NON-LINEAR DYNAMICS OF CABLE-STAYED MASTS

Transcription

1 Copyright 29 by ABCM Janary 4-8, 21, Foz do Igaç, R, Brazil NON-LINEAR DYNAMICS OF CABLE-STAYED MASTS Zenon J. G. N. del rado, School of Ciil Engineering, Federal Uniersity of Goiás, UFG , Goiânia, GO, Brazil alo B. Gonçales, Elher Chaes Caralho, Department of Ciil Engineering, Catholic Uniersity, UC-Rio , Rio de Janeiro, RJ, Brazil Abstract. Cable-stayed strctres are widely sed to bild towers and bridges, coer wide spans and in off-shore strctres, among others. In this work, the non-linear finite element method, sing an pdated Lagrangian formlation, is sed to stdy the non-linear ibrations of cable-stayed masts sbjected to axial time dependent loads. The non-linear eqations are soled sing the Newton-Raphson method associated to an arc-length techniqe and the Newmark method is sed to calclate the time responses of the system. Validation examples are presented and the inflence of initial geometric imperfections and cable tensioning is stdied when stayed towers are sbjected to dynamic loads. Using the Bdianski s criterion, the loss of stability nder sdden and harmonic loads is also analyzed. Obtained nmerical reslts show the inflence of both cable tensioning and cable positioning on the non-linear behaior of the system and cold be sed as a tool for an analysis of the nonlinear dynamics of the strctre preios to design. Keywords: Cable-stayed strctres, Non-linear oscillations, Non-linear finite element, Dynamic instability. 1. INTRODUCTION Cable-stayed trss and tbe masts and towers are widely sed in seeral engineering areas with applications in ciil, off-shore, mechanical, telecommnications and aero-space engineering. The efficiency of these strctres to spport axial loads is de to the stay cables and their behaior is characterized by large displacements associated with high load bearing ratios. As cable-stayed strctres show large displacements, high non-linearities are associated with their static and dynamic behaior. Therefore, the knowledge of their non-linear behaior is of interest to engineers and scientist. The cables of a cable-stayed strctre work solely in tension. The cables mst not only hae sfficient capacity to carry the dead loads, bt mst also hae enogh resere capacity to carry the lie loads. De to their efficiency and different engineering applications, the analysis of cable-stayed strctres has been object of seeral inestigations in the last decades. Among the most important stdies we can mention Nees (199), who presented a finite element program to stdy the non-linear static and dynamic behaior of cable-stayed bridges. Using a three-dimensional model, he showed that, de to their non-linearity, the cables strongly inflence the strctral response of the system. Kahla (1997), sing a three dimensional finite element model, stdied the non-linear response of cablestayed towers. The obtained reslts demonstrated that, dring the non-linear response, the strctre failed de to the compression forces generated by the cables. X et al (1997) proposed a three dimensional finite element to stdy the dynamic response of the stayed towers of the Tsing Ma bridge. They showed that there is a high dynamic interaction between towers and cables which affects the natral freqencies of the system. Wahba et al (1998a, 1998b), sing the nonlinear finite element method and an analytical model, stdied the non-linear static and dynamic response of stayed towers. Obtained reslts showed that the analytical model presents lower displacements if compared with those obtained by the finite element method. Experimental models were also stdied and reliable agreement between nmerical the model was obsered. Kahla (2) stdied the effect of cable failre on the dynamic response of stayed towers conclding that, if the failre occrs in certain cables, the chance of failre of the whole strctre is increased. Millar and Barghian (2), sing two finite element codes, stdied the static and dynamic response of strctres that displays dynamic jmps. They conclded that non-linear static problems can be analyzed as dynamic systems withot dmping. Cheng et al. (22), sing an adanced non-linear finite element formlation, stdied the aerostatic stability of stayed bridges showing the high non-linearity of the response de to lateral winds. Chan et al. (22) performed a second order analysis of imperfect stayed colmns and showed that the bckling load of the colmns can be increased by the pre-tensioning of the stay cables. Yan-Li et al. (23), sing a discrete model, analyzed the ibrations of stayed masts nder wind loads, finding a good agreement between experimental and nmerical reslts. Freire et al. (25) stdied the non-linear effects of a stayed bridge sing both linear and non-linear finite elements. The obtained reslts show that the cable cratre increases the non-linearity of the system mainly when large displacements generate axial tensions on the cables. asqetti (23) stdied the bckling and ibration characteristics of cable-stayed towers sing a simplified sdof model. Finally, Orlando (26) stdied the non-linear dynamics and control of tower-pendlm system nder harmonic loads. A detailed parametric analysis of the non-linear oscillations showed that a non-linear pendlm absorber can increase or decrease the ibration amplitdes of the tower.

