H = H = cons tan t d z dz H + q 1 + ( ) = dx dx which is the cassic catenary equation for the defected profie z of the arc. () (3) The boundary conditions at the cabe supports A and B are: x=, z = x = z, = c Appying the boundary conditions and integrating equation (3) twice, we obtain: (4) H βx z = coshα cosh( ) q α (5) β ( c / ) α = sinh 1 + β β sinh (6) β = q H (7) The behaviour of a catenary segment of incined cabe is described by equations (5) to (7) in terms of an as yet unknown horizonta force H. If β were specified, the non-inear equation (5) coud be soved numericay for β and hence. H 747

4 3. DESCRIPTION OF CABLE CORROSION Cabes for cabe-stayed bridges are susceptibe to corrosion during their service ife. Because the accuracy in describing cabe state is critica in the evauation of the tensie forces of the cabes, which in turn are responsibe for the strength of the whoe structure, cabe corrosion deterioration shoud therefore be taken into account to mode the actua behaviour of cabes, athough corrosion probems associated with cabe-supported structures tend to be unique and compex (Hopwood and Havens, 1984). We consider mechanisms of cabe corrosion deterioration that resut in the reduction in cabe cross-sectiona area. We make this assumption here for the reason that simpified soutions can be deveoped that can satisfy the engineering requirements. Let denote the cross-sectiona area of the incined cabe in a perfect state, and due to cabe corrosion. Now we can define the effective area A ~ as: * A A be the impaired area * A % = A A (8) Then the corresponding corrosion ratio D can be defined in the form: * A A D A A = = A% (9) Figure. Equivaent-strain for corrosion cabe Since incined cabe can ony be oaded by tension, the norma stresses of an incined cabe in a perfect state and in a corrosion state may be defined by Cauchy stress σ and effective stress σ ~ respectivey. We obtain: T σ = ~ T σ = ~ (1) A A Since σ A = % σ A% and according to equation (9) we obtain: ~ σ σ = (11) 1 D By introducing Lemaitre s (1971) equivaent-strain principe, which is a usefu method for evauating damage in meta structures subjected to tension and dynamic oading, the constitutive reationship of incined cabe in a corrosion state can be determined by the Cauchy stress of cabe in a perfect state. Using the principe, it can be proved that corroded cabe subjected to Cauchy stress σ is equivaent to perfect cabe subjected to effective stress ~ σ on condition that the corresponding strains are the same, as shown in Figure. The principe can be written as: 748

5 σ ~ σ σ ε = ~ = = (1) E E (1 D) E Then from equations (11) and (1), the effective moduus of easticity E ~ for corroded cabe can be arrived at by: A E% = % E (13) A Based on the above discussion, the corrosion characteristic of incined cabe has been taken into consideration by rationa discounting of the moduus of easticity, and a convenient way to determine the actua cabe state of existing cabe structures in engineering has been deveoped. 4. DETERMINATION OF TENSION AND DEFORMATION OF CORROSION CABLE According to the geometry of the differentia ength as shown in Figure 1, the stretched ength S of the incined cabe is obtained from equation (5) as foows: dz βc S dx β Aso, since the tension must be directed aong the tangent to the arc: = ds = 1+ ( ) dx = sinh β + ( ) (14) H T = (15) cosθ Based on Lemaitre s equivaent-strain principe, the catenary deformation is cacuated as: T H q ε = = = EA % EA % cosθ EA % βcosθ Thus the increased ength of the incined cabe is: S q coth β βc S S = εds = sinh β + ( ) + 1 4EA % β β Substituting equation (14) in equation (17) yieds: βc q coth β βc sinh β + ( ) S = sinh β + ( ) + 1 β 4EA % β β Rewriting equation (17) as a function of β, we obtain: (16) (17) (18) sinh β q coth β βc F( β) = c + ( ) S sinh β + ( ) + 1 β 4EA % β β (19) 749