2 Copyright 29 by ABCM Janary 4-8, 21, Foz do Igaç, R, Brazil In this work, the non-linear finite element method, sing an pdated Lagrangian formlation, is sed to stdy the non-linear ibrations of cable-stayed masts sbjected to axial time dependent loads. The non-linear eqations are soled sing the Newton-Raphson method associated to an arc-length techniqe and the Newmark method is sed to calclate the time responses of the system. Validation examples are presented and the inflence of initial cable tensioning is stdied when stayed towers are sbjected to dynamic loads. Using the Bdianski s criterion, the loss of stability nder sdden and harmonic loads is also analyzed. Obtained nmerical reslts show the great inflence of both cable tensioning and cable positioning on the non-linear behaior of the system. 2. MATHEMATICAL FORMULATION The present formlation is based on preios works by Sileira (1995), Galão (2), Olieira (22), Campos Filho (24) and Caralho (28) who implemented finite element models to analyze the geometric non-linear behaior of plane strctral frames and cable system. This formlation is based on the strain-displacement relations sing a complete Green-Lagrange tensor. The colmn is modeled sing beam-colmn elements and the cables, sing trss elements. The material of all elements is considered to be linear and elastic. 3. NUMERICAL RESULTS Consider a perfect clamped-free colmn with internal diameter =.475 m, external diameter =.5 m and elasticity modls E= 1.18e8 kn/m 2, respectiely. The colmn is clamped at the base and spported by two inclined cables with =6 o, cross-section diameter =.18 m, elasticity modls E=1.e8 kn/m 2 and a pre-tensioning force T= 1 kn. The colmn is sbjected to an axial load and a pertrbing moment M as shown in Fig. 1(a). Figre 1(b) shows the inflence of stay cables on the post-critical behaior of an axially loaded colmn. When no cables are considered in the analysis, after the critical load, the system displays a stable post-critical path with a small initial cratre. If the two cables are considered, the ale of the critical load increases more than seen times, bt the system displays in this case an nstable post-bckling behaior with a sharp decrease in the load carrying capacity, being the minimm post-critical load, associated with a fold bifrcation, lower than the critical load of the colmn withot cables. So, when the colmn reaches the critical loads it jmps to a post-bckling configration associated with large displacements. Figre 1(c) displays the deformed configration of the cable-stayed for selected eqilibrim positions along the non-linear post-bckling path identified by capital letters in Fig. 1(b). So, based on the theory of elastic stability, a high imperfection sensitiity is expected for this strctral system. L=1 m L²/EI Colmn with Cables Colmn withot Cables C oints of the Deformed /L D E B A C E B D A Figre 1. erfect cable-stayed colmn. (a) Geometric characteristics. (b) ost-critical paths: lateral displacement s. load parameter. (c) Deformed configrations along the post-bckling path. Figre 2 shows the effect of initial cable tensioning on the post-bckling paths of the axially loaded colmn. If different tensioning is considered, say T1=9 kn and T2=1 kn, initial geometric imperfections on the colmn are indced. Figre 2(a) displays the initial deformed configration of the colmn de to different initial cable tensioning and the geometric and load parameters sed in the analysis. Figre 2(b) displays the ariation of the lateral displacement of the top of the colmn,, with the load parameter, while figre 2(c) shows the ariation of the axial displacement. When compared with the perfect case (eqal cable tensioning) the nonlinear eqilibrim path shows a rather different behaior. The colmn looses stability at a limit point which is mch lower than the critical load of the perfect system. The decrease in the critical load is of abot 6%. This illstrates the high imperfection sensitiity of this strctral system and the deleterios effect of asymmetric cable tensioning on the non-linear response.

3 Copyright 29 by ABCM Janary 4-8, 21, Foz do Igaç, R, Brazil.2643 m 1 m T1.593 m L²/EI T2 T1 = T2 5 m T1 < T /L /L L²/EI T1 = T2 T1 < T2 Figre 2. Imperfect cable-stayed colmn. (a) Geometric characteristics. (b) ost-critical paths - ertical displacements (c) ost-critical paths - lateral displacements 1 m 8 m 7 m p 1 m 6 m p f (Hz) Cables 4 Cables Figre 3. Inflence of nmber of cables and inclination angle on the natral freqency. (a) Colmn with two cables. (b) Colmn with for cables. (c) Variation of natral freqency. Figre 3 shows the inflence of the nmber and inclination of the stay cables on the lowest natral freqencies of the colmn. Two configrations are analyzed, considering two and for stay cables, as shown in Figres 3(a) and 3(b) respectiely. In both cases the height of the attaching point remains constant and we ary the distance of the cable spport to the colmn, ths arying the inclination angle,. The reslts show that these parameters hae an important inflence on the lowest natral freqency. For the colmn with two cables, as the ale of angle increases, the natral freqency increases reaching a maximm ale at =4 o, after this ale, as the angle increases, the natral freqency is redced tending to zero for =9 o. When for stay cables are considered, the maximm freqency occrs at =45 o. Comparing both cres, it is possible to obsere that the natral freqency increases as the nmber of stay cables increase. Now consider the colmn with two stay cables and sbjected to a sddenly applied axial load and a harmonic axial load with freqency Ω as illstrated in Fig. 4. ( )

4 Copyright 29 by ABCM Janary 4-8, 21, Foz do Igaç, R, Brazil M =,1 kgf = a cos(ωt) M =,1 kgf L=1 m L=1 m (a) (b) Figre 4. Cable-stayed colmn with (a) Sdden axial load and, (b) Harmonic axial load. Consider initially the cable-stayed colmn of Fig. 4(a), sbjected to a sddenly applied axial load. Fig. 5 displays the time responses of the stayed colmn for increasing ales of axial load. Zero initial conditions are considered in the analysis. As shown in Fig. 5(a), for certain ales of, the damped response conerge to a pre-bckling static configration. As increases, see Figs. 5(b) and 5(c), the colmn displays increasing ibration amplitdes and jmps to a post-bckling configration. Figre 6 shows the loss of stability of the cable-stayed colmn for increasing ales of axial load, sing the Bdianski s criterion. The critical load is / cr.95. Displacement - (m) Displacement - (m) Displacement - (m) Figre 5. Time responses for increasing axial load. (a) =,92 cr ; (b) =,96 cr ; (c) =,97 cr. cr critical bckling load. Fig. 4(b) shows the colmn sbjected to an axial harmonic load with amplitde a and freqency Ω. Fig. 7 shows the lateral time responses of the stayed colmn, considering a forcing freqency eqals to the natral freqency (ω o ) of the system and increasing ales of the forcing amplitde. As shown in Figs. 7(a) and 7(b), for small ales of the forcing amplitde, the colmn displays small amplitde lateral oscillations. Howeer, in Fig. 7(c), for a = 2,4 cr, the lateral displacement of the colmn grows exponentially, indicating loss of instability of the Mathie type. Figre 8 shows the ariation of the maximm lateral displacement of the colmn as a fnction of the forcing amplitde. In Fig. 8(a) the forcing freqency Ω = ω o while in Fig. 8(b), Ω = 2ω o. For Ω = ω o, the critical load, according with Bdianski s criterion, / cr 2.3. For Ω = 2ω o, / cr.7. This case corresponds to the main parametric instability region, being the dynamic bckling load mch lower than the static critical load.

5 Copyright 29 by ABCM Janary 4-8, 21, Foz do Igaç, R, Brazil max / cr Figre 6. Bdianski cre for stayed cre sbjected to sdden axial load. Displacement - (m) Displacement - (m) Displacement - (m) Figre 7. Time responses for increasing axial harmonic load. (a) a = 1,4 cr ; (b) a = 2,2 cr ; (c) a = 2,4 cr. cr critical bckling load. max a / cr a / cr Figre 8. Bdianski cre for stayed cre sbjected to harmonic axial load. (a) Ω = ω o, (b) Ω = 2ω o 4. CONCLUSIONS In this work, the non-linear finite element method, sing an pdated Lagrangian formlation, is employed to stdy the non-linear ibrations of cable-stayed masts sbjected to axial time dependent loads. The non-linear eqations are soled sing the Newton-Raphson method associated to an arc-length techniqe and the Newmark method is sed to calclate the time responses of the system. Using the Bdianski s criterion, the loss of stability nder sdden and harmonic loads is also analyzed. As obsered, the behaior of the system is highly inflenced by cable tensioning and load characteristics, which generates lower or max