6 Figure 3. Graph of function F( β ) Equation (19) coud be soved directy for β using numerica procedures. Figure 3 is obtained by appying four equaised increments, and a modified Newton-Raphson iteration was performed after each increment to achieve a convergence toerance of.1. It is easy to verify that F() and F (+ ) +, and there is just one positive root in β (, + ) from Descartes rue of signs, which is the required vaue of F (β ). Once we obtain a soution to equation (19), equation (7) can be used to determine H. With reference to Figure 1(a), static equiibrium equations wi be invoved in determining the horizonta and vertica components of cabe tension at two supports. q H = () β V B = H βc coth β βc β + ( sinh ) βc βc V A = H coth β + sinh β + ( ) () H B = (3) H H A = H (4) and therefore the tension of the incined cabe is: H βc βx βc βx T = sinh β + ( ) cosh( β ) + sinh( β ) (5) sinh β When the cabe-end position is not changed reative to the vertica and horizonta distance, the cabe shape and cabe stress can be cacuated directy by appying the convergence iterative soution method in this paper. Based on the approach presented, oading techniques associated with the behaviour of arge-dispacement incined cabe during the appication of oads are then obtained. We shoud first take the unstressed cabe state as the initia state, during which there is no cabe weight oading and the incined cabe is suspended in a straight ine between two fixed supports. If the first step increment of oading is appied, c and wi be obtained by the coordinate difference of the cabe-end position. We then obtain iterative soution β, and the deformed ength of the cabe as we as the cabe tension can consequenty be determined. To improve the accuracy of the convergence iterative soution method presented in this paper, it is recommended that dead oad be appied in sma increments. In this way the effect of cabe sag caused by cabe weight oading can be incuded without any approximations. (1) 75

7 On the other hand, cabes woud become sack and stress reaxation woud occur if the reative cabe-end position is changed during service ife. The divergence probem wi be encountered if Ernst s equivaent moduus is used in this case. Such probems can be avoided by adopting the method presented with the incrementa oading scheme. If the cabe-end position is taken as the current reference configuration after the previous increment oad has been appied, we may search for a dispacement increment due to the current added increment oad. Therefore the actua cabe-end position after the current increment oad has been appied is an iterative soution, which can be as theoreticay accurate as required. 5. CONCLUSIONS By introducing Lemaitre s equivaent-strain principe, the corrosion characteristic of incined cabe is taken into consideration by rationa discounting of the moduus of easticity. A convenient engineering method has been deveoped and expored to determine the actua cabe state of existing cabe structures. The scheme presented for the cacuation of arge cabe dispacements makes it possibe to sove the divergence probem of stress reaxation of cabes caused by the use of other techniques, and provides a new approach to structura heath monitoring of ong-span cabe-stayed bridges. 6. REFERENCES [1] Ahmadi-Kashani, K, Deveopment of cabe eements and their appications in the anaysis of cabe structures. PhD thesis, University of Manchester, Institute of Science and Technoogy (UMIST), UK. [] Ernst, JH, Der E-modu von Seien unter Berücksichtigung des Durchganges. Der Bauingenieur, 4, pp [3] Gambhir, ML and Batcheor, Finite eements for cabe anaysis. Int. J of Structure, 6, pp [4] Hopwood, T and Havens, JH, Corrosion of cabe suspension bridges. In co-operation with Transportation Cabinet, Commonweath of Kentucky and Fed. Highway Administration, U.S. Dept. of Transportation, Kentucky Transportation Research Program, Univ. of Kentucky, Lexington, USA. [5] Jayaraman, HB and Knudson, WC, A curved eement for the anaysis of cabe structures. Computers & Structures, 14, pp [6] Kwan, ASK, A new approach to geometric noninearity of cabe structures. Computers & Structures, 67, pp [7] Lemaitre, J., Evauation of dissipation and damage in metas subjected to dynamic oading. Proc. of ICM1, Kyoto, [8] Leonard, JW, Tension Structures, McGraw-Hi, New York. [9] Mitsugi, J, Static anaysis of cabe networks and their supporting structures. Computers & Structures, 51, pp

Technical Data for Profiles. Groove position, external dimensions and modular dimensions

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