6 Copyright 29 by ABCM Janary 4-8, 21, Foz do Igaç, R, Brazil higher instability loads. Then nmerical reslts show the great inflence of both cable tensioning and cable positioning on the non-linear behaior of the system. 5. ACKNOWLEDGEMENTS This work was made possible by the spport of the Brazilian Ministry of Edcation CNq and FAERJ-CNE. 6. REFERENCES Campos Filho, Erlande da Costa, 24. Análise do Comportamento Não-Linear de Estrtras Estaiadas lanas Master Thesis, UFG, Federal Uniersity of Goiás, Goiânia. 19 p. Caralho, Elher Chaez de, 28, Análise da Instabilidade Dinâmica de Estrtras Estaiadas. Máster Thesis, UFG, Federal Uniersity of Goiás, Goiânia. 121 p. Chan, S. L. et al., 22. Stability Analysis and arametric Stdy of re-stressed Stayed Colmns. Engineering Strctres, No. 24, pp Cheng, J. et al., 22. Adanced Aerostatic Stability Analysis of Cable-Stayed Bridges Using Finite-Element Method. Compters & Strctres, No. 8, pp Freire, A. M. S. et al., 26. Geometrical Nonlinearities on the Static Analysis of Highly Flexible Steel Cable-Stayed Bridges. Compters & Strctres, No. 84, pp Galão, A. S., 2. Formlações Não-Lineares de Elementos Finitos ara Análise de Sistemas Estrtrais Metálicos Reticlados lanos. Master Thesis Uniersity Federal of Oro reto, Oro reto. 168 p. Kahla, Nabil Bem, Nonlinear Dynamic Response of a Gyed Tower to a Sdden Gy Rptre. Engineering Strctres, Vol. 19, No. 11, pp Millar, Malcolm A. and Bargian, Majid, 2. Snap-Throgh Behaior of Cables in Flexible Strctres. Compters & Strctres, No. 77, pp Nees, Francisco de Assis das, 199. Vibrações de Estrtras Aporticadas Espaciais. Doctorate Thesis, COE, Uniersity Federal of Rio de Janeiro, Rio de Janeiro. 168 p. Olieira,. A., 22. Análise Estática Não-Linear de Cabos Sspensos Utilizando o Método Dos Elementos Finitos. Master Thesis, Uniersity Federal of the araná, Critiba. 93 p. Orlando, D., 26. Absorsor endlar para Controle de Vibrações de Torres Esbeltas. Master Thesis, UC, ontifical Catholic Uniersity of Rio de Janeiro, Rio de Janeiro. 168 p. asqetti, E., 23, Análise da Instabilidade Estática e Dinâmica de Torres Estaiadas", Master Thesis, UC, ontifical Catholic Uniersity of Rio de Janeiro, Rio de Janeiro. 168 p. Sileira, R. A. M., Análise de Elementos Estrtrais Esbeltos com Restrições Unilaterais de Contato. Doctorate Thesis, UC, ontifical Catholic Uniersity of Rio de Janeiro, Rio de Janeiro. 212 p. Wahba, Y. M. F. et al., 1998.b. Dynamic Response of Gyed Masts. Engineering Strctres, Vol. 2, No. 12, pp Wahba, Y. M. F. et al., 1998.a. Ealation of Non-Linear Analysis of Gyed Antenna Towers. Compters & Strctres, No. 68, pp X, Y. L. et al., Modal Analysis of Tower-Cable System of Tsing Ma Long Sspension Bridge. Engineering Strctres, Vol. 19, No. 1, pp Yan-Li, H. et al., 23. Nonlinear Discrete Analysis Method for Random Vibration of Gyed Masts Under Wind Load. Jornal of Wind Engineering and Indstrial Aerodynamics, No. 91, pp RESONSIBILITY NOTICE The athors are the only responsible for the printed material inclded in this paper